Estereología

Practical Stereology2nd Edition

John C. RussMaterials Science and Engineering Department

North Carolina State UniversityRaleigh, NC

Robert T. DehoffMaterials Science and Engineering Department

Florida State UniversityGainesville, FL

Published by Plenum Press, New York, NYISBN 0-306-46476-4

Introduction

Since the first edition of this book was published more than a decade ago, a revolution in stereological techniqueshas occurred. The new emphasis is on design-based techniques in which new types of probes are used to extractspecific kinds of information from structures, without the traditional requirements that they be isotropic, uniformor random. The literature on new methods for sampling structures using oriented plane and line probes isextensive and continuously expanding, and there is some danger of overlooking the utility of the traditional andclassical methods which retain important areas of applicability.

In rewriting the second edition of this book, we have tried to treat each of the three major classes of stereologicaltools. Most general and easily applied are those that measure global microstructural parameters such as volumefraction, mean intercept length, surface area and curvature, and line length. The classical model-based methodsthat make geometric assumptions about the structures of interest are covered along with the simulation methodsthat are used to generate the required models. And finally the new design-based techniques are introduced, usingvertical sections and cycloids, structured sampling strategies, and oriented surface probes. Some chaptersemphasize the procedures for applying stereological measurement techniques, while others emphasize thederivation of the underlying relationships. The chapters are intended to be largely self-contained and so someduplication of topics occurs.

All of these methods can be applied relatively simply by understanding what is being measured and practicing thestep-by-step methods of implementation. It is not required that the user be able to derive the techniques from basicgeometric probability (which is fortunate). The most important requirement for the user is to become comfortablewith what aspects of structure are important and how they can be measured. In most cases, the resultingmeasurements are used to compare one structure to another or to correlate the geometric properties of the structurewith its history of development or its properties and performance. This must be done statistically, and so there isalso a brief but useful chapter on the statistical tools of primary importance. Again, these are introduced aspractical tools for usage and not derived.

The basic stereological tools are equally useful in a broad variety of applications. They are used in biological andmedical research, materials science, geology, food science, and other fields. There are some minor differences ofemphasis based on how sampling and sectioning can be accomplished with these different types of materials, andsome differences in terminology used to describe the results, but the underlying methods are the same.Metallurgists may discuss grains where biologists see cells, but the idea of a space-filling tesselation of subunits isthe same. Although our own field is materials science, we have tried to use a mixture of terminologies throughoutin hopes of communicating the ideas to a broad spectrum of potential users of these methods.

We are indebted to Dr. Jeanette Norden, from the Department of Cell Biology at Vanderbilt University, who hascollaborated with us in teaching this material at a series of annual workshops at North Carolina State Universityfor more than fifteen years. These workshops consider in depth both computer-based image processing andmeasurement, and the stereological interpretation of the measurements. It has been our experience that these twotopics are generally taught quite separately and individual researchers are often not familiar with both areas, whichis unfortunate. Our collaboration in teaching the workshops has led to the writing of this book, which we believecombines our two somewhat different points of emphasis to provide a balanced treatment of the topics.

John C. Russ Robert T. DehoffRaleigh, NC Gainesville, FLDecember 1999 December 1999

Table of Contents

Chap. 1 Introduction 1Elements of microstucture 1Geometric properties of features 4Typical stereological procedures 6Fundamental relationships 7Intercept length and grain size 11Curvature 12Second order stereology 13Stereology of single objects 15

Chap. 2 Basic Stereological Procedures 17What stereology is 17How stereology works 17Why stereology works 22Ground rules for applying stereology 25

Chap. 3 Geometry of microstructures 27The qualitative microstructural state 29The quantitative microstructural state 31

Metric properties 31Topological properties 32Ratios of global properties 34

The topographic microstructural state 35Gradients: variation with position 35Anisoptropies: variation with orientation 36Associations 37

Chap. 4 Classical stereological measures 39Two Dimensional Structures; Area Fraction from the Point Count 39Volume Fraction from the Point Count 44Two Dimensional Structures; Feature Perimeter from the Line Intercept Count 50Three Dimensional Structures: Surface Area and the Line Intercept Count 57Three Dimensional Microstructures; Line Length and the Area Point Count 63

Chap. 5 Less common stereological measures 71Three Dimensional Features: Topological Properties and the Volume Tangent Count 71Three Dimensional Features: the Mean Caliper Diameter 78Mean Surface Curvature and Its Integral 80The Sweeping Line Probe in Two Dimensions 89Edges in Three Dimensional Microstructures 93

Chap. 6 Sample design in stereology 99Population of point probes 99Population of lines in two dimensions 102Line probes in three dimensions 103Planes in three-dimensional space 104Disectors in three-dimensional space 105Sampling strategies in 3D 106

Chap. 7 Procedures for IUR sampling 110Volume fraction 112Sampling planes 115Isotropic planes 116

Isotropic line probes 120Volume probes -the Disector 124Networks 126

Chap. 8 Statistical interpretation of data 130Sources of variability in experiments 130Distributions of values 133The mean, median and mode 135The central limit theorem and the Gaussian distribution 136Variance and standard deviation 138Testing distributions for normality - skew and kurtosis 140Some other common distributions 143Comparing sets of measurements - the t test 145Nonparametric comparisons 147Linear regression 153Nonlinear regression 157

Chap. 9 Computer-assisted methods 159Getting the image to the computer 159Display and storage 162Image processing 166

Contrast manipulation 167Correcting nonuniform brightmess 171Reducing image noise 173Rectifying image distortion 174Enhancement 176

Overlaying grids onto images 179Basic stereological calculations 183Appendix: grid generation routines 185

Chap. 10 Computer measurement of images 193Measurement using grids 193Measuring area with pixels 198Measurement parameters - size 205Other feature measurements: shape and position 210Image processing to enable thresholding and measurement 212Image processing to extract measurable information 217Combining multiple images 220Summary 224

Chap. 11 Geometric modeling 225Methods: analytic and sampling 225Sphere intercepts 228Intercept lengths in other bodies 229Intercept lengths in three dimensions 233Intersections of planes with objects 237Bertrand’s paradox 240The Buffon needle problem 241Appendix: simulation routines 243

Chap. 12 Unfolding size distributions 246Linear intercepts in spheres 246Plane intersections 249Other shapes 253Simpler methods 256

Lamellae 258Chap. 13 Anisotropy and Gradients 260

Grain structures in rolled metals 260Boundary orientation 266Gradients and neighbor relationships 268Distances and irregular gradients 273Alignment 275

Chap. 14 Finite section thickness 276Projected images 276Bias in stereological measurements 280Measurements within sections 283

Chap. 15 Three-dimensional imaging 288Limitations of stereology 288Serial methods for acquiring 3D image data 288Inversion to obtain 3D data 289Stereoscopy as a 3D technique 292Visualization 295Processing 299Measurement 301

References 304

Chapter 1 - Introduction

Elements of microstructure

Stereology is the science of the geometrical relationships between a structure that exists in three dimensions andthe images of that structure that are fundamentally two-dimensional (2D). These images may be obtained by avariety of means, but fall into two basic categories: images of sections through the structure and projection imagesviewed through it. The most intensive use of stereology has been in conjunction with microscope images, whichincludes light microscopes (conventional and confocal), electron microscopes and other types. The basic methodsare however equally appropriate for studies at macroscopic and even larger scales (the study of the distribution ofstars in the visible universe led to one of the stereological rules). Most of the examples discussed here will useexamples from and the terminology of microscopy as used primarily in the biological and medical sciences, and inmaterials science.

Image analysis in general is the process of performing various measurements on images. There are manymeasurements that can be made, including size, shape, position and brightness (or color) of all features present inthe image as well as the total area covered by each phase, characterization of gradients present, and so on. Most ofthese values are not very directly related to the three-dimensional (3D) structure that is present and represented inthe image, and those that are may not be meaningful unless they are averaged over many images that represent allpossible portions of the sample and perhaps many directions of view. Stereological relationships provide a set oftools that can relate some of the measurements on the images to important parameters of the actual 3D structure. Itcan be argued that only those parameters that can be calculated from the stereological relationships (using properlymeasured, appropriate data) truly characterize the 3D structure.

Figure 1. Diagram of a volume (red), surface (blue) and linear structure (green) in a 3D space.

Chapter 1 - Basic Stereological Concepts Page # 1

What are the basic elements of a 3D structure or microstructure? Three-dimensional space is occupied by features(Figure 1) that can be:

1. Three-dimensional objects that have a volume, such as particles, grains (the usual name for space-filling arraysof polyhedra as occur in metals and ceramics), cells, pores or voids, fibers, and so forth.

2. Two-dimensional surfaces, which include the surfaces of the 3D objects, the interfaces and boundaries betweenthem, and objects such as membranes that are actually of finite thickness but (because they are much thinnerthan their lateral extent) can often be considered as being essentially 2D.

3. One-dimensional features, which include curves in space formed by the intersection of surfaces, or the edges ofpolyhedra. An example of a one-dimensional (1D) structure in a metal or ceramic grains structure is the networkof “triple lines” formed by the meeting of three grains or grain boundaries. This class also includes objectswhose lateral dimensions are so small compared to their length that they can be effectively treated as 1D.Examples are dislocations, fibers, blood vessels, and even pore networks, depending on the magnification.Features that may be treated as 3D objects at one magnification scale may become essentially 1D at a differentscale.

4. Zero-dimensional features, which are basically points in space. These may be ideal points such as the junctionsof the 1D structures (nodes in the network of triple lines in a grain structure, for example) or the intersection of1D structures with surfaces, or simply features whose lateral dimensions are small at the magnification beingused so that they are effectively treated as points. An example of this is the presence of small precipitateparticles in metals.

Figure 2. Light microscope image of a metal (low carbon steel) showing the grain boundaries(dark lines produced by chemical etching of the polished surface).


In the most common type of imaging used in microscopy, the image represents a section plane through thestructure. For an opaque specimen such as most materials (metals, ceramics, polymers, composites) viewed in thelight microscope this is a cut and polished surface that is essentially planar, perhaps with minor (and ignored)relief produced by polishing and etching that reveals the structure (Figure 2).

For most biological specimens, the image is actually a projected image through a thin slice (e.g. cut by amicrotome). The same types of specimens (except that they are thinner) are used in transmission electronmicroscopy (Figure 3). As long as the thickness of the section is much thinner than any characteristic dimensionof the structure being examined, it is convenient to treat these projected images as being ideal sections (i.e.,infinitely thin) as well. When the sections become thick (comparable in dimension to any feature or structurepresent) the analysis requires modification, as discussed in Chapter 14.

Figure 3. Transmission electron microscope image of rat liver. Contrast is produced by a combinationof natural density variations and chemical deposition by stains and fixatives.

When a section plane intersects features in the microstructure, the image shows traces of those features that arereduced in dimension by one (Figure 4). That is, volumes (three-dimensional) are revealed by areas, surfaces(two-dimensional) by lines, curves (one-dimensional) by points, and points are not seen because the section planedoes not hit them. The section plane is an example of a stereological probe that is passed through the structure.There are other probes that are used as well - lines and points, and even volumes. These are discussed in detail


below and in the following chapters. But because of the way microscopes work we nearly always begin with asection plane and a 2D image to interpret.

Figure 4. Sectioning features in a three dimensional space with a plane, showing the area intersection with avolume (red), the line intersection witha surface (blue) and the point intersection with a linear feature (green).

Since the features in the 2D image arise from the intersection of the plane with the 3D structure, it is logical toexpect that measurements on the feature traces that are seen there (lower in dimension) can be utilized to obtaininformation about the features that are present in 3D. Indeed, this is the basis of stereology. That is, stereologyrepresents the set of methods which allow 3D information about the structure to be obtained from 2D images. It ishelpful to set out the list of structural parameters that might be of interest and that can be obtained usingstereological methods.

Geometric properties of features

The features present in a 3D structure have geometric properties that fall into two broad categories: topological andmetric. Metric properties are generally the more familiar; these include volume, surface area, line length andcurvature. In most cases these are measured on a sample of the entire specimen and are expressed as “per unitvolume” of the structure. The notation used in stereology employs the letters V, S, L, and M for volume, surfacearea, length, and curvature, respectively, and denotes the fact that they are measured with respect to volume usinga subscript, so that we get

VV the volume fraction (volume per unit volume, a dimensionless ratio) of a phase (the general stereologicalterm used for any identifiable region or class of objects, including voids)


SV the specific surface area (area per unit volume, with units of m-1) of a surfaceLV the specific line length (length per unit volume, with units of m-2) of a curve or line structureMV the specific curvature of surfaces (with units of m-2), which is discussed in detail later in Chapter 5.

Other subscripts are used to indicate the measurements that have been made. Typically the probes used formeasurement are areas, lines and points as will be illustrated below. For example, measurements on an image arereported as “per unit area” and have a subscript A, so that we can have

AA the area fraction (dimensionless)LA the length of lines per unit area (units of m-1)PA the number of points per unit area (units of m-2)

Likewise if we measure the occurrence of events along a line the subscript L is used, givingLL the length fraction (dimensionless)PL or NL the number of points per unit length (units of m-1)

And if we place a grid of points on the image and count the number that fall on a structure of interest relative to thetotal number of points, that would be reported as

PP the point fraction (dimensionless)

Volumes, areas and lengths are metric properties whose values can be determined by a variety of measurementtechniques. The basis for these measurements is developed in Chapters 2 through 4. Equally or even moreimportant in some applications are the topological properties of features. These represent the underlying structureand geometry of the features. The two principle topological properties are number NV and connectivity CV, bothof which have dimensions of m-3 (per unit volume). Number is a more familiar property than connectivity.Connectivity is a property that applies primarily to network structures such as blood vessels or neurons in tissue,dislocations in metals, or the porosity network in ceramics. One way to describe it is the number of redundantconnections between locations (imagine a road map and the number of possible routes from point A to point B). Itis discussed in more detail in Chapter 3.

The number of discrete objects per unit volume is a quantity that seems quite simple and is often desired, but is nottrivial to obtain. The number of objects seen per unit area NA (referring to the area of the image on a section plane)has units of m-2 rather than m-3. NA is an example of a quantity that is easily determined either manually or withcomputer-based image analysis systems. But this quantity by itself has no useful stereological meaning. Thesection plane is more likely to intercept large particles than small ones, and the intersections with particles that arevisible do not give the size of the features (which are not often cut at their maximum diameter). The relationshipbetween the desired NV parameter and the measured NA value is NV = NA/<D> where <D> is the mean particlediameter in 3D. In some instances such as measurements on man-made composites in which the diameter ofparticles is known, or of biological tissue in which the cells or organelles may have a known size, this calculationcan be made. In most cases it cannot, and indeed the idea of a mean diameter of irregular non-convex particleswith a range of sizes and shapes is not intuitively obvious.

Ratios of the various structural quantities listed above can be used to calculate mean values for particles orfeatures. For instance, the mean diameter value <D> introduced above (usually called the particle height) can in


fact be obtained as MV/(2πNV). Likewise the mean surface area <S> can be calculated as SV/NV and the meanparticle volume <V> is VV/NV. These number averages and some other metric properties of structures are listed inTable 1. The reasoning behind these relationships is shown in Chapter 4.

Table 1. Ratios of properties give useful averages

Property Symbol Relation

Volume <V> m3 <V> = VV/NV

Surface <S> m2 <S> = SV/NV

Height <D> m1 <D> = MV/2π•NV

Mean Lineal Intercept <λ> m1 <λ> = 4•VV/SV

Mean Cross-Section <A> m2 <A> = 2π•VV/MV

Mean Surface Curvature <H> m-1 <H> = MV/SV

Typical Stereological Procedures

The 3D microstructure is measured by sampling it with probes. The most common stereological probes are points,lines, surfaces and volumes. In fact, it is not generally practical to directly place probes such as lines or points intothe 3D volume and so they are all usually implemented using sectioning planes. There is a volume probe (calledthe Disector) which consists of two parallel planes with a small separation, and is discussed in Chapters 5 and 7.Plane probes are produced in the sectioning operation. Line probes are typically produced by drawing lines orgrids of lines onto the section image. Point probes are produced by marking points on the section image, usuallyin arrays such as the intersections of a grid.

There probes interact with the features in the microstructure introduced above to produce “events,” as illustrated inFigure 4. For instance, the interaction of a plane probe with a volume produces section areas. Table 2 summarizesthe types of interactions that are produced. Note that some of these require measurement but some can simply becounted. The counting of events is very efficient, has statistical precision that is easily calculated, and is generallya preferred method for conducting stereological experiments. The counting of points, intersections, etc., is doneby choosing the proper probe to use with particular types of features so that the events that measure the desiredparameter can be counted. Figure 5 shows the use of a grid to produce line and point probes for the features inFigure 4.

Table 1.2. Interaction of probes with feature sets to produce events

3D Feature Probe Events Measurement

Volume Volume Ends Count

Volume Plane Cross-section Area

Volume Line Chord intercept Length

Volume Point Point intersection Count

Surface Plane Line trace Length

Surface Line Point intersection Count

Line Plane Intersection points Count


a

b

Figure 5. Sampling the section image from Figure 4 using a grid: a) a grid of lines produces line segmentson the areas that can be measured, and intersection points with the lines that can be counted; b) a grid of

points produces intersection points on the areas that can be counted.

With automatic image analysis equipment (see Chapter 10) some of the measurement values shown in Table 2 mayalso be used such as the length of lines or the area or intersections. In principle, these alternate methods providethe same information. However, in practice they may create difficulties because of biased sampling by the probes(discussed in several chapters), and the precision and accuracy of such measurements are hard to estimate. Forexample, measuring the true length of an irregular line in an image composed of discrete pixels is not very accuratebecause the line is “aliased” by consisting of discrete pixel steps. As another example, area measurements incomputer based systems are performed simply by counting pixels. The pixels along the periphery of features aredetermined by brightness thresholding and are the source of measurement errors. Features with the same area butdifferent shapes have different amounts of perimeter and so produce different measurement precision, and it is noteasy to estimate the overall precision in a series of measurements. In contrast, the precision of countingexperiments is well understood and is discussed in Chapter 8.

Fundamental relationships

The classical rules of stereology are a set of relationships that connect the various measures obtained with thedifferent probes with the structural parameters. The most fundamental (and the oldest) rule is that the volumefraction of a phase within the structure is measured by the area fraction on the image, or VV = AA. Of course, thisdoes not imply that every image has exactly the same area fraction as the volume fraction of the entire sample. All


of the stereological relationships are based on the need to sample the structure to obtain a mean value. And thesampling must be IUR - isotropic, uniform and random - so that all portions of the structure are equallyrepresented (uniform), there is no conscious or consistent placement of measurement regions with respect to thestructure itself to select what is to be measured (random), and all directions of measurement are equallyrepresented (isotropic).

It is easy to describe sampling strategies that are not IUR and have various types of bias, less easy to avoid suchproblems. For instance, if a specimen has gradients of the amount or size of particles within it, such as more of aphase of interest near the surface than in the interior, sampling only near the surface might be convenient but itwould be biased (nonuniform). If the measurement areas in cells were always taken to include the nucleus, theresults would not be representative (nonrandom). If the sections in a fiber composite were always taken parallel tothe lay (orientation) of the fibers, the results would not measure them properly (nonisotropic).

If the structure itself is perfectly IUR then any measurement performed any place will do, subject only to thestatistical requirement of obtaining enough measurements to get an adequate measurement precision. But few real-world specimens are actually IUR, so sampling strategies must be devised to obtain representative data that do notproduce bias in the result. The basis for unbiased sampling is discussed in detail in Chapter 6, and some typicalimplementations in Chapter 7.

The fundamental relationships of stereology are thus expected value theorems that relate the measurements that canbe made using the various probes to the structural parameters present in three dimensions. The phrase expectedvalue (denoted by < >) means that the equations apply to the average value of the population of probes in the 3Dspace, and the actual sample of the possible infinity of probes that is actually used must be an unbiased sample inorder for the measurement result to give an unbiased estimate of the expected value. The basic relationships usingthe parameters listed above are shown below in Table 3. These relationships are disarmingly simple yet verypowerful. They make no simplifying assumptions about the details of the geometry of the structure. Examples ofthe use and interpretation of these relationships are shown below and throughout this text.

Table 3. Basic relationships for expected values

Measurement Relation Property

Point count <PP> = VV Volume fraction

Line intercept count <PL> = SV/2 Surface area density

Area point count <PA> = LV/2 Length density

Feature count <NA> = MV/2π = NV•<D> Total curvature

Area tangent count <TA> = MV/π Total curvature

Disector count <NV> = NV Number density

Line fraction <LV> = VV Volume fraction

Area fraction <AA> = VV Volume fraction

Length per area <LA> = (π/4)•SV Surface area density

It should also be noted that there may be many sets of features in a microstructure. In biological tissue we may beinterested in making measurements at the level of organs, cells or organelles. In a metal or ceramic we may have


several different types of grains (e.g., of different chemical composition), as well as particles within the grainsand perhaps at the interfaces between them (Figure 6). In all cases there are several types of volume (3D) featurespresent, as well as the 2D surfaces that represent their shared boundaries, the space curves or linear features wherethose boundaries intersect, and the points where the lines meet at nodes. In other structures there may be surfacessuch as membranes, linear features such as fibers or points such as crystallographic defects that exist as separatefeatures.

Figure 6. Example of a polyhedral metal grain (a) with faces, edges (triple lines where three faces fromadjacent grains meet) and vertices (quadruple points where triple lines meet and four adjacent grainstouch); (b) shows the appearance of a representative section through this structure. If particles form alongthe triple lines in the structure (c) they appear in the section at the vertices of the grains (d). If particlesform on the faces of the grains (e) they appear in the section along the boundaries of the grains (f).

Faced with the great complexity of structures, it can be helpful to construct a feature list by writing down all of thephases or features present (and identifying the ones of interest), and then listing all of the additional ones thatresult from their interactions (contact surfaces between cells, intersections of fibers with surfaces, and so on).Even for a comparatively simple structure such as the two-phase metal shown in Figure 7 the feature list is quiteextensive and it grows rapidly with the number of distinct phases or classes of features present. This is discussedmore fully in Chapter 3 as the “qualitative microstructural state.”


Figure 7. An example microstructure corresponding to a two-phase metal. Color-coding is shown tomark a few of the features present: blue = β phase, red = αβ interface; green = ααα triple points,

yellow = βββ triple points.

Consider the common stereological measurements that can be performed by just counting events when anappropriate probe is used to intersect these features. The triple points can be counted directly to obtain number perunit area NA, which can be multiplied by 2 to obtain the total length of the corresponding triple lines per unitvolume LV. Note that the dimensionality is the same for NA (m-2) and LV (m/m3).

Figure 8. A grid (red) used to measure the imagefrom Figure 7. There are a total of 56 gridintersections, of which 9 lie on the β phase (bluemarks). This provides an estimate of the volumefraction of 9/56 = 16% using the relationship PP =VV. The total length of grid line is 1106 µm, andthere are 72 intersections with the αβ boundary(green marks). This provides an estimate of thesurface area of that boundary of 2•72/1106 = 0.13µm2/µm3 using the relationship SV = 2•PL. Thereare 8 points representing βββ triple points (yellowmarks) in the area of the image (5455 µm2). Thisprovides an estimate the length of triple line of2•8/5455 = 2.9•10-3 µm/µm3 using the relationshipLV = 2•PA. Similar procedures can be used tomeasure each of the feature types present in thestructure.


Other measurements are facilitated by using a grid. For example, a grid of points placed on the image can be usedto count the fraction of points that fall on a phase (Figure 8). The point fraction PP is given by the number ofevents when points (the intersections of lines in the grid) coincide with the phase divided by the total number ofpoints. Averaged over many fields, the result is a measurement of the volume fraction of the phase VV.

Similarly, a line probe (the lines in the same grid) can be used to count events where the lines cross theboundaries. As shown in Figure 8, the total number of intersections divided by the total length of the lines in thegrid is PL. The average value of PL (which has units of m-1) is one half of the specific surface area (SV, area perunit volume, which has identical dimensionality of m2/m3 = m-1).

Chapters 4 and 5 contain numerous specific worked examples showing how these and other stereologicalparameters can be obtained by counting events produced by superimposing various kinds of grids on an image.Chapter 9 illustrates the fact that in many cases the same grids and counting procedures can be automated usingcomputer software.

Intercept length and grain size

Most of the parameters introduced above are relatively familiar ones, such as volume, area, length and number.Surfaces within real specimens can have very large amounts of area occupying a relatively small volume. Themean linear intercept λ of a structure is often a useful measure of the scale of that structure, and as noted in the

definitions is related to the surface-to-volume ratio of the features, since λ = 4•VV/SV. It follows that the mean

surface to volume ratio of particles (cells, grains, etc.) of any shape is <S/V> = 4/λ.

The mean free distance between particles is related to the measured intercept length of the region between particles,with the relationship L = λ (VVβ/VVα) where β is the matrix and α the particles. This can also be structurally

important, for example, in metals where the distance between precipitate particles controls dislocation pinning andhence mechanical properties. To illustrate the fact that stereological rules and geometric relationships are notspecific to microscopy applications, Chandreshakar (1943) showed that for a random distribution of stars in spacethe mean nearest neighbor distance is L = 0.554•NV-1/3 where NV is the number of points (stars) per unit volume.For small features on a 2D plane the similar relationship is L = 0.5•NA-1/2 where NA is the number per unit area;this will be used in Chapter 10 to test features for tendencies toward clustering or self-avoidance.

A typical grain structure in a metal consists of a space filling array of more-or-less polyhedral crystals. It has longbeen known that a coarse structure consisting of a few large grains has very different properties (lower strength,higher electrical conductivity, etc.) than one consisting of many small grains. The size of the grains varies within areal microstructure, of course, and is not directly revealed on a section image. The mean intercept length seems tooffer a useful measure of the scale of the structure that can be efficiently measured and correlated with variousphysical properties or with fabrication procedures.

Before there was any field known as stereology (the name was coined about 40 years ago) and before theimplications of the geometrical relationships were well understood, a particular parameter called the “grain sizenumber” was standardized by a committee of the American Society for Testing and Materials (ASTM). Although it


does not really measure a grain “size” as we normally use that word, the terminology has endured and ASTMgrain size is widely used. There are two accepted procedures for determining the ASTM grain size (Heyn, 1903;Jeffries et al., 1916), which are discussed in detail in Chapter 9. One method for determining "grain size" actuallymeasures the amount of grain boundary surface SV, and the other method measures the total length of triple lineLV between the grains. The SV method is based on the intercept length, which as noted above gives the surface tovolume ratio of the grains.

Curvature

Curvature of surfaces is a less familiar parameter and requires some explanation. A fuller discussion of the role ofsurface curvature and the effect of edges and corners is deferred to Chapter 5. The curvature of a surface in threedimensions is described by two radii, corresponding to the largest and smallest circles that can be placed tangent tothe surface. When both circles lie inside the object, the surface is locally convex. If they are both outside theobject, the surface is concave. When one lies inside and the other outside, the surface is a saddle. If one circle isinfinite the surface is cylindrical and if both are infinite (zero curvature) the surface is locally flat. The meancurvature is defined as 1/2 (1/R1 + 1/R2).

The Gaussian curvature of the surface is 1/(R1•R2) which integrates to 4π over any convex surface. This is basedon the fact that there is an element of surface area somewhere on the feature (and only one) whose surface normalpoints in each possible direction. As discussed in Chapter 5, this also generalizes to non-convex but simplyconnected particles using the convention that the curvature of saddle surface is negative.

MV is the integral of the local mean curvature over the surface of a structure. For any convex particle M = 2πD,where D is the diameter . MV is then the product of 2π<D> times NV, where <D> is the mean particle diameterand NV is the number of particles present. The average surface curvature H = MV/SV, or the total curvature of thesurface divided by the surface area. This is a key geometrical property in systems that involve surface tension andsimilar effects.

For convex polyhedra, as encountered in many materials’ grain structures, the faces are nearly flat and it mightseem as though there is no curvature. But in these cases the entire curvature of the object is contained in the edges,where the surface normal vector rotates from one face normal to the next. The total curvature is the same 2πD. Ifthe length of the triple line where grains meet (which corresponds to the edges between faces) is measured asdiscussed above, then MV = (π/2)•LV. Likewise for surfaces (usually called muralia) in space, the total curvatureMV = (π/2) • LV where the length is that of the edge of the surface. For rods, fibers or other linear features thetotal curvature is MV= π•LV; the difference from the triple line case is due to the fact that the fibers have surfacearea around them on all sides.

Curvature is measured using a moving tangent line or plane, which is swept across the image or through thevolume while counting events when it is tangent to a line or surface. This is discussed more in Chapter 5 as itapplies to volumes. For a 2D image the tangent count is obtained simply by marking and counting points where aline of any arbitrary orientation is tangent to the boundary. Positive tangent points (T+) are places where the localcurvature is convex and vice versa. The integral mean curvature is then calculated from the net tangent count as


MV = π(T+ – T-)/A. Note that for purely convex shapes there will be two T+ and no T- counts for each particleand the total mean curvature HV is 2πNA.

Second order stereology

Combinations of probes can also be used in structures, often called second-order stereology. Consider the case inwhich a grid of points is placed onto a field of view and the particles which are hit by the points in the grid areselected for measurement. This is called the method of point-sampled intercept lengths. The point samplingmethod selects features for measurement in proportion to their volume (points are more likely to hit large thansmall particles). For each particle that is thus selected, a line is drawn through the selection point to measure theradius from that point to the boundary of the particle. If the section plane is isotropic in space, these radial lines aredrawn with uniformly sampled random orientations (Figure 9). If the section plane is a vertical section asdiscussed in Chapters 6 and 7, then the lines should be drawn with sine-weighted orientations. If the structure isitself isotropic, any direction is as good as another.

Figure 9. Point sampled linear intercepts. A grid (green) is used to locate points within features, fromwhich isotropic lines are drawn (red) to measure a radial distance to the boundary.

The volume of the particle vi = (4/3)•π<r3> where <> denotes the expected value of the average over manymeasurements. This is independent of particle shape, except that for irregular particles the radius measured shouldinclude all segments of the particle section which the line intersects. Averaging this measurement over a smallcollection of particles produces a mean value for the volume <v>V = (4/3)•π<<r3>> where the subscript V


reminds us that this is the volume-weighted mean volume because of the way that the particles were selected formeasurement.

If the particles have a distribution of sizes, the conventional way to describe such a distribution is fN(V) dV wheref is the fraction of number of the particles whose size lies between V and V+dV. But we also note that the fractionof the volume of particles in the structure with a volume in the same range if fV(V)dV. These are related to eachother by

fVdV = Vf N (V)dV (1.1

This means that the volume-weighted mean volume that was measured above is defined by

< v >V = V ⋅ fV (V)dV0

Vmax

∫ (1.2

and if we substitute equation (1.1) into (1.2) we obtain

< v >V = V 2

0

Vmax

∫ ƒ N (V )dV=< v2 >N (1.3

The consequence of this is that the variance σ2 of the more familiar number weighted distribution can be computed

for particles of arbitrary shape, since for any distribution

σ 2 =< v2 > N − < v >2N (1.4

This is a useful result, since in many cases the standard deviation or variance of the particle size distribution is auseful characterization of that distribution, useful for comparing different populations as discussed in Chapter 8.Determining the volume-weighted mean volume with a point-sampled intercept method provides half of therequired information. The other needed value is the conventional or number-weighted mean volume. This can bedetermined by dividing the total volume of the phase by the number of particles. We have already seen how todetermine the total volume using a grid count. The number of particles can be measured with the disector,discussed in Chapter 7. So it is possible to obtain the variance of the distribution without actually measuringindividual particles to construct the distribution function.

There is in fact another way to determine the number-averaged mean volume of features <v>N without using thedisector. It applies only to cases in which each feature contains a single identifiable interior point (which does not,however, have to be in the center of the feature), and the common instance in which it is used is when this is thenucleus of a cell. The method (called the “Nucleator”) is similar to the determination of volume-weighted meanvolume above, except that instead of selecting features using points in a grid, the appearance in the section of theselected natural interior points is used. Of course, many features will not show these points since the section planemay not intersect them (in fact, if they were ideal points they would not be seen at all). When the interior point ispresent, it is used to draw the radial line. As above, if the section is cut isotropically or if the structure is isotropicthan uniform random sampling of directions can be used, and if the surface is a vertical section then sine-weightedsampling must be employed so that the directions are isotropic in 3D space as discussed in Chapters 6 and 7.


The radial line distances from the selected points to the boundary are used as before to calculate a mean volume<v>N = (4/3)•π<<r3>> which is now the number-weighted mean. The technique is unbiased for feature shape.The key to this technique is that the particles have been selected by the identifying points, of which there is one perparticle, rather than using the points in a grid (which are more likely to strike large features, and hence produce avolume-weighted result).

Stereology of single objects

Most of the use of stereological measurements is to obtain representative measures of 3D structures from samples,using a series of sections taken uniformly throughout a specimen, and the quantities are expressed on a per-unit-volume basis. The geometric properties of entire objects can also be estimated using the same methods providedthat the grid (either a 2D array of lines and points or a full 3D array as used for the potato in Figure 7.4 of Chapter7) entirely covers the object.

Figure 10. A leaf with a superimposed grid. Thegrid spacing is 1/2 inch and 39 points fall on theleaf, so the estimated area is 39•(0.5)2 = 9.75 in2.This compares to a measured area of 9.42 in2 usinga program that counts all of the pixels within the leafarea.

In two dimensions this method can be used to measure (for example) the area of an irregular object such as a leaf(Figure 10). The expected value of the point count in two dimensions is the area fraction of the object, or <PP> =


AA. For an (n × n) grid of points this is just <PP> = /n2 where is the number of points that lie on thefeature. The area fraction

AA = A

n2l2 where l is the spacing of the grid. Setting the point fraction equal to thearea fraction gives A = l2 P . This means that the number of points that lie on the feature times the size of onesquare of the grid estimates the area of the feature. Of course, as the grid size shrinks this is just the principle ofintegration. It is equivalent to tracing the feature on graph paper and counting the squares within the feature, or ofacquiring a digitized image consisting of square pixels and counting the number of pixels within the feature.

When extended to three dimensions, the same method becomes one of counting the voxels (volume elements). Ifthe object is sectioned by a series of N parallel planes with a spacing of t, and a grid with spacing l is used oneach plane, then the voxel size is t ⋅l2

. If the area in each section plane is measured as above then the volume isthe sum of the areas times the spacing t, or V = t ⋅l2 PT where <PT> is the total number of hits of grid pointson all N planar sections. This method, elaborated in Chapter 4, is sometimes called Cavalieri’s principle, but willalso be familiar as the basis for the integration of a volume as V = A ⋅dz∫ .

Measurements of the total size of an object can be made whenever the sampling grid(s) used for an objectcompletely enclose it, regardless of the scale. The method can be used for a cell organelle or an entire organ. Theappropriate choice of a spacing and hence the number of points determines the precision; it is not necessary thatthe plane spacing t be the same as the grid spacing l .

Summary

Stereology is the study of geometric relationships between structures that exist in three-dimensional space but areseen in two-dimensional images. The techniques summarized here provide methods for measuring volumes,surfaces and lines. The most efficient methods are those which count the number of intersections that varioustypes of probes (such as grids of lines or points) make with the structure of interest. The following chapters willestablish a firm mathematical basis for the basic relationships, illustrate the step-by-step procedures forimplementing them, and deal with how to create the most appropriate sampling probes, how to automate themeasuring and counting procedures, and how to interpret the results.


Chapter 2 - Basic Stereological Procedures

What Stereology Is

As stated in Chapter 1, stereology is a collection of tools that make measurements of geometric properties of realworld microstructures practical.

In its typical application, information about the structure under study is available as a collection of images preparedfrom that structure. Microscopes and other imaging devices usually present us with images that are twodimensional. The visual information they contain is a biased sample of the three dimensional structure. The imagemay be obtained from a section viewed in reflection, a slice viewed in transmission, or the projection of somemore or less rough external surface of the structure. The geometry of features in the image may be quantified bymeasuring one or more geometric properties that may be defined for individual features, or for a set of features.

Although most uses of stereology seek geometric information about microstructures, its application is not limitedto microstructures. Stereology has been used to study the geometry of rock structures in mine roofs, inastronomy, in geology, in agronomy, but the analysis of microstructures is its forte. Thus, every field of endeavorthat deals with microstructures has found stereology useful, including materials science, mineralogy, physiology,botany, anatomy, pathology, histology, and a variety of other -ologies in the life sciences.

With modern image analysis software it is possible to define and measure several dozen such geometric propertiesof features in a two dimensional image. Indeed, because a picture is worth “a thousand words,” the centralproblem in image analysis is the reduction of a few megabytes of information that constitutes a grey scale or colorimage to a few meaningful and useful numbers. Stereology provides one answer to the question, "Of all of thesedozens of numbers that can be defined and measured on an image, which are useful to me?" The specific answerto that question depends explicitly upon the application under consideration, but stereology limits the numbers thathave useful meaning for the three dimensional microstructure sampled.

If your interest lies in obtaining quantitative information about the three dimensional structure that the imagesamples then the answer to that question is limited. Only a small subset of the geometric properties that can bemeasured in a two dimensional image have stereological meaning, that is, are unambiguously related to geometricproperties of the three dimensional structure which the image samples. Only those relatively few image propertiesthat are identified in the fundamental equations of stereology have potential for yielding quantitative threedimensional information.

How Stereology Works

Insight into the geometry of a three dimensional microstructure is acquired by sampling the structure with probes.Typically the structure is probed by an array of plane sections, or by a collection of line probes, point probes orsmall thin volume elements called disector probes. Other probes have been devised for specific applications andwill be treated in later chapters, but these four probe types, points, lines, planes and volumes, are classical instereology. Operationally, these probes are generated by sectioning the structure with a plane and preparing therevealed image for observation, as shown in Figure 1. In a given field of view a plane probe is delineated by

Chapter 2 - Basic Stereological Procedures Page # 17

framing an area for examination in the field of view. Thus, a field on a microstructural section is viewed as asample of the population of planes that can be constructed in the three dimensional space that the specimenoccupies. Line and point probes are obtained in such a field by superimposing a grid consisting of an arrangementof lines and points, as shown in Figure 1. A variety of grids have been devised for these purposes, and exampleswill be examined and recommendations made in later chapters.

Figure 1. Point, line and plane probes are constructed by superimposing a grid of lineson a plane section through the microstructure.

The disector volume element probe, as shown in Figure 2, consists of a pair of planes spaced closely enough sothat unambiguous inferences may be made about how features appearing on the two planes are connected in thevolume between the planes.

These probes interact with features in the image to produce outcomes or events of stereological interest. Figure 1shows a microstructure composed of two feature sets. Call the white areas α features and the shaded areas ß

features.

1. A point on the grid may exhibit two outcomes in this structure: it may lie in the α feature set, or in the ß

set1.

1The third possibility, that the point lies on the boundary between the features, arises because in real microstructures the boundaries

and grid lines have finite widths. This possibility will be discussed in Chapter 4.


2. Lines in the grid interact with traces of the αß boundary (an interface surface in three dimensions) that

separates the two feature sets; here the event of interest is the intersection of the line probe with such atrace.

3. The area of the plane delineated by the boundary of the grid interacts with α and ß features in the structure;

the event of interest is the intersection of the plane probe with such three dimensional features.4. A small volume of the structure called a disector is contained between matched fields delineated on two

plane sections a small known distance apart interacts with α or ß features in the volume; the event of

interest is the appearance of the top of a particle within this volume.

Figure 2. A disector probe is constructed by comparing the microstructural features observed on twoclosely spaced plane sections, to characterize the volume contained between them. In this example twofeatures continue through both planes, two features appear only on the upper plane, and three features

appear only in the lower plane, indicating that their tops must lie in the volume between the planes.

This brief list of probe/event combinations is by no means an exhaustive list of the interactions that are of interestin stereology; others will be developed in appropriate chapters. However, these combinations are most frequentlyapplied and serve to illustrate how stereology works.

In a typical application the set of probes of a given class is scanned over a field and the events of interested arenoted and simply counted. For example:


1. The points in the grid are scanned and the number of points that lie in the ß phase is counted and recordedfor the field. (In Figure 2.1 four points lie in the ß phase.)

2. The lines in the grid are systematically scanned along their length and the number of intersections with αß

interface is counted and recorded (34 intersections in Figure 1);3. The area of the field is scanned systematically and the number of ß features is counted (24 features in Figure

1)2.4. Comparisons between features on the top (reference plane) and bottom (look-up plane) of the disector

shown in Figure 2 reveals the appearance of three new particle tops within the volume sampled.

Each of these simple counting experiments is repeated on a number of fields that appropriately sample the threedimensional structure. What is meant by an appropriate sample is a central issue in stereology. Indeed, as will beclearly shown in this development, the measurements of stereology are trivially simply; the fundamentalmathematical relationships that connect these measurements with three dimensional geometry are also triviallysimple. The design and collection of an appropriate sample of probes is the hard part of stereology.

Counts for a given probe/event measurement are usually reported as normalized counts by dividing the number ofcounts by the total quantity of probe scanned in the field. For the examples cited above for Figure 1:

1. The count of points that lie in the ß phase is normalized by dividing by the total number of points in the grid(25 points in Figure 1, to give a point fraction PP of 4/25 or 0.16;

2. The count of the number of intersections of lines is normalized by dividing by the total length of line probesscanned in the field (34 intersection points on the 40 µm of probe line length in Figure 1) to give a lineintercept count PL of 0.85 (counts per micrometer) or 0.85 x 104 = 8500 (counts per cm);

3. The count of features contained in the delineated area of the field is normalized by dividing by that area (24features in the 160 µm2 of area probed in Figure 1) to give the feature count NA of 0.15 (counts per µm2)or 0.15 x 108 = 1.5 x 107 (counts per cm2).

4. Each new particle appearance noted in the disector volume identifies a point which is the upper bound ofthat particle in the direction perpendicular to the planes of the disector. Since each particle has a singleupper bounding point, dividing this count by the volume contained within the disector provides an estimateof NV, the number of features per unit volume in the structure. If in Figure 2 the area of the countingframe outlined on the top frame is 130 µm2 and the distance between the planes is 6 µm, the volume of thedisector is 780 µm 3. The corresponding estimate of number density is 3/780 = 0.0038 (particles/µm3) =3.8 x 109 particles per cm3.

For a given measurement, counts are recorded on a number of fields under the microscope, on photomicrographs,or on a digitized image. The mean value and standard deviation of the counts from this sample of probes selectedfor measurement are computed using standard statistical formulae (Chapter 8). The standard deviation is the basisfor estimating the precision of the estimate of the mean value and deciding whether a sufficient number of fieldshave been be examined. The mean value of the counts in the sample is used to estimate the expected value for

2Some of the features in the area intersect the boundaries of the counting frame. Adopting the rule that features that intersect two

adjacent edges of the frame boundary are counted (in the figure, the left and bottom edges) while those that intersect the other two edges

are excluded provides an unbiased feature count.


those counts for the entire population of probes. The central objective of sample design in stereology, (i.e., theselection of fields to be included and how they are to be measured) is to devise a sample that provides an unbiasedestimate of this expected value or population mean for all such probes that can be constructed in three dimensionalspace.

The connection between expected values of counts for a given probe/event combination and a geometric propertyof the three dimensional structure is obtained by applying the appropriate fundamental relation of stereology.These equations, elegant in their simplicity, have the status of expected value theorems. The simplest of theserelations,

< PP >= V v (2.1

where VV is the volume fraction occupied by the phase being counted and the brackets around <PP> signify theexpected value for this normalized count, in this case the point fraction. This equation may be read "The expectedvalue of the fraction of the population of points that exist in the volume of the structure under study that lie in thephase of interest is equal to the fraction of the volume of the structure occupied by that phase." For the ß phasesampled in Figure 1, the expected value of PPß is VVß=0.16. If the set of fields included in the count isappropriately selected, the mean value of PP for this sample is an unbiased estimator of the expected value of PP

for the population of points in the three dimensional space occupied by the specimen and thus of the volumefraction of the ß phase in the three dimensional structure.

Other fundamental relations are equally simple. The expected value of the line intercept count for the population ofline probes in space is proportional to the surface area density of the interfaces being counted:

< PL >= 12 SV (2.2

where SV is the surface area of per unit volume of the set of interfacial features in the three dimensional structurewhose intersections were counted in the experiment. In Figure 1, the line intercept count estimates the surface areaper unit volume of the αß boundaries separating the α and ß feature sets. Sv = 2 PL = 2 • 8500 = 17000

(cm2/cm3).

The feature count is related to the number density NV of features in the structure and their average caliper diameter<D>:

< NA >= NV < D > (2.3

This latter relation is limited in its application to structures with features that are convex bodies, although it isotherwise independent of their shape. For features of more general shape <NA> measures a property called theintegral mean curvature, discussed in Chapter 5.


Similarly, three dimensional features or particles that are convex have a single "upper bound point" in anydirection. The expected value of the number of points per unit volume of disector probes is a direct estimate of thenumber of such points per unit volume in the structure:

< NV >= NV (2.4

The expected value relationships of stereology are powerful because the require no assumptions about thegeometry of the features being characterized. That point bears repeating. The expected value of counts made onany of the probe/event combinations in an appropriately designed sample of probes provide an unbiased estimateof the corresponding geometric property of the three dimensional feature set without geometric assumptions.

The typical stereological experiment begins by devising a procedure for selecting a set of fields to be examined thatprovides an appropriate sample of the population of probes to be used in the measurement. Each field is examinedby overlaying an appropriate grid containing those probes. Interactions of the probes with features of interest inthe structure are noted and counted. The mean of these counts for the sample of probes is computed andnormalized. This sample mean is then used to estimate the population mean for the same count for the full set ofprobes in three dimensions. The result is inserted in the appropriate fundamental relationship to yield an estimateof the corresponding geometric property for the three dimensional structure. The standard deviation of the samplemeasurements is used to compute the precision of this estimate.

Why Stereology Works

Why do simple counts of the number of probe/event occurrences on an appropriate sample of probes reportgeometric properties of the three dimensional structure they sample? This question goes to the core of stereology,which is contained in its collection of fundamental relationships. In this overview brief arguments are presentedwith the intention of making these relationships plausible; more rigorous derivations of these relationships arepresented in later sections.

Keep in mind that the stereological relationships are expected value theorems. This means that the result applieswhen the whole population of probes is included in the sample. The task is to design a sample of that populationwhich will yield an unbiased estimate of that expected value.

Consider the population of points in three dimensional space. In a two phase α + ß structure this population of

points fills the volumes occupied by the α and ß phases. If points are uniformly distributed in space the number of

points in the a part of the structure is proportional to the volume of the α part. The number of points in the total

specimen is proportional to the total volume of the specimen. The volume fraction of the α phase may be thought

of as the ratio of the number of points in the α phase to the number of points in the structure. Equation (2.1)

follows. If a sample of points is drawn uniformly and randomly from this population, i.e., without bias, then thefraction of points in α in this sample will estimate the fraction of points in the population, which in equation (2.1)

is shown to be the volume fraction of the α phase in the structure.


Figure 3. The set of vertical line probes (a) intersect an element of surface area dS if their positions liewithin the projected area dA on the horizontal plane (b). The relationship between dA and dS depends

on the angle Φ between the line probe and the surface normal.

To make equation (2.2) plausible consider the population of vertical lines in three dimensional space, as shown inFigure 3. An individual member of this set of lines may be located by its point of intersection P with a horizontalplane. Now focus on an element (a small piece) of the αß surface dS somewhere in the three dimensional

structure. In the application of equation (1.2) the event of interest is the intersection of a line probe with such asurface in space. In Figure 3, dA is the area of the projection of dS on the horizontal plane. The event of interest,i.e., an intersection with dS, occurs for the subset of vertical line probes that pass through the area dA on thehorizontal plane. The number of lines that produces an intersection with dS is evidently reported by the number ofpoints in the horizontal plane that lie within its projected area on that plane, which is evidently determined by thearea dA. The projected area dA is simply related to the area of the surface element dS:

dA = dS cosΦ (2.5

where Φ is the angle between the direction that is perpendicular to the surface element in the three dimensional

structure and the direction of the line probes. Averaging this result over the full population of lines in threedimensional space averages the cosine of Φ over the sphere of orientation. In Chapter 4 this average is shown to

be 1/2. Thus parallel line probes in space sense the projected area of surfaces (on a plane that is perpendicular tothe direction of the set of lines) they encounter in the structure. Including all orientations of lines in the probe setreports (one half of) the true area of the surfaces in three dimensions.

Equation (2.3) derives from interactions of plane probes with features in the structure. To make this resultplausible focus on the subset of the population of plane probes that all share the same normal direction, (i.e., are


all parallel to each other). For example, the set of horizontal planes in space is illustrated in Figure 4. Anindividual member of this set of planes may be uniquely identified by its point of intersection with the verticalaxis. The population of planes in this set is thus the population of points on the vertical axis. A feature in the ßphase will be observed on a plane probe if the probe intersects the feature. A plane intersects the feature in Figure2.4 if it lies within the vertical interval bounded by the two planes that are tangent to the top and bottom of the ßfeature. The number of planes that satisfy this condition is the same as the number of points that lie within theinterval on the vertical axis between the two tangent planes, since each plane is uniquely represented by its pointon this axis. Assuming the population of points along this line is uniformly distributed, the number of points inthis interval is proportional to the length of this interval. The length of the interval is equal to the tangent diameterD (i.e., the distance between the two tangent planes) in the vertical direction. Thus the feature count on horizontalplanes senses tangent diameters in the vertical direction of the three dimensional features. If the sample of planesincludes the full set of orientations of plane probes then the tangent diameters will be averaged over the set oforientations in space. Equation (2.3) follows.

Figure 2.4. The set of horizontal planes intersect a particle if their positions on the vertical axis lie betweenthe top and bottom tangent planes, i.e., within the caliper diameter D in the vertical direction.

The disector probe is a direct sample of the three dimensional volume of the structure. Equation (2.4) simplyrecognizes that the expected value of the NV for the population of disectors is the same as the value for thestructure.

Point probes sample features according to their volume; line probes sense the area of surfaces and interfaces; planeprobes select features according to their tangent diameter; disector probes select features according to theirnumber. The demonstrations in this section show that these counting measurements sense directly theircorresponding geometric properties. That is the basis for the fundamental relationships in stereology.


Ground Rules for Applying Stereology

The fundamental relationships of stereology make no geometric assumptions. The interactions of probes withfeatures are free of such assumptions precisely because the probe/event combinations that are counted sensespecific geometric properties directly, regardless of how the features that display those properties are arrayed inthe structure. However the connecting relationships are expected value theorems relating the average value ofcounts for the entire population of probes to a corresponding geometric property. In a given experiment somesubset of this population of probes is selected for inclusion in a sample. The mean of the sample is used toestimate the mean of the probe population. In order to insure that this sample mean is an unbiased estimate of theexpected value it is essential to obey the ground rules for stereological sample design.

The ground rules for sample design center around the word uniform. The fundamental relationships assume thatthe elements of the population of each class of probes, - points, lines, planes, disectors, etc., - are uniformlydistributed in space. Positions of probes are uniformly distributed in space in the derivation of these equations.Probes that have directions, like lines and planes, have in addition orientations that are uniformly distributed overthe sphere of orientations in three dimensional space, i.e., are isotropically distributed.

If the structures being probed are uniformly distributed in space and isotropic in orientation, then any set ofsample probes can be used. However, it is difficult to know a priori whether this condition is met, and for manyreal structures of interest it is not. A discussion of anisotropy and gradients is presented in Chapters 3, 6 and 13.

For statistical purposes it is also convenient that sample probes be selected randomly from these uniformdistributions because this simplifies the statistical interpretation of results. In some cases it has been shown thatrelaxation of this requirement samples the structure more efficiently.

The typical design of the selection of fields, grids and probes for examination in a stereological experiment strivesto attain an Isotropic, Uniform, Random sample of the corresponding population of probes. In seeking anunbiased estimate of the expected value of an event count the primary requirements are that the probe sample be"IUR".

Summary

Stereology is the methodology that is required to obtain quantitative estimates of geometric properties of featuresin three dimensional structures from measurements that are made on essentially two dimensional images.

The structure is sampled with point, line, plane, disector or other probes. Events that result from interactions ofthese probes with features in the structure are counted. Normalized averages of these counts are used to estimatethe corresponding average count for the full population of probes. Fundamental relations of stereology connectthese expected values for the probe population to a geometric property of the structure being sampled.


Point probes sense the volume of three dimensional features. Line probes sense the area of interfaces in thestructure. Plane probes sense the average tangent diameter of features. Disector probes sense the number offeatures in the structure.

The relationships of stereology are geometrically general because each kind of probe senses its correspondinggeometrical property directly. The result is thus independent of how the features that exhibit the property aredistributed in the structure.

Unbiased estimation of the stereological counting measurements requires that the selection of probes employed inany experiment be chosen Isotropically, Uniformly and Randomly from the corresponding population of thoseprobes.


Chapter 3 - Geometry of Microstructures

The generic term that is used to describe the physical elements that make up a microstructure in this text applies theconcept of a phase borrowed from classical thermodynamics. Each phase in a microstructure is a set of threedimensional features. To belong to the same phase the features must have the same internal physical properties.Usually this means features of one phase have the same chemical makeup and the same atomic, molecular, crystalor biological structure. The collection of parts of the structure that belong to the same phase is one example of afeature set.

Microstructures are space filling, not-regular, not-random arrangements of the feature sets of phases in threedimensional space. A microstructure may be a single phase tessellation1 consisting of four feature sets, as shownin Figure 1: polyhedral cells that have faces, edges and vertices arranged to fill the three dimensional space.Alternatively a microstructure may consist of two phases - two distinguishable feature sets - labeled, for instance,α and ß, as shown in Figure 2, either or both of which may be cell structures, that fit together with precision to fill

up the space occupied by the structure. Microstructures frequently consist of several distinguishable phases eachcontributing to the total collection of feature sets in the system, as shown in Figure 3. In some cases, voids orporosity may be present and is also treated as a measurable phase.

Figure 3.1. Idealized illustration of a space filling tesselation with a cell removed toshow faces (red), edges (triple lines, blue) and vertices (quadruple points, green).

In the description of a microstructure it may be useful to focus on the properties of a particular feature set such asthe ß phase and its boundaries, edges and vertices if it has them. Each member of the feature set has its owncollection of geometric properties. For example, particles of the ß phase each have their own volume, surface areaand lineal dimensions such as tangent diameter. Properties of the whole collection of features in the set are calledglobal properties. Examples include the total volume, or boundary surface area, or number of particles in the set.

1A tesselation is a subdivision of space into cells which then fill that space.

Chapter 3 - Geometry of Microstructure Page # 27

A more complete description of the structure might incorporate information about the spatial distribution of the ßphase in the context of other feature sets in the structure.

Figure 2. Particles of a second phase may distribute themselves at vertices (a),along triple lines (b), on the faces (c), or within the cell volume (not shown)

in a space filling structure. The white phase is α, the colored phase is β.

To facilitate the organization of an exercise in microstructural characterization, this chapter introduces three levelsof characterization of microstructures:

1. The qualitative microstructural state;2. The quantitative microstructural state; and3. The topographic microstructural state.

The first of these levels of description is a list of the classes of feature sets that exist in the structure. The secondlevel makes the description quantitative by assigning numerical values to geometric properties that are appropriateto each feature set. The third level of description deals with nonuniformities in the spatial distribution that mayexist in the structure.

Figure 3.3. Microstructures may consist of a number of phases. In this example, based on a superalloymetal, there are five: a matrix phase (white background), a precipitate (red) distributed through the matrix,

and three different phases (blue, yellow, and green) distributed along the undulating cell boundary.


The Qualitative Microstructural State

Microstructures consist, not only of the three dimensional features that make up the phases in the structure, butalso of two dimensional surfaces, one dimensional lines and zero dimensional points associated with thesefeatures. The qualitative microstructural state is a list of all of the classes of feature sets that are found in thestructure. Surfaces, edges and points arise from the incidence of three dimensional particles or cells. For examplethe incidence of two cells in space forms a surface. The kind of surface is made explicit by reporting the class ofthe two cells that form it. If both belong to the α feature set, the surface is an αα surface; if one is from the α set

and the other is a member of the collection of ß cells, the interface is an αß interface. And so on. Triple lines are

formed by the incidence of three cells, (e.g., an αßß triple line results from the incidence of an a cell with two ß

cells in space). Quadruple points require the incidence of four cells, and are so labeled, (e.g., ααßß).

The qualitative microstructural state can be assessed or inferred by inspection of a sufficient number of fields torepresent the structure. In many cases, one field will be enough for this qualitative purpose. In making thisassessment keep in mind that the process of sectioning the structure reduces the dimensions of the features by one,as shown in Figure 4. Sections through three dimensional cells of volume features appear as two dimensionalareas on the section. Sections through two dimensional surfaces appear as one dimensional lines or curves on thesection. Lineal features in space appear only as points of intersection with the sectioning plane; triple lines in a cellstructure intersect as triple points. Point feature sets in three dimensions, such as quadruple points, are in generalnot observable on a sectioning plane.

Figure 4. The dimension of each feature is reduced byone when a three dimensional structure is sectioned bya plane. In the illustration, phases α (an array oftransparent grains or cells) and β (colored), surfacesαα (grain or cell boundaries) and αβ, and triple linesααα and ααβ are sectioned so that volumes arerepresented by areas, surfaces by lines, and lines bypoints. Points in the original volume are not observablein the plane.

Exhaustive lists of possible feature classes for single phase and two phase structures are given in Table 3.1. Thefeatures that may exist in a three phase structure are listed in Table 3.2. Examples of each feature class arepresented on the sections shown in Figure 4.


Table 3.1. Feature classes that may exist in one and two phase microstructures.

Feature Class Single Phase (α) Two Phase (α+ß)

Volumes α α,ß

Surfaces αα αα, αß, ßß

Triple Lines ααα ααα,ααß,αßß,ßßß

Quadruple Points αααα αααα, αααß,ααßß,αßßß,ßßßß

Total Number of Classes 4 14

Table 3.2. Feature classes that may exist in a three phase microstructure.

Feature Class Three Phases (α+ß+ε)

Volumes α,ß,e

Surfaces αα, αß, αε, ßß, ßε. εεTriple Lines ααα, ααß, ααε, αßß, αεε, ßßß, ßßε, ßεε, εεε, αßεQuadruple Points αααα, αααß, αααε, ααßß, ααεε, αßßß, αεεε, ßßßß, ßßßε,

ßßεε, ßεεε, εεεε, ααßε, αßßε, αßεεTotal Number of Classes 34

Characterization of any given microstructure should begin with an explicit list of the feature sets it contains. Thisexercise is trivial for single phase structures. For systems with two or more phases this exercise has threepurposes:

1. To force you to think in terms of the three dimensional geometry of the structure;2. To identify in the list of possible features those that are absent;3. To provide an explicit basis for eliminating from characterization those feature sets that may not be of

interest.

For example, if the ß phase is porosity, then it contains no internal boundaries. This means that ßß surfacefeatures are absent, and any triple lines and quadruple points that contain two or more ß's in their designation arealso absent. The list of feature sets contained in such a structure is: α, ß; αα, αß; ααα, ααß; αααα, αααß.

There are no features described by ßß; αßß, ßßß; ααßß, αßßß, ßßßß.

The description of the qualitative microstructural state may be further expanded by ascribing qualitative aspects ofshape, scale, topology or topography. Shape is frequently conveyed qualitatively by comparing structural featureswith familiar objects: alveoli in the lung are "bunches of grapes"; solidifying grains are "dendritic" or tree like, asare neural cells; a structure may be equiaxed, plate-like or rod-like.


The Quantitative Microstructural State

Associated with each of the classes of feature sets listed in the last section is one or more geometric properties.Use of stereology to estimate values for these properties constitutes specification of the quantitative microstructuralstate. Properties that are stereologically accessible have the useful attribute that they have unambiguous meaningfor feature sets of arbitrary shape or complexity. These geometric properties may be associated with individualfeatures in the feature set, or as global properties of the whole feature set.

Metric Properties

As an example, consider a dispersion of particles of ß in an α matrix, as shown in Figure 5. In three dimensions

individual particles of ß each have a value for their volume. Over the collection of all of the ß particles there issome distribution of values of volume. Further, taken together, the entire collection of ß particles has a globalproperty which is its total volume. Stereological measures are generally related to these global properties of the fullset of features. They are usually reported in normalized units as the value of the property per unit volume ofstructure. Thus, the total volume of the set of ß particles divided by the total volume of the structure that containsthem is reported as the volume fraction, written VV, of the ß phase. The volume fraction occupied by a feature setmay be quantitatively estimated through the point count, the most used relationship in stereology, see Chapters 2and 4.

Figure 5. A simple two-phase microstructure provides a sample of the volumesand surfaces that exist in the three-dimensional structure.


Surfaces that exist as two dimensional feature sets in the three dimensional microstructure possess the propertyarea. Each feature in a surface or interfacial feature set in the structure has a value of its area. The feature set as awhole has a global value of its surface area. The concept of the area of a surface has unambiguous meaning forfeature sets of arbitrary size, shape, size distribution or complexity. Cell faces in a tessellation have an area. Thecollection of aß interfaces on the set of ß particles in an α matrix has an area. The surface that separates particles

from the gas phase in a stack of powder has an area. The normalized global property measured stereologically isthe surface area per unit volume, written SV, sometimes called the surface area density of the two dimensionalfeature set. The surface area may be estimated quantitatively with the line intercept count described in Chapter 2and 4.

Lines or space curves that exist in the three dimensional microstructure, such as the ααα or ααß lines in the

structure in Figure 4, possess a length. Individual line segments, such as edges in a cell network, each have alength. The full feature set has a global value of its length, reported as LV, the length per unit volume instereological measurements. Many feature sets, such as fibers in composite materials, axons in neurons, capillaryblood vessels, or plant roots, approximate lineal features. Length density of each of these feature sets may beestimated stereologically by applying the area point count on plane probes through the structure as presented inChapters 2 and 4.

Topological Properties

Line length, surface area and volume are called metric properties because they depend explicitly on the dimensionsof the features under examination. Geometric properties that do not depend upon shape, size, or size distributionare called the topological properties. Most familiar of these properties is the number of disconnected parts of afeature set. The ß phase particles in Figure 6 may be counted in principle to report their number. The surfacesbounding these particles also may be counted. In this case the number of disconnected parts in the collection of αß

surfaces is the same as the number of particles in the three dimensional ß feature set. (This will not be true if theparticles are hollow spheres, for example. Then each particle is bounded by two surfaces and the number ofsurfaces is twice the number of particles.) The number of edges in the network of triple lines in a tessellation iscountable. The normalized value of this generic topological property is the number density, NV.

A less familiar topological property is the connectivity of a feature set. Connectivity reports the number of extraconnections that features in the set have with themselves. To visualize the connectivity of a three dimensionalfeature determine how many times the feature could be sliced with a knife without dividing it into two parts, asshown in Figure 6. The sphere, the potato or the blob in the top row in Figure 6 all have connectivity zero, sinceany cut will divide any of these features into two parts. Features that may be deformed into a sphere withouttearing or making new joints are “topologically equivalent to a sphere” and are said to be “simply connected”.

The features in the second row in Figure 6 can all be cut once without separating the feature into two parts. Thesefeatures all have connectivity equal to one and are topologically equivalent to a torus. Connectivities of theremaining features are given on the figure. A microstructure that consists of a network, such as a powder stack, orthe capillaries in an organ, may have a very large value of connectivity. The normalized property is theconnectivity per unit volume, CV.


Figure 3.6. Connectivity, C, is the maximum number of cuts that can be madethrough a three-dimensional feature without separating it into two parts.

A third topological property of a feature set, called the Euler (pronounced “oiler”) characteristic, χ, is the

difference between the number of connected parts and the global connectivity of the feature set:

χ = N − C (3.1

This combination is a useful measure of the topological properties that combines both rudimentary concepts. It ismore directly accessible stereologically, as will be shown in Chapter 5. Measurements of the topologicalproperties require a sample of the three dimensional volume of the structure either by serial sectioning andreconstructing the three dimensional feature set or by applying the disector, see Chapters 2 and 5.

Table 3.3 reviews the primary geometric properties of three dimensional microstructures that can be estimated withstereology. Values for each may be estimated and appropriately assigned among the list of feature sets contained inthe qualitative microstructural state to provide the quantitative microstructural state for the microstructure.

Some additional geometric properties may be defined that involve the concept of the local curvature at a point on asurface, or at a point on a line. These concepts require more detailed development. They are unfamiliar and not as


widely used as those listed in Table 3.3. For the interested reader they are developed in detail in Chapter 5, alongwith the stereological methods that permit their estimation.

Table 3.3. Geometric Properties of Feature Sets

Features Geometric Property Symbol/Units

Topological Properties

α, ß... Number Density NV m-3

α, ß... Connectivity Density CV m-3

Metric Properties

α, ß... Volume Fraction VV m3/m3

αα, aß... Surface Area Density SV m2/m3

ααα, ααß... Length Density LV m/m3

Ratios of Global Properties

Ratios of selected global properties also provide some useful measures of microstructural geometry. Arepresentative measure of the scale (i.e., size) of the features in the feature set is the mean lineal intercept, <λ>.

This property is the average length of lines that intersect the features in the set. It is the mean surface - to - surfacedistance through the three dimensional features in the structure. It is given by

< λ >= 4VS

=4VV

SV

(3.2

This property must be applied knowledgeably in interpreting its meaning. For a collection of same size spheres itis equal to 2/3 the sphere diameter.

λ = 4VS

=4 4

3 π R3( )4π R2 = 4

3R (3.3

If the spheres have a size distribution, this property is related to a ratio of the third to the second moment of thedistribution function; it is not the mean particle radius (Underwood, 1970a). For a collection of rod shapes <λ>

reports the diameter and provides no information about their length (<λ> = 3/2 D). For plates, it is primarily

determined by the plate thickness (<λ> = 2τ ) and provides no information about dimensions in the plane of the

plate.

Other accessible ratios of global properties report the mean cross sectional area of particles, <A>, and the averagemean surface curvature, <H>, which may be a useful quantity when capillarity or surface tension effects play arole in shaping the microstructure. A more detailed presentation of the measurement and meaning of these ratios isgiven in Chapter 5


The Topographic Microstructural State

Many if not most microstructures exhibit nonuniformities in the distribution and disposition of the feature sets thatmake them up. Quantitative measures of these nonuniformities lead to the specification of the topographicalmicrostructural state.

Gradients: Variation with Position

If a metal rod is heated and then dropped into a water bath the outside quenches while the inside cools moreslowly. The change in microstructure that accompanies this heat treatment will yield a structure that varies from thesurface to the axis of the rod because different cooling rates produce different microstructures. Figure 7 shows anaxial section through a cylindrical sample which has a radial gradient of the second phase with the amountdecreasing from the surface toward the axis of the cylinder. Other properties, like the number density or surfacedensity will also vary along the radius of the rod. Variations of geometric properties with position in the structureare called gradients.

Figure 7. An axial section through a cylindrical sample with a gradient in theamount of the β phase decreasing radially from the surface to the center.

Such spatial variations are common in biological structures because the function of an organ usually requires thestructure to vary from the surface to the core. If the positional variations may be captured in a plane, then globalvalues of the properties may be obtained on a single properly chosen sample plane which averages these variations


appropriately. If the spatial variation is more complex, estimation of global properties requires a carefully designedsample of sections and fields for observation, see Chapter 6. As an alternative you may wish to characterize thegradient, i.e., to quantify the variation of, say, the volume fraction of the ß phase with distance from the surface.In this case the sample design requires estimation of VV at each of a selected set of positions along the radius byviewing a number of fields that lie along a plane that is parallel to the surface. In designing such a samplingstrategy it is necessary that you have a clear concept of the geometry of the spatial distribution of the feature set.

Anisotropies: Variations with Orientation

Muscle fibers tend to be aligned along the macroscopic axis of the muscle. Similarly, if a polycrystalline metal baris deformed directionally as in rolling or extruding, the grains are stretched out in the deformation direction andflattened normal to that direction, as shown in Figure 8. Wool fibers in felt tend to lie in the same plane, althoughthey are (ideally) uniformly distributed in orientation within that plane. This tendency for features to be more orless aligned with one or more preferred directions in space is given the generic term of anisotropy.

Figure 8. Grains in a polycrystal that has been directionally deformedare elongated in the direction of deformation.

The anisotropy of a collection of particles resides in the surfaces that bound the particles. To visualize this attributeof structures in three dimensions imagine that the surface bounding the particles is made up of a large number ofsmall patches of equal area. The direction associated with a patch is described by the vector that is perpendicular tothe tangent plane at the patch, i.e., locally perpendicular to the surface, called the local surface normal. If thesevectors are distributed uniformly over the sphere that describes all possible orientations in three dimensional spacethen the feature set is said to be isotropic. If the distribution of vectors is not uniform on the sphere, but tends tocluster about certain directions, the feature set exhibits anisotropy. This nonuniformity with respect to orientation


in space can be quantified using appropriate sampling strategies that involve making line intercept counts onoriented line probes, as discussed in Chapter 4. Such nonuniformities may also be appropriately averaged byusing a sampling strategy based upon the method of vertical sections (Baddeley et al., 1986), also discussed inChapter 4.

Collections of space curves or other linear features in space may also exhibit anisotropy. This anisotropy may alsobe quantified by making area point counts on oriented plane probes (Gokhale, 1992), see Chapter 4.

Associations

In a truly uniform structure the spatial distribution of any one feature set is not influenced by other feature sets thatmay exist in the system. In real microstructures there is always some tendency for relations between the variousconstituents in the system. There are associations among elements of different feature sets. Mitochondria may tendto be in contact with cell walls. Particles of the ß phase may tend to reside at grain boundaries in the matrix, or atgrain edges or corners, as shown in Figures 2 and 9a. In a three phase structure γ particles (nucleoli) may be

completely contained within ß particles (nuclei) and have no contact with the α matrix, as shown in Figure 9b.

Figure 9. Illustration of association tendencies in microstructures: (a) all of the β particles (orange)are on ααα triple points, and thus on the grain edges in three dimensions (Figure 2b);

(b) all of the blue nucleoli (γ) are contained within the yellow nuclei (β).

Quantitative measures of tendencies for positive or negative associations of elements in a microstructure are basedon comparisons of measured values of properties that report the interactions with values computed from a modelthat assumes the distributions are in fact independent. For example, the ß particles in Figure 9a form triple lines ofthe type ααß when they lie on the αα grain boundary. An area point count (PA ) measures the total length of these

triple lines per unit volume of structure. A line intercept count (PL) reports the area of αα grain boundary and,


separately, the area of αß boundary. The length of ααß triple line that would result if the interactions between the

two interfacial feature sets were random can be computed. Comparison of the measured length of ααß triple line

with that expected for the random structure gives a quantitative measure of the tendency for ß particles to beassociated with αα grain boundaries.

Inclusions or other small particles in the structure may appear as clustered, random or ordered (uniformlydistributed). A variety of measures of this tendency have been proposed; most are based on comparisons betweenthe observed properties and those of some model spatial distribution of points that can be easily visualized.

Summary

There are at three levels of characterization in the description of the geometric state of a microstructure.

The qualitative microstructural state is simply a list of the three, two, one and zero dimensional features that existin the structure at hand.

Each of the feature sets in the list has geometric properties that have unambiguous meaning for features ofarbitrary complexity. These properties may be visualized for individual features in the microstructure or as globalproperties for the whole feature set. There are two classes of geometric properties associated with these sets:

Topological properties including number density NV, connectivity density CV, and Euler characteristic χV;

Metric properties including volume fraction VV, surface area density SV, length density LV, and curvaturemeasures to be developed in Chapter 5.

Evaluation of one or more of these geometric properties constitutes a step in the assessment of the quantitativemicrostructural state.

Real microstructures exhibit variations of some properties with macroscopic position in the structure. Somefeature sets may display variation with orientation in the macroscopic specimen. Proper sample design mayprovide appropriate averages of global properties. Alternatively these gradients and anisotropies may be assessedquantitatively. Comparison of appropriate combinations of these properties with predictions from random oruniform models for the structure provide measures of tendencies for feature sets to be positively or negativelyassociated with each other.


Chapter 4 - Classical Stereological Measures

This chapter reviews the most commonly used stereological measurements. More detailed discussions andderivations may be found in the collection of traditional texts in the field (Saltykov, 1958; DeHoff and Rhines,1968; Underwood, 1970; Weibel, 1978; Kurzydlowski and Ralph, 1995; Howard and Reed, 1998). Each sectionof the chapter focuses on a manual stereological measurement. In each case, the procedure is illustrated with amicrostructure and a superimposed grid. Each figure lists:

1. The probe population that is required in the measurement;2. The specific probe that is used in the illustration;3. The event that results when this probe interacts with the microstructure;4. The measurement to be made and its specific result;5. The stereological relationship that connects the measurement to the geometric property of the structure;6. The calculation of the appropriately normalized version of the measured result;7. The calculation of the geometric property of the microstructure that is estimated from this measurement.

A discussion of the corresponding procedure used in computer-based image analysis follows the descriptions ofthe manual measurements.

The chapter begins with the measurement of the area of features in a two dimensional structure since this is easilyvisualized. Measurement of volume fraction, surface area density and line length are then reviewed.

Two Dimensional Structures; Area Fraction from the Point Count

Figure 1 shows a two dimensional structure consisting of two feature sets. Label the white background α and the

colored regions ß. The fraction of the area of the structure occupied by the ß phase is AAß. This area fraction ismeasured using point probes. The population of points in this two dimensional world is sampled bysuperimposing a grid of points on the structure. An outline of the measurement process is given in the caption ofFigure 1.

The square 5x5 grid in Figure 1 constitutes a sample of 25 points (the intersections in the grid) out of thispopulation of all of the points that could be identified in the area of the specimen. The event of interest that resultsfrom the interaction of the probe sample with the structure is, "the point hits the ß phase". The actual measurementis simply a count of these events, i.e., the number of points in the 5x5 grid in Figure 1 that lie within the ß areas.In Figure 1, this count gives 8 points in the ß phase; that count is the result for this placement of the probe samplein this field of the microstructure. This point count is related to the area fraction of the ß phase in this twodimensional structure by the fundamental stereological relationship,

< PP >β = AAβ

(4.1

The left side of this equation, <PP>, is the "expected value" of the fraction of points that hit the ß phase in asample set of grid placements; the right hand side is the area fraction of ß in the microstructure.

Chapter 4 - Classical Stereological Measures Page # 39

Figure 1. Measurement of the area fraction of the dark phase AAProbe population: Points in two dimensional spaceThis sample: 25 points in the gridEvent: Points lie in the phaseMeasurement: Count the points in the phaseThis count: 8 points in the phaseRelationship: <PP>=AANormalized count: PP=8/25=0.32Geometric property: AA=0.32

In the example of Figure 1, the point fraction is 8/25 = 0.32. In practice the grid will be placed on a number offields, each producing a particular point count for beta hits. This distribution of number of hits has a mean valueP , and a corresponding sample mean value for the point fraction, P P = P

PT, where PT is the total number of

points in the grid, 25 in this example. Following usual statistical practice, this sample mean value, P P , is use toestimate the expected value of the point fraction <PP> in the ß phase for the population of points in the structure,and, through equation (3.1), the area fraction of the ß phase. The area fraction of the α phase in this structure maybe evaluated by counting hits in α, or simply by subtracting AAß from 1.

Figure 2 shows a three phase structure composed of α, ß and γ areas. In this example, a 5x5 grid is placed on the

structure. The procedure described in the preceding section may be applied separately to hits in α, hits in ß and

hits in the γ phase. This placement of the grid gives 10 hits on the α phase, 8 on the ß particles and 7 hits on γ.


The corresponding point fractions of the three phases are obtained by dividing by PT = 25. Estimates for the threearea fractions for this single placement of the grid are AAα = 0.40; AAß = 0.32; AAγ = 0.28. Statistically validresults will require replication of these counts on a number of appropriately chosen fields.

Figure 2. Measurement of the area fractions in a three phase structure AAα, AAβ, AAγ

Probe population: Points in two dimensional spaceThis sample: 25 points in the gridEvent: Points lie in each phase, α, β, γMeasurement: Count the points in each phaseThis count: 10 points in α, 8 points in β, 7 points in γRelationship: <PP>=AANormalized count: PPα=10/25=0.40; PPβ=8/20.32; PPγ =7/25=0.28Geometric property: AAα=0.40; AAβ=0.32; AAγ =0.28

Area fractions of constituents can be measured directly in computer assisted image analysis. This is astraightforward application of the point count applied with a very high density of points. In this case, each pixel inthe image is a point probe. The phase of interest in the count is segmented from the rest of the image on the basisof its grey shade range or its color, perhaps along with some other geometric characteristics of the phase, such assize or shape. Numerical gray shades or color values for those pixels that satisfy the conditions set up to identify


the phase are set to black, and the remaining pixels are set to white. The "detected image" is thus a "binary image".The computer then simply counts black pixels in this binary image. The point fraction is the ratio of this count tothe total number of pixels in the image. These counts can be made essentially in real time.

It might seem that the pixel count in an image analysis system would provide a much greater precision in theestimate of AA than a manual count of a comparatively very limited number of points. However, analysis hasshown that the manual count may give about the same precision, as reflected in the standard deviation of thecounts for a collection of fields. This is partly because much of the variation in AA derives from differences fromone field to another. Also, there is a great deal of redundant information in the pixel count, since many pixels lie inany given individual feature.

The point count method is most efficient when the grid spacing is such that adjacent points rarely fall within thesame feature, cell or region in the image (they are then said to be independent samples of the structure). Anadvantage of this method is that when the hits produced by the point grid are all independent, the number of hitscan be used directly to estimate the measurement precision, as discussed in Chapter 8.

The most difficult step in image analysis is detection or segmentation, i.e., making the computer "see" the phase tobe analyzed as the human operator sees it. There are always pixels included in the discriminated phase that theoperator can "see" are not part of the phase, and pixels not included that "should be". Detection difficultiesincrease with the complexity of the microstructure. Such problems frequently can be minimized with more carefulsample preparation and additional image processing and more steps in the analysis. In any experimental situation itis necessary to balance the effort required to develop an acceptable sample preparation strategy and discriminationalgorithm against the inconvenience of the manual measurement which incorporates the most sophisticated ofdiscrimination systems, the experienced human operator with independent knowledge about the sample and itspreparation.

In these procedures for estimating area fraction of a phase it is not necessary to know the actual dimensions of thegrid, since the measurement involves only ratios of counts of points. The scale of the grid relative to themicrostructure, and the number of points it contains, do not influence the expected value relationship, equation(4.1). These choices do influence the precision of the estimate obtained, through the spread in the number ofcounts made from one grid placement to the next, but not the estimated expected value. For example, as a limitingcase, if the grid were very small in comparison with the features in the structure then most of the time the entiregrid would either lie within a given phase, or be outside of it. For a 5 x 5 grid, counts for most placements of thegrid would give a count of 25 or zero. The mean number of counts would still be the area fraction for a largenumber of observations, but the standard deviation of these counts would be comparatively large. Acorrespondingly large number of fields would have to be viewed to obtain a useful confidence interval on theestimate of the mean.

The point count can also be used to estimate the total area of a single feature in a two dimensional structure. Thesolution to this problem is an example of a sampling strategy that is pervasive in the design of stereologicalexperiments called the "systematic random sample". Figure 3 shows a single particle of the ß phase. Design a gridof points with outside dimensions large enough to include the whole particle. In order to estimate the area of thefeature with a point count it is necessary to calibrate the dimensions of the grid at the actual magnification of the


image. A stage micrometer may be used to measure the overall dimensions of the grid and compute the spacingbetween grid points. Let the spacing thus measured be l0 , and the total number of points in the grid be PT. For

the 6 x 8 point grid in Figure 3, there are 48 grid points. Calibration of the magnification gives the grid spacing as

l0 = 5.7 µm. The population of points that the grid samples is contained in the area AT given by PT · l02 ( = 48

· (5.7)2 = 1560 µm2 in Figure 3) .

Figure 3. Area of a feature in two dimensionsProbe population: Points in two dimensional spaceThis sample: 6 × 8 = 48 points in the gridCalibration l0 = 5.7 µmEvent: Points lie in the featureMeasurement: Count the points in the featureThis count: 20 points in the featureRelationship: A = l0

2 PGeometric property: AA=(5.7µm) 2•20 = 650 µm2

Focus on the square box in the upper left corner of the area in Figure 3. A specific placement of the grid may bespecified by locating the upper left grid point Q at any point (x,y) within this box. A given choice of (x,y)specifies the location of the remaining 48 probe points in the grid. Imagine that the point (x,y) is moved to surveyall of the points in the corner box. Then the remaining grid points systematically sample all of the points in the areaAT, i.e., the population of points of interest.


The number of points P that lie within the feature is counted for a given placement of the grid (20 points hit theparticle in Figure 3). The point fraction PP for that placement of the grid may be computed as P/PT. If the point Qat (x,y) that locates the grid is chosen uniformly from the points in the corner box then the point fraction for thissample value is an unbiased estimate of the expected value for the population <PP> and equation (4.1) applies.Statistically speaking, the point fraction in this experiment is an unbiased estimate of the fraction of the area AT

occupied by the ß particle. The expected value relation is

< PP >= AA = A

AT

= APT l0

2 (4.2

The number of points PT in the grid in this experiment is fixed. The number of points that hit the area of thefigure, P, varies from trial to trial. Thus

< PP >= PT

(4.3

where is the expected value of the number of points observed within the feature on each trial. Insert thisresult into equation (4.2) and solve for A

A = (PTl0

2 ) < PP >= (PTl02 )

PT

= l02 (4.4

This simple result is the statistical equivalent of tracing the figure on graph paper and counting the number ofsquares that lie in the figure. The area of the ß feature is seen to be the area associated with a given point ( l0

2)

times the number of points that hit the figure. This is approximately true for any single placement of the grid. Theargument that leads to this perhaps obvious result demonstrates that the area is exactly equal to the area of a gridsquare times the expected value of the number of hits on the feature. This transforms the simple geometricapproximation into a statistical result, with the attendant potential for replication of the experiment, evaluation ofstandard deviation and estimate of the precision of the result. For the ß feature in Figure 3, the 20 hits for theplacement of the grid shown estimates the area of the feature to be 650 µm2.

Volume Fraction from the Point Count

The most commonly measured property of three dimensional feature sets is their volume, usually reported as thevolume fraction, VV, of the phase. This property may be estimated using either plane, line or point probes; thesimplest and most commonly used measurement relies on point probes. The population of points to be sampled bythese probes is the set of points contained within the volume of the specimen in three dimensional space. Pointprobes are normally generated by first sectioning the sample with a plane, and generating a grid of points on theplane section. As in the two dimensional structure described in the last section, the event of interest is "point hitsthe ß phase" where "ß phase" is taken to mean the set of features at the focus of the analysis. These points aresimply counted. The stereological relation that connects this point count with the volume of the structure is


< PP >β = VVβ

(4.5

Visualize a specimen composed of two phases α and ß. This structure is revealed by sectioning it with planes and

examining fields on these sections, as shown in Figure 4. The population of points in the three dimensionalspecimen is probed by the 5 x 5 grid of points superimposed on this structure. For this sample, 5 points lie within

the ß phase. This count is replicated on a series of fields on the set of sectioning planes. The mean P and

standard deviation σP of these counts are computed. The mean point fraction, P / PT , is used to estimate the

expected value for the population of points, <PP>, and, through equation (4.5), the volume fraction occupied bythe ß phase. For the field shown in Figure 4, this estimate is 5/25 = 0.25.

Figure 4. Measurement of the volume fraction VV of a phase in three dimensionsProbe population: Points in three dimensional spaceThis sample: 25 points in the gridEvent: Points lie in the phaseMeasurement: Count the points in the phaseThis count: 5 points in the phaseRelationship: <PP>=VVNormalized count: PP=5/25=0.25Geometric property: VV=0.25


In order to obtain an estimate of the volume fraction of ß in the three dimensional structure represented by theplane section in Figure 4 it is necessary to repeat this measurement on a number of fields that are chosen torepresent the population of points in three dimensions and average the result. Table 4.1 provides an example ofresults that might be derived from a set of observations of 20 such fields. The mean number of counts on thesefields is 7.60 and the standard deviation of the set of 20 observations is found to be 1.5.

The standard deviatoin of the population mean in this experiment is given by σ P = σ P

n where n = 20, the

number of readings in this sample. P ± 2σ P is the 95% confidence interval associated with this set of readings.

The result in Table 4.1 may be interpreted to mean that the probability is 0.95 that the expected value of P for thepopulationof points lies within the interval 7.60 ± 0.68. Since each field was sampled with 25 points, thenormalized point count and its confidence interval is obtained by dividing both numbers by 25. There is thus an0.95 probability that the expected value of the point fraction lies in the interval 0.277 to 0.231. Since the expectedvalue of the point fraction estimates the volume fraction by equation (4.5), this range is also the confidenceinterval for the estimate of the volume fraction.

Table 4.1. Point Counts from the Structure in Figure 4.

P σ P σ P P ± 2σ P P P ± 2σ P PVV ± 2σ VV

7.60 1.51 0.34 7.60 ±0.68 0.304 ± 0.027 0.304 ± 0.027

Figure 5 shows a structure consisting of three phases, α, ß and γ. In this structure the small γ particles lie within

the ß phase; there are no γ particles in the α matrix. This is a very common structure in life science applications

where an organelle (γ) is a part of the ß cells. The α phase is not of interest in the experiment; the fraction of the

volume of the ß cells that are occupied by γ organelles is the object of this example. There are 70 points in the

grid. For the field shown, 30 points hit the ß phase and 8 hit γ; the remaining 32 points are in α. These counts are

replicated on a number of fields. The mean number of hits in each phase, P β and P α , are computed, along withtheir standard deviations. Corresponding point fractions are obtained by dividing by PT = 70 for this case. Theresulting point fractions are used to estimate their corresponding expected values and hence the volume fractions,VVß, VVγ , and, by difference, VVα by applying equation (4.5).

The volume fraction of the structure occupied by the ß cells including the γ organelles contained within them is the

sum VVß + VVγ . To find the fraction of that volume occupied by γ organelles, take the ratio, VVγ / (VVß + VVγ ).

For the single field shown the estimates are: VVγ = 8/70 = 0.11; VVß = 30/70 = 0.43 and VVα 0.46. The fractionof the volume of the ß regions occupied by γ is 0.11/(0.43+0.11) = 0.20.

In this analysis it is important to obtain valid estimates of the volume fractions of the three phases separately, andthen combine the results to obtain the desired comparison. As an alternate (incorrect) procedure, imagine takingcounts of ß and γ for each field, Pß and Pγ . Add the ß and γ counts, (Pß + Pγ ). Next, take the ratio [Pγ / (Pß +

Pγ )]. Average this ratio over a number of fields to estimate the fraction of ß cells occupied by γ organelles. This


procedure does not give the same result as that described in the previous paragraph because the average of a sumof ratios is not the same as the ratio of the averages. To obtain valid estimates it is important to measure thevolume fractions of the various phases with respect to the structure as a whole, and subsequently manipulate theseresults to provide measures of relative volumes among the phases.

Figure 5. Small particles of the γ phase (blue) are located within β features (green) in thisthree phase structure. There are no γ particles in the α phase (white). A 70 point grid isused to estimate the fraction of the volume of β occupied by γ by separately estimating

the volume fractions of the β and γ phases in the structure.Probe population: Points in three dimensional spaceThis sample: 70 points in the gridEvent: Points lie in α, β or γ phaseMeasurement: Count the points in each phaseThis count: 30 points in β, 8 points in γRelationship: <PP>=VVNormalized count: PPβ=30/70=0.43; PPγ =8/70=0.11Geometric property: VVβ=0.43; VVγ =0.11;

VVβ,γ=0.11/(0.43+0.11)=0.20

Figure 6 illustrates the use of the point count to estimate the total volume of a single feature in a three dimensionalmicrostructure. A three dimensional array of points is used to sample the feature. Visualize a box large enough tocontain the feature. N equally spaced planes are sliced through the feature; let h be the distance between planes. N= 5 and h = 11.5 µm in Figure 6. Each plane has an (m x n) grid of points with a calibrated grid spacing l0

imposed on it. Grids with (5 x 5) points with spacing of 5.2 µm are shown in Figure 6). In this way a threedimensional grid of points with (N x m x n) points (5 x 5 x 5) = 125 in Figure 6 is constructed which includesthe entire feature. The box associated with each grid point has dimensions (v0 = l0 • l0 • h), (5.2 • 5.2 • 11.5

= 311 µm3 in the figure. Counts are made of the number of points that hit the feature on each of the five


sectioning planes. The sum of these five counts is the total number of points in the three dimensional grid that hitthe particle. In Figure 6 the total number of hits is (1 + 3 + 2 + 3 + 1 = 10 hits).

Figure 6. The Cavalieri principle is used to estimate the volume of a single feature bycounting points on a series of sectioning planes. The spacing between the planes is h.

Probe population: Points in three dimensional spaceThis sample: 5 x 5 x 5 = 125 points in the gridCalibration: h = 11.5 µm, l0 =5.2 µm

v0 = l0 ⋅l0 ⋅ h =5.2 • 5.2 • 11.5 = 311 µm3

Event: Points lie in the featureMeasurement: Count the points in the featureThis count: 1+3+2+3+1=10 points in the featureRelationship: Vβ=PTv0<PP>β=v0β

Geometric property: Vβ=311 µm3 • 10 = 3110 µm3

Visualize a small box with the dimensions v0 = l0 • l0 • h at the upper left rear corner of the large box that

contains the feature. The upper, left, back corner of the three dimensional grid will be located at some point Q =(x,y,z) within this small box. For any given choice of Q, the positions of the remaining (N x m x n) points in thegrid is determined. They provide a systematic sample of the population points within the containing box. As Q ismoved to all of the points in the small corner box, the grid of points samples all of the population of points withinthe containing box. Thus, a random choice for the position of Q from its uniform distribution of possible points inthe small box produces a systematic random sample of the population of points in the containing box. The fractionof points in the three dimensional grid thus provides an unbiased estimate of the expected value for the populationof points in the containing box and equation (4.5) may be applied:

< PP >β = VVβ = V β

VT

= V β

PT v0

(4.6

The number of points that hit the feature varies from trial to trial. Since the total number of points PT is the same inall placements of the three dimensional grid, the expected value of this point count estimates the point fraction


< PP >β = β

PT

(4.7

Insert this result into equation (4.6) and solve for Vß:

V β = PTv0 < PP >β = PT v0β

PT

= v0 β(4.8

Thus, the volume of the ß feature is the volume associated with an individual point ( l0 • l0 • h) times the

expected value of the number of hits that points in the grid make with the feature. For a single position of the grid,v0 Pß provides an estimate of the volume of the feature. For the example illustrated in Figure 6, a total of 10 hitsare noted. The volume associated with a grid point was computed earlier to be 311 µm3. Thus the volume of thefeature is estimated at (10)(311) = 3,110 µm3.

The structured random sample is a much more efficient procedure than random sampling in which grid points areindependently placed in the volume. In the latter case, points will inevitably cluster in some areas producingoversampling, while being sparse in others producing undersampling. It will take on the average nearly threetimes as many independent random points to reach the same level of precision as with the use of the structuredapproach, but the same answer will still result, namely that the point fraction that hit the phase or structuremeasures the volume.

The point counting procedure to estimate the volume of a three dimensional object is an example of the oldest ofthe stereological procedures based upon the "Cavelieri Principle" (Howard & Reed, 1998a). The object to bequantified is sliced into a collection of slabs of known thickness. Some method is used to measure the area of eachslab, such as the point count described in an earlier section. An alternate procedure might use a planimeter, an areameasuring mechanical instrument used by cartographers before the advent of the computer, to measure theindividual cross section areas. If the slabs are of uniform thickness, and uniform content, weighing each slab,together with a calibration of the weight per unit area, could be used. The underlying principle takes the volume ofeach slab to be its cross sectional area times its thickness, so that the volume of the object is the sum of thevolumes of the slabs

V = Aihi=1

N

∑ =< A > ( Nh) (4.9

where <A> is the average cross section area of the slabs and (N h) is the total height of the object.


Two Dimensional Structures; Feature Perimeter from the Line Intercept Count

Figure 7. Use of the line intercept count to estimate the total lengthof the boundary lines of a two dimensional feature set.

Probe population: Lines in two dimensional spaceThis sample: Four horizontal lines in the gridCalibration: L0=17.7µm, total probe length = 4 • 17.7 = 70.8 µmEvent: Line intersects the feature boundary lineMeasurement: Count the intersectionsThis count: 17 intersectionsRelationship: <PL>=(π/2)•LANormalized count: PL=17 counts / 70.8 µm = 0.24 counts/µmGeometric property: LA = (π/2) PL = (π/2)•0.24 = 0.38 (µm/µm 2)

Each of the collection of features in the two dimensional structure shown in Figure 7 has a boundary, and eachboundary has a length commonly referred to as its perimeter. The normalized parameter, LA, is the ratio of thetotal length of boundaries of all of the features in the specimen divided by the area that the specimen it occupies.This perimeter length per unit area can be estimated by probing the structure with a set of line probes representedin Figure 7 by the four horizontal lines in the superimposed grid. The event of interest in the measurement is "lineintersects boundary". A simple count of these events forms the basis for estimating LA through the fundamentalstereological formula (Underwood, 1970a):


< PL >= 2π

LA (4.10

PL in this equation is called the "line intercept count"; it is the ratio of the number of intersections counted to thetotal length of line probe sampled (only the horizontal lines in the grid were used in this example). Each of thelines in Figure 7 is found by calibration to be 17.7 micrometers (µm) long. The total length of line probe sampledin this placement of the grid is 4 x 17.7 = 70.8 µm. Since there are 17 intersections marked and noted in Figure 7,for this example PL is 17/70.8 = 0.24 counts / µm probed. The left hand side of equation (4.10) is the expectedvalue of this measurement for the population of lines that can be constructed in two dimensional space. Invertingequation (4.10) gives the estimate of the perimeter length per unit area for the features shown in Figure 7:

LA = π2

PL = π2

0.24 = 0.38 µm

µm2

(4.11

The area within the grid shown in Figure 7 is (17.7)2 = 313 µm2. A rough estimate of the boundary length of thefeatures contained in the grid area is thus 0.38 (µm/µm 2) • 313 µm2 = 119 µm. There are 8.5 particles in the areaof the grid. (Particles that lie across the boundary are counted as 1/2.) A rough estimate of the average perimeterof particles may be estimated by dividing the total boundary length in the area by the number of particles: 119 µm /8.5 = 14 µm. Inspection of the features in Figure 7 indicates that this result is plausible.

Each member of the population of lines in two dimensional space has two attributes: position, and orientation. Theset of lines in the grid used for a probe in Figure 7 represent a few different positions in the population of lines,but only a single orientation. In order for a sample of line probes to provide an unbiased estimate of a value for thepopulation of probes it is necessary that the probe lines uniformly sample all orientations of the circle. Arepresentative sample of the population of orientations of lines could be obtained by rotating the stage by uniformincrements between measurements. A much more direct strategy, which guarantees uniform sampling of lineorientations, uses test line probes in the shape of circles to make the PL measurement.

Figure 8 shows a microstructure with phase boundaries that tend to be aligned in the horizontal direction. InFigure 8a and 8b a square grid of points is superimposed on this structure. Calibration shows that the grid issquare 23.7 µm on a side. The set of five horizontal lines are used to sample the population of lines in twodimensional space in Figure 8a. The total length of lines probed in the horizontal direction is (5 • 23.7 = 118.5µm). The events, line intersects boundary, are marked with red markers in Figure 8a. There are 8 intersections,giving a line intercept count in the horizontal direction of 8/118.5 = 0.068 (counts/µm). If the set of verticallines in the grid are used as a line probe (Figure 8b), the resulting count is 62; for this direction, PL = 62/118.5 =0.52 counts/µm. It is clear that the line intercept count is different in different directions in this aligned structure,and that this reflects the anisotropy of the structure.

Indeed, counts in different directions may be used to characterize the anisotropy quantitatively. Replications on acollection of fields produce mean and standard deviations of the counts made in each direction. A polar plot of PL

as a function of θ, the angle the line direction makes with the horizontal direction, as shown in Figure 9, called the


"rose of the number of intersections" [Saltykov, 1958; Underwood, 1970b] provides a graphical representation ofthis anisotropy in a two dimensional structure.

Figure 8a. Use of horizontal line probes to measure the total projected length ofboundaries of two dimensional features on the vertical axis LA,proj(vert)

Probe population: Horizontal lines in two dimensional spaceThis sample: Five horizontal lines in the gridCalibration: L0=23.7µm, total probe length = 5 • 23.7 = 118.5 µmEvent: Line intersects the feature boundary lineMeasurement: Count the intersectionsThis count: 8 intersectionsRelationship: <PL>(horz) = LA,proj(vert)

Normalized count: PL=8 counts / 118.5 µm = 0.068 counts/µmGeometric property: LA,proj(vert) = PL (horz) = 0.068 (µm/µm 2)


Figure 8b. Use of vertical line probes to measure the total projected length ofboundaries of two dimensional features on the horizontal axis LA,proj(horx)

Probe population: Vertical lines in two dimensional spaceThis sample: Five vertical lines in the gridCalibration: L0=23.7µm, total probe length = 5 • 23.7 = 118.5 µmEvent: Line intersects the feature boundary lineMeasurement: Count the intersectionsThis count: 62 intersectionsRelationship: <PL>(horz) = LA,proj(vert)

Normalized count: PL=62 counts / 118.5 µm = 0.52 counts/µmGeometric property: LA,proj(vert) = PL (horz) = 0.52 (µm/µm 2)

The true length of boundaries in this anisotropic structure may be estimated by applying test lines that are made upof circular arcs, as shown in Figure 8c. The population of orientations in this set of circular line probes provides auniform sample of the population of line orientations in two dimensional space. This line grid is equivalent to eightcircles, each with a calibrated diameter of 23.7/4 = 5.92 µm. The total length of this collection of line probes is


thus 8 [ π • (5.92µm)] = 148.9 µm. The number of intersections with these line probes marked in Figure 8c is 55,giving a line intercept count PL = 55 / 148.9 = 0.37 (counts / µm). Note that this result is within the rangebetween the result for horizontal lines (0.068) and the vertical lines (0.52). Equation (4.10) gives thecorresponding estimate for the true perimeter length: LA = (π/2) 0.37 = 0.58 µm / µm 2.

Figure 8c. Use of circular line probes to measure the true total length ofboundaries in two dimensional features, LA

Probe population: Lines in two dimensional spaceThis sample: Set of lines equivalent to 8 circlesCalibration: d=23.7/4=5.9µm, total probe length = 8 • π • d = 148.9 µmEvent: Line intersects the feature boundary lineMeasurement: Count the intersectionsThis count: 55 intersectionsRelationship: <PL> = (π/2) LANormalized count: PL=55 counts / 148.9 µm = 0.37 counts/µmGeometric property: LA = (π/2) PL = 0.58 (µm/µm 2)


The number of features per unit area in Figure 8 is counted to be 21 / (23.7)2 = 0.037 (1/µm2). A rough estimateof the average perimeter of features in this structure is LA / NA = 0.58 / 0.037 = 16 µm. Inspection of Figure 8shows that this is not an unreasonable estimate of the average particle perimeter.

Figure 9. Rose-of-the-number-of-intersections for a two-dimensional anisotropic structure

Most image analysis programs provide a measurement of the perimeter of individual features directly on thedigitized image. Boundary pixels are identified in the detected binary image. The boundary line is approximated asa broken line connecting corresponding points in adjacent boundary pixels. The perimeter of the feature is thencomputed as the sum of the lengths of the line segments that make up this broken line. The attendantapproximation may produce significant errors. The magnitude of these errors depends upon how many differentdirections are used to construct the broken line boundary.

Figure 10. Illustration of the bias in measuring perimeter of a circle in a digitized image: a) four directionsgives the perimeter of a square; b) eight directions gives that of an octagon.


Consider the circle shown in Figure 10. Its digitized image is represented by the blue plus red pixels.Identification of which pixels lie on the boundary may vary with the choice of the number of neighbors that arerequired to be "non-ß". The broken line used to compute the perimeter is illustrated for two different cases. InFigure 10a only four directions are used in constructing the perimeter. It is easy to see that the line segmentsconnecting neighboring pixels can be combined to make the square that encloses the circle. If the diameter of thecircle is d, then its actual perimeter is (π d). The reassembled broken lines make a square with edge length d; itsperimeter is (4 d). This overestimates the perimeter of the circle by a factor of 4/π = 1.27. That is, use of thisalgorithm gives a built in bias for every particle of about + 27%. In Figure 10b the segments assemble into anoctagon; the ratio of perimeter of the broken line to the circle is computed to be 1.05. Thus, even for this moresophisticated algorithm, evaluation of the perimeter of simple particles has a bias of about +5%.

Constructing an n-sided polygon with sides running in 16 possible directions (a 32-gon) further improves theaccuracy of the perimeter. For example, measuring circles varying from 5 to 150 pixels in diameter using thistechnique gives results that are within 1% as shown in Table 4.2. However, the bias is always positive (themeasured value is longer than the true value). Another new method for perimeter measurement is based onsmoothing the feature outline and interpolating a super-resolution polygon using real number values (fractionalpixels) for the vertices (Neal et. al., 1998). This method has even better accuracy and less bias, but is not yetwidely implemented. The presence of any bias in measurements is anathema to stereologists and so they generallyprefer to use the count of intercepts made by a line grid with the outlines.

Table 4.2 - Perimeter measurements using a 32-sided polygon

Diameter (pixels) Actual Perimeter Measured Perimeter Error (%)

5 15.7078 15.6568 -0.326

10 31.4159 31.8885 +1.504

15 47.1239 47.2022 +0.166

20 62.8319 63.4338 +0.958

30 95.2478 95.4562 +0.219

40 125.6637 126.6030 +0.747

50 157.0796 158.1100 +0.656

75 235.6194 236.6480 +0.437

100 314.1593 314.6880 +0.168

125 392.6991 393.1660 +0.119

150 471.2389 471.8700 +0.134

A simple experiment will permit the assessment of a given software in its measurement of perimeter. Generate, byhand or computer graphics, a collection of black circles or different sizes. Acquire the image in the computer.Detect the collection of circles. Instruct the software to measure the area A and perimeter P of each feature andcompute some measure of the "circularity" of the features, such as the formfactor 4πA/P2. This parameter has avalue of 1 for a perfect circle. Measured values may be consistently smaller than 1.00, and may vary with size ofthe circle in pixels. Use of the manual line intercept count to estimate perimeter does not suffer from this bias.


Three Dimensional Structures: Surface Area and the Line Intercept Count

Figure 11. A three-dimensional specimen with internal surfaces.

Figure 11 shows a specimen that contains internal interfaces. Usually interfaces of interest in microstructures arepart of the boundaries of particles, e.g., the αß interface in a two phase structure, but this is not a requirement for

estimating their surface area, reported as the surface area of interface per unit volume of structure. This property,SV, is sometimes referred to as the surface density of the two dimensional feature set imbedded in a threedimensional structure. Line probes are used to sense the area of surfaces. A sample of the population of lines inthree dimensional space is usually constructed by sectioning the structure with a plane to reveal its geometry, thensuperimposing a grid of lines on selected fields in that plane. The events to be counted are the intersections of theprobe lines with traces of the surfaces revealed on the plane section. The line intercept count, PL, described in thepreceding section in the context of analyzing a two dimensional structure, i.e., the ratio of the number ofintersection points to the total length of probe lines in the sample, is made for the grid in each field examined. Thegoverning stereological relationship is

< PL >= 12 SV (4.12

In the preceding section the structure under analysis was viewed as a two dimensional microstructure, and theproperties evaluated were two dimensional (perimeter, area). If indeed the microstructures shown are plane probesamples of some three dimensional microstructure, then the features observed have the significance of sectionsthrough three dimensional features. To obtain an unbiased estimate of the expected value of PL for the populationof lines in three dimensional space, the probes in the sample must be chosen uniformly from the population ofpositions and orientations of lines in three dimensional space, and the geometric properties estimated are those ofthe three dimensional microstructure sampled.

In this context, the line intercept count obtained from Figure 7, PL = 0.24 (counts/µm) takes on new meaning. Ifthe field chosen and the line probes are a properly representative sample of the population of lines in threedimensions, then this value is an unbiased estimate of the expected value <PL> for that population. This requires a


carefully thought out experimental design that prepares and selects fields and line directions to accomplish thistask. For the single sectioning plane represented in Figure 7, this will only be valid if the surfaces bounding thefeatures in the figure are themselves distributed uniformly and isotropically. Then inversion of equation (4.12)then gives SV = 2 (0.24) = 0.48 µm2 / µm3 for the surface density of the αß interface in this structure.

It is difficult to visualize the meaning of this value of surface density for the particles in Figure 7. As an aid inseeing whether this result is plausible, estimate the mean lineal intercept for the ß particles in the structure byapplying equation (3.2) in Chapter 3. The points in the grid may be used to estimate the volume fraction of the ßphase by applying equation (4.5). For the placement shown, 3 points hit the ß phase, giving a rough estimate ofVVß = 3/16 = 0.19. This result may be combined with the surface density estimate to calculate a rough value of themean lineal intercept of the ß particles:

< λ >β =4VV

β

SVαβ = 4 ⋅0.19

0.48= 1.6µm (4.13

The grid points in Figure 7 are about 6 µm apart. The mean lineal intercept averages longest dimensions withmany small intercepts near the surface of the particles. Thus, this very rough estimate appears to be reasonable.

The structure in Figure 8 may also be viewed as a section through a three dimensional anisotropic microstructure.This structure is squeezed vertically and elongated horizontally. However, a single orientation of sectioning planeis insufficient to establish the nature of this anisotropy in three dimensions. Indeed, out of the population of lineprobes in three dimensional space only those that lie in the section plane of Figure 8 are sampled. The observationthat PL is different in different directions takes on a three dimensional character, as does the anisotropy that thisdifference measures. The concept of the rose of the number of intersections as a quantitative description of theanisotropy in the microstructure extends to three dimensional microstructure. The three dimensional rose is a plotof the line intercept count as a function of longitude and latitude on the sphere of orientation.

Figure 12. Orientation in three-dimensional space is specified by a point on a sphere represented byspherical coordinates (θ,φ). The population of line orientations is sampled without bias if the line probesused are cycloids on vertical sections (see text). The dimensions of the cycloid can be expressed in terms

of the height, h, of the circumscribed rectangle (sometimes referred to as the minor axis).


An understanding of the meaning of such a construction may be aided by visualizing the meaning of the directedline intercept count. Consider the subpopulation of lines in three dimensional space that all share the sameorientation, specified by the longitude, θ, and the co-latitude (angle between the direction and the north pole), φ,

as shown in Figure 12. The line intercept count may be performed on a set of directed line probes taken from thissubpopulation. This directed count, PL(θ,φ) measures the total projected area, A(θ,φ) of surface elements in the

structure on the plane that is perpendicular to the direction of the test lines. Thus, in Figure 8, the low value forcounts for horizontal line reflects the fact that the features have a small footprint when projected on the planeperpendicular to this direction. The large number of counts for vertical line probes derives from the large projectedarea of these particles on the horizontal plane. The governing stereological relationship is (Underwood, 1970b):

< PL θ,φ( ) >= SV < cosα > (4.14

where <cos α> is the average value of the angle between the direction of the test probes and the directions normal

(perpendicular) to all of the surface elements in the structure.

The circular test lines used for the two dimensional analysis of the structure in Figure 8 provide an ingenious andeconomical mechanism for automatically averaging over the population of line orientations in two dimensionalspace. However in three dimensional space line probe orientations are uniformly distributed over the sphere oforientation. Averaging over the circle of orientations in a given plane does not provide an unbiased sample of thepopulation of line orientations in three dimensions, as is demonstrated by the shaded regions in Figure 12a.

Circular test probes in a plane sample the set of orientations in a great circle on the sphere, such as the circlecontaining the pole in Figure 12a. This subset of line orientations in a great circle is uniformly sampled in alldirections by a circular line probe in the plane. This implies that the length of line probes with orientations withinfive degrees of the pole (green cap) is the same as then length sampled within five degrees of the equator (redstripe). The green cap is clearly a very much smaller fraction of the area of the sphere of orientation than is the redstripe. The green cap represents a much smaller fraction of the set of line orientations in three dimensions thandoes the red stripe. But the circular test probe samples these regions with the same length of line probe. Thus,such a probe design over-samples orientations near the pole and under-samples them near the equator. This designdoes not provide a uniform sample of the population of line orientations in three dimensions, and will produce abiased estimate of expected values for the line intercept count.

An unbiased sample of this population is provided by a sample design known as the method of vertical sections(Baddeley et al., 1986), illustrated in Figure 13. A reference direction is chosen in the macroscopic specimen,shown in Figure 13a. In the remainder of the analysis, this is called the "vertical direction". Sectioning planes areprepared which contain (are parallel to) this reference direction, as shown in Figure 13b. These sections arechosen so that they represent the variations in the structure with position, and with orientation in "longitude", i.e.,around the vertical direction.

In a typical experiment, a longitude direction is chosen at random and a section is cut that contains this directionand the vertical direction. Two other vertical sections, 120° away from the first, are also prepared. This provides a


random systematic sample of the longitude orientations. Fields are chosen for measurement in these sections thatare uniformly distributed with respect to position on the vertical plane. The vertical direction is contained in eachfield, as shown in Figure 13c, and is known.

Figure 13. The stereological experimental design based on vertical sections givesan unbiased sample of the population of line orientations in three-dimensional space.

In order to provide an unbiased sample of the population of line orientations on the sphere, a grid is constructed ofline probes that have the shape of a cycloid curve1, shown in Figure 13d. This curve has the property that thefraction of its length pointing in a given direction decreases as the tangent rotates from the vertical direction in amanner that is proportional to the sine of the angle from the vertical. The need for this sine-weighting is indicatedin Figure 12b. Two equal intervals in the range of φ values are shown, one at the pole (green) and the other at the

equator (red). The cycloid test line exactly compensates for the sine weighting required of the φ direction with

longer length of lie probe near the equator (red segment) and shorter near the pole (green segment). This sineweighting of the line length produces an unbiased sample of the population of orientations in three dimensionalspace. A grid consisting of cycloid test lines with known dimensions, so that the total length of lines probed isknown, forms the basis for an unbiased PL count in three dimensional space.

1The cycloid curve is generated by a point on the rim of a wheel as the wheel rolls along a horizontal road. Cycloid segments used inthe method of vertical sections as shown in Figure 4.13 correspond to the curve traced out by the first half turn of the wheel.


Figure 14. Line intercept count made on a presumed vertical section using a cycloidtest line grid gives an unbiased estimate of the surface area per unit volume, SV.

Arrow indicates the vertical direction in the specimen.Probe population: Lines in three dimensional spaceThis sample: Vertical section with cycloid line probes; 12 cycloid units

joined together in three linesCalibration: L0=28.8 µm. Each unit has height h=28.8/6 = 4.8 µm

length of each cycloid = l = 2•h = 9.6 µmtotal probe length = 12• l = 115.1 µm

Event: Cycloid probe intersects the feature boundary lineMeasurement: Count the intersectionsThis count: 24 intersectionsRelationship: <PL> = (1/2) SVNormalized count: PL=24 counts / 115.1 µm = 0.21 counts/µmGeometric property: SV = 2 PL = 0.42 (µm 2/µm3)

Figure 14 reproduces the anisotropic structure in Figure 8 assuming it is a representative vertical section through athree dimensional structure. The height of the box containing the grid is calibrated at 28.8 µm. The grid is madeup of 12 cycloid segments joined together in rows of four. The height of each cycloid segment is (28.8/6) = 4.8


µm. The length of a cycloid segment is twice its height, or 9.6 µm. (Figure 12 has the dimensions of a cycloidsegment.) Thus, the total length of lines in the grid is (12 segments x 9.6) = 115.1 µm. A total of 24 intersectionswith aß boundaries are marked and counted. The line intercept count, PL = 24/ 115.1 = 0.21 (counts/µm). This isan unbiased estimate of the expected value <PL> for lines in three dimensional space. It may thus be used toprovide an estimate of the surface area density through the equation:

SVαβ = 2 < PL >αβ = 2 ⋅0.21 1

µm( ) = 0.42 µm2

µm3

(4.15

In the analysis of Figure 8 as a two dimensional structure, line intercept counts were made in the vertical andhorizontal directions (Figure 8). Results of this analysis gave PL,h = 0.059 (1/µm) and PL,v = 0.45 (1/µm).Interpret these counts as directed probes in the three dimensional population of lines. According to equation(4.14), these counts measure the total projected area of particles on a plane perpendicular to the probe direction,and are related to the average of the cosine of the angle that surface elements make with the probe direction.Measurement with the cycloid test line grid provides an estimate of the unbiased value for SV. These results maybe combined to provide a first estimate of the average of the cosines of surface element normals with the horizontaldirection and the vertical direction in Figure 8:

< cosα >h =PL,h

SV

= 0.0590.42

= 0.14

< cosα > v =PL,v

SV

= 0.450.42

= 1.09

(4.16

The maximum value of the cosine function is of course 1.00. The estimate for the vertical projection violates thislimting value. This impossible result arises because counts were obtained from a single placement of the gridsinvolved, and these counts are only estimates of their expected values. Further, the variance of the ratio of twostatistics, like PL,v / SV is the sum of the variances of the individual statistics; the precision of the estimate from asingle field is low. Nonetheless, it is clear from these results (as from a casual inspection of the structures, thatmost of the surface elements in this structure have normal directions near the vertical where α is near zero and cos

α is near 1.0.

In order to help visualize the geometric significance of the surface area value extimated above it is useful tocompute the mean lineal intercept of the features in the structure. This requires an estimate of the volume fractionof the ß phase. Apply the point count to the grid in Figure 8: there are three hits on ß particles. Thus, a very roughestimate of VVß is 3/25 = 0.12. Insert this result into the expression for the mean lineal intercept given in equation(3.2), Chapter 3.

< λ >= 4 ⋅ 0.120.42

= 1.2µm (4.17

For plate-like structures such as those shown in Figure 8, <λ> reports the thickness of the plate. This value is

plausible for the average plate thickness in this structure.


Three Dimensional Microstructures; Line Length and the Area Point Count

Figure 15. Illustration of the use of a plane probe to intersect linear structures in a volume.Lineal features in a three-dimensional specimen reveal themselves on the plane probe

as a collection of points. Counting the number of intersections provides a tool tomeasure the total length of the structures as shown in Figure 16.

Figure 15 shows a specimen with some lineal features as part of its microstructure. The triple lines in a cellstructure or grain structure described in Chapter 3 provide a common example of such lineal structural features.Dislocations in crystals observed in the transmission electron microscope provide another example. Edges ofsurfaces are also lineal features. Tubular structures, such as capillaries, or plant roots, may be approximated ascollections of space curves. The total length of a one dimensional feature set may be estimated from observationson a set of plane probes. Line features intersect plane probes in a collection of points, as shown in Figure 16.These points are counted over a field of known area. The area point count PA is the ratio of the number of pointsof emergence of these line features in the area, normalized by dividing by the area. If the collection of fieldsincluded in the measurement provides a uniform sample of the population of positions and orientation of planes inthree dimensional space, then this count provides an unbiased estimate of LV, the length of line features in unitvolume of structure:

< PA >= 12 LV (4.18

Construction of a representative sample of the population of planes in three dimensional space is a challenge, sincesectioning the sample to form a plane probe divides the sample. For the special situation in which the linealfeatures are contained in a transparent medium and can be viewed in projection a procedure similar to the methodof vertical sections is available (Gokhale, 1992).


Figure 16. Point count made on a section through a structure containinglinear features (shown here with finite cross-sectional area for clarity).Probe population: Planes in three dimensional spaceThis sample: Square area contained within the measurement frameCalibration: L0=40.0 µm; probe area A0 = (40)2 = 1600 µm2

Event: Plane intersects the linear features (observed as a small feature)Measurement: Count the intersectionsThis count: 13 intersectionsRelationship: <PA> = (1/2) LVNormalized count: PA=13 counts / 1600 µm2 = 0.0081 counts/µm2

Geometric property: LV = 2 PA = 2 • 0.0081 = 0.016 (µm/µm 3) = 16 (km/cm3)

Figure 17 shows a single phase cell structure (e.g., grains in a polycrystal, or botanical cells). The triple points onthis section result from the intersection of the plane probe with the triple lines in the three dimensional structure.Calibration shows that the square area to be sampled is 31.1 µm on a side; the area thus probed in this field is(31.12 = 967 µm2). The triple points in this area are marked and counted. The grid facilitates counting the triplepoints by dividing the area into sixteen small squares, which are counted systematically. There are 37 triple pointsin the area. Thus, PA = 37/ 967 = 0.038 (counts/µm2). Assuming this field is an unbiased sample of the


population of planes in the specimen, equation (4.18) may be used to estimate the length of triple lines in this cellstructure:

LV = 2 < PA >= 2 ⋅ 0.038 = 0.076 µm

µm3 (4.19

In more familiar units, this length corresponds to 76 (km/cm3). This at first surprisingly large result is not unusualfor line lengths in microstructures.

Figure 17. Section through a cell structure; a count of the triple points measuresthe length of triple lines in the three-dimensional structure, LV.

Probe population: Planes in three dimensional spaceThis sample: Square area contained within the gridCalibration: L0=31.1 µm; probe area A0 = (31.1)2 = 967 µm2

Event: Plane intersects the triple line (observed as triple point)Measurement: Count the intersectionsThis count: 37 intersectionsRelationship: <PA> = (1/2) LVNormalized count: PA=37 counts / 967 µm2 = 0.038 counts/µm2

Geometric property: LV = 2 PA = 2 • 0.038 = 0.076 (µm/µm 3) = 76 (km/cm3)


Figure 4.18. A two phase microstructure in which the a phase is a cell structure. The length per unitvolume, LV, of three kinds of triple lines may be characterized: ααα, ααβ, and ααα (occupied)

Probe population: Planes in three dimensional spaceThis sample: Square area contained within the gridCalibration: A0=182.6 µm 2

Event: Plane intersects each class of triple lineMeasurement: Count the intersectionsThis count: ααα=19, ααβ=39, ααα(occ)=15Relationship: <PA> = (1/2) LVNormalized count: PAααα=19 / 182.6 µm 2 = 0.10 counts/µm2

PAααβ=39 / 182.6 µm 2 = 0.21 counts/µm2

PAααα(occ)=15 / 182.6 µm 2 = 0.08 counts/µm2

Geometric property: LVααα = 2 PAααα = 2 • 0.10 = 0.20 (µm/µm 3)LVααβ = 2 PAααβ = 2 • 0.21 = 0.42 (µm/µm 3)LVααα(occ) = 2 PAααα(occ) = 2 • 0.08 = 0.16 (µm/µm 3)

Figure 18 shows a two phase structure (α + ß) with particles of the ß phase distributed along the αα interfaces

and ααα triple lines. This is inferred from the qualitative observation that the ß particles are all situated at the

former ααα triple lines. One can visualize three different types of triple lines in this structure through their

corresponding triple points: ααα, ααß, and ααα triple points that are not present because they are occupied by ß

particles (ααα’). These three classes of triple points are marked with respectively red, green and blue markers in

the field be analyzed in Figure 18. These results are tabulated in Table 4.3.


Table 4.3. Triple point counts in Figure 18. Calibrated probe area is 182.6 µm2.

Category Mark Counts PA (1/µm2) LV (µm / µm3)

ααα red 19 0.10 0.20

ααß green 39 0.21 0.42

ααα(occ) blue 15 0.08 0.16

The total length of ααα triple line, occupied plus unoccupied, is 0.36 (µm / µm3). The fraction of the ααα triple

line occupied by ß particles is 0.16 / 0.36 = 0.44. Since the volume fraction of ß is small (probably only a fewpercent, since there are no hits on ß for the points in the grid in Figure 18) this observation that almost half of thecell edge network in three dimensional space is occupied by the ß phase measures the evidently strong tendency ofß to be associated with the cell edge network. Further, although the ß phase is sparse, the triple line length in thecategory ααß is about the same as the total length as that of the cell edge network.

Invoking equation (4.18) in these last two examples presumes that the fields chosen are unbiased, thoughextremely limited, samples of the population of plane probes in three dimensional space. The realization of thiscondition borders upon intractable, particularly with respect to acquiring planes whose normals sample thepopulation of orientations on the sphere. The method of vertical sections devised for line probes has the advantagethat each vertical plane section contains all of the line orientations from vertical to the equator. A plane probethrough an opaque specimen has a single orientation; each orientation contained in the sample requires a differentsectioning plane, and each section cuts the sample into two parts. Preparation of a series of plane probes withnormals that are distributed in some sine weighted fashion with respect to an overarching vertical direction in thespecimen may be possible if a large volume of specimen material is available (the Orientator, shown in Chapter 7),but it requires a significant effort. A simpler sample design that may accomplish this is shown in Chapter 6.

If the sample is transparent, so that the lineal features may be viewed as a projected image, then a strategy similarto the method of vertical sections for lines may be applied. Figure 19 shows the projected image of a thick slab ofa microstructure with internal lineal features. A probe line on the projection plane corresponds to a probe plane,defined by that line and the projection direction, that passes through the transparent structure. Intersection pointsbetween the probe line and the lines in the projected image have a one-to-one correspondence to intersections ofthe lineal features in the structure with the associated probe plane. The area probed is L0 • t, where L0 is the lengthof the line probe on the image and t is the specimen thickness. The PA count is the number of intersections withthe probe line, divided by this area.

If the lineal structure is not isotropic, PA will be different for different directions of the probe line in the projectionplane. Vertical and horizontal lines give different counts in the structure shown in Figure 19. The statistical testcompares the number of counts using the square root of the number as the standard deviation. In the example, 30vertical counts is clearly greater than 12 horizontal counts (30±√30 or 30±5.5 compared to 12±√12 or 12±3.5). Inorder to obtain an unbiased sample of the population of orientations of planes in space, choose a vertical directionfor the specimen. Prepare slices of known thickness that contain this vertical direction and uniformly sample theequatorial circle of longitudes; these are called "vertical slices". To obtain a uniform sampling of the latitude angle


(angle from the pole) it is again necessary to sample more planes with orientations near the equator than near thepole. The orientation distribution of plane normals must be sine weighted.

Figure 19. A microstructure consisting of linear features viewed as a projectionthrough a section of thickness t = 2 µm.

Probe population: Planes in three dimensional spaceThis sample: Planes represented by their edges which appear as lines on the grid

(vertical and horizontal lines represent two sets of planes)Calibration: L0=20 µm, A0 = L0 • T • 5 planes = 20 • 2 • 5 = 200 µm2

Event: Linear features intersect the horizontal or vertical planesMeasurement: Count the intersections with the horizontal and vertical grid linesThis count: Vertical (red arrows) = 30

Horizontal (green arrows) = 12Relationship: <PA> = (1/2) LVNormalized Count: PA = (30+12)/200 µm2 = 0.21 (counts/µm2)Geometric Property: LV = 2 PA = 0.42 µm / µm 3

This can be accomplished by using cycloid shaped line probes on the projected image, as shown in Figure 20. Inthis application, the cycloids in the grid must be rotated 90° from the direction used in the line probe sample designbecause the normals to the cycloid curve (representing the plane normals in the structure) must be sine weighted.


Each cycloid curve, when combined with the projection direction, generates a cycloid shaped surface with normalsthat provide a sine weighting of the area of the surface as a function of colatitude orientation. If PL is the lineintercept made on these cycloid line probes in the projected image, then the length of lineal features in unit volumeof the structure is given by

LV = 2 < PA >= 2t

< PL > (4.20

Figure 20. A line probe on a projected image represents a surface probe in the volumebeing projected (a). A cycloid-shaped line probe (b) produces a cycloidal surface probe,

which uniformly samples the latitudinal angle of the probe. Note that the cycloid isrotated 90 degrees with respect to the orientation used in Figure 13.

This result also forms the basis for estimating the total length of a single lineal object suspended in threedimensional space, like a the whole root structure of a plant, a wire frame object or the set of capillaries of anorgan. In this application it must be possible to view the whole lineal structure from any direction. Choose avertical direction, then a vertical viewing plane. The plane of observation must have an area A0 large enough toview the entire object. Visualize a box with the area of the projection plane and depth t large enought to contain thewhole object. The volume of the box is A0 • t. Construct a grid of cycloid test lines with major axis parallel to thevertical direction on the projection plane. The total length LT of these probes is calibrated and known. Countintersections of the cycloid line probes with the projected lineal features. If PL is the number of intersections perunit length of line probes, then manipulation of equation (4.20) gives

LV = LA0t

= 2t

< PL >= 2t

LT

L = A0

LT

(4.21


where is the expected value of the number of intersections that form between the cycloid grid and theprojected image of the structure. Thus, a simple count of the number of intersections between the cycloid grid andlines in the projected image, replicated and averaged over a number of orientations for the vertical projection plane,provides an unbiased estimate of the actual length of the feature in three dimensions.

Summary

Normalized geometric properties, volume fraction VV, surface area density SV and length density LV areaccessible through simple manual counting measurements applied to interactions between grids that act asgeometric probes and corresponding features in the three dimensional microstructure. Correspondingmeasurements performed with image analysis software may automate these measurements if the features to beanalyzed can be detected satisfactorily. Geometric properties of single features can also be estimated if theobservations can encompass the entire feature.

The connecting relationships make no assumptions about the geometry of the microstructure. It may be simple orcomplex, and exhibit anisotropies and gradients. Each of the relationships involves the expected value of counts ofsome event that result from the interaction of probe and microstructure. The typical stereological experiment isdesigned to yield an unbiased estimate of the corresponding expected value. This requires a sample design thatguarantees that the population of probes, encompassing all possible positions and, for lines and planes, allorientations on the sphere, will be uniformly sampled.

Uniform sampling of all positions requires selection of fields that accomplish that goal. Uniform sampling oforientations can be accomplished with appropriately configured test probes. In two dimensional structures,circular test lines automatically sample all orientations uniformly. The method of vertical sections uses cycloids toguarantee an unbiased sample of orientations of line probes in three dimensional space. A similar procedure, withthe cycloids rotated ninety degrees, and applied to projected images, gives isotropic sampling of the population oforientations of planes in three dimensions.


Chapter 5 - Less Common Stereological Measures

This chapter deals with manual stereological measurements that provide access to geometric properties that arerelated to the curvature of the objects of analysis. It begins with the concept of the spherical image of surfaces andits relation to the number and connectivity of the features the surfaces enclose. The probe for this analysis is aplane that is visualized to sweep through the three dimensional structure. This probe is implemented through thedisector, or more generally by serial sectioning. The concept of integral mean curvature is then developed withsome insight into the meaning of this rather abstract notion. Plane probes yield a feature count in the plane whichmeasures integral mean curvature. The same information may be obtained in a more general way through the areatangent count. Polyhedral features have edges and corners which contribute to the spherical image of the featureand its integral mean curvature. The latter property provides access to the mean dihedral angle at edges through acombination of counting measurements.

Three Dimensional Features: Topological Properties and the Volume Tangent Count

The topological properties, number NV and connectivity CV, provide rudimentary information about threedimensional feature sets. A fundamental property of surfaces that is related to these topological properties derivesfrom the concept of the spherical image, Ω of the surface.

Figure 1. The spherical image of a point P on a surface (a) is a point P’ on the unit sphere (b).The spherical image of a patch of surface dS (c) is a patch dΩ on the unit sphere (d).

If a surface is smoothly curved, every point P on it has a tangent plane, as shown in Figure 1a. (If the surface isnot smoothly curved, i.e., has edges and corners, then they also contribute to the spherical image.) A tangentplane at P also has a vector, pointing outward from the surface, called the local surface normal. This vectorrepresents the local orientation of the patch of surface at P. This orientation can be mapped in longitude andcolatitude as a point P’ on the sphere of orientation, as shown in Figure 1b. The point P’ on the unit sphere is

Chapter 5 - Less Common Stereological Measures Page # 71

called the spherical image of the point P on the surface. It represents the local direction of the surface. For a smallpatch on a curved surface the collection of its normal directions map as a small patch on the unit sphere, as shownin Figure 1c,d. This is the spherical image of the patch.

Figure 2. Convex bodies (a) are composed of convex surface elements. Non-convex (b,c) bodies haveconvex and saddle surfaces, and may also have concave surface elements. Holes in multiply connected

closed surfaces (d) contribute a net of -4π spherical image for each hole.

Now think about a convex surface (no bumps, dimples or saddles), as shown in Figure 2a. The spherical imagesof the collection of normal directions for such a surface cover the sphere of orientation exactly once. For everypoint that represents a direction on the sphere there is a point on the surface that has a normal in that direction. Nonormal direction is represented more than once. Thus, the spherical image of any convex body covers the unitsphere exactly once, as shown in Figure 2a, and may be evaluated as the area of the unit sphere, which is 4πsteradians1. This result is the same for every convex body, no matter what its shape or size. The spherical imagethus has the character of a topological property, at least for convex bodies, i.e., it has a value that is independentof the metric properties of the body. For a collection of N convex bodies, the spherical image is 4πN.

1A steradian is a unit of solid angle represented by an area on the sphere of orientations analogous to the unit, radian, which applies toangles in a plane. The circle in a plane subtends 2π radians (the circumference of a circle with unit radius). The sphere in threedimensions subtends a solid angle of 4π steradians.


Particles with smooth bounding surfaces that are not convex, as shown in Figure 2b and c, may have threedifferent kinds of surface patches: convex, concave, and saddle surface, as shown in the figure. At any point on aconvex patch of surface, the tangent plane is outside the surface. At points on a concave patch, the tangent plane isinside the surface. At any point on a saddle surface patch the tangent plane lies partly inside the surface, partlyoutside. Saddle surface patches (curved outward in one direction and inward in another like a potato chip) arerequired to smoothly transition between convex and concave elements, or to surround holes in the particle, as inthe inside of a doughnut, as shown in Figure 2c. Patches of each kind of surface have their own collection ofnormal directions which may be mapped on the unit sphere.

It can be shown , and this appears remarkable at first sight, that if the parts of area of the unit sphere that arecovered by the convex and concave patch images are counted as positive and the parts covered by saddle imagesare counted as negative, then the result, called the net spherical image,

Ωnet = Ω convex + Ωconcave − Ω saddle = 4π(1− C ) (5.1

where C is the connectivity of the particle. For simply connected features (no holes), C = 0, and Ωnet = 4π. Thisis a generalization of the result for convex bodies described above. Where the shape departs from a simple convexclosed surface, the extra convex and concave spherical image is exactly balanced by the (negative) spherical imageof saddle surfaces on the feature, so that the unit sphere again is covered (algebraically speaking) exactly once. Ifthe feature has holes in it, every hole requires a net of (- 4π) steradians of saddle surface, which thus gives rise toequation (5.1). For a collection of N particles in a structure, with a collective connectivity of C, the net sphericalimage of the collection of features is

Ωnet = 4π(N − C) (5.2

The difference (N - C) is called the Euler characteristic of the particles in the structure.

The probe used to measure these properties is the sweeping tangent plane. Figure 3 shows a set of particles in aspecimen. Imagine that the plane shown at the top of the figure is swept through the structure from top to bottom.Such a probe can be realized in practice by confocal microscopy in which a plane of focus is translated through atransparent three dimensional specimen. For opaque specimens where this is not possible, the sweeping planeprobe may be visualized by producing a set of closely spaced plane sections (serial sections) through the structureand examining the changes that occur between successive sections. The disector probe (Sterio, 1984) is theminimum serial sectioning experiment, employing information from two adjacent planes.

The events of interest to the measurement of spherical image are the formation of tangents with the particlesurfaces as the plane sweeps through the particles in the structure. Three kinds of tangent events may occur: at aconvex surface element (++); at a concave element (--), or at an element of saddle surface (+-). These threesituations are pictured in Figure 3. These events are marked and counted separately. The result is normalized bydividing by the volume swept through by the sweeping probe. The spherical image per unit volume of each classof surface in the structure is given by


ΩV ++ = 2πT

V ++

ΩV −− = 2πT

V −−

ΩV +− = 2πT

V +−

(5.3

Combining equations (5.1) , (5.2) and (5.3) leads to a simple relation between the Euler characteristic of acollection of particles and the volume tangent counts

NV − CV = 14π Ω

V ++ + ΩV −− −Ω

V +−( ) = 14π 2πT

V++ + 2πTV −− − 2πT

V +−( ) (5.4

which further simplifies to

NV − CV = 12 TVnet

= 12 T

V ++ + TV−− − T

V +−( ) (5.5

Figure 3. The sweeping plane probe forms tangents with convex (++), concave (––)and saddle (+–) surface elements.

Probe population: Sweeping plane in three dimensional spaceThis sample: The specimenCalibration: None requiredEvent: Plane forms tangents with convex, concave and

saddle surfaceMeasurement: Count each class of tangent formedThis count: T++ = 16; T–– = 1; T+– = 11Relationship: N – C = (1/2) Tnet = (1/2) (T++ + T–– – T+–)Geometric property: Actual count: N – C = (6 – 3) = 3

Tangent count: N – C = (1/2)•(16+1–11) = 6/2 = 3


For a collection of convex bodies CV = 0, as are TV-- and TV+-. Every particle has exactly two convex tangents,and counting tangents is evidently equivalent to counting particles. More generally, for a collection of simplyconnected (not necessarily convex) particles, no matter what their shape, CV = 0, and the net tangent count givesthe number of particles in the structure. At the other extreme, for a fully connected network (one particle) NV <<CV, and the net tangent count (which will be mostly saddle tangents) gives (minus) the connectivity. The casewhere both number and connectivity are about the same order of magnitude is fortunately rare in microstructures.In this case the separation of number and connectivity is a challenge. A sufficient volume of the sample must beexamined, either in a confocal microscope or by serial sections, to encompass many whole particles so thatbeginnings and ends of specific particles may be identified and counted to determine NV. This information maythen be combined with the measurement of the Euler characteristic to estimate the connectivity.

Figure 4. The simple disector analysis measures the number density, NV,of convex features in three dimensional space.

Probe population: Set of disector planes in three dimensional spaceCalibration: A0=64 µm2; h = 0.5 µm, V 0 = 32 µm3

Event: Feature in counted unbiased frame on the reference planedoes not appear on the look-up plane

Measurement: Count these eventsThis count: N=6Relationship: NV = N / V0Geometric property: NV = 6 / 32 µm3 = 0.19 µm –3 = 1.9•1011 per cm3


The disector is a minimum serial sectioning experiment composed of two closely spaced planes. The distancebetween the planes, h, must be measured so that, together with area of the field analyzed on the plane, the volumecontained in the disector sample is known. The spacing h must be small enough so that tangent events that occurin the volume between the planes can be inferred unambiguously. Usually this means that the spacing must be afraction (perhaps 1/4 or 1/5) of the size of the smallest features being analyzed. Figure 4 shows the two planes ofthe disector viewed side by side for a simple structure composed of convex bodies. An unbiased counting frame2

with area, A0, is delineated on the first plane, called the reference plane. The second plane is called the look-upplane. Each feature on the reference plane is viewed in conjunction with the corresponding region on the look upplane. If the particle still persists on the look-up plane it is not marked. If there is no particle section observed inthe region then it is inferred that the particle came to an end between the planes. The particle is marked on thereference plane to indicate that a convex tangent has occurred within the volume of the disector. Counting themarks is equivalent to counting particle ends. Since in this simple case of convex particles each particle has onlyone end, this is equivalent to counting particles. If N particle ends are counted, then N/(h • A0) is an estimate ofNV. If the disector is chosen to sample the population of disector volumes uniformly then this provides anunbiased estimate of NV.

Figure 5 a,b (continued...)

2A potential bias in counting particles in a plane probe frame arises because some particles will intersect the boundary of the frame.Further, large particles are more likely to intersect the frame boundary than small ones. The bias can be eliminated by assigning acount of 1/2 to particles that cross the frame. A more general strategy involving the concept of an unbiased frame is described in detailin a later section of this chapter.


cFigure 5. The sweeping tangent analysis analysis using the disector measures the Euler characteristic offeatures in three dimensional space. The events are most easily seen when the two images are overlaid(Figure 5c), and are marked with arrows (red = T++, green = T––, blue = T+–). Note that features

are not counted if any part of them touches that touch the exclusion line in either image.Probe population: Set of disector planes in three dimensional spaceCalibration: A0= (18 µm)2 = 324 µm2; h = 0.5 µm, V 0 = 162 µm3

Event: Features in counted unbiased frame on the each planeindicate various tangent events in the volume

Measurement: Count these eventsThis count: T++ = 3 (red)

T–– = 3 (green)T+– = 5 (blue)

Relationship: NV – CV = 1/2 • (TV++ + TV

–– – TV+–)

Geometric property: NV – CV = 1/2 • (3 + 3 – 5) / 162 µm3 = 0.0031 µm –3

= 3.1•109 cm–3

If the look-up and reference planes are now reversed, particles on the second plane that do not appear on the firstimply that a convex tangent has formed within the volume of the disector. On the average, the number of counts inthe second case will be equal to that in the first, but this will not be true for individual disector fields.


Figure 5 shows a more complex structure with particles that have concave as well as convex boundary segmentson a section. The presence of concave boundary elements on the plane probes results from sections through saddlesurface elements and/or concave elements in the three dimensional structure. The sweeping plane probe isexpected to form tangents of all three types of surfaces. A tangent is formed with a convex surface element (T++ )if an isolated particle on the reference plane does not appear on the look-up plane, and also if an isolated particleon the look-up plane does not appear on the reference plane. A tangent with a concave surface element (T–– ) isindicated if a hole on the reference plane disappears on the look-up plane, and also if a hole seen on the look-upplane disappears on the reference plane. If one feature on the reference plane becomes two on the look-up plane,or two on the reference plane becomes one on the look-up plane, then a tangent with an element of saddle surface(T+–) is inferred. The three classes of tangent planes inferred from the disector sample in Figure 5 are indicatedwith red (++), green (––) and blue (+–) markers. The analysis of the disector is presented in the figure caption.The Euler characteristic estimated from this single observation is 0.0031 µm–3. If it is assumed that these particlesare simply connected (CV=0), the number of particles can be estimated to be 0.0031 µm–3. Relative amounts ofeach of the three classes of spherical image may also be estimated from the tangent counts by applying equations(5.3).

Three Dimensional Features: the Mean Caliper Diameter

The notion of the diameter has a straightforward meaning for a sphere. For more general convex bodies, the"diameter" of a particle is different in different directions. Figure 6 illustrates the concept of the caliper diameter,D, of a convex particle. Choose a direction and visualize the sweeping plane probe moving along that direction.The probe will form two tangents with the particle. The perpendicular distance between these two tangent planes isthe caliper diameter of the particle in that direction. This diameter has a value for each direction in the population oforientations. If the value of the diameter is averaged over the hemisphere of orientation (the other hemispheregives redundant information), the mean caliper diameter of the particle is obtained. This concept may be applied toconvex bodies of any shape. It provides one measure of the particle size of features in the structure.

Figure 6. The mean caliper diameter, D, is the distance between parallel tangent planes in any directionon a convex particle. This distance is sensed by plane probes in three-dimensional space.


An estimate of the mean caliper diameter may be obtained using a plane probe through the three dimensionalstructure. Consider the subpopulation of plane probes that all have the same normal direction. Individual planes inthis subpopulation are located by their position along the direction vector. In order for a plane probe to produce anintersection with a particle in the structure, its position must lie between the planes that are tangent to the ends ofthe particle in that orientation. The event of interest is that the plane probe intersects the particle. This event willproduce a two dimensional feature, a section through the particle, observed on the plane probe. The number ofthese events, i.e., the number of these two dimensional features that appear on the probe, is counted. The count isnormalized by dividing by the area of the probe sample to give NA, the number of features per unit area of probescanned. The governing stereological relationship is,

NA = NV D (5.6

The number of particles per unit volume can be estimated separately using the disector probe as described in thelast section. With this information, the mean caliper diameter of particles in the structure can be evaluated from theexpected value of a feature count on plane probes. This result is limited to convex bodies, but the shapes of theparticles are not otherwise constrained. In order to obtain the mean caliper of particles averaged over orientation itwill be necessary to sample the population of plane orientations in three dimensions. The difficulties in obtainingan unbiased sample of orientations of plane probes have already been discussed.

For a single orientation of plane probes, this count provides a measure of the mean caliper diameter of particles inthe direction perpendicular to the probe planes. This result can be visualized as an application of equation (4.14) inChapter 4, which is the stereological basis for measuring the length of lineal features in a three dimensionalstructure. Visualize a subpopulation of parallel planes that share some normal direction given by coordinates (θ,φ)

on the orientation sphere. Replace each convex particle with a stick that is the same length as the caliper diameterin the direction perpendicular to those planes D(θ,φ). Then each particle section on the plane probe may be

visualized as resulting from an intersection of a plane probe with one of these sticks. NA for the particle sections isthe same as PA for the collection of sticks for this structure. The feature count thus measures the collective lengthof these sticks, or more precisely, the collective lengths of the caliper diameters of the particles in the structure, inthe direction that is perpendicular to the plane probes.

N A θ ,φ( ) = NV D θ,φ( ) (5.7

For a collection of particles that show a tendency to be elongated or flattened in a particular direction in spacemeasurement of NA on planes with different orientations provides one measure of the anisotropy of the particleshapes. Ratios of NA counts on planes oriented in different directions directly yield ratios of the mean caliperdiameters in those directions:

NA θ1 ,φ1( )N A θ2 ,φ2( ) =

NV D θ1,φ1( )NV D θ2 ,φ 2( ) =

D θ1,φ1( )D θ2 ,φ 2( ) (5.8


Note that these caliper diameters are measured in the direction that is parallel to the normal to the plane probe.

Thus a feature count on a section, when combined with a disector measurement of NV gives access to the meancaliper diameter of convex particles, one convenient measure of particle "size". If the particle shapes areanisotropically arranged in space, then a feature count on an oriented plane probe provides a measure of the meanparticle diameter in the direction perpendicular to the probe.

Mean Surface Curvature and Its Integral

This section introduces curvature related geometric properties that are significantly more abstract than thestraightforward geometric properties defined and measured in previous sections. These hard-to-visualizeproperties are introduced here because they have geometric meaning that is general for structures of arbitrarygeometry, and they can be measured (they are what is measured) by the feature count when the features are notlimited to being convex, as they were in the previous section. For some special classes of features (plates ormuralia, or rods or tubules) these geometric properties have a meaning that can be visualized and put to conceptualuse. For more general kinds of three dimensional features, although the meaning is abstract, at the very least themeasurement provides an additional descriptor of the geometry of the system.

The central concept, the mean curvature at a point on a smooth surface, is the geometric property of surfacesthrough which capillarity effects operate. If surface energy or surface tension plays a role in the problem beinginvestigated, access to a measure of the mean curvature provides geometric information that has directthermodynamic meaning (DeHoff, 1993).

Figure 7. Curvature at a point P on a curve in two dimensional space is the reciprocal ofthe radius of a circle that “passes through three adjacent points” on the curve at P.

In order to present the concept of curvature at a point on a surface, it is first useful to recall the concept ofcurvature of a plane curve in two dimensional space. Figure 7 shows a curve in the xy plane. To define thecurvature at a point P place two other points, A and B, on the curve. Construct a circle through these three points.


This circle has a center, O, and a radius r. Now let A and B approach P. The point O moves and the radiuschanges. In the limit as A and B arrive at P, the center approaches the center of curvature for the point P, and theradius becomes the radius of curvature. This limiting circle, which just kisses the curve, is called the osculatingcircle. The curvature of the curve at P is the reciprocal of this radius. It can also be shown that the curvature is therate of rotation of the tangent to the curve as the point P moves along the arc length, s:

k = 1r

= dθds

(5.9

This two dimensional concept is the basis for defining the curvature at a point on a surface in three dimensionalspace.

Figure 8. Curvature at a point P on a surface is reported by two principle normal curvatures κ1 and κ2measured on two orthogonal normal planes through P. Of all the possible planes through P that

contain the normal direction (a), the two with the minimum and maximum radii (b) givethe principle normal curvatures.

Figure 8a shows a smooth surface. Focus on the point P on this surface. There is a tangent plane at P, and a localsurface normal, perpendicular to that plane. Imagine a plane that contains this normal direction and intersects thissurface at P. The intersection of this "normal plane" with the surface is a plane curve that passes through P. Theradius of curvature, r, and the curvature, k, of that curve may be defined at P using the concepts developed in theprevious paragraph.

Next visualize another normal plane that makes some arbitrary angle with the first plane. The curve of intersectionof this plane with the surface also has a value of curvature at P. If the piece of surface is a piece of a sphere, these


two curvatures will be the same. However, for an arbitrarily shaped surface, the curvature values on two differentnormal planes will be different. As the normal plane is rotated, the curvature of the intersecting curve at P changessmoothly. As the normal plane is rotated through half a circle, there will be some direction for which the curvaturehas a maximum value, and another direction for which its value is a minimum. In differential geometry, whichdeals with the geometry of entities in the vicinity of a point on the entity, it is shown that the directions for whichthe maximum and minimum values occur are 90° apart. These two directions in the tangent plane are called theprinciple directions. The curvatures in these two directions, κ1 and κ2, are called the principle normal curvatures

at the point P, corresponding to radii r1 and r2. In differential geometry it is shown that these two curvature valuescompletely describe the local geometry near a point on a smooth surface. The local configuration for an element ofsurface around a point P is shown in Figure 8b.

For a spherical surface with radius R, κ1 = κ2 = 1/R at every point on the surface. For a right circular cylinder of

radius r, the maximum curvature occurs on a plane perpendicular to the axis: κ1 = 1/r. The minimum curvature

occurs on a plane containing the cylinder axis, which intersects the surface in a straight line so that κ2 = 0. The

values of κ1 = 1/r and κ2 = 0 are the same at all points on the cylindrical surface. For smooth surfaces with

arbitrary geometry, κ1 and κ2 vary smoothly from point to point and have some distribution of values over the

surface area.

For a surface that encloses a three dimensional volume, i.e., for a closed surface, these curvatures may beassigned a sign. If the curvature vector points toward the inside of the surface it is defined to be positive; if itpoints outward, it is negative. The possible combinations of signs of the curvature give rise to the three classes ofsurface elements described earlier in this chapter:

Convex surface (++) - if both curvatures point inward and are thus positive;Concave surface (––) - if both point outward and are thus negative;Saddle surface (+–) - if one points inward and the other points outward.

These three classes of surfaces were illustrated in Figure 2. In general, if a surface has any departure from asimple convex shape, extra convex lumps, or concave dimples, it must possess patches of saddle surface thatconnect these smoothly. At the boundaries of saddle surface one of the curvatures changes sign (passes from +– to ++, or from +– to ––), so that one of the principle curvatures passes through zero. Such points occuronly along the lines bounding saddle patches (or on the surface of a cylinder), and are called parabolic points.

Two combinations of the principle normal curvatures find widespread application in geometry, topology, physicsand stereology:

The mean curvature, defined by H = 12 (κ1 + κ2 ) , the algebraic average of the two curvatures, and

The Gaussian curvature, defined by Κ = κ1 ⋅ κ2 .

Both have a value at each point on a surface and thus vary smoothly over the surface.

The mean curvature, H, is important for two distinct reasons. Capillarity effects play a crucial role in the formationand performance of most microstructures. These effects arise from the physical action of curved surfaces on theirsurroundings. The geometric property of the surfaces that determines the nature of these capillarity phenomena is


shown in thermodynamics to be the local mean curvature, H. Thus an important component of the behavior ofsurfaces involves the local mean curvature, its distribution and its average value over the surfaces involved.

The second compelling reason for interest in this geometric property has to do with its integrated value for thecollection of surfaces in a microstructure, called the integral mean curvature. This property has a single value for aclosed surface. It may also be evaluated for a collection of surfaces that make up a microstructure. Visualization ofits meaning is elusive. Consider an element (a small patch) of surface with incremental area dS; suppose H is thevalue of mean curvature at that element. Compute the product HdS for the element. Then add these values togetherfor all of the surface elements that bound the particle. The resulting quantity, defined mathematically by

M ≡ HdSS∫∫ (5.10

has units of length. H has units of length-1 and dS has units of length2. (The double integral signs ∫∫ signifies thatthe local incremental value HdS is summed over the whole area of the surface, S.) This abstract geometricproperty has geometric meaning that can be visualized for particular classes of three dimensional features.

As a simple example, apply the concept to a spherical particle of radius R. The principle curvatures are both equalto (1/R), and have the same value at every point on the sphere. Thus, for every point on a sphere the meancurvature is simply 1/R:

H = 12

1r1

+ 1r2

= 1

21R

+ 1R

= 1R

(5.11

Put this value into the definition of M:

Msphere = H dSS∫∫ = 1

RdS =

S∫∫

1R

dS = 1R

Ssphere = 1R

4πR 2 = 4πRS∫∫ (5.12

This result generalizes through the Minkowski formula; for general convex bodies,

Mconvex = 2πD (5.13

where D is the mean caliper diameter defined in the previous section.

We have seen that for a collection of convex bodies the feature count on a plane probe, NA, reports the product ofthe number density and the mean caliper diameter. For non-convex bodies, i.e., for the general case, ageneralization of the feature count reports the integral mean curvature, M. In the discussions up to now, theclasses of features considered produced sections that are free of holes on the plane probe. However, in the generalcase of arbitrarily shaped particles in three dimensional space, some plane probes will produce features with holesin them. In this most general kind of structure, the feature count generalizes to the Euler characteristic of theparticle sections, which is defined to be the number of particles minus the number of holes (N - C). It is useful to


think of this as a count of the net number of closed loops bounding particles in the structure, where loopsenclosing particles are counted as positive and those enclosing holes are negative. The fundamental stereologicalrelationship is

NA net= NA − CA = 1

2πMV (5.14

where MV is the integral mean curvature of the surfaces bounding the particles in the three dimensional structuredivided by the volume of the structure. Thus, no matter how complex the features in three dimensions are, the netfeature count on a plane probe provides an unbiased estimate of this abstract property, the integral mean curvature.It is the three dimensional property that the feature count measures.

Applying Minkowski’s formula for convex bodies, equation (5.13), to equation (5.14):

NA convex= 1

2πMVconvex

= 12π

NV M Vconvex= 1

2πNV 2πD = NV D (5.15

recovers equation (5.6), now seen to be a special case of equation (5.14).

Figure 9 reproduces the microstructure in the left side of the disector in Figure 4 so that a feature count may bemade and illustrated. In this case, the value of the mean curvature MV estimated to be 4.45 µm/µm 3 can be used toestimate the mean caliper diameter. The value of NV obtained from the disector analysis in Figure 4 is 0.19 µm–3.Apply the Minkowski formula, equation (5.13), to compute:

D =MV

2πNV

= 4.452 ⋅π ⋅ 0.19

= 3.72µm (5.16

Figure 10 reproduces the microstructure on the left side of the disector in Figure 5. A net area loop count estimatesthe integral mean curvature for this structure of general geometry.

The integral mean curvature is a key geometric property of a collection of plate shaped particles, or, moregenerally muralia, i.e., "thick surfaces", features that are small in one dimension and extensive in the other two, asshown in Figure 11a. For such features it can be shown that M measures the length of the perimeter of the edge ofthe the plate or muralia, L (DeHoff, 1977):

Mmuralia = π2 L (5.17

At the other extreme of feature shapes, the integral mean curvature reports visualizable information for rod or moregenerally for tubule features. A feature satisfies the definition of a tubule if it is small in two of its dimensions andextensive in the third, as shown in Figure 11b. For this class of features,

Mtubules = πL (5.18


where L is the length of the tubule.

Figure 9. A feature count measures the integral mean curvature, MV, of the boundingsurfaces in three dimensional space.

Probe population: Planes in three dimensional spaceThis sample: The unbiased frame in the fieldCalibration: L0 = 6.9 µm; A0= (6.9 µm) 2 = 47.9 µm 2

Event: Feature lies “within” the unbiased frameMeasurement: Count the featuresThis count: N=34Relationship: <NA> = (1/2π) • MVNormalized count: NA = 34 counts/47.9µm2 = 0.71 counts/µm2

Geometric property: MV = (2π) • 0.71 = 4.45 µm/µm 3

Figure 12 shows a plane probe section through a microstructure consisting of muralia. The vast majority ofsections will appear as curved strips in the microstructure. Orientations and positions of the probe that produce fatsections are not very likely. For muralia equation (5.17) yields,

NA muralia= 1

4 LVmuralia(5.19


where LV muralia is the total perimeter of plates or length of edges of muralia in unit volume of structure. If it isassumed that Figure 12 is an unbiased sample of the population of positions and orientations of plane probes inthe structure, the feature count, NA = 0.074 (1/µm2) gives, for the total length of feature edges in the threedimensional structure = 0.295 (µm/µm 3).

Figure 10. Net feature count for non-convex features in the area measures the integral mean curvature,MV, of the surfaces bounding the features in the three dimensional microstructure.

Probe population: Planes in three dimensional spaceThis sample: The unbiased frame in the fieldCalibration: L0 = 17.7 µm; A0= (17.7 µm) 2 = 313 µm2

Event: Feature (particle or hole) lies “within” the unbiased frameMeasurement: Count the featuresThis count: 8 particles; 2 holesRelationship: <NA>net = <NA>+ – <NA>– = (1/2π) • MVNormalized count: NA net = 8 - 2 counts/313µm2 = 0.019 counts/µm2

Geometric property: MV = (2π) • 0.019 = 0.120 µm/µm 3

Most sections through tubules are small and equiaxed, as shown in Figure 13. Orientations and positions whichproduce a very long section only occur for planes that are nearly parallel to the local axis of the tubule, and areclose to it in position. Such sections will be relatively rare, particularly if the radius is small. From equation (5.18)the expected value of the feature count gives


NA tubules= 1

2 LVtubules(5.20

where LV tubules is the collective length of the rods or tubules in unit volume. For the example shown in Figure13, NA = 0.29 (1/µm2) which estimates a total tubule length of 0.58 (µm/µm 3)

Figure 11. Muralia (a) are “thick surfaces; a flat muralium is a plate. Tubules (b) are “thick space curves

A plane probe intersects particles in three dimensional space to produce a collection of two dimensional features onthe sectioning plane. The number of such features, normalized by dividing by the area of the probe, isproportional to the integral mean curvature of the surfaces bounding the three dimensional particles in the structuredivided by the volume of the specimen (equation 5.14). Integral mean curvature has simple geometric meaning forconvex bodies, for which it reports the mean caliper diameter, platelets or muralia, for which it reports the totallength of edge, and tubules, for which it reports the total tubule length. These results assume the set of planeprobes in the sample are drawn uniformly from the population of positions and orientations of plane probes inthree dimensional space. Oriented plane probes give information about these lengths in the direction that isperpendicular to the probes.


Figure 12. A feature count on a section through muralia measures the integral mean curvature, MV, or themuralia surfaces in three dimensional space. In this case, MV is simply related to the total length of edges

of the muralia, LV muraliaProbe population: Planes in three dimensional spaceThis sample: Area within the unbiased frameCalibration: A0 = 122 µm2

Event: Feature lies “within” the unbiased frameMeasurement: Count the featuresThis count: N=9Relationship: <NA> = (1/2π) • MV muralia = 1/4 LV muraliaNormalized count: NA = 9 counts/122 µm2 = 0.074 counts/µm2

Geometric property: LV muralia= 4 NA = 4 • 0.074 = 0.295 µm/µm 3


Figure 13. A feature count is used to estimate MV, the integral mean curvature. For tubules MV isproportional to the LV tubules length of the tubules.

Probe population: Planes in three dimensional spaceThis sample: Area within the unbiased frameCalibration: L0 = 6.9 µm; A0= (6.9 µm) 2 = 47.9 µm 2

Event: Feature lies “within” the unbiased frame (indicated by color)Measurement: Count the featuresThis count: N=14Relationship: <NA> = (1/2π) • MV tubules = 1/2 LV tubulesNormalized count: NA = 14 counts/47.9 µm2 = 0.29 counts/µm2

Geometric property: MV tubules = 2πNA = 1.84 µm/µm 3

LV muralia= 2•NA = 0.58 µm/µm 3

The Sweeping Line Probe in Two Dimensions

Figure 14 shows a feature in two dimensional space. Any point P on the boundary has a tangent direction and,perpendicular to the tangent and pointing outward, a normal direction. Mapping these directions onto a unit circlecreates the circular image of the points, and a small arc of the boundary ds has a small range of normal directionsthat map as a segment of arc on the unit circle, dθ, as shown in Figure 14a. The circular image of a convex feature

maps point for point on the unit circle exactly once. Thus, no matter what the shape of the particle boundary, solong as it is convex, its circular image is exactly 2π radians, the circumference of the unit circle.


Figure 14. The circular image of a convex feature in a plane is equivalent to the unit circle. Every point onthe feature periphery has a normal direction that corresponds to one point on the circle, as shown by thecolored vectors in Figure 14a. A segment dS on the feature maps to a segment of arc dθ on the circle. Anon-convex feature has regions of negative curvature as shown in Figure 14b. The angular range whichthese cover (shown in green) exactly cancels the extra positive curvature (shown in blue and orange),

again leaving a net circular image of 2π.

If the boundary shape departs from convex, as shown in Figure 14b, then part of it will consist of convex arcsegments and part will be made up of concave arc segments. Mapping the rotations of the boundary normal on theunit circle as the point P moves aroung the perimeter of the particle will produce ranges of overlap as shown inFigure 14d. However, if concave segments are defined to contribute a negative circular image, then the netrotation of the normal vector around the perimeter remains exactly 2π radians because the point chosen to beginand end the map is the same point. Thus, like the spherical image of a surface discussed earlier, the circular imageof the boundary of a two dimensional feature also has the character of a topological property. It is equal to 2πradians, no matter what is the size and shape of the particle enclosed by the boundary.

In a two dimensional structure particles of the ß phase are said to be "multiply connected" if they have holes inthem. The connectivity of a two dimensional particle, C, is equal to the number of holes in the particle. Theboundary of each of these holes, no matter what their size or shape, contributes a net circular image of (-2π)because overall, a hole contributes a net of 2π of concave arc. Thus, the net circular image of a two dimensionalparticle with C holes in it is

θnet = θ+ − θ – = 2π(1− C) (5.21

For a collection of N features with a total of C holes in them,


θnet = 2π(N − C) (5.22

The difference in the topological properties (N - C) is called the Euler characteristic of the two dimensionalcollection of particles. These concepts for a two dimensional structure mirror those presented earlier for thespherical image and the topological properties of three dimensional closed surfaces.

It was shown earlier that a volume tangent count, based on a sweeping plane probe through a three dimensionalstructure, measured the spherical image of the surface bounding particles in the volume. An analogous probe andmeasurement applies to the boundaries of two dimensional features in two dimensions. Figure 15 illustrates theconcept of the sweeping line probe in two dimensions. The line in the figure is swept across the field. In theprocess the moving line forms tangents with elements of the αß boundary. Tangents formed with elements of the

boundary that are convex (T+ ) with respect to the ß phase and elements that are concave (T– ) may be markedseparately and subsequently counted. Dividing these counts by the area of the field included in the count gives thearea tangent count. This is the two dimensional analogue of the volume tangent count described in a previoussection.

The area tangent count applied to boundary elements of a collection of features measures the circular image ofthose features:

θ A+ = πTA+ ; θ A− = πTA−

θ A net= π TA+ − TA−( ) = πTAnet

(5.23

Combine this result with equation (5.22) to show that the Euler characteristic of a collection of two dimensionalfeatures with holes is simply related to the net area tangent count:

NA − CA = 12 TAnet

= 12 TA+ − TA–( ) (5.24

For a collection of convex features CA = 0 (there are no holes) and TA – = 0, so that NA = 1/2 • TA +; the resultthat every particle has two tangents is self evident in this case. If the features are simply connected, i.e., have noholes, then every bounding loop has two terminal convex tangents and, for each concave tangent there is abalancing extra convex tangent. The net tangent count still gives two per particle. If there are holes, equation 5.24gives the Euler characteristic of the collection of features.


Figure 15. An area tangent count is used to estimate the Euler characteristic and integral mean curvature.A horizontal line is swept down across the image and the locations of tangents with convex

and concave boundaries are counted.Probe population: Sweeping line in two dimensional spaceThis sample: Area within the frameCalibration: L0 = 20 µm; A0= (20 µm)2 = 400 µm2

Event: Line forms tangents with convex and concave boundariesMeasurement: Count each class of tangent formedThis count: T+ = 45 (red)

T– = 19 (green)Relationship: <TAnet>= (1/2) • (T+ – T–) = (1/π) • MVNormalized count: NA – CA = (1/2) • (45 - 19) /400 µm2 = 0.0325 counts/µm2

Geometric property: MV = 0.204 µm/µm 3

If the two dimensional structure in this discussion results from probing a three dimensional microstructure with aplane then the expected value of the Euler characteristic on the section is an estimator of the integral meancurvature of the corresponding collection of aß surfaces in the volume, according to equation (5.14). Thus, the nettangent count provides an unbiased estimate of the integral mean curvature:


T Anet = 2 N A net= 1

π MV (5.25

It may appear that the tangent count merely provides the same information that a feature and hole count couldprovide, since in a two dimensional structure the features are visible in the field and can be separately marked andcounted. The tangent count would appear to give redundant information. However, there are some validarguments for considering replacing the feature count with the tangent count:

1. Tangents occur at a point; a point is either inside the boundary of the field, or outside it. Bias due toparticles intersecting the boundary of the field is not a factor in the tangent count;

2. In some structures, e.g., lamellar structures, or structures in which both phases occupy about the same areafraction, features of, say, the ß phase may wander in and out of the boundary of the field several times sothat it is not possible to make the feature count.

3. In a three phase structure (α + ß + ε) part of the boundary of ß particles is αß boundary, and part may be ßεboundary. In this case it is possible make separate area tangent counts of these two kinds of interface andassess the circular image (and the integral mean curvature) of each.

4. If the boundary on a section has vertices, separate application of the tangent count to smooth segmentsversus vertices provides a measure of the dihedral angle at edges in the three dimensional structure thesection samples. This is described in more detail in an example below.

5. Since separate counts are obtained for convex and concave elements of boundary the circular images ofconvex and concave arc in the plane probe structure may be separately estimated. While this informationdoes not have a simple relation to the geometry of the parent three dimensional microstructure (DeHoff,1978) it may be useful information in some applications.

Thus, the net tangent provides an unbiased estimator of MV in all of its applications.

Edges in Three Dimensional Microstructures

Because microstructures are space filling, triple line structures, such as those described in the qualitativemicrostructural state discussion in Chapter 3, are common. A triple line results when three "cells", some or all ofwhich may be the same phase, are incident in three dimensional space. Three surfaces also meet at a triple line,namely the three pairwise incidences of the cells involved. From the viewpoint of any particular cell, the triplelines it touches are edges of that polyhedral shaped body. More generally an edge is a geometric feature whichresults from the intersection of two surfaces that bound a feature to form a space curve, as shown in Figure 16. Inaddition to the properties of space curves (length, curvature and torsion) an element of edge has a dihedral angle.The two surfaces that meet at an element of edge each have a local normal vector. The dihedral angle, χ in Figure

16, is the angle of rotation between these two surface normals3. In general this angle varies from point to pointalong the edge.

The dihedral angle also plays an important role in the physical behavior of structures in which surface tensionplays a role. In the thermodynamics of surfaces that meet at triple lines it is shown that the dihedral angles at thethree edges are determined by the relative surface energies of the three surfaces that meet there. For example, the

3Like surfaces, edges may be convex (ridge with a maximum); concave (valley with a minimum) or saddle (ridge with a minimum orvalley with a maximum) in character. Valley edges by definition have a negative value of χ.


tendency of one phase to spread over a surface between two other phases, i.e., to "wet" the surface, is measuredby this dihedral angle. This property plays an important role in soldering, adhesion, liquid phase penetration,welding, powder processing and the shape distribution of phases at interfaces.

Figure 16. An edge is formed by two surfaces meeting to form a space curve. The angle between thesurface normals at a point P on an edge, χ, is called the dihedral angle.

An element of length of an edge may be thought of as a limiting case of an element of surface for which one of theradii of curvature goes to zero, i.e., sharpens to an angle. With this point of view it can be shown that for apolyhedral particle the edges make their own contribution to the integral mean curvature of the boundary of aparticle:

Medges = 12 χ ⋅dL

L∫ (5.26

where the integration is over the length of edge in the structure. To lend some credence to this assertion that thisproperty is in fact the contribution to M due to edges it may be shown that this result may be used in conjunctionwith Minkowski’s formula for convex bodies, equation (5.13), to compute the mean caliper diameter ofpolyhedral shapes. Consider the cube with edge length e shown in Figure 17a. The dihedral angle at all points onthe twelve edges is π/2. The total length of edge is 12e. The surfaces (faces) all have zero curvature, so that thecontribution of the surfaces to M is zero. Combine equation 5.26 with the Minkowski formula:

Mcube = M faces + Medges = 0 + 12

π2 L = π

4 12e = 3πe = 2πD cube

D cube = 32 e

(5.27


Figure 17. The integral mean curvature for features with edges contains a contribution from edges. For thecube (or any flat faced polyhedron), all of the integral mean curvature resides at the edges. A feature with

curved faces and edges has both contributions.

This is the same result that is obtained by evaluating the caliper diameter of a cube as a function of orientation andaveraging over orientation.

Similar analysis of other shapes provides an efficient way to calculate D averaged over orientation. This isgenerally far easier than the modelling approach using Monte-Carlo sampling introduced in Chapter 11, but thelatter method can also produce the distribution of intercept lengths or areas which are needed to unfold particle sizedistributions.

Figure 17b shows a cylinder with radius r and length l. In this case there are contributions to M from both thecurved surface and the edges. On the curved face of the cylinder the curvature in the axial direction is zero, and themean curvature everywhere has the value (1/2r). The dihedral angle at every point on the edges is π/2. Evaluate Mfor a cylinder:

Mcyl = M faces + Medges = H ⋅ dS + 12

S∫∫ χ ⋅dL

L∫

= 12r

S∫∫ dS + 1

2π2

L∫ dL = 1

2r 2πrl + π4 2 ⋅2πr

Mcyl = πl + π2 r = 2πD cyl

D cyl = 12 l + π

2 r

(5.28


Again, this is the same result that is obtained if D is computed as a function of orientation and averaged over the

hemisphere. Note that for the limiting cases of a plate (r >> l), D = π2 r , while for a rod (l >> r), D = 1

2 l .

Figure 18. The mean dihedral angle at edges, <χ> on the β (colored) phase of this structure can bemeasured with a triple point count combined with a tangent count.

Probe population: Planes in three dimensional space and sweeping lineprobes in two dimensions

Calibration: L0 = 14.7 µm; A0= (14.7 µm) 2 = 215 µm2

Event: Plane intersects ααβ triple line;Sweeping line forms tangent at ααβ triple line

Measurement: Count the triple points; count the tangentsThis count: 49 triple points, 32 tangentsRelationship: <χ> = π TA / PANormalized count: PA = 45 counts/215 µm2 = 0.23 counts/µm2

TA = 32 counts/215 µm2 = 0.15 counts/µm2

Geometric property: <χ> = π • 0.15 / 0.23 = 2.05 radians = 117 degrees


Edges can also be treated as a limiting case of a curved surface in the sense that the integral mean curvature of anedge can be measured by the tangent count, equation (5.24):

TA edges= 1

π MVedges= 1

π12 χ ⋅dL

LV

∫ (5.29

where χ is the dihedral angle measured at any point along the edge. An average dihedral angle may be defined,

χ =

χ ⋅ dLLV

∫dL

LV

∫=

2MVedges

LV

(5.30

MV edges can be measured with the tangent count at edges, using equation (5.29), and LV can be measured bycounting points of intersection of the edge line with the plane probe. Thus, the average dihedral angle for anyspecific type of edge can be estimated from the ratio

χ =2π⋅ TAedges

2PA

=πTAedges

PA

(5.31

Figure 18 shows a two phase structure with ααß triple lines. A sweeping line probe is visualized to move from

the top to the bottom of the field. Tangents with the ααß edge are marked and counted. The total number of ααß

triple points is also counted. With the usual assumptions about the sample plane, these properties are normalizedand used to estimate MV edge and LVααß. These computations are indicated in the caption to Figure 18. The ratiois then used to estimate the average dihedral angle on ααß edges of the ß particles. The result, 2.05 radians or 117

degrees, is the average value of the angle between the surface normals at the edge. This result may be used, forexample, in the assessment of the relative surface energies of the αß and αα interfaces in this system.

Summary

The disector probe is required to obtain information about the topological properties, number (NV) andconnectivity (CV) of surfaces bounding three dimensional features. This is achieved by comparing the structureson a reference plane and a look-up plane in order to infer the occurrence of tangents with the surfaces formed by aplane that is visualized to sweep the volume between the planes of the disector. Counts of the three kinds oftangents measure the spherical image of convex, concave and saddle surfaces in the structure. The Eulercharacteristic, (NV - CV), may be estimated from these tangent counts in the disector probe

NV − CV = 12 TVnet

= 12 TV++ + TV−− − TV+−[ ]


For most structures the connectivity is small in comparison with the number of disconnected parts of thestructure, and the disector probe provides the primary strategy for estimating number density in microstructures.

If the three dimensional particles are convex, then a feature count on a plane probe estimates the product of thenumber density and mean caliper diameter:

NA = NV D

Feature counts on oriented plane probes measure the caliper diameter in the direction of the plane probe normal.

N A θ ,φ( ) = NV D θ,φ( )

For more general structures, the (net) feature count (or equivalently, the net area tangent count) reports the integralmean curvature of the surface in the structure:

NA net= NA − CA = 1

2 TAnet= 1

2 TA+ − TA− = 12π M

Particles of arbitrary shape in the three dimensional structure may, for some positions and orientations of a planeprobe, produce sections with holes in them. In this general case, the feature count is interpreted as the number offeatures in the section minus the number of holes. This net number of features can also be determined from thearea tangent count, which visualizes a line that sweeps across the field of view and forms tangents with convexand concave elements of the boundaries of particle sections. In the general case, either of these counts reports thevalue of integral of the mean surface curvature of the boundaries of particles in the three dimensionalmicrostructure. For convex bodies, M is proportional to the mean caliper diameter. For muralia (plates) it reportsthe length of perimeter of the features in three dimensions. For tubules (rods), M reports their total length in thevolume.

If the features in the three dimensional collection of particles have edges, then these edges contribute to the integralmean curvature of the boundaries of the particles so that

MVtotal= MVsurfaces

+ MVedges= H ⋅ dS

SV

∫∫ + 12 χ ⋅dL

LV

∫

where χ is the dihedral angle between the surface normals at the each point on the edge. This result may be used to

estimate the average dihedral angle along a collection of edges in the structure.

χ =πTAedges

PA


Chapter 6 - Sample Design in Stereology

The fundamental relations in stereology are expected value theorems that relate measurements that are made ongeometric probes passed through a three dimensional microstructure and corresponding geometric properties ofthe three dimensional structure that the probes sample. Implementing any of these relations requires that a sampleof the population of appropriate probes be produced and the appropriate measurements be made upon the set ofprobes in the sample. The mean value of this set of sample measurements is computed. This mean value is thentaken as an estimate of the expected value of the measurement on the population of probes from which the sampleset is drawn.

It is highly desirable that this estimate be unbiased. If this is not guaranteed by the choice of the set of probesincluded in the sample set then a great deal of effort may be expended to produce an estimate of a geometricproperty which may be highly precise, but wrong.

In order for the set of probes in the sample to provide an unbiased estimate of the expected value for thepopulation it is required that the sample set be drawn uniformly from the population of probes. This is the centralrule in the design of the probe set included in a stereological experiment. This condition must be satisfied becauseit is the central assumption made in the derivation of each of the fundamental relations that lie at the heart of themethodology of stereology. In order to design a sample of probes that is selected uniformly from the population ofthose probes it is necessary to understand what the population is, for each class of probes used in stereologicalmeasurements.

This chapter presents the population for each of the most important probes that are used in stereology: points,lines, planes and volumes. Once the population can be visualized for a particular probe class, the ground rules forobtaining a sample of probes drawn uniformly from each population will become evident. Schemes and strategiesfor obtaining a uniformly drawn sample set will then be discussed for each probe class.

Population of Point Probes

A point on a one dimensional line is specified by assigning a value to a variable x that reports the distance fromsome conveniently defined origin on the line. The population of points on a line is the set of values that thisvariable may be assigned. For some interval of length L0, which may be viewed as the "specimen" in this onedimensional world, the population of points in the specimen is the set of values of x that correspond to pointswithin L0. Since the possible numerical values for x are considered to be distributed uniformly, the set of points isdistributed uniformly within L0.

This concept may be extended to a two dimensional "specimen" which encompasses the total area A0 of interest inan experiment. A point within the specimen is defined by constructing a convenient Cartesian coordinate systemwith x and y axes and assigning particular values to x and y. The population of points in the specimen is the set ofall possible combinations of values for x and y that lie within A0. Again, since the numerical values of x and y aredistributed uniformly, the set of points in the population are uniformly distributed over the area of the specimen.

Chapter 6 - Sample Design in Stereology Page # 99

In order to specify a point in three dimensional space it is necessary to assign a value to three variables, e.g., theCartesian coordinates (x, y, z). Each point in the volume V0 of the macroscopic specimen is given by itscorresponding three coordinates. The population of points that lie within the specimen is the collection of allpossible values of the triplet (x, y, z) that describe points contained within its boundaries. These points areuniformly distributed in the volume of the specimen.

Since the points in these Cartesian spaces represent a continuum there is a conceptual problem in counting howmany points are contained in a given domain in any of these that spaces. Even the smallest finite domain containsan infinite number of points. Indeed domains of different sizes and shapes each contain an infinite number ofpoints. There is an infinite number of points in the interval from x = 0 to x = 1 on the x-axis, but there is also,conceptually, an infinite number of points in the interval from 0 to 1/2. Intuitively it seems that there should be"twice as many" points between 0 and 1 as there are between 0 and 1/2.

a b

cFigure 1. Measures of line length (a), area (b) and

angle (c) as discussed in the text.

This problem is solved by introducing the concept of the "measure of the set of points" in a domain in any space.The measure of the set of points in an interval on a line is the length of the interval assuming that the density of


points is uniform along the line. Thus, the measure of the set of points contained in a one dimensional "specimen"of length L0 is equal to L0. The measure of the set of points in some short segment of length λ contained within

the specimen, as shown in Figure 1a, is equal to the length λ. With these concepts, the fraction of points in the

specimen that lie within the segment λ is the ratio of these measures: fλ = λ/L.

In two dimensional space the measure of the set of points in any domain is equal to the area A of that domain. Thepopulation of points in the two dimensional specimen in Figure 1b has a measure equal to A0. The set of pointsthat lie within the ß phase particles in this figure has measure Aß, the total area of this collection of particles. Thefraction of points that lie within the ß phase is the ratio of these measures: Aß/A0 = AAß, the area fraction occupiedby the ß phase. Thus for the entire population of points in the specimen the point fraction equals the area fraction,which is the basis for equation (4.2) in Chapter 4.

In three dimensional space the measure of the set of points in a domain is the volume V of the domain. For aspecimen this measure is the volume V0 of the specimen. The measure of the set of points within a set of ßfeatures contained in the specimen is their volume Vß. The fraction of points in the specimen that lie within the setof ß features is the ratio of these measures, Vß/V0. This is the basis for the classic volume fraction relation inChapter 4, equation (4.5)

In a stereological experiment a sample is drawn from the population of points in the specimen. The total number ofpoints in the sample is PT and Pß is the number of these points that lie within the ß feature set. The ratio of thesetwo counts is then used to estimate the expected value for this ratio of these measures for the populations of pointsin the specimen. This estimate will be unbiased if and only if the points in the sample are drawn uniformly fromthe population of points in the volume of the specimen, V0.

As example, suppose there were a gradient in the volume of the ß phase so that the volume fraction of ß increasedas the specimen was traversed from top to bottom. If in the experimental sample of point probes was confined to asingle plane near the top of the specimen then the mean of this sample will underestimate VV since there is less ßphase near the top than the bottom. This sample provides a biased estimate of the expected value of PP. If it wereknown a priori that the gradient in ß was one dimensional and in the vertical direction, then a plane in the verticaldirection would capture the gradient and sample it uniformly. Armed with the a priori information in this case,points drawn in a single appropriately chosen plane could produce an unbiased estimate of PP.

If the ß particles are themselves uniformly positioned over the area of the specimen then the sampling problem issimplified. For example, all of the points in the sample could come from one region in the specimen and theresulting mean value would be an unbiased estimate of the population expected value. Uniformity (or the absenceof gradients) may be characteristic of some man-made materials, but most specimens will contain gradients instructure. The assumption of uniformity must be verified before it is used in an implementation that simplifies thestereological design.

Thus, in the general case the positional variation of the volume of the ß feature set is either unknown, or of such anature that it can’t be captured in a single plane. In the absence of a clear knowledge that the volume of the featuresin the sample is uniformly distributed, or has a simple, one dimensional spatial variation, the sample of pointprobes must be distributed uniformly throughout the volume of the specimen. In this general case an unbiased


estimate for PP is obtained only if the set of points chosen for inspection in the sample is uniformly distributedover the volume V0 of the macroscopic specimen.

A specimen exhibiting gradients in VV does not necessarily require a great deal more effort to estimate PP thandoes a specimen in which the ß phase is uniformly distributed. It is only necessary to plan ahead (design) incutting up the specimen for preparation and inspection. For example, the specimen may be diced into a number ofchunks. Pieces selected randomly and uniformly from these chunks (applying the fractionator Howard & Reed,1998b principle described below) may all be mounted in a single mount. A single plane sectioned through thisaggregate would provide fields of view that uniformly sample the volume V0 of the specimen. Grids imposed onfields selected for measurement would provide an unbiased sample of the population of points in V0.

This sampling strategy, which inspects points that are uniformly drawn from the population of points in thespecimen, gives an unbiased estimate of the expected value for the structure no matter how the features aredistributed.

The Population of Lines in Two Dimensions

The only attribute that points have is position x, (x,y), or (x,y,z). A line drawn in a plane has both position, p,and orientation, θ. In two dimensions an orientation of a line is represented by a point on a unit circle of

orientations, as shown in Figure 1c. The population of orientations is the set of points on the unit circle. Themeasure of the set of orientations in two dimensional space given by the measure of the set of points on thecircumference of the unit circle: 2π. The measure of the set of orientations contained in some angular range α, as

shown in the figure, is the length of the arc on the unit circle that contains the set of points that correspond to thedirections encompassed by α. The fraction of orientations that lie in α is the ratio of these two measures, α/2π.

For a line probe, in which adding 180 degrees to the angle of the line produces the same result, the orientations aremeasured over half the unit circle and the fraction of orientations contained within some angular range α is α/π.

Figure 2. An array of lines with a given orientation in a plane (a),and the measure of a range of angles (b) as discussed in the text.


Consider now the subset of the population of lines in two dimensional space that all have the same direction, asshown in Figure 2a. To aid in describing the population of positions of lines, construct a convenient Cartesiancoordinate system with origin O and x and y axes. Construct a line OA through the origin in a directionperpendicular to that of the subset of oriented lines. An individual line in this subset of lines is located uniquely bya position on the line OA. The measure of the set of lines that have the orientation shown and are contained insome interval BC on OA is the measure of the set of points in this interval, and is therefore the length BC.

Thus, a given line in two dimensional space is given by a pair of numbers: (p, θ). The population of lines is

represented by the values of θ from 0 to π and, for a given value of θ, a range of p values that encompass the

specimen. The population of orientations may be guaranteed to be uniformly sampled by simply using circles asline probes as illustrated in Chapter 4, Figure 8.c. If the set of circles contained in the sample is distributeduniformly with respect to position over the specimen area then the population of lines is guaranteed to be sampleduniformly in both position and orientation. Observations obtained from such a sample will provide an unbiasedestimate of the expected value of some measure derived from the population of lines in two dimensional space.

Line Probes in Three Dimensions

As in two dimensions, a line in three dimensional space has both orientation and position. An orientation may bedescribed by a point on the unit sphere. The coordinates of such a point are given by two angles, as they are on aglobe that maps the earth, as shown in Figure 2b (see also Chapter 4, Figure 12). The longitude, θ, is represented

by a polar circle that passes through a point on the circle of the equator. The angle θ is measured relative to a

reference direction defined to be 0 longitude (through Greenwich on the globe). The latitude φ may be specified by

the angle between the point (θ, φ) on the sphere and the north pole. (By convention on the globe latitudes are

specified by the compliment of φ, i.e. by the angle along a longitude at φ from the equator to the point (θ, φ). In

this text latitude will be specified by the angle from the pole, sometimes called the colatitude.)

A continuous range of orientations in space maps as some domain on the unit sphere called its spherical image, asdiscussed in Chapter 5. The measure of the set of orientations represented by this domain is the measure of the setof points on the unit sphere contained in its spherical image, which is an area on the unit sphere, as shown inChapter 5, Figure 1. In terms of the variables (θ, φ) a small patch of this area is given by sinφ ⋅dθ dφ . The sin

f factor arises because the arcs bounding this element in latitude are arcs of a small circle (as opposed to a greatcircle), which has a radius of ρ sin φ where ρ=1 is the radius of the sphere. Integration of this function over the

range of longitudes and latitudes corresponding to some domain of orientations on the sphere gives the sphericalimage of the patch. For example, integration over the northern hemisphere gives the area of the hemisphere 2π.

Now consider the subset of lines in three dimensional space that all share the same orientation, as shown in Figure3a. This positional aspect to defining a line in three dimensional space may be visualized by constructing a planethat is perpendicular to the orientation of the line set. A Cartesian coordinate system in this plane facilitatesdefining the location of a particular line. A particular line is then represented by a point (x,y) on this plane. A setof lines that lie in some region on this plane (e.g., the set of lines that intersect a particular particle of ß in thespecimen) is represented by the set of points (x,y) contained in that domain. The measure of this set of orientedlines is thus the measure of this set of points, which is the area of the domain on the plane.


Figure 3. Arrays of parallel lines (a) and planes (b) in 3D space as discussed in the text.

Thus, a line in three dimensional space requires four variables for its complete specification: two orientationvariables (θ, φ) and, for a given (θ, φ) two position variables (x,y). The measure of a set of orientations is an area

on the unit sphere and, for a given orientation, the measure of a set of positions is an area on the plane which isperpendicular to the orientation direction. In selecting a set of lines to be included in a sample from this four-variable population it is necessary that the two sets of points represented in the construction be sampled uniformly.

A sampling strategy, called the method of vertical sections, provides a simple and automatic procedure forguaranteeing such a uniform sample of positions and orientations of line probes from the population of lines inthree dimensional space.

Planes in Three Dimensional Space

Like a line, a plane in three dimensional space has an orientation and a position, as shown in Figure 3b. Theorientation of a plane is unambiguously described by the direction of a line that is perpendicular to the plane, calledthe normal to the plane. Since the plane normal is a line, the measure of a set of plane orientations is identical withthat described for lines in three dimensional space, i.e., the area of the spherical image of the plane normals in theset.

Specification of the orientation of a plane in space by the angles (θ, φ) does not complete the description of the

plane. There is an infinite set of parallel planes that all share a given orientation. A specific plane in such a set ofparallel planes may be specified by constructing a convenient Cartesian coordinate system and constructing the lineOA through its origin that is parallel to the direction given by (θ, φ). The position of a plane is given by its

distance p from the origin along that line. The measure of the set of planes in some interval on OA is the measureof the set of points that lie in that interval. We have seen earlier that the measure of the set of points in an intervalon a line is the length of the interval.


Thus, a particular plane in three dimensional space is described by three numbers in some convenient Cartesiancoordinate system: (p, θ, φ). The measure of the set of orientations is an area on the unit sphere of orientations

and the measure of the set of positions is a length on a line in the direction of the plane normal. In designing asample of a set of planes in a stereological experiment it is required that their orientations and positions beuniformly chosen from these measures.

Producing a uniform sample of orientations and positions of plane probes in three dimensional is perhaps the mostchallenging of the sampling problems in stereology. Chapter 7 describes procedures that have been used.

Disectors in Three Dimensional Space

The disector sample consists of a rectangular prism with areas A of the top and bottom faces and a knownthickness, h, as shown in Figure 4. It is used to sample a volume V = A0h. An individual disector may bespecified by the coordinates (x, y, z) of a convenient point in the box, e.g., one of its corners. There are noorientation variables involved in this specification. Since for every disector there is one such point and for everypoint in space there is a unique disector, the population of the set of disectors in space is identical with thepopulation of the set of points in three dimensional space.

Figure 4. The populations of disectors in a three dimensional space is identical to the population of points,since each disector may be located by the position of one of its corners. The volume analyzed by each is A0h.


A uniform sample may be selected from the population by selecting points that locate the disector from the uniformpopulation of points in the specimen.

Sampling Strategies in Stereology

In order for the mean of a sample of stereological measurements to provide an unbiased estimate of the expectedvalue of that measurement for the probe population that corresponds to the measurement it is necessary that the setof probes chosen for examination be drawn uniformly from the population of those probes in three dimensionalspace. The population of probes may be described by position variables, and by orientation variables. For pointprobes and disector probes the set of sample probes must be drawn uniformly from the population of points thatoccupy the specimen. For line and plane probes the sample must also be drawn uniformly from the set of pointson the unit sphere of orientation. Once an appropriate sample of probes has been obtained, the correspondingcounts are made and the fundamental relations of stereology are then applied to estimate the geometric propertythat is of interest in the experiment.

Producing such a uniform sample may at first appear to be a daunting task. One might imagine that it wouldrequire a large number of fields to produce such a sample, or to prepare a number of sections from several placesin the specimen. However, the sampling strategies discussed in this section do not require measurements onsignificantly more fields than might be needed to produce an estimate for a structure which is itself uniform. It isonly necessary to plan ahead in sectioning the specimen and collecting a random, uniform set of the pieces. Insome cases this sample set of pieces can all be mounted together in one mount for subsequent preparation andmeasurement.

An additional principle which is important in fostering inclusion of a representative uniform sample of probesfrom the their population is "Do more less well" (Gundersen & Osterby, 1981). It is better to choosemagnification and grid scales so that a few counts are made on a large number of fields rather than a larger numberof counts on a few fields. This strategy not only includes more regions of the specimen in the sample but, byencouraging collection of a few counts on the typical field, helps to minimize counting errors.

A primary strategy used to sample uniformly with respect to position has been called the fractionator (Howard &Reed, 1998b). The specimen is cut into a number N of pieces; depending upon the application, these pieces maybe sliced, then cut into sticks or rods and or into smaller pieces. Some preliminary experiments determine roughlywhat fraction, f, of these pieces will be included for mounting and examination. These pieces are chosen randomlyand uniformly from the population of pieces using a random systematic sampling strategy similar to that discussedin Chapter 4.

Spread out the pieces in a row on a table. Suppose N = 29, and on the basis of preliminary study it is decided thata total of about six of these pieces will form a convenient mounting for analysis. Imagine the pieces on the tableare numbered 1 through 29. An exhaustive collection of samples from the set of pieces may be obtained byselecting every fifth piece starting with, respectively,


A 1 6 11 16 21 26B 2 7 12 17 22 27C 3 8 13 18 23 28D 4 9 14 19 24 29E 5 10 15 20 25 -

Sample set E has only 5 members because 29 is not divisible by 5, but that still satisfies our choice of "about six"pieces for analysis. Choose a random number from 1 to 5 to select sample A, B, C, D or E for analysis. Supposethe number comes up 4, so that sample D is chosen. Then the pieces

D 4 9 14 19 24 29will be mounted for preparation and subsequent analysis.

If the 29 pieces were laid out on the table without paying attention to where they came from in the specimen,sample D (or for that matter, any of the samples) provides a random sample of the population of pieces. Everypiece in the set of 29 pieces has an equal probability of being included in the sample. If on the other hand thepieces are laid out in some order with respect to position in the original specimen (e.g., slices from "top" to"bottom") then the selection of one sample, D, constitutes a random systematic sample of the population of, in thiscase, slices. In this case every sample (A, B, C, D, E) has an equal probability of being included in themeasurement.

This difference has been found to be important in the design scheme for stereological experiments. In the simplerandom sample there is some chance that the pieces in a given sample chosen for measurement may all come fromnear the top of the specimen, or be otherwise localized. In the random systematic sample each sample set haspieces that are uniformly distributed from top to bottom. Both samples produce unbiased estimates of populationexpected values as required by the stereological relationships.

However experience has shown that the variance associated with the set of sample means is very much smaller (bya factor of 2 or 4 or perhaps more Gundersen & Jensen, 1987) for the random systematic sample than for thesimple random sample. This result has important practical implications for the level of effort required to obtain agiven precision in the statistical experiments of stereology. The precision of an estimate of a mean value is givenby the confidence interval assigned to the estimate of the mean. The confidence interval is a range about theestimated mean defined by

C. I .= x ± 2σ x (6.1

where σ x is the standard error of the mean computed from the sample. This "95% confidence interval" means

that the probability that the true mean of the population lies within this interval is 0.95. The precision with whichthe mean of the population is estimated is thus determined by σ x . This property is computed from the standard

deviation sample measurements, σ x , given by

σ x = σ x

n(6.2


where n is the number of readings (fields) in the sample. The smaller the confidence interval, the more precise isthe estimate. Precision can be increased in two ways:

1. Increasing the number of fields n measured in the sample: (Taking four times as many measurements cutsthe confidence interval in half, making the estimate "twice as precise");

2. Decreasing the standard deviation of the sample, which is an estimate of the square root of the variance ofthe population.

Applying the strategy of the random systematic sample decreases the variance (in comparison with a simplerandom sample), decreasing σ x and thus σ x and the confidence interval for a sample of a given size, n. Halving

the variance cuts the confidence interval in half.

Applying the strategy of the random systematic sample permits attainment of a given precision with fewer counts,or, for the same number of counts, increases the precision. This sampling strategy is now widely used instereology because it significantly increases the efficiency of these techniques. Implementation of this strategyrequires only conscious planning of the choices made in designing a sample for analysis, i.e., where the specimenshould be cut and how the pieces should be chosen for preparation and examination.

This fractionator approach can be implemented for each of the stereological probes that have been described.

1. In using the Cavalieri method to estimate the volume of an object the pieces may be slices obtained bymicrotoming the sample.

2. The point count may be also applied to this fractionated sample of slices through the structure. On a given slicefields chosen for measurement may be chosen from the set of possible fields by another fractionating process.

3. The method of vertical sections may be applied to a fractionated sample of pieces obtained from the specimenby cutting it as you would to make french fries from a potato. This is illustrated in Figure 5. The vertical directionin the sample is the long dimension of these stick shaped pieces. Rotating the sticks randomly or using asystematic random rotation of the pieces about the vertical axis when placing them in the mounting powder withlong (vertical) axes parallel provides a uniform sample of longitudinal directions as well as a clear indication of thevertical direction in every field. Using a cycloid test line uniformly samples latitude directions. A single sectionreveals structure that uniformly samples position in the x, y, and z directions and orientation in θ and φ. Random

systematic sample of all of the possible fields that may be constructed on this one section provides a uniformisotropic random sample for analysis.

4. It mat be possible to produce an isotropic, uniform random sample by dicing the specimen into small pieces thatare equiaxed, (e.g., cubes), fractionating them and mounting them with uniform random rotations as they areplaced in the mounting medium. Then any plane section through the structure will produce an IUR sample ofpoints, lines and planes. The practical difficulty associated with such a method is contained in the phrase"mounting them with random uniform rotations".


Figure 5. A procedure for obtaining IUR vertical sections: a) the specimen is cut into vertical samples; b) astructured random procedure for selecting an appropriate subsample of these is used; c) the selected prisms

are rotated either randomly or by structured random angles; d) the rotated prisms are embedded andsectioned. On the resulting section image the use of cycloid grids will provide isotropic, uniform and

random sampling of orientations in the microstructure.


These examples are not practical in all cases and are presented here to illustrate the principle that, once the groundrules are clearly formulated for obtaining a "representative" sample from a population of probes, a little ingenuityapplied in the original design of the experiment may solve the sampling problem in stereology without a great dealof additional effort expended in making the measurements. Chapter 7 discusses some of the practical samplingprocedures that may be used.

Summary

Each of the stereological relationships assumes that the population of probes used in the experiment is uniformover the domain of the specimen. Knowledgeable design of stereological experiments requires that the nature ofthese probe populations be clearly understood. Each class of probes has a population that is described by thevariables necessary to define an individual probe.

The population of point probes in two dimensions is the set of all possible position pairs (x,y) in thespecimen. The measure of the set of point probes in a given region is the area of that region.

The population of point probes in three dimensions is the set of all possible position triplets (x,y,z) in thespecimen. The measure of the set of point probes in a given region is the volume of that region.

The population of line probes in three dimensions is the set of all possible points (θ, φ) on the sphere of

orientation and, for a given such orientation, the set of points (x, y) on a plane perpendicular to thatorientation. The measure of the set of orientations in a region of orientations is an area on the sphere oforientation. For a set of lines, the range of orientations is an area on the hemisphere of orientation. Themeasure of the set of positions of lines with a given orientation is an area on the plane perpendicular to thatorientation.

The population of plane probes in three dimensions is the set of points (θ, φ) on the unit sphere or

hemisphere designated by the normal to any given plane, and the set of positions p of points along a line inthat direction. The measure of the set of orientations is an area on the sphere and the measure of the set ofpositions of planes is a length along their normal directions.

The population of disectors in three dimensional space is the same as the population of points, since eachdisector may be uniquely located by one of its corner points.

The fractionator strategy envisions cutting the specimen into a large number of pieces then, using a simple randomor a random systematic method, selects a uniformly representative small sample from the population of pieces foranalysis.

With careful planning in the design of the sample it is possible to satisfy the isotropic uniform random samplingrequirement of stereology without significant additional effort. Chapter 7 discusses some representativeimplementations of such strategies.


Chapter 7 - Procedures for IUR Sampling

Chapter 6 emphasized the requirement that sampling of a three-dimensional structure with points, lines or planesbe isotropic, uniform and random (IUR). Accomplishing this is not easy, and much of the thrust of recentstereological development has been to explore sampling strategies that provide such results. It is useful tosummarize here just what IUR means (without recapitulating the detailed discussion in Chapter 6). Point probesdo not have to be isotropic, because the points themselves have no direction. It is sufficient to disperse a set ofpoints through the structure with an equal probability of sampling all regions (e.g., not points that are concentratednear one end or near the surface), and which are randomly placed with respect to the structure.

The idea of randomness is not always an easy one to test. If the structure itself has no regular spacings orrepetitive structure, a regular grid of points can be used (and often is, for convenience in counting and forefficiency of sampling). But if the structure does have some regularity, then the use of a regular grid of points willproduce a biased result. Figure 1 shows a cross section of a man-made fiber composite. While not perfectlyregular in their arrangement, the fibers are clearly not randomly distributed in the microstructure. Attempting tomeasure the volume fraction of fibers with a grid of points would produce a biased result because that grid and thearrangement of the fibers would interfere to produce a result in which points systematically hit or missed thefibers. For an image such as this one a routine that placed points at random on the image would be required.Typically this is done by generating two uniform random numbers using a computer pseudo-random numbergenerator (a mathematical routine seeded by the system clock; we will not consider the intricacies of such routineshere). The two numbers are typically provided as decimal fractions between 0 and 1, and can conveniently bescaled to the dimensions of the image to locate points in X, Y coordinates.

Figure 1. Cross section image of a man-made fiber composite, with an approximately regular array of fibers.

Chapter 7 - Procedures for IUR sampling Page # 111

Such random sampling can be used for any structure of course, even one that has no regularities. But there is amore efficient procedure that produces equivalent results. It converges on the same answer as more and morepoints are sampled and converges more rapidly than the random sampling approach (Gundersen & Jensen, 1987).This is equivalent to saying that it has a better precision for a given amount of effort. The method (described inearlier chapters) is called structured random sampling, which at first seems to be an oxymoron. It can be applied tomany types of sampling with points, lines and planes, but will be explained first as it applies to the point samplingof an image.

Volume fraction

To determine the volume fraction of a phase in a structure, assuming that the sampling plane is representative (wewill see below how to deal with that requirement), the fundamental stereological relationship is VV = PP, or thevolume fraction equals the point fraction. The precision of a counting measurement is a standard deviation equal tothe square root of the number of events counted. For instance, if asked to estimate the number of cars per hourtraveling along a highway we could count the cars and measure the time required for 100 cars to pass. The numberof cars divided by the time T (expressed as a fraction of one hour) would give the desired answer. The standarddeviation of the number counted is 10, so the relative standard precision would be 10/100 = 10%. By counting1000 cars we could reduce the standard deviation to √1000 = 31.6, or a relative standard precision of 3.16%.Counting 10,000 cars would give a relative standard precision of 1%.

Usually in a given experiment we know what precision is required or desired. Consider for the moment a desiredprecision of 10%. This requires counting 100 events as described above. For point counting, this means that thereare 100 points in the grid or random distribution that hit the phase of interest. If the phase represents (e.g.) 25% ofthe structure, this would require placing 400 points onto the image to achieve 100 hits. However, there is oneadditional assumption hidden in this procedure: each of the points is required to be an independent sample of thestructure. This means that the points should be separated from each other by enough distance that they do notrepeat the same measurement, which is usually taken to mean that they do not lie closer together than acharacteristic distance in the structure such as the mean intercept length.

However, random points generated by the procedure outlined above will not all conform to that assumption. Asshown in Figure 2, random distributions of points on an image inevitably produce some points that are quite closetogether. These are not all independent samples, and so the actual precision of the measurement is not as goodwould be predicted based on the number of points. The clustering of points produced by a random (Poisson)generation process is used in another context to study the degree to which features in images have tendenciestoward clustering or self-avoidance. The standard deviation of the PP value obtained using random points will begreater (worse) than with a regular grid by a factor of e (2.718). Notice that this does not mean that the resultobtained with such a random point array is biased or in error, just that it takes more work (more points have to besampled) to obtain an equivalent measurement precision.

The use of regular grids of points to count PP is widely used, of course. Grids of points are available as reticles tofit within microscope eyepieces, or applied to photographic prints as transparent overlays, or superimposed on


images using computer displays. This is not so much a consequence of the greater measurement efficiency as arecognition of the ease of counting such points manually by having the eye track along a regular grid.

Figure 2. An example of 200 random points. Note the presence of clusters of points thatoversample some regions while large gaps exist elsewhere.

The proper way to use such a grid while still obeying the requirements of random sampling according to thestructured random sampling method is as follows:1. Determine, based on the desired level of precision and the estimated volume fraction of the target phase, the

number of points to be sampled in the image. Note that in many real cases not all of the points will be sampledin a single image - there will be many fields and many section planes in order to meet the requirements ofuniform sampling of the entire structure. There is also the restriction against using too many points on oneimage based on the requirement that the points must be separated far enough to be independent samples,meaning as noted above that multiple points do not fall into the same region of the structure.

2. The resulting number of points form a regular grid. Typically these grids are square so that the number ofpoints is a perfect square (25, 49, 100, etc.) with fairly small numbers used for purely manual countingbecause larger numbers foster human counting errors. In the example shown in Figure 3, the grid is 5x5=25points.


3. Generate a pair of random numbers that define the X,Y coordinates for the placement of the first point of thegrid within a square on the image. This will automatically place all of the other grid points at the same relativeoffset.

4. Perform the counting of PP.5. For other fields of view, generate new random numbers for the offset and repeat steps 3 and 4, until enough

fields of view, section planes, etc. have been examined to produce uniform sampling of the structure andenough points have been counted to produce the desired precision.

Figure 3. A 5x5 grid placed on an image area. The upper left corner is displaced randomlywithin the red square to accomplish structured random sampling of the area.

This procedure is a straightforward application of structured random sampling. Several examples will be detailedbelow that extend the same method to other situations and types of probes. In all cases note that it is important toestimate beforehand the number of images (sections planes and fields of view) that must be examined in order toobtain the desired precision. This requires an estimate of the quantity being measured, in this example the volumefraction. Typically this parameter is known at least approximately from previous sampling of the structure, priorresults on other similar specimens, or independent knowledge about the objects of interest.


Sampling planes

The same procedure as detailed above can be applied to the selection of the sampling planes themselves. Forexample, consider the problem of determining the volume fraction of a phase within an irregular 3D object asshown in Figure 4. The measurement in this case must measure the volume of the object as a whole as well as thatof the phase in order to obtain the volume fraction. If the phase is expected to be about 20% of the volume, and ameasurement precision of 5% is desired, this means that about 400 points should hit the phase and hence 2000points should hit the object. If the object is 5 inches long and about 3 inches in diameter, a rough estimate of itsvolume can be made by assuming a more-or-less ellipsoidal shape. The volume of an ellipsoid is about half (π/6 =0.524) of the volume of the bounding box. This means that grid should have about 4000 points. A cubical gridspacing of 0.2 inches will produce a total of 15x15x25 = 5625 points, quite suitable for this measurement.

Figure 4. A representative irregular object (potato) to be sampled as discussed in the text,and one section to be used for counting with a portion of the grid superimposed.

This sampling grid can be achieved by sectioning the specimen every 0.2 inches and placing a grid with thatspacing onto each section for counting. The placement of the grid uses the method described above. A similarmethod is used to position the planes: A single random number can be used to determine the position of the firstplane within the first 0.2 inches of the specimen, and then all of the other planes are regularly spaced with respectto this first one. In this way structured random sampling is extended to all three dimensions.

This represents a fairly large number of points and a high sampling density; most stereological experiments usemuch lower sampling density and achieve lower precision on each specimen, but average over many specimens.In the example of Figure 4, a lower sampling density has been used for clarity. the planes are spaced 1/2 inchapart and the grids have a 1/2 inch spacing, producing a total of 360 points, of which about 180 can be expected tofall within the object and hence about 36 within the phase of interest. This corresponds to a standard deviation dueto counting statistics of ±6, or a relative precision of 6/36 = 17%.


In many cases the sections of the specimen are obtained with a microtome, in which case the procedure is to takeevery Nth section from the series to obtain the desired spacing, starting with a randomized initial section.Likewise, if the examination procedure is to use M fields of view on each section, the procedure is to create a gridof field positions which is offset by a random location on each section. The procedure can also be extended tosampling specimens within a population in the same way.

Isotropic planes

Unlike point probes, lines and planes have a direction and so it is also necessary to make their orientation isotropicwithin the structure. This is much more difficult to accomplish (and may be very important, as few naturalstructures are themselves isotropic). Line probes are used to sample surfaces and volumes, either by counting theintersections of the line probes with surfaces or measuring the intercept length with the volumes. Becausecounting is a much easier and hence preferred technique, and the precision of counting experiments can bepredicted as discussed above, most modern stereological experiments emphasize counting of intercepts and so wediscuss here the use of line probes for that purpose.

Lines cannot in general be passed through structures without first cutting planes on which to draw the lines. If allof the planes were cut as in the preceding experiment as a series of parallel sections, it would not be possible todraw lines in any direction except those included in the planes. Hence the most commonly used and efficientmethod of cutting parallel plane sections through a specimen cannot be used when line probes are to be employedfor surface area measurement.

It is instructive to consider first the procedures for placing an IUR set of planes through an object. This is requiredwhen the measurement probe is a plane, for instance when measuring the size and shape of plane intersectionswith features or counting the intersections of the plane with features in the volume. Plane probes are primarilyused for counting the number of intersections per unit area with linear structures such as fibers, which are rarelyisotropic in space, in order to measure the total length of the linear structures. IUR planes can also be used fordrawing line probes through structures, although we will see later that there is an easier way to accomplish thiswhen the purpose is simply to count intersections of the lines with surfaces.

In order for sampling to be uniform, it will still be necessary to obtain a series of sections that cover the extent ofthe feature. There is a tendency in visualizing the orientation of section planes to deal with those that pass throughthe geometric center of the object being sectioned, which of course does not provide uniform sampling of theextremities. Since many real structures are not only anisotropic but also have gradients, the need for uniformsampling is very important.

Using a series of section planes (distributed as discussed above using the principles of structured randomsampling) naturally limits the sampled planes to a single orientation, because once cut they cannot be reassembledto permit sampling again in another orientation. This usually means that multiple specimens must be used to obtainmany different sampling orientations, which in turn requires that specimens within a population must have somedefinable or discernible axis of orientation of their own that can be used as a basis for selecting orientations. Inbiological systems, which is the primary field of application of these techniques, this is usually straightforward.For materials or geological specimens there are also natural axes (typically perpendicular to deposited planes or


parallel to a cooling gradient or deformation direction) but these may be more difficult to ascertain beforesectioning.

In any case, assuming that there is some natural axis, one approach is to define a different axis orientation forsampling each member of the population using a scheme that distributes the orientations randomly and uniformlyin three dimensions to achieve isotropic sampling. This is not so easy to do for an arbitrary number oforientations. We have already seen the efficiency of the structured random sampling approach, so a method basedon random tumbling of the specimen is not desired. Visualization of the problem may be assisted by realizing thateach direction can be mapped as a point onto a half-sphere of orientations. Is there an arrangement of such pointsso that they are regularly and equally spaced over the surface of the sphere?

Figure 5. The five platonic solids (regular polyhedra).

Figure 6. Orientation vectors for the dodecahedron, plotted on the sphere of orientation.

The arrangements which are exactly equal and regular correspond to the surface normals of the regular polyhedra,of which there are only five (Figure 5): the tetrahedron with four triangular faces, cube with size square faces,octahedron with eight triangular faces, dodecahedron with twelve pentagonal faces, and icosahedron with twentytriangular faces. Taking just the faces whose normal directions point into the upper half of the sphere oforientations, the number of orientations is half of the number of faces (except for the tetrahedron). For instance,the dodecahedron has 12 faces but they provide only six sampling orientations. If one of these solids correspondsto the desired number of orientations to be used for sampling, a useful strategy would be to embed each entireobject to be sectioned into such a structure, the number of such objects corresponding to the number of faces


(e.g., six organs from a population placed into six dodecahedra). This would place the natural axis at somerandom orientation, which would however fall within the area of one of the faces. Identify this face, and select aunique and different face for each of the polyhedra. Figure 6 shows the normal directions for the dodecahedron.Use the surface normal of the selected face as a sectioning axis for that object, using the usual procedure aspresented above to determine the number of sections to be used and the location of fields of view within eachsection.

This procedure is fairly easy to implement but is appropriate only when there are multiple objects representing apopulation which is to be measured. When a single object must be sampled isotropically with planes, anequivalent procedure is to first subdivide the object into portions, for example cutting it into six (for thedodecahedron) or ten (for the icosahedron) parts. Then each part can be embedded into the regular polyhedron andsectioned as described above. The danger in this approach is that orientations of structure within the object mayvary from place to place, and since the sampling orientation also varies this can result in measurement bias.Correcting such bias requires performing the procedure on several specimens with different orientations selectedfor each portion. In this case, it is just as efficient to use the first approach with sectioning of each specimen in oneorientation.

Figure 7. Vertical sections cut to include the vertical axis of a specimen. Each section includesthe vertical direction (arrow) and is rotated randomly to a different azimuthal angle.

If it is not required that the specimen be uniformly sampled at each orientation, then another sampling method canbe employed to obtain section planes that are isotropic. First cut a so-called “vertical” section that includes thespecimen’s natural axis (or some identifiable direction) and is uniformly random with respect to orientation, asshown in Figure 7. Then cut a surface perpendicular to this surface but at an angle that is not uniformly randombut is instead sine-weighted. Sine weighting is accomplished by generating a uniform random number in the range-1..1 and then calculating the angle whose sine has that value = Arc Sin (Random). This angle varies from -90 to+90 degrees and represents the angle from vertical. It is also easy to implement this procedure by creating a grid ofradial lines that are not uniformly spaced but instead have the appropriate sine weighting, and then to select oneusing a uniform random number from 0 to 99 (Figure 8). This procedure is called the Orientator (Mattfeldt, quoted


in Gundersen et al., 1988). Figure 9 shows the process applied to the object from Figure 4. It produces sectionplanes that are isotropically uniform. Of course, this produces only a single section that passes through the object,and so does not provide spatially uniform sampling.

Figure 8. The “Orientator”. Place a horizontal cut surface (perpendicular to the assigned vertical directionin a specimen) along the 0-0 line with a vertical cut (parallel to the vertical direction and uniformly randomwith respect to rotation). Generating a random number from 0-99 to select a cutting direction as shown.

Repeating this procedure produces planes that are isotropic (uniformly sample orientations).


a

b

Figure 9. Implementation of the orientator: a) cutting a vertical section;b) cutting the examination plane using the sine-weighted grid.

Isotropic line probes

Fortunately, there are few stereological procedures that require IUR planes as the sampling probes. IUR lines aremuch easier to generate. If the generated planes in the specimen were themselves IUR then drawing lines on themthat were also IUR would produce line probes having an IUR distribution in 3D space. Such a grid of lines wouldbe produced for example by drawing a set of parallel lines with an appropriate spacing, shifting them by a randomfraction of that spacing, and rotating them to an angle given by a uniform random number between 0 and 180degrees. The structured random sampling approach lends itself directly to the positioning of the lines in this way.Note that a grid of radial lines with uniform angular spacing drawn from the center of the field of view or of thespecimen does not satisfy the IUR requirement because it is not uniform across the area - more of the lines samplepositions near the center than at the periphery.


Given the difficulty of drawing the IUR planes in the first place, another easier approach is generally used. Itrelies on the idea of a vertical section (Baddeley et al., 1986), the same as mentioned above. This is a plane thatincludes some natural axis of the specimen or some readily identifiable direction within it. It does not matterwhether this orientation is actually “vertical” in a geocentric sense, and the name comes from the fact that theplacement of images of the vertical section plane for viewing and measurement often positions this direction in thevertical orientation.

Figure 10. A series of planes cut parallel to one vertical section.

A vertical section plane can be cut through a specimen parallel to the vertical direction without necessarily passingthrough the center of the specimen. In fact, while most diagrams and illustrations of vertical sectioning tend todraw the plane as passing through the center, for IUR sampling it is of course important that this restriction not bepresent. A series of parallel vertical section planes (Figure 10) with one uniformly random rotation angle about thevertical axis can be cut using the same principles of uniform random sampling discussed above by calculating anappropriate plane spacing and a random offset of a fraction of that spacing. The uniform random sampling mustthen also be performed by cutting a similar set of section planes on additional specimens (or portions of the samespecimen) at angles offset from the first chosen angle.

On all of the vertical section planes cut at different rotational angles, directions near the vertical direction aresampled. Plotted onto a sphere of directions (as shown in Figures 11 and 12), it can be seen that these lines clusternear the north pole of the figure while directions at low latitudes near the equator are sparsely sampled. Thecompensation for this is to use the same sine-weighting as discussed above. By generating lines with an angle


from the vertical calculated as Arc Sine (R) where R is a uniform random number between -1 and +1, thedirections are spread uniformly over the latitudes on the sphere (Figure 12).

Figure 11. Radial lines drawn on a vertical section; top) uniformly distributed every 15 degrees(these lines are not isotropic in 3D space); bottom) the same number of lines drawn

with sine-weighting, which does produce lines isotropic in 3D space.

Figure 12. Direction vectors in space mapped as points on a half-sphere of orientations: a) uniformlydistributed angles at every 15 degrees along each vertical section corresponding to Figure 11a; the vectors

are close together near the north pole. b) sine-weighted angles along the same vertical sectioncorresponding to Figure 11b; the same number of directions are now uniformly distributed.

This approach fits well with structured random sampling because the number of angles in the horizontal (uniform)direction, the vertical (sine-weighted) direction, the number of lines and their spacing can all be calculated basedon the required precision. However, it requires more horizontal orientations and hence more vertical section cutsthan a modified procedure that uses cycloids rather than straight lines. This obviously does not apply to


experiments in which intercept lengths are to be measured, but is quite appropriate for the counting of intersectionsbetween the line probes (the cycloids) and surfaces within the structure.

Cycloids are the correct mathematical curve for this application because they have exactly the same sine weightingas used to generate the straight lines in the method above. A cycloid is the path followed by a point on the rim of arolling circle, and can be generated using the mathematical procedure shown in Figure 13. The curve has only asmall fraction of line with vertical orientation and considerably greater extent that is near horizontal, and exactlycompensates for the vertical bias in the vertical section planes. Distributing a series of cycloidal arcs (in all fourpossible orientations obtained by reflection) across the vertical section and shifting the grid of these linesaccording to the usual structured random sampling guidelines produces isotropic uniform random sampling inthree dimensions. Note that the grid of cycloids (Figure 14) is not square, but wider than it is high; the shiftingusing random offsets must displace the grid by a random fraction of the grid size in each direction. As usual, thesize of the grid must be such that in few cases will more than one line intersect the same element of structure. Inmany examples of this technique the statement is made that as few as three sets of planes at angles of 60 degreesare sufficient for many purposes. Obviously, this premise can be tested in any particular experiment by samplingat higher density and seeing whether the results are affected.

Figure 13. A cycloidal arc and the procedure for drawing it. The are has a width of π,a height of 2 and a length of 4 (twice its height).

For projected images through transparent volumes, it is possible to generate IUR surfaces using cycloids. Asdescribed in detail in Chapter 14 on finite section thickness, a set of cycloidal lines drawn on the projected imagerepresent surfaces through the volume. Choosing a vertical direction (which can be arbitrary) for the volume androtating it about that axis while collecting projected images causes the same bias in favor of directions near thevertical axis as that produced by vertical sectioning. Elements of area of these surfaces have orientations thatcompensate for the bias so that isotropic sampling of the structure is obtained. As shown in the examples in thatchapter, the result is an ability to obtain isotropic uniform random sampling of intersections of linear features tomeasure NA, from which LV can be calculated.


Figure 14. A grid of cycloidal arcs. Placed on vertical sections (the vertical direction in thespecimen is vertical on the figure), the lines isotropically sample directions in 3D space.

Volume probes - the Disector

One of the first developments in the so-called “new” stereology that emphasizes design of probes and samplingstrategies rather than “classical” methods such as unfolding of size distributions based on shape is the Disector(Sterio, 1984). The disector requires comparing two parallel section planes to detect features that are present inone but not the other. But while it is implemented using section planes, it is actually a volume probe (the volumebetween the planes). Since volumes have no orientation isotropy is not an issue, although requirements foruniform random sampling remain (and can be satisfied using the same methods described above).

The initial and still primary use of the disector is to determine the number of objects per unit volume. Point, lineand plane probes cannot accomplish this without bias because they are more likely to intersect large features thansmall ones. As noted in the first chapter, the number of objects present in a region is a topological property, andcannot be determined by probes of lower dimension than the volume. The disector provides a surprisingly simpleand direct way to count objects that is unbiased by feature size and shape. It relies on the fact that features can becounted by finding some unique characteristic that occurs only (and always) once per feature. In this case thatcharacteristic is taken to be the topmost point (in some arbitrary direction considered to be “up”) of each feature.

For illustration, consider counting people in a room. For most of them the topmost point would be the top of thehead, but in some cases it might be the tip of an raised hand, or the nose (someone lying on his back on the floor).Regardless of what the point is, there is only one. Counting those points gives the number of people. Of course,in this example the procedure must be able to look throughout the volume of the room. When three-dimensionalimaging is used, as discussed in Chapter 15, this is the method actually used. The disector provides an efficientmethod using just sets of two parallel planes.

The disector can be implemented either with physical sectioning (e.g., microtoming) to produce two thin sections,or by sequential viewing of two polished surfaces of an opaque material, or by optical sectioning of a transparent


volume using confocal microscopy. In all cases the spacing between the two sections must be known, which canbe difficult particularly in the case of microtomed sections. For polished opaque materials one method foraccomplishing this is to place hardness indentations (which have a known shape, typically either spherical, conicalor pyramidal) in the surface and measure the change in size with polishing, from which the depth of polishing canbe determined. These hardness indentations also help to align the two images. Similar fiducial marks can be usedfor the microtomed sections. Optical sectioning is generally the easiest method (when applicable) because spacingcan be measured directly and image alignment is not required.

The key to the disector is to have the two parallel images be close enough together (small spacing between them)that the structure between them is relatively simple. No entire feature of interest can be small enough to hide in thatvolume, and as we will see no branching of networks can have branch lengths smaller than that distance. It mustbe possible to infer what happens in the volume by comparing the two images. Only a small number of basictopological events can be allowed to occur, which can be detected by comparing the images:1. A feature may continue from one plane to the other with no topological events (the size of the intersection can

change, but this is not a topological change).2. A feature may end or begin between the planes, appearing in one but not the other.3. A feature may branch so that it appears as a single intersection in one plane and as two (or more) in the other.4. Voids and indentations within a feature may also continue, begin or end, or branch.

Figure 15. Diagram of the disector (cases illustrated are discussed in the text).

Figure 15 shows a diagram illustrating several of these possibilities. The critical assumption is made thatfamiliarity with the structures will enable identification by a human to detect features in the two sections that arematched. For automatic analysis the planes must be spaced closely enough together that feature overlaps can beused to identify feature matches. Features of type 6 (in the figure) are considered to be continuations of the sameobject with no topological events occurring. Events of type 3 and 4 represent the start or end of a feature


(depending on which of the two planes is taken to be the floor and which the ceiling). Events of type 1 and 2represent simple branching. The type 5 event reveals the bottom of an internal void.

If the spacing between the planes is small so that no complex or uninterpretable events can occur in the volumebetween them, then in real images most objects will continue through the two planes (type 6) and these non-eventsare ignored. For counting features, events of type 3 and 4 are of interest. The number of these events divided bytwo since we are now counting both beginnings and endings of features, and divided by the volume between theplanes (the area of the images times their spacing) gives the number of features per unit volume directly. As notedabove, this value is unbiased since feature size or shape does not affect the count (Mayhew & Gundersen, 1996).

Figure 16. Guard frame and exclusion lines for the disector. Only the red features are considered.

The method is only simple for convex objects. When features may branch, or are long and slender so that theymay cross and re-cross through the sampled volume, it becomes necessary to keep track of all of the parts of thefeature so that it is only counted once. Since the images are finite in area, attention must be given to the problemsthat the edges of the image introduce. As shown in Figure 16, this is accomplished by defining “exclusion edges”and a guard region around the active counting area so that features are ignored if they extend across the exclusionedge (Gundersen et al., 1988). Of course, as noted above it is necessary to follow features that branch or extendlaterally to detect any crossing of the exclusion edges (this is why the exclusion edges are extended as shown bythe arrows in the figure). The requirement for a small spacing between planes to eliminate confusion aboutconnectivity means that only a few topological events are detected, so that a large area or many fields of view arerequired to obtain enough counts for useful statistical precision.

Networks

As noted above, the disector can be used to count the number per unit volume (NV) when the objects are convex,or at least relatively compact and well separated. Objects that are long, twisted, branched and intertwined create


difficulties in identifying the multiple sections that occur as part of the same object. On the other hand, the disectoris also very useful for dealing with an extended network such as the pore structure in a ceramic, blood vessels orneurons in the body, etc. The topological property of interest for these structures is the connectivity or genus ofthe network. This is a measure of how branched the structure is, or more specifically of the number ofconnections (per unit volume) or paths that are present between locations.

Topological properties such as number of objects and connectivity of pore networks require volume probes. Theuse of full three-dimensional imaging (Chapter 15) offers one approach to this, but the disector offers another thatis elegant and efficient. It will be useful first to digress slightly and revisit some aspects of topology and genus,and define some terms and quantities, which were introduced more comprehensively in earlier chapters.

Points on the surfaces of objects in three-dimensional space have normal vectors (perpendicular to the localtangent plane) that identify their orientation. Each such vector can be represented by a point on the surface of asphere, which is called the “spherical image” of the point. A patch or region on the curved surface of the objectcorresponds to a patch of orientations on the sphere. For the surface of a convex, closed object every direction onthe sphere corresponds to one (and only one) point somewhere on the surface of the object. Since the sphericalimage of the convex surface exactly covers (or is mapped onto) the unit sphere, its spherical image value is 4π (thearea of the unit sphere), independent of any details about the shape of the body. The spherical image is animportant topological property since it has a value that is independent of size and shape.

Figure 17. Convex, concave and saddle curvaturesof surface elements.


For bodies that are not totally convex we must recall the idea of negative spherical image introduced in Chapter 5.This is the projection of points on saddle surfaces (Figure 17), which have two principle radii of curvature withdifferent signs. With this convention, the total spherical image of any simply-connected closed surface is 4π (andthe total spherical image of N objects would be 4Nπ). If the body has a hole through it (for instance a donut ortorus, Figure 18), the surface around the hole is saddle surface and covers the sphere exactly once, and the surfaceon the outside is convex and also covers the sphere once, so the new spherical image of a donut is 0.

Figure 18. A torus; the green shaded area is saddle surface with a negative spherical image,the red surface is convex with a positive spherical image.

The hole in the donut changes the topological quantity called the genus of the object. It becomes possible to makeone cut through the object without disconnecting it into separate parts. Every hole introduced into the object adds aspherical image of -4π, so the general result is that the spherical image of an object is related to the genus orconnectivity by the relationship Spherical Image = 4π•(1–C). The total spherical image of a set of N objects with atotal connectivity of C would be 4π•(N–C). We are usually primarily interested in the two extreme cases whenC=0 (a set of separate, simply connected objects which need not be convex but have no holes) or N=1 (a singleextended network whose genus we wish to determine).

Imagine a plane sweeping through the volume of the structure and note the occurrences when the plane ismomentarily tangent to the surface. There can be three different kinds of tangent events as illustrated in Chapter 5(DeHoff, 1987; Zhao & MacDonald, 1993; Roberts et al., 1997):1. The plane may be tangent to a locally convex surface (both radii of curvature point inside the object). This is

called a T++ event and the total number of them per unit volume is denoted TV++.2. The plane may be tangent to a locally concave surface (both radii of curvature point outside the object). This is

called a T–– event and the total number of them per unit volume is denoted TV––.3. The plane may be tangent to a local patch of saddle surface (the two radii of curvature lie on opposite sides of

the plane). This is called a T+– event and the total number of them per unit volume is denoted TV+–.

The disector can be used to count these tangent events. The appearance of features in the two planes allows us toinfer that a tangent event of one type or another occurred in the volume between the planes. Referring back to thediagram in Figure 15, types 1 and 2 correspond to T+– events, types 3 and 4 to T++ events, and type 5 to a T––


event. The total number of counts per unit volume (the product of the area of the image and the spacing betweenthe planes) can be obtained by counting.

The sum of these tangent counts (TV++ + TV–– – T V+–) is called the net tangent count TV, and the total sphericalimage of the structure is just 2π•TV. Consequently the difference between the number of features present N andthe connectivity or number of holes C is just N – C = TV/2. For a network structure with N=1, this gives theconnectivity of the structure directly. Because topological properties and the volume probe used cannot beanisotropic, orientation considerations do not arise in using the disector (but of course averaging of samples toobtain uniform representation of the structure is still necessary).


Chapter 8 - Statistical Interpretation of Data

Introduction - sources of variability in measurement

Figure 1. Effects of accuracy and precision in shooting at a target.

It is commonplace to observe that repeated measurements of what seems to be the same object or phenomenon donot produce identical results. Measurement variation arises from a number of sources, but one root cause is oftenthe finite precision of the measuring tool. If a simple yardstick is used to measure carpet, we expect to obtain aresult no better than perhaps 1/4 inch, the smallest division on the measurement scale, but this is entirely adequate

Chapter 8 - Statistical Interpretation of Data Page # 130

for our purpose. If we try to measure the carpet to a higher resolution, say 1/64 inch, we are likely to find that theroughness of the edge of the carpet causes uncertainty about just where the boundary really is, and measurementsat different locations produce different answers. To push this particular example just a bit farther, if themeasurement is performed without taking care that the ruler is perpendicular to the edges of the carpet, themeasurement will be biased and the result too large.

The limitations of the measurement tool and the difficulty in defining just what is to be measured are two factorsthat limit the precision and accuracy of measurements. Precision is defined in terms of repeatability, the ability ofthe measurement process to duplicate the same measurement and produce the same result. Accuracy is defined asthe agreement between the measurement and some objective standard taken as the "truth." In many cases, the latterrequires standards which are themselves defined and measured by some national standards body such as NIST orNPL (for example, measures of dimension). Arguing with a traffic officer that your speedometer is very precisebecause it always reads 55 mph when your car is traveling at a particular velocity will not overcome his certaintythat the accuracy of his radar detector indicated you were actually going 60 mph (arguing about the accuracy of thecalibration of the radar gun may in fact be worthwhile- the calibration and standardization of such devices can berather poor!). Figure 1 suggests the difference between precision and accuracy - a highly precise but inaccuratemeasurement is generally quite useless.

In a few cases, such as counting of events, it is possible to imagine an absolute truth, but even in such anapparently simple case the ability of our measurement process to yield the correct answer is far from certain. Inone classic example of counting errors, George Moore from the National Bureau of Standards (before it becameNIST) asked participants to count the occurrences of the letter "e" in a paragraph of text. Aside from minorambiguities such as whether the intent was to count the capital "E" as well as lower case, and whether a missing"e" in a misspelled word should be included, the results (Figure 2) showed that many people systematicallyundercounted the occurrences (perhaps because we read by recognizing whole words, and do not naturally dwellon the individual letters) while a significant number counted more occurrences than were actually present.

A computer spell checker program can do a flawless job of counting the letter "e" in text of course, provided thatthere are no ambiguities about what is to be counted. This highlights one of the advantages that computer-assistedmeasurements have over manual ones. But the problem remains one of identifying the things that are to becounted. In most cases involving images of microstructure (particularly of biological structures which tend to bemore complex than man-made ones) this is not so easy. The human visual system, aided by knowledge about theobjects of interest, the system in which they reside, and the history of the sample, can often outperform acomputer program. It is, however, also subject to distractions and results will vary from one person to another orfrom one time until the next. Tests in which the same series of images are presented to an experienced observer ina different orientation or sequence and yield quite different results are often reported.

In addition to the basic idea that measurement values can vary due to finite precision of measurement, naturalvariations also occur due to finite sampling of results. There are practically no instances in which a measurementprocedure can be applied to all members of a population. Instead, we sample from the entire collection of objectsand measure a smaller number. Assuming (a very large assumption) that the samples are representative of thepopulation, the results of our measurements allow us to infer something about the entire population. Achieving a


representative sample is difficult, and many of the other chapters deal to greater or lesser degrees with howsampling should be done in order to eliminate bias.

Nearly all laboratories where there is occasion to use extensive manualmeasurement of micrographs, or counting of cells or particles therein, havereason to note that certain workers consistently obtain values which areeither higher or lower than the average. While it is fairly easy to count allthe objects in a defined field, either visually or by machine, it is difficultto count only objects of a single class when mixed with objects of otherclasses. Before impeaching the eyesight or arithmetic of your colleagues, seeif you can correctly count objects which you have been trained to recognizefor years, specifically the letter 'E.' After you have finished reading theseinstructions, go back to the beginning of this paragraph and count all theappearances of the letter 'e' in the body of this paragraph. You may use apencil to guide your eyes along the lines, but do not strike out or mark E'sand do not tally individual lines. Go through the text once only; do notrepeat or retrace. You should not require more than 2 minutes. Please stopcounting here!

Please write down your count on a separate piece of paper before you forget itor are tempted to "improve" it. You will later be given an opportunity tocompare your count with the results obtained by others. Thank you for yourcooperation!

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Figure 2. The "e" counting experiment, and some typical results. The true answer is 115,and the mean (average) from many trials is about 103.

The human population affords many illustrations of improper sampling. It would not be reasonable to use thestudent body of a university as a sample to determine the average age of the population, because as a groupstudents tend to be age selected. It would not be reasonable to use the population of a city to represent an entirenation to assess health, because both positive and negative factors of environment, nutrition and health caredistinguish them from rural dwellers. In fact, it is very difficult to figure out a workable scheme by which to


obtain a representative and unbiased sample of the diverse human population on earth. This has been presented asone argument against the existence of flying saucers: If space aliens are so smart that they can build ships to travelhere, they must also understand statistics; if they understand the importance of proper random sampling, wewould not expect the majority of sightings and contact reports to come from people in the rural south who drivepickup trucks and have missing teeth; Q.E.D.

For the purposes of this chapter, we will ignore the various sources of bias (systematic variation or offset inresults) that can arise from human errors, sampling bias, etc. It will be enough to deal with the statisticalcharacterization of our data and the practical tests to which we can subject them.

Distributions of values

Since the measurement results from repeated attempts to measure the same thing naturally vary, either when theoperation is performed on the same specimen (due to measurement precision) or when multiple samples of thepopulation are taken (involving also the natural random variation of the samples), it is natural to treat them as afrequency distribution. Plotting the number of times that each measured value is obtained as a function of thevalue, as was done in Figure 2 for the "e" experiment, shows the distribution. In cases where the measurementvalue is continuous rather than discrete, it is common to "bin" the data into ranges. Such a distribution plot is oftencalled a histogram.

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17 18 19 20 21 22 23Figure 3. Histogram plots of 200 data points in 4,12 and 60 bins


Usually for convenience the bin ranges are either scaled to fit the actual range of data values, so as to produce areasonable number of bins (typically 10 to 20) for plotting, or they are set up with boundary limits that arerounded off to an appropriate degree of precision. However this is done, the important goal is to be sure that thebins are narrow enough and there are enough of them to reveal the actual shape of the distribution, but there arenot so many bins that there are only a few counts in each, which as we will see means that the heights of the binsare not well defined. Figure 3 shows cases of plotting the same data with different numbers of bins. A usefulguide is to use no more bins than about N/20 where N is the number of data points, as this will put an average of20 counts into each bin. Of course, it is the variation in counts from one bin to another that makes distributionsinteresting and so there must be some bins with very few counts, so this offers only a starting point.

Plotting histogram distributions of measured data is a very common technique that communicates in a moresuccinct way than tables of raw data how the measurements vary. In some cases, the distribution itself is the onlymeaningful way to show this. A complicated distribution with several peaks of different sizes and shapes is noteasily reduced to fewer numbers in a way that retains the complexity and permits comparison with otherpopulations.

Usually the reason that we perform the measurements in the first place is to have some means of comparing a setof data from one population with a set of data from another group. This may be to compare two groups that areboth being measured, or to compare one with data taken at another time and place, or to compare real data withsome theoretical model or design specification. In any case, plotting the two histograms for visual comparison isnot a very good solution. We would prefer to have only a few numbers extracted from the histogram thatsummarize the most important aspects of the data, and tools that we can employ to compare them.

The comparisons will most often be of the form that can be reported as a probability - usually the probability thatthe observed difference(s) between the two or more groups are considered to be real rather than chance. This is asurprisingly difficult concept for some people to grasp. We can almost never make a statement that group A isdefinitely different from group B based on a limited set of data that have variations due to the measurementprocess and to the sampling of the populations. Instead we say that based on the sample size we have taken thereis a probability that the difference we actually observe between our two samples is small enough to have arisenpurely by chance, due to the random nature of our sampling.

If this probability is very high, say 30 % or 50 %, then it is risky to conclude that the two samples really aredifferent. If it is very low, say 1% or 5 %, then we may be encouraged to believe that the difference is a realreflection of an underlying difference between the two populations. A 5 % value in this case means that only onetime in twenty would we expect to find such a large difference due solely to the chance variation in themeasurements, if we repeated the experiment. Much of the rest of this chapter will present tools for estimatingsuch probabilities for our data, which will depend on just how great the differences are, how many measurementswe have made, how variable the results within each group are, and what the nature of the actual distribution ofresults is.


Let us consider two extreme examples by way of illustration. Imagine measuring the lengths of two groups ofstraws that are all chosen randomly from the same bucket. Just because there is an inherent variation in themeasurement results (due both to the measurement process and the variation in the straws), the average value ofthe length of straws in group A might be larger than group B. But we would expect that as the number of strawswe tested in each group increased, the difference would be reduced. We would expect the statistical tests toconclude that there was a very large probability that the two sets of data were not distinguishable and mightactually have come from the same master population.

Next, imagine measuring the heights of male and female students in a class. We expect that for a large test sample(lots of students) the average height of the men will be greater than that for the women, because that is what isusually observed in the adult human population. But if our sample is small, say twenty students with 11 men and9 women, we may not get such a clear-cut result. The variation within the groups (men and women) may be aslarge or larger than that between them. But even for a small group the statistical tests should conclude that there isa low probability that the samples could have been taken from the same population.

When political polls quote an uncertainty of several percent in their results, they are expressing this same idea in aslightly different way. Most of us would recognize that preferences of 49 and 51% for two candidates are notreally very different if the stated uncertainty of the poll was 4 percentage points, whereas preferences of 35 and65% probably are different. Of course, this makes the implicit assumption that the poll has been unbiased in howit selected people from the pool of voters, in how it asked the questions, and so forth.

Expressing statistical probabilities in a meaningful and universally understandable way is not simple, and this iswhy the abuse of statistical data is so widespread. When the weatherman says that there is a 50% chance ofshowers tomorrow, does this mean that a) 50% of the locations within the forecast area will see some moment ofrain during the day; b) every place will experience rain for 50% of the day; c) every location has a 50% probabilityof being rained upon 50% of the time; or d) something else entirely?

The mean, median and mode

The simplest way to reduce all of the data from a series of measurements to a single value is simply to averagethem. The mean value is the sum of all the measurements divided by the number of measurements, and is familiarto most people as the average. It is often used because of a belief that it must be a better reflection of the "true"value than any single measurement, but this is not always the case; it depends on the shape of the distribution.Notice that for the data on the "e" experiment in Figure 2, the mean value is not a good estimate of the true value.

There are actually three parameters that are commonly used to describe a histogram distribution in terms of onesingle value. The mean or average is one; the median and mode are the others. For discrete measurements such asthe "e" experiment the mode is the value that corresponds to the peak of the distribution (108 in that example). Fora continuous measurement such as length, it is usually taken as the central value within the bin that has the highestfrequency. The meaning of the mode is simple - it is the value that is most frequently observed.


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Figure 4. Distribution showing the mean, median and mode.

The median is the value that lies in the middle of the distribution in the sense that just as many measured values arelarger than it as are smaller than it. This brings in the idea of ranking in order rather than looking at the valuesthemselves, which will be used in several of the tests in this chapter.

If a distribution is symmetric and has a single peak, then the mean or average does correspond to the peak of thedistribution (the mode) and there are just as many values to the right and left so this is also the median. So, for asimple symmetrical distribution the mean, median and mode are identical. For a distribution that consists of asingle peak but is not symmetric, the mode lies at the top of the peak, and the median is always closer to the modethan the mean is. This is summarized in Figure 4.

When distributions do not have very many data points, determining the mode can be quite difficult. Even thedistribution in Figure 2 is "noisy" so that repeated experiments with new groups of people would probably showthe mode varying by quite a lot compared to the mean. The mean is the most stable of the three values, and it isalso the most widely used. This is partly because it is the easiest to determine (it is just the average), and partlybecause many distributions of real data are symmetrical and in fact have a very specific shape that makes the meana particularly robust way to describe the population.

The central limit theorem and the Gaussian distribution

One of the most common distributions that is observed in natural data has a particular shape, sometimes called the"normal" or "bell" curve; the proper name is Gaussian. Whether we are measuring the weights of bricks, humanintelligence (the so-called IQ test), or even some stereological parameters, the distribution of results often takes onthis form. This is not an accident. The Central Limit Theorem in statistics states that whenever there are a verylarge number of independent causes of variation, each producing fluctuations in the measured result, then


regardless of what the shape of each individual distribution may be, the effect of adding them all together is toproduce a Gaussian shape for the overall distribution.

The Gaussian curve that describes the probability of observing a result of any particular value x, and which fits theshape of the histogram distribution, is given by equation (8.1).

G(x,µ,σ) =1

σ 2π⋅ e

−( x−µ )2

2⋅σ2(8.1

The parameters µ and σ are the mean and standard deviation of the distribution, which are discussed below. The

Gaussian function is a continuous probability of the value x, and when it is compared to a discrete histogramproduced by summing occurrences in bins or ranges of values, the match is only approximate. As discussedbelow, the calculation of the µ and σ values from the data should properly be performed with the individual values

rather than the histogram as well, although in many real cases the differences are not large.

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Figure 5. Distributions of normally distributed data with differing means and standard deviations.

Figure 5 shows several histogram distributions of normally distributed (i.e. Gaussian) data, with the Gaussiancurve described by the mean and standard deviation of the data superimposed. There are three data setsrepresented, each containing 100 values generated by simulation with a Gaussian probability using a randomnumber generator. Notice first of all that several of the individual histogram bins vary substantially from thesmooth Gaussian curve. We will return to this variation to analyze its significance in terms of the number ofobserved data points in each bin later.


There are certainly many situations in which our measured data will not conform to a Gaussian distribution, andlater in this chapter we will deal with the statistical tests that are properly used in those cases. But when the data dofit a Gaussian shape, the statistical tools for describing and comparing sets of data are particularly simple, welldeveloped and analyzed in textbooks, and easily calculated and understood.

Variance and standard deviation

In Figure 5, notice that the three distributions have different mean values (locations of the peak in the distribution)and different widths. Two of the distributions (b and c) have the same mean value of 50, but one is much broaderthan the other. Two of the distributions (a and b) have a similar breadth, but the mean values differ. For theexamples in the figure, the means and standard deviations of the populations from which the data were sampledare:

Figure µ σ5.a 53.0 3.05.b 50.0 3.05.c 50.0 1.0

The mean value, as noted above, is simply the average of all of the measurements. For a Gaussian distribution,since it is symmetrical, the mode and median are equal to the mean. The only remaining parameter needed todescribe the Gaussian distribution is the width of the distribution, for which either the standard deviation (σ in

equation 8.1) or the variance (the square of the standard deviation) is used. These descriptive parameters for adistribution of values can be calculated whether it is actually displayed as a histogram or not.

The standard deviation is defined as the square root of the mean (average) value of the square of the differencebetween each value and the mean (hence the name "root-mean-square" or rms difference). However, calculating itaccording to this definition is not efficient. A procedure for doing so with a single pass through the data is shownin Figure 6 (this also includes several parameters discussed in the next section).

// assume data are held in array value[1..N]for i = 1 to 5 do

sum[i] = 0.0;for j = 1 to N

temp = 1.0for i = 1 to 5 do sum[i] = sum[i] + temp;

temp = temp * value[j];

mean = sum[2]/sum[1];variance = sum[3]/sum[1]-mean*mean;std_dev = sqrt(variance);skew = ((sum[4]-3*mean*sum[3])/sum[1]+2*mean**3)/variance**1.5;kurtosis = ((sum[5]-4*sum[4]*mean+6*sum[3]*mean**2)/sum[1]-

3*mean**4)/variance**2;

Figure 6. Example computer procedure to calculate statistical parameters


Because it has the same units as the mean, the standard deviation is often used to denote the spread of the data inthe form (e.g.) 50.0 ± 3.0 cm, the units in which the value was measured. For data that are normally distributed,this means that 68% of the measured values are expected to fall within the range from 47.0 (50 – 3.0) to 53.0 (50+ 3.0) cm, 95% within the range 44.0 (50 – 2 • 3.0) to 56.0 (50 + 2 • 3.0) cm, and 99% within the range 41.0(50 – 3 • 3.0) to 59.0 (50 + 3 • 3.0) cm.

The values 68, 95 and 99% come directly from integrating the mathematical shape of the Gaussian curve withinlimits of ±1, 2 and 3 times the standard deviation. As written in equation (8.1), the integral of the Gaussian curveis 1.0 (meaning that the probability function is normalized and the total probability of all possible measurementvalues being observed is unity, which makes the statisticians happy). The definite integral between the limits of±1, 2 and 3 σ gives results of 0.6827, 0.9545, and 0.9973, respectively. Tables of the value and integral of the

Gaussian function are widely available in statistics books, but will not be needed for the analytical tests discussedin this chapter.

The variance is just the square of the standard deviation and another measure of the distribution width. It is used insome statistical tests discussed later on. Because in most cases the analyzed data set is but a sample of thepopulation of all possible data, the estimated variance of the population is slightly greater than that calculated in theprocedure of Figure 6 for the data set. The variance of the population is N/(N–1) times the sample variance, whereN is the number of values in the analyzed data set.

The mean value calculated from the data values sampled from a larger population is also only an estimate of thetrue value of the entire population. For example, the data shown in Figure 5b come from a simulated populationwith a mean of exactly 50, but the sample of 100 data points have a calculated mean value of 50.009. We alsoneed to be able to evaluate how well our calculated mean value estimates that of the population, and again theresult depends on how many data points we have taken. The standard error of the mean is just σ / √N where σ is

the calculated standard deviation of the sample and N is the number of values used. It is used in the same way asthe standard deviation, namely we expect to find the true population mean within 1, 2 or 3 standard errors of thecalculated sample mean 68, 95 and 99 % of the time, respectively. It is important to understand the differencebetween the standard deviation of our data sample (σ) and the standard error of the calculated mean value µ.

The mean and standard deviation are also used when measurement data are combined. Adding together two sets ofdata is fairly straightforward. If one set of measurements on one population sample determine a mean and standarddeviation for a particular event as µ1 and σ1, and a second set of measurements on another sample determine the

mean and standard deviation for second event as µ2 and σ2, then the mean probability of either event is the sum of

the two individual means (µ1 + µ2) and the standard deviation is the square root of the sum of the two variances(σ12 + σ22)1/2. To give a concrete example, if the fraction of vehicles passing on the highway is determined in one

time period as 50 ± 6.5 buses per hour, and in a second time period we count 120 ± 8.4 trucks per hour, then thecombined rate of "large vehicles" (busses plus trucks) would be estimated as 170 ± 10.6. The 170 is simply 50 +120 and the 10.6 is the square root of 112.8 = (6.5)2 + (8.4)2.

The situation is a bit more complicated and certainly much more unfavorable if we try to combine two events todetermine the difference. Imagine that we counted 120 ± 8.4 trucks per hour in one time period and then 75 ± 7.2


eighteen-wheel transports (a subclass of trucks) in a second period. Subtracting the second class from the firstwould give us the rate of small trucks, and the mean value is estimated simply as 45 per hour (= 120 – 75). Butthe standard deviation of the net value is determined just as it was for the case of addition; multiple sources ofuncertainty always add together as the square root of the sum of squares. The standard deviation is thus 11.1 (thesquare root of 122.4 = (8.4)2 + (7.2)2) for an estimated rate of 45 ± 11.1, which is a very large uncertainty. Thishighlights the importance of always trying to measure the things we are really interested in, and not determiningthem by difference between two larger values.

Testing distributions for normality - skew and kurtosis

In order to properly use the mean and standard deviation as descriptors of the distribution, it is required that thedata actually have a normal or Gaussian distribution. As can be seen from the graphs in Figure 5, visual judgmentis not a reliable guide (especially when the total number of observations is not great). With little additionalcomputational effort we can calculate two additional parameters, the skew and kurtosis, that can reveal deviationsfrom normality.

The skew and kurtosis are generally described as the third and fourth moments or powers of the distribution. Justas the variance uses the sum of (x – µ)2, the skew uses the average value of (x – µ)3 and the kurtosis uses theaverage value of (x – µ)4 divided by σ4. Viewed in this way, the mean is the first moment (it describes how far

the data lie on the average from zero on the measurement scale) and the variance is the second moment (it uses thesquares of the deviations from the mean as a measure of the spread of the data). The skew uses the third powersand the kurtosis the fourth powers of the values, according to the definitions:

Skew = m3/m23/2 (3rd moment)Kurtosis = m4/m22 (4th moment)

where mk = ∑(xi – µ)k/N . An efficient calculation method is included in the procedure shown in Figure 6.

The skew is a measure of how symmetrical the distribution is. A perfectly symmetrical distribution has a skew ofzero, while positive and negative values indicate distributions that have tails extending to the right (larger values)and left (smaller values) respectively. Kurtosis is a measure of the shape of the distribution. A perfectly Gaussiandistribution has a Kurtosis value of 3.0; smaller values indicate that the distribution is flatter-topped than theGaussian, while larger values result from a distribution that has a high central peak. A word of caution is needed:some statistical analysis packages subtract 3.0 from the calculated kurtosis value so that zero corresponds to anormal distribution, positive values to ones with a central peak, and negative values to ones that are flat-topped.

It can be quite important to use these additional parameters to test a data set to see whether it is Gaussian, beforeusing descriptive parameters and tests that depend on that assumption. Figure 7 shows an example of four sets ofvalues that have the same mean and standard deviation but are very different in the way the data are actuallydistributed. One set (#3 in the figure) does contain values sampled from a normal population. The others do not -one is uniformly spaced, one is bimodal (with two peaks), and one has a tight cluster of values with a fewoutliers. The skew and kurtosis reveal this. The outliers in data set #1 produce a positive skew and the clusteringproduces a large kurtosis. The kurtosis values of the bimodal and uniformly distributed data sets are smaller than


3. For the sample of data taken from a normal distribution the values are close to the ideal zero (skewness) and 3(kurtosis).

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Data Set # Mean=µ Variance=σ2 Skew Kurtosis1 100 140 1.68 5.062 100 140 0 1.123 100 140 –0.352 2.9294 100 140 0 1.62

Figure 7. Four data sets with the same mean and standard deviation values,but different shapes as revealed by higher moments (skew and kurtosis).

But how close do they actually need to be, and how can we practically use these values to test for normality? Asusual, the results must be expressed as a probability. For a given value of skewness or kurtosis calculated for anactual data set, what is the probability that it could have resulted due to the finite sampling of a population that isactually normal? The graphs in Figures 8 and 9 show the answers, as a function of the number of data valuesactually used. If we apply these tests to the 100 data points shown in Figure 5b we calculate a skew value of 0.09and a kurtosis of 2.742, well within the values of 0.4 and range of 2.4 - 3.6 that the graphs predict can happenone time in twenty (5%). Consequently there is no reason to expect that the data did not come from a normalpopulation and we can use that assumption in further analysis. As the data set grows larger, the constraints onskew and kurtosis narrow.


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Figure 9. The probability that the kurtosis will exceed various values for randomsamples from a Gaussian distribution, as a function of the number of observations.


Some other common distributions

The Gaussian or normal curve is certainly not the only distribution that can arise from simple physical processes.As one important example, many of the procedures we use to obtain stereological data are based on counting, anda proper understanding of the statistics of counting is therefore important. Consider a simple event countingprocess, such as determining the number of cars per hour that pass on a particular freeway. We could count for anentire hour, but it seems simpler to count for a short period of time and scale the results up. For example, if wecounted the number of cars in a 5 minute period (e.g., 75 cars) and multiplied by 12, it should give the number ofcars per hour (12*75 = 900). But of course, the particular five minute period we chose cannot be perfectlyrepresentative. Counting for a different 5 minute period (even assuming that the average rate of traffic does notchange) would produce a slightly different result. To what extent can we predict the variation, or to put it anotherway how confident can we be that the result we obtain is representative of the population?

When we count events, it is not possible to get a negative result. This is immediately a clue that the results cannothave a Gaussian or normal distribution, because the tails of the Gaussian curve extend in both directions. Instead,counting statistics are Poisson. The Poisson function is

P(x,µ) =µ⋅ x

x!e−µ (8.2

where µ is the mean value and x! indicates factorial. Notice that there is no σ term in this expression, as there was

for the Gaussian distribution. The width of the Poisson distribution is uniquely determined by the number ofevents counted; it is the square root of the mean value.

This simple result means that if we counted 75 cars in our 5 minute period, the standard deviation that allows us topredict the variation of results (68% within 1 standard deviation, 95% within 2, etc.) is simply √75 = 8.66. Thefive minute result of 75±8.66 scales up to an hourly rate of 900±103.9, which is a much less precise result thanwe would have obtained by counting longer. Counting 900 cars in an hour would have given a standard deviationof √900 = 30, considerably better than 104. On the other hand, counting for 1 minute would have talliedapproximately 900/60 = 15 cars, √15 = 3.88, and 15±3.88 scales to an estimated hourly rate of 900±232.8 whichis much worse. This indicates that controlling the number of events (e.g., marks on grids) that we count is vitallyimportant to control the precision of the estimate that we obtain for the desired quantity being measured.

Figure 10 shows a Poisson distribution for the case of µ=2 (a mean value of two counts). The continuous line forthe P(x,µ) curve is only of mathematical interest because counting deals only with integers. But the curve doesshow that the mean value (2.0) is greater than the mode (the highest point on the curve), which is always true forthe Poisson distribution. So the Poisson function has a positive skew (and also, incidentally, a kurtosis greaterthan three).

Fortunately, we usually deal with a much larger number of counts than 2. When the mean of the Poisson functionis large the distribution is indistinguishable from the Gaussian (except of course that the standard deviation σ is

still given by the square root of the mean µ). This means that the function and distribution becomes symmetricaland shaped like a Gaussian, and consequently the statistical tools for describing and comparing data sets can beused.


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Figure 10. Poisson function and distribution for a mean of 2.

However, there is one important arena in which the assumption of large numbers is not always met. When adistribution of any measured function is plotted, even if the number of bins is small enough that most of themcontain a large number of counts, there are usually some bins at the extremities of the distribution that have only afew counts in them. If these bins are compared from one distribution to another, it must be based on theirunderlying Poisson nature and not on an assumption of Gaussian behavior. Comparing the overall distributionsby their means and standard deviations is fine as a way to determine if the populations can be distinguished.Trying to detect small differences at the extremes of the populations, such as the presence of a small fraction oflarger or smaller members of the population, is much more difficult.

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Figure 11. Linear and logarithmic scale histogram plots of the same set of log-normally distributed values.

Another distribution that shows up often enough in stereological measurement to take note of is the log-normaldistribution. This is a histogram in which the horizontal axis is not the measured value but rather the logarithm ofthe measured value, but the resulting shape of the distribution is normal or Gaussian. Note that the bins in thedistribution are still of uniform width, which means they each cover the same ratio of size values (or whatever themeasurement records) rather than the same increment of sizes. This type of distribution is often observed for


particle sizes, since physical processes such as the grinding and fracturing of brittle materials produce thisbehavior. Figure 11 shows a histogram distribution for log-normally distributed data; on a linear scale it ispositively skewed.

For a thorough discussion of various distributions that occur and the tools available to analyze them, see P. R.Bevington (1969) “Data Reduction and Error Analysis for the Physical Sciences” McGraw Hill, New York

Comparing sets of measurements - the t-test

As mentioned at the beginning of this chapter, one common purpose of statistical analysis is to determine thelikelihood that two (or more) sets of measurements (for example, on different specimens) are the same ordifferent. The answer to this question is usually expressed as a probability that the two samples of data could havecome from the same original population. If this probability is very low, it is often taken as an indication that thespecimens are really different. A probability of 5% (often written as α=0.05) that two data sets could have come

from the same population and have slightly different descriptive parameters (mean, etc.) purely due to samplingvariability is often expressed as a 95% confidence level that they are different. It is in fact much more difficult toprove the opposite position, that the two specimens really are the same.

When the data are normally distributed, a very efficient comparison can be made with student's t-test. Thiscompares the mean and standard deviation values for two populations, and calculates a probability that the twodata sets are really drawn from the same master population and that the observed differences have arisen by chancedue to sampling. Remember that for a Gaussian distribution, the mean and standard deviation contain all of thedescriptive information that is needed (the skew and kurtosis are fixed).

Given two normal (Gaussian) sets of data, each with ni observations and characterized by a mean value µ i and astandard deviation σi, what is the probability that they are significantly different? This is calculated from the

difference in the mean values, in terms of the magnitude of the standard deviations and number of observations ineach data set, as shown in equation (8.3); ν is the number of degrees of freedom, which is needed to assess the

probabilities in Figure 12.

T =µ1 −µ 2

σ12

n1− σ2

2

n2

ν =

σ12

n1+

σ22

n2

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σ12

n1

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n1 −1+

σ22

n2

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n2 −1

(8.3

The larger the difference of means (relative to the standard deviations) and the larger the data sets, the more likelyit is that the two sets of observations could not have come from a single population by random selection. Theprocedure is to compare the value of the parameter T to the table of student's t values shown by the graph inFigure 12 for the number of degrees of freedom and probability α. If the magnitude of T is less than the table

value, it indicates that the two data sets could have come from the same population, with the differences arisingsimply from random selection. If it exceeds the table value, it indicates the two sets are probably different at the


corresponding level of confidence 100 • (1 – α); for example, α=0.01 corresponds to a 99% confidence level that

the two groups are not the same.

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Figure 12. Critical values for t-test comparison as a function of the number of degrees of freedom.

The value of α is given for a double-sided test (the two means are "different"). In a single sided test (deciding

whether one mean is "greater" or "less" than the second), the value of α is halved. Notice that the curves of critical

values level off quickly as the number of degrees of freedom increases. For typical data sets with at least tens ofobservations in each, only the asymptotic limit values are needed. Hence a value of T greater than 1.282 indicatesthat the groups are not the same with a confidence level of 90% (α=0.10), and values greater than 1.645 and

1.960 give corresponding confidence values of 95% (α=0.05) and 99% (α=0.01) respectively. Common and

convenient use is made of the easily remembered approximate test value T≥2.0 for 99% confidence.

Comparing the two sets of data in Figure 5a and 5b using the t-test gives the following results:Data Set µ σ n

a 52.976 3.110 100b 50.009 3.152 100

The calculated value of T is 57.83, the number of degrees of freedom is large enough to consider just theasymptotic values, and since T exceeds 1.96 the probability that the two data sets are significantly different isgreater than 99%.

The Analysis of Variance or ANOVA test is a generalization of the t-test to more than 2 groups, still making theassumption that each of the groups has a normal data distribution. The ANOVA test compares the differencesbetween the means of several classes, in terms of their number of observations and variances. For the case of twodata sets, it is identical to the t-test.


To perform the ANOVA test, we calculate the following sums-of-squares terms from the observations yij (i=class,j=observation number). yi* is the mean of observations in class i, and ymean is the global average. There are ni

observations in each class, k total classes, and t total observations.

SST (total sum of squares of differences) = ∑∑ (yij - ymean)2

SSA (sum of squares of difference within the classes) = ∑∑ (yi* – ymean)2 = ∑ ni • (yi* – ymean)2 (8.4SSE (difference between the total variation and that within classes) = SST – SSA

From these, an F value is calculated as F = (SSA/n1) / (SSE/n2)where the degrees of freedom are n1 = k – 1 and n2 = t – k

n2 n1=3 n1=5 n1=10 n1=40 n1=∞

3 9.28 9.01 8.79 8.59 8.535 5.41 5.05 4.74 4.46 4.3610 3.71 3.33 2.98 2.66 2.5440 2.84 2.45 2.08 1.69 1.51∞ 2.60 2.21 1.83 1.39 1.00

Figure 13. Critical values of F for α=0.05 (ANOVA test).

This value of F is then used to determine the probability that the observations in the k classes could have beenselected randomly from a single parent population. If the value is less than the critical values shown in tables, thenthe difference between the groups is not significant at the corresponding level of probability. Figure 13 showscritical values for F for the case of α=0.05 (95% confidence that not all of the data sets come from the same parent

population).

Nonparametric comparisons

The t-test and ANOVA test are flawed if the data are not actually Gaussian or “normal” since the mean andstandard deviation do not then fully describe the data. Nonparametric tests do not make the assumption ofnormality and do not use “parameters” such as mean and standard deviation to characterize the data. One set ofmethods is based on using the rank order of the measured data rather than their actual numerical values. As appliedto two data sets, the Wilcoxon test sorts the two sets of values together into order based on the measurementvalues as shown schematically in Figure 14. Then the positions of the observations from the two data sets in thesequence are examined. If all of the members of one set are sorted to one end of the stack, it indicates that the twogroups are not the same, while if they are intimately mixed together the two groups are consideredindistinguishable. This test is also called the Mann-Whitney test, which generalized the original method to dealwith groups of different sizes.


A

A

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BBBB

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Sorted Object

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Indistinguishable Distinguishable

Figure 14. Principle of the Wilcoxon or Mann-Whitney test based on rank order.

This idea is much the same as examining the sequence of red and black cards in a deck of playing cards to decidewhether it has been well shuffled. The binomial theorem allows calculation of the probability of any particularsequence of red and black cards occurring. Shuffles that separate the red and black cards are much less likely tooccur than ones with them mixed together, and will therefore happen only rarely by chance. To illustrate theprocedure, consider the two extreme cases of data shown in Figure 15. Case A has the two groups well mixed andCase B is completely segregated. The sum of rank positions of the groups are tallied and whichever is smaller isthen used to calculate the test statistic U.

U = W1 − n1 ⋅n1 +1( )

2(8.5

Rank Case A Case B1 1 12 2 13 1 14 2 15 1 16 2 27 1 28 2 29 1 210 2 2

Case W1=Rank Sum n1 U αA 25 5 10 0.6015B 15 5 0 0.0090

Figure 15. Example of two extreme cases for Wilcoxon comparison.

where W i is the sum of rank values and ni is the number of observations in the groups. If the value of U is lessthan a critical value that depends on the number of observations in the two data sets and on a the probability of


chance occurrence by random sampling of values from a single population, then the two groups are considered tobe different with the corresponding degree of confidence. In the example, the two groups in Case A are notdistinguishable (there is a 60% probability that they could have come from the same population) and those in CaseB are (there is a 0.9% probability that they came from the same population). Figure 16 shows the critical testvalues of U for α=0.01 and α=0.05 (99 and 95% confidence respectively) for several values of ni.

Figure 16. Critical values for the Wilcoxon (Mann-Whitney) test.

The Kruskal-Wallis test is a generalization of the Wilcoxon or Mann-Whitney test to more than two groups, inmuch the same way that the ANOVA test generalizes the t-test. The data sets are sorted into one list and their rankpositions used to calculate the parameter H based on the number of groups k, the number of objects n (andnumber in each group ni), and the rank order of each observation R (summed for those in each group). This testvalue H is compared to the critical values (which come from the chi-squared distribution) for the number ofdegrees of freedom (k–1) and the probability α that this magnitude of H could have occurred purely by chance for

observations all sampled randomly from one parent group.

H = n • (n + 1)

12 • ∑i=1

k

( ni

Ri2

) – 3 • (n + 1)

(8.6

Figure 17 shows a tiny data set to illustrate the principle of the Kruskal-Wallis test. Three sets of measurementvalues are listed with the groups identified. Sorting them into order and summing the rank order for each groupgives the values shown. Based on the number of degrees of freedom (df=k–1), the probability that the calculatedvalue of H could have occurred by chance in sampling from a single parent population is 43.5%, so we would notconclude that the three data sets are distinguishable. Figure 18 shows a table of critical values for H for severalvalues of df and different values of α. H must exceed the table value for the difference to be considered significant

at the corresponding level of confidence.


Length Group24.0 116.7 122.8 119.8 118.9 123.2 219.8 218.1 217.6 220.2 217.8 218.4 319.1 317.3 319.7 318.9 318.8 319.3 317.3 3

Class ni Rank sum1 5 612 6 63.53 8 65.5

k 3df 2n 19H 1.663α 0.4354

Figure 17. Kruskal-Wallis example (3 groups of data).

df α 0.05 0.025 0.011 3.841 5.024 6.6352 5.991 7.378 9.2103 7.815 9.348 11.3454 9.488 11.143 13.2775 11.070 12.832 15.0866 12.592 14.449 16.8127 14.067 16.013 18.4758 15.507 17.535 20.0909 16.919 19.023 21.66610 18.307 20.483 23.209

Figure 18. Critical value table for the Kruskal-Wallis test.

The Wilcoxon (Mann-Whitney) and Kruskal-Wallis tests rely on sorting the measured values into order based ontheir numerical magnitude, but then using the rank order of the values in the sorted list for analysis. Sorting ofvalues into order is a very slow task, even for a computer, when the number of observations becomes large. Forsuch cases there is a more efficient nonparametric test, the Kolmogorov-Smirnov test, that uses cumulative plotsof variables and compares these plots for two data sets to find the largest difference.


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Figure 19. Cumulative plot compared with usual differential histogram display.

As shown in Figure 19, the cumulative distribution plot shows the fraction or percentage of observations that havea value (length in the example) at least as great as the value along the axis. Cumulative plots can be constructedwith binned data or directly from the actual measurement values. Because they are drawn with the vertical axis inpercent rather than the actual number of observations, it becomes possible to compare distributions that havedifferent numbers of measurements.

Data Set Length ValuesA 1.023, 1.117, 1.232, 1.291, 1.305, 1.413, 1.445, 1.518,

1.602, 1.781, 1.822, 1.889, 1.904, 1.967B 1.019, 1.224, 1.358, 1.456, 1.514, 1.640, 1.759, 1.803,

1.872

2.01.81.61.41.21.00.0

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Greatest Difference

Data Set A

Data Set B

n1 14n2 9Max.Diff. 0.214A (α=0.05) 1.22

Test Value 0.521

Figure 20. Example of applying the Kolmogorov-Smirnov test.


Figure 20 shows the step-by-step process of performing the Kolmogorov-Smirnov test. The two data sets areplotted as cumulative distributions showing the fraction of values as a function of length. Once the plots have beenobtained, the greatest vertical difference between them is located. Since the vertical axis is a fraction, there are nounits associated with the difference value. It does not matter where along the horizontal axis the maximumdifference occurs, so the actual magnitude of the measurement values is unimportant.

The maximum difference is compared to a test value calculated from the number of observations in the groups ni

S = A ⋅n1 + n2

n1 ⋅n2

1

2(8.7

where the parameter A is taken from the table in Figure 21 to correspond to the desired degree of confidence. Forthe specific case of α=0.05 (95% probability that the two data sets did not come from sampling the same

population), A is 1.22 and the test value is 0.521. Since the maximum difference of 0.214 is less than the testvalue, the conclusion is that the two data sets cannot be said to be distinguishable.

α 0.10 0.05 0.025 0.010

A 1.07 1.22 1.36 1.52

Figure 21. Critical Values for the Kolmogorov-Smirnov test.

For large data sets, the Kolmogorov-Smirnov test is a more efficient nonparametric test than the Wilcoxon becausemuch less work is required to construct the cumulative distributions than to sort the values into order. Figure 22shows an example of a comparison of two large populations (n1=889, n2=523). The greatest difference in thecumulative histograms is 0.118 (11.8%). For a confidence level of 99%, α=0.01 and A=1.52, so the test value S

is 0.84. Since the measured difference is greater than the test value, we conclude that the two populations aredistinguishable and not likely to have come from the same population.

For a complete discussion of these and other nonparametric analysis tools, see J. D. Gibbons (1985)“Nonparametric Methods for Quantitative Analysis, 2nd Edition” American Sciences Press, Columbus, OH.Nonparametric tests are not as efficient at providing answers of a given confidence level as parametric tests, but ofcourse they can be applied to normally distributed data as well as to any other data set. They will provide the sameanswers as the more common and better-known parametric tests (e.g., the t-test) when the data are actuallynormally distributed, but they generally require from 50% more to twice as many data values to achieve the sameconfidence level. However, unless the measured data are known to be Gaussian in distribution (such as count datawhen the numbers are reasonably large), or have been tested to verify that they are normal, it is generally safer touse a nonparametric test since the improper use of a parametric test can lead to quite erroneous results if the datado not meet the assumed criterion of normality.


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Maximum DiameterFigure 22. Overlay comparison of cumulative plots of two populations for the Kolmogorov-Smirnov test.

Linear regression

When two or more measurement parameters are recorded for each object, it is usually of interest to look for somerelationship between them. If we recorded the height, weight and grade average of each student we might searchfor correlations. It would be interesting to determine the probability that there is a real correlation between the firsttwo variables but not between them and the third one. Again, we will want to express this as a confidence limit or


a probability that any observed correlation did not arise from chance as the data were selected from a truly randompopulation.

Figure 23. Presentation modes for data correlatingtwo variables: a) point plot, b) flower plot, c) two-way histogram.

The first tool used is generally to plot the variables in two (or more) dimensions to look for visual trends andpatterns. Figure 23 shows some of the common ways that this is done. The point plot works well if the individualpoints representing data measured from one object are not so dense that they overlap. Flower plots are producedby binning the data into cells and counting the number within each cell, while the two-way histogram shows quitegraphically where the clusters of points lie but in doing so hides some parts of the plot. With interactive computergraphics, the scatter plot can be extended to handle more than two dimensions by allowing the free rotation of thedata space, and color coding of points allows comparison of two or more different populations in the same space.

Since the human visual system, abetted by some prior knowledge about the problem being studied, is very good atdetecting patterns in quite noisy data, this approach is often a very useful starting point in data analysis. However,humans are often a bit too good at finding patterns in data (for instance, constellations in the starry sky), and sowe would like to have more objective ways to evaluate whether a statistically significant correlation exists.

Linear regression makes the underlying assumption that one variable (traditionally plotted on the vertical y axis) isa dependent function of the second (plotted on the horizontal x axis), and seeks to represent the relationship by anequation of the form y = m • x + b, which is the equation for a straight line. The process then determines thevalues of m and b which give the best fit of the data to the line in the specific sense of minimizing the sum of


squares of the vertical deviations of the line from the points. For N data points xi, yi the optimal values of m and bare

m =N • xi yi − xi • yi∑∑∑

N • xi2 − xi∑( )2∑

b = yi − m • xi∑∑N

(8.8

This is conventional linear regression. When it is used with data that are not all of the same value (weightedregression), or where the data have been previously transformed in some way (for example by taking thelogarithm, equivalent to plotting the data on log paper and fitting a straight line), the procedure is a bit morecomplex and beyond the intended scope of this chapter. Figure 24 shows a computer procedure that will determinem and b and also calculate the standard deviation of both values.

// assumes data are held in arrays x[1..N], y[1..N]sumx = 0;sumy = 0;sumx2 = 0;sumy2 = 0;sumxy = 0;for i = 1 TO N

sum = sum+1;sumx = sumx+x[i];sumy = sumy+y[i];sumx2 = sumx2+x[i]*x[i];sumy2 = sumy2+y[i]*y[i];sumxy = sumxy+x[i]*y[i];

dx = sum*sumx2-sumx*sumx;dy = sum*sumy2-sumy*sumy;d2 = sum*sumxy-sumx*sumy;m = d2/dx; //slopeb = (sumy-a*sumx)/sum; //interceptss = 1.0/(sum-1)*(sumy2+sum*b*b+a*a*sumx2-2*(b*sumy-

a*b*sumx+a*sumxy));sigma_b = sqrt(ss*sumx2/dx); //intercept standard deviationsigma_m = sqrt(sum*ss/dx); //slope standard deviationr2 = d2/sqrt(abs(dx*dy)); //correlation coefficient

Figure 24. Example computer procedure to calculate linear regression

But how good is the line as a description of the data (or in other words how well does it fit the points)? Figure 25shows two examples with the same number of points. In both cases the procedure outlined above determines abest fit line, but the fit is clearly better for one data set than the other. The parameter that is generally used todescribe the goodness of fit is the correlation coefficient R or R2. This is a dimensionless value that can varybetween zero (no correlation whatever - any line would be as good) and one (a perfect fit with the line passingexactly through all of the points). The sign of the R value can be either positive (y increases with x) or negative (ydecreases as x increases) but only the magnitude is considered in assessing the goodness of fit. R is calculatedfrom the product of two slopes, one treating y as a dependent function of x (minimizing the squares of the verticaldeviations of the line from the points) and one treating x as a dependent function of y (minimizing the sum ofsquares of the horizontal deviations).


R =N • xiyi − xi • yi∑∑∑

N • xi2 − xi∑( )2∑ • N • yi

2 − yi∑( )2∑(8.9

For the examples in Figure 25, the "good fit" data have an R value of 0.939 and the "poor fit" data a value of0.271. How are we to assess these values? It depends upon the number of points used in calculating the fit as wella the magnitude of R, and again we use α as the probability that a fit with the same R might occur by chance if we

were fitting the line to truly random points like salt grains sprinkled on the table. For the case of 12 data points,the value of 0.939 would be expected to occur only once in 10000 times (α=0.01%), while a value of 0.271

would be expected to occur nearly 2 times in 5 (α=39.4%). We would consider the first to be statistically

significant but not the second.

Figure 25. Data sets with high and low correlation coefficients.

Figure 26 shows the relationship for evaluating the fit. For any given number of points, an R value above thecorresponding line means that the fit is significant with the corresponding level of confidence (i.e. did not occurby chance with a probability α). As the number of points increases, the value of R required for any selected

degree of confidence is reduced. Alpha is the probability that the apparent fit arose by chance from uncorrelatedvalues. The R2 value is a measure of the percentage of the variation in the y (dependent variable) values that is“explained” is a statistical sense by the fit to the independent variable x.

Linear regression can be extended relatively straightforwardly to deal with more than two variables. Multipleregression tries to express one dependent parameter z as a linear combination of many others (z=a0 + a1•x1 +a2•x2 + ...). The procedure for efficiently determining the ai values is very similar to the matrix arithmetic usuallyemployed to calculate analysis of variance for multiple parameters. If the xi parameters are actually values of asingle parameter raised to successive powers, then the relationship is z=a0 + a1•x1 + a2•x2 + ... and we havepolynomial regression. An excellent reference for regression methods is N. Draper, H. Smith (1981) “AppliedRegression Analysis, 2nd Edition” J. Wiley & Sons, New York.


Figure 26. Probability of significance of values of the linear correlation coefficient R for N observations.

Stepwise multiple regression starts with a list of independent variables x and adds and removes them one at a time,keeping only those which have a values that are statistically significant. We will not cover the details of thecalculation here but many computer statistics packages provide the capability. A closely related technique providedby quite a few programs plots all of the data points in a high dimensionality space (corresponding to the number ofmeasured parameters) and then finds the orientation of axes that best projects the data onto planes that give highcorrelation coefficients. The "principal components analysis" method is another way, like stepwise multipleregression, that identifies which of the available independent variables actually correlate with the dependentvariable.

The main difference is that stepwise regression treats the independent variables separately while principalcomponents analysis groups them together algebraically. Neither method tells the user whether the correlationreveals any physical significance, whether the assumed dependent variable is actually causally related to theindependent one, or vice versa, or whether both may be dependent on some other (perhaps unmeasured)parameter. Attempting to infer causality from correlation is a common error that provides some of the moreegregious examples of the misuse of statistical analysis.

Nonlinear regression

The previous section dealt with the mathematics of fitting straight lines to data, and referenced methods that fitpolynomials and other fixed mathematical relationships. These all assume, of course, that the user has some ideaof what the appropriate relationship between the independent and dependent variables is (as well as correctlyidentifying which is which). The assumption of linearity is convenient from a calculation standpoint, but there arefew reasons to expect genuine physical circumstances to correspond to it.

When there is no reason to expect a particular functional form to the data, we would still like to have a tool toassess the degree to which the data are correlated (one increases monotonically with variation in the other).Spearman or rank correlation accomplishes this in a typical nonparametric way by using the rank order of thevalues rather than their numerical values. The data are sorted into rank order and each point's numerical values forthe x and y parameters is replaced by the integer rank position of the value for the corresponding parameter. Theserank positions are then plotted against each other (as illustrated in Figure 27) and the R value calculated and


interpreted in the usual way. This method also has advantages in analyzing data that are strongly clustered at oneend of a plot or otherwise distributed nonuniformly along a conventional linear regression line.

-1.5

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Figure 27. Comparison of regression using actual values (a) and rank order of values (b).


Chapter 9 - Computer-Assisted Methods

There are relatively few applications in which completely automatic computer-based methods are easily employedto perform the various counting and measuring operations that yield stereological data. In practically all casessome human effort is required at least to oversee the image processing operations that make it ready for automaticmeasurement, to verify that the features that are of interest are actually extracted from the more complex image thatis acquired. In quite a few cases, the computer is used only to tally human counting operations (e.g., mouseclicks) while the human powers of recognition are used (or sometimes misused) to identify the features of interest.Even in this mode, the use of the computer is often worthwhile for image acquisition and processing to enhancevisibility of the structure, and to keep the records accurately for later analysis. This chapter discusses the use of acomputer to aid manual analysis, while the next one deals with computer-based measurements.

Getting the image to the computer

A typical computer-assisted system consists of the components shown in Figure 1 (Inoue, 1986). In the majorityof cases, the camera is a conventional video camera, usually a solid state camera. It may be either color ormonochrome ("black and white") depending on whether the application deals with color images. In most cases,biological tissue uses colored stains to delineate structures whereas in materials science color is less useful. Thecamera is connected to a "frame grabber" (analog to digital converter, or digitizer board) installed in the computer.This reads the analog voltage signal from the camera and stores a digital representation in the form of pixels withinthe computer. This is then displayed on the computer screen, stored as desired, printed out, and used forcomputation.

Optics

Camera Digitizer Memory Display

Storage

Printout

Processing

User

Figure 1. Block diagram of a typical computer-based imaging system.

There are many choices involved in most of these operations. Video cameras come in two principal flavors, tubetype and solid state. The vidicon is a classic example of the former, the CCD of the latter. Tube cameras have acontinuous light sensitive coating that is capable of higher resolution than the array of discrete transistorsfabricated on the chip in a solid state camera, and have a spectral response that matches the human eye andconventional photography well. But most of the other advantages belong to the solid state camera. It has very littledistortion (especially if electrical fields from a nearby transformer are present to distort the electron beam scanningin the tube camera), is inherently a linear device in terms of output signal versus light intensity, does not suffer

Chapter 9- Computer Assisted Measurements Page # 159

from "bloom" in which bright areas spread to appear larger than they actually are, can be mounted in anyorientation required by the microscope, and is generally less expensive for equivalent performance because of theeconomics of manufacturing high-end consumer cameras in quantity.

Color sensitivity can be achieved in three ways: by splitting the incoming light to fall on three separate detectors(tube or CCD) each filtered to see just the red, green or blue; by using filters to sequentially capture the red, greenand blue images; or by depositing a filter on the solid state chip that allows columns of transistors to detect thedifferent colors (there are other more efficient geometries possible but they are more expensive to fabricate). Thelatter method gives only about one-half the lateral resolution of the former (it is acceptable for broadcast videobecause the color information has lower resolution than the brightness variations). The single chip color camera isnot so satisfactory for most technical applications, but much less expensive. Even with a monochrome camera, theuse of a filter to eliminate infrared light, to which the solid state cameras are sensitive, can be very importantbecause the imaging optics will not focus the light beyond the visible range to the same plane as the useful image.

Figure 2. The effect of changing the size of a pixel in a grey scale and a color image (clockwisefrom upper left pixels are doubled in size) is to obscure small features and boundaries.

The signal from the video camera is the same as used for standard broadcast television, and effectively limits thehorizontal resolution of the image to less than 330 points and the grey scale resolution to less than 60 grey levels.Figure 2 illustrates the effect of changing pixel size on image representation, and Figure 3 shows the effect ofvarying the number of grey levels used. Digitizers often store the image in an array 640 pixels wide by 480 highbecause this corresponds to the dimensions of many computer screens, and use 256 grey levels because itcorresponds to the size of one byte of memory, but this is partially empty magnification in both cases. If colorinformation is being used, the composite signal produced by low end cameras is greatly inferior in colorbandwidth to component (also called S-video) signals with separate wires for luminance and chrominance. Theconsequence is blurring of color boundaries, which directly affects image measurements. Some high end camerasprovide separate red, green and blue (R G B) signals.


Figure 3. The effect of changing the number of bits used for grey scale and color plane representation is toposterize the image. Depending on the quality of the image reproduction on screen or book page, the segmentsof the grey scale image with a large number of discrete steps may not be fully resolved. The color image shows

that colors are significantly altered with only 2, 4 or 8 levels for red, green and blue planes.

Video digitizers are called frame grabbers because they record a single video frame lasting 1/30 (in Europe 1/25)of a second. Standard video is broadcast using interlace, so the frame actually consists of two fields, one with theodd number scan lines (1, 3, 5, 7, ...) and then 1/60th of a second later the even scan lines (2, 4, 6, 8, ...). Forrapidly changing images, this can cause additional distortion of features. But in most applications the image isstationary and the rapid speed of frame grabbing is not needed; indeed it causes a definite loss of quality becauseof electronic noise. Averaging several video frames, or (with more specialized cameras not used for standardvideo) collecting light for a longer period of time, produces much better quality images with more distinguishablegrey levels.

Special cameras with many more transistors that can record images with more than 1000x1000 pixels areavailable, but cost more than standard video cameras. Cooled cameras (to reduce electrical noise) with a large pixelarray can be used for computer imaging, but in addition to their cost suffer from relatively slow image transfer.Most of these perform the digitzation within the camera and transmit digital values to the computer, and the timerequired to send millions of values limits either the portion of the image or the number of updates that can beviewed per second, so that focusing and field selection can be difficult. The great advantage of digital cameras ofthis type is that with cooling of the chip and electronics, thermal noise is reduced and it becomes practical tointegrate the photons striking the transistors for many seconds. This allows capturing very dim images;astronomical image and fluorescence microscopy are the two major applications.

There is another group of digital cameras that are aimed more at the consumer and general photography marketthan at the microscopy market. At the lower end of the price range, these have built in lenses that are not easily


removed or adapted to microscope connection, and arrays that produce full color digital images of about 320x240resolution (sometimes the images can be made larger, but it is empty magnification). Recall that this is about thesame as the resolution of a typical consumer level video camera connected to a frame grabber (indeed, the cameraCCD is identical to that used in the video camera), but the cost is very low and no frame grabber is required. Theinterface to the computer is generally a relative slow (several seconds for image transfer) serial link, and many ofthe cameras come with cables and software for both Macintosh and Windows platforms. These offer some veryinteresting possibilities for capturing images that are to be used for interactive marking of points to count. Foroccasional use, simply attaching the camera so that it looks through the microscope eyepiece may be satisfactory.The principal limitation of the low-end units is distortion of the image due to the limited optics.

Higher end units consist of a camera back that can be attached to standard lens mounts, which does allow theirconnection to microscopes. These may either capture full color images in a single pass, or use color filters tocapture three images (red, green, and blue) sequentially which are then recombined in the computer. These caneither be three separate filters inserted with a filter wheel, or a more costly LCD filter that has no moving parts andchanges it transmission characteristics to produce the desired band pass. The image quality with such a setup canbe quite high, approaching the resolution of 35 mm film.

Cameras in the bottom end of the range are available at this writing from many of the companies in the“traditional” photographic camera and consumer electronics businesses, such as Casio, Olympus, Kodak, etc.Higher performance units are sold by Sony, Nikon, Kodak, Polaroid and others. The evolution of consumer“handy-cam” cameras to ones with digital capability is also underway. The latest generation of these units usedigital tape to record the images (and sound) instead of recording the analog signals. Many of these units arealready capable of being computer controlled and it seems only a matter of time until the images can also be read,either from the tape or from the camera directly into computer memory. This would approximately double theresolution of images from these cameras, since the principal current limitations are in the electronics and not theCCD or optics.

There is another type of digital camera that does not require the large CCD array (which is the source of much ofthe cost of most units). The MicroLumina camera from Leaf/Scitex uses three parallel linear CCD arrays (one forred, one for green, one for blue) which are scanned across the film plane in the camera back (it mounts whereverNikon backs can be attached) to record the image. With 12 bits each (a range of 212 or 4096:1) and a fullresolution of 3400x2700 pixels, these images rival film performance at a cost considerably below the cameras thatuse a large CCD array. They are not much more costly than a good quality color video camera and frame grabberboard. The limitation, of course, is that the image capture time is over a minute (somewhat dependent on computerand hard disk speed) so the image must be a stationary one with adequate brightness, since each point is read foronly a momentary time interval.

Display and storage

One of the reasons that transferring the image to the computer in digital form is desirable is the flexibility withwhich it can be displayed on the computer screen (Foley & Van Dam, 1984). Some frame grabber boards alsofunction as the display boards for the computer and directly show the image on the screen, but in most cases the


image is held in memory and transferred to the display memory under program control. This allows for variousmodifications and enhancements that can increase the visibility of details, to assist the human operator inrecognizing, locating and marking the features of interest. The processing possibilities are summarized below.Even without such steps, simple modification of the display brightness to maximize contrast range, adjust itnonlinearly (“gamma” adjustments) to increase contrast in either the bright or dark areas, adjust colors for changesin lighting, substitute colors (“pseudo color”) for grey scale values to make small variations visible, are moreeasily and reproducibly accomplished in the digital realm than by using analog circuits. They can also beaccomplished without altering the original data.

Storing images for archival or reference purposes is one area where great care is needed in assembling a system.Images are large. A full scanned color image from the MicroLumina camera mentioned above occupies more than26 megabytes of storage, either in ram memory or on disk. Obviously, this means that a computer with adequatememory to hold the entire image for user access, a large display to show many pixels at a time, and a fastprocessor to be able to access all of the data, are desirable. But computer systems have advanced rapidly in all ofthese areas and are expected to continue to do so. A system with a fast Pentium or Power PC processor, 128 Megof ram or more, a display screen with at least 800 and perhaps more than 1000 pixel width and capable of “truecolor” (8 bits or 256 levels each for R, G and B), and a reasonably large (several gigabyte) hard disk can be had ina laptop at this writing.

The major questions have to do with long term storage of the images. Any hard disk will be filled quickly withimages if many are to be kept. Less expensive, slower access, more permanent storage is therefore desirable. Tapesystems have been widely used for computer disk backup, but because access is sequential and the tapesthemselves are subject to stretching and wear if they are used often, this does not seem attractive for an imagestorage facility. Writable and rewritable CD-ROM disks are widely regarded as the most suitable medium at themoment, with the latest generation of CDs providing expanded storage on a single disk from 600 Meg by about anorder of magnitude, enough to store a useful number of images on a single platter. Access speed is good, but theonly practical way to use CD-R (as opposed to CD-RW) is to write all of the disk at one time, which means thatconventional hard disk storage of the same size is required to assemble the files to be written.

The greatest problem with archiving images is having some way to find them again when they are needed. Ifpictures can be identified based on some independent scheme such as patient identification in medical imaging,date and sample number in forensics, etc., then the problem is relatively simple. In a research setting things getmore complicated. It is very difficult to set up a data base that uses words to describe all of the importantcharacteristics of each particular image. The dictionary of words must be very consistent and fairly small so that aword search of the data based will locate all of the relevant images. The discipline to create and maintain such adata base is hard to instill, and consistency between operators is poor.

It would be more desirable to actually have the ability to see the image, since recollection and recognitionprocesses in human memory are extremely powerful. But archival storage implies that the images themselves areoff-line, and only the data base with some means of identification is available to the user. A partial solution to thisdilemma is offered by image compression. A very much smaller (in terms of storage requirements) version of animage can be produced using encoding algorithms that take advantage of redundancies in the pixel values within


the image, or discard some frequency components. Compression methods that achieve useful compression ratiosof 50:1 or higher are “lossy,” meaning that some of the data in the original image are permanently lost in thecompressed version. These can be very important for measurement purposes - small features are missing, colorand brightness values are distorted, and boundaries are blurred and shifted.

We will want to ensure that our archival storage system does not use a lossy compression method for the originalversions of the images. But a compressed version may be quite suitable for inclusion in the image data base tohelp the user locate a particular image that cannot be completely or adequately described in words. Thecompression methods have been developed for digital video, putting movies onto CDs, transmission of imagesover the Internet, and other similar purposes. The emphasis has been on keeping those bits of the image that seemto be most important to human vision and discarding things that the human does not particularly notice. A casualglance at an original image and a compressed version that requires only 1 or 2 percent of the storage space doesnot reveal substantial differences, which only become apparent when side-by-side higher magnification displaysare used.

The compression methods most often used for still color or monochrome images are JPEG (Joint PhotographersExpert Group) and fractal compression. Fractal compression breaks the image into small blocks and seeksmatches between blocks of different sizes. As these are found, the larger blocks can be discarded and replaced byduplicating the smaller ones. Even for compression ratios of 100:1 or more, the method offers fair visual fidelity.Because of patent issues, and because it is a very slow process that requires serious computation to compress theoriginal images (decompression to view them after compression is very fast, however), fractal methods are not sowidely used.

Figure 4. Comparison of a 2x enlarged fragment of a light microscope image before and afterit has been JPEG compressed with a ratio of 40:1. Color shifts, blurring or omission of lines,

and block-shaped artefacts are all present.


The JPEG method also breaks the image into blocks. Each 8 pixel wide square is transformed into frequencyspace using a discrete cosine transform, and the magnitude of the various frequency terms examined. Ones that aresmall, below a threshold set to accomplish the desired level of compression, are discarded. Repetitions orredundancies in the remaining terms are eliminated and the values stored in a sequence that allows the image to bereconstructed as the data are received rather than needing the entire image to be reconstructed at once (the methodwas developed to deal with image transmission in digital high resolution video). Both compression anddecompression use similar computer routines, and are also supported by specialized hardware that can perform theprocess in real time. Many standard programs for image handling on standard computers perform JPEGcompression and decompression entirely in software. The speed is quite satisfactory for the intended use forvisual recognition in an image data base, as long as the original uncompressed images are still available in casemeasurements are to be made. However, as shown in Figure 4, highly compressed images are not suitable formeasurement purposes.

Transferring images to a computer also opens up many possibilities for printing them out, for example to includein reports. The quality of printers has steadily improved, and in particular dye-sublimation (also called dye-diffusion) printers offer resolution of at least 300 pixels per inch with full color at each point. This should not beconfused with laser printers, for which the "dots per inch" (dpi) specification may be higher. Because the "dots"from a laser printer are either on or off, it takes an array of them to create the illusion of grey scale or (for colorversions) colors. The effective resolution is consequently much poorer than the dpi specification, and even themagazine or book images printed by high end image setters with dot resolution of well over 1200 dpi do not standup to close scrutiny as photographic representations. Furthermore, color aliasing is particularly difficult to control.

Figure 5. Additive (RGB) colors which begin with a black background and combine to produceother colors and white, with subtractive (CMY) colors which begin with a white background

and combine to produce other colors and black.

Color printers work by depositing cyan, magenta and yellow inks or powders (the complementary colors to red,green and blue) onto white paper. Figure 5 compares the behavior of additive (RGB) and subtractive (CMY)colors. By combining the inks in various proportions, a color "gamut" is achieved that is never as wide as additivecolor displays that can appear on video screens, particularly with respect to fully saturated primary colors. Adding


black ink (CMYK printing) does produce somewhat improved dark tones. In the dye-sublimation process, theinks from three (or four) separate sources are combined by diffusion into a special coating on the print stock thatallows them to mix and produce a true color instead of relying on the juxtaposition of colored dots to fool the eye.Some ink-jet printers also allow for blending of the inks on paper, rather than simply depositing them side byside. While not as high in resolution as a photographic print, such hardcopy results of images are relatively fastand not too expensive to produce. The main drawback is that every copy must be printed out separately. On theother hand, computer programs to process the images or at the very least to conveniently label them are widelyavailable.

It is possible, of course, to record the computer display onto film. This can either be done directly using an imagesetter (which performs any necessary color separations and directly exposes film to produce the masters forprinting), or a film recorder. The latter records the full color image onto a single piece of color film using a highresolution monochrome monitor. The image is separated into red, green and blue components and three exposuresare superimposed through corresponding color filters. This is much more satisfactory than trying to photographthe computer screen, because it avoids the discrete red, green and blue phosphor dots used on the color videodisplay, which trick the human eye into seeing color when they are small but can become noticeable when theimage is enlarged. It also overcomes geometric distortions from the curved screens, as well as glare from reflectedlight and other practical problems.

Image processing

Image processing is an operation that starts with an image and ends with another corrected or enhanced image, andshould not be confused with image analysis. The latter is a process that starts with an image and produces aconcise data output, reducing the very large amount of data that is required to store the original image as an arrayof pixels to a much smaller set of (hopefully relevant) data. Image processing includes a family of operations thatmodify pixel values based on the original pixels in the image. It usually produces as its output another image thatis as large as the original, in which the pixel values (brightness or color) have changed.

The overview of processing methods presented here is necessarily brief. A much more complete discussion ofhow and why these procedures are used can be found in John C. Russ (1998) "The Image Processing Handbook,3rd Edition" CRC Press, Boca Raton FL (ISBN 0-8493-2532-3. All of the examples shown in this chapter andthe next were created using Adobe Photoshop, an inexpensive image processing program that supports most kindsof image capture devices and runs on both Macintosh and Windows platforms, using the Image Processing ToolKit, a set of plug-ins for Adobe Photoshop distributed on CD-ROM that implements the various processing andmeasurement algorithms from The Image Processing Handbook. Information on the CD is available over theworld-wide-web (http://members.aol.com/ImagProcTK). There are numerous books available that describe themathematics behind the steps involved in image processing, for instance Castleman, 1979; Gonzalez & Wintz,1987; Jain, 1989; Pratt, 1991; Rosenfeld & Kak, 1982.

The steps used in image processing may usefully be categorized in several ways. One classification is byoperations that correct for various defects that arise in image capture (a common example is nonuniformillumination), operations that improve visibility of structures (often by relying on the particular characteristics of


human vision) and operations that enhance the visibility or measurability of one type of feature by suppressinginformation from others (an example is enhancing boundaries or texture in an image while reducing the brightnessdifference between regions). We will examine operations in each of these categories in this chapter and the next.

Another basis for classification is how the operations function. This classification cuts across the other, with thesame functional procedures capable of being used for each of the different purposes.Point processes replace each pixel value (color or brightness) with another value that depends only on the originallocal value. Examples are increases in contrast and pseudo-coloring of grey scale images.Neighborhood operations modify the pixel value based on the values of nearby pixels, in a region that is typicallya few pixels across (and is ideally round, although often implemented using a small square region for ease ofcomputation). There are two subsets of these operations: ones that use arithmetic combinations of the pixel values,and rank operations that rank the values in the neighborhood into order and then keep a specific value from theranked results.Global operations typically operate in frequency space, by applying a Fourier transform to the image and keepinginformation based on the orientation and frequency of selected wave patterns of brightness in the image. These areparticularly useful for removing period noise that can arise in image capture, or detecting periodic structures suchas high resolution atom images in the presence of noise.

The most common imaging defects are:Inadequate contrast due to poor adjustment of light intensity and camera settings;Nonuniform brightness across the image resulting from lighting conditions, variations in sample thickness (intransmission viewing) or surface orientation (particularly in the SEM), or from optical or camera problems such asvignetting;Noisy images, due to electronic faults or more commonly to statistical fluctuations in low signal levels (especiallyin SEM X-ray imaging, fluorescence microscopy, autoradiography, etc.)Geometrical distortions resulting from a non-normal viewing orientation or non-planar surfaces (particularly in theSEM and TEM because of their large depth of field), or from nonlinear scan generators (particularly in the atomicforce microscope and its cousins)Requirement for several different images taken with different lighting conditions (different polarizations, differentcolor filters, etc.) or other signals (e.g. multiple X-ray maps in SEM) to be combined to reveal the importantdetails.It might be argued that some of these are not really imaging defects but rather problems with specimens orinstruments that might be corrected before the images are acquired, and when possible this is the preferred course.But in reality such corrections are not always possible or practical and images are routinely acquired that havethese problems, which need to be solved before analysis.

Contrast manipulation

The brightness range of many images is greater than the discrimination capability of the human eye, which candistinguish only about 20 grey levels at a time in a photograph. The contrast of a stored image can be manipulatedby constructing a transfer function that relates the stored brightness value of each pixel to the displayed value. Astraight line identity relationship can be varied in several ways to enhance the visibility of detail. These


manipulations are best understood by reference to the image histogram, which is a plot of the number of pixelshaving each of the possible brightness levels. For a typical stored grey scale image this will be 256 values fromwhite to black, and for a typical color image it will be three such plots for the red, green and blue planes, or theequivalent conversion to HSI space.

Figure 6. A light microscope color image separated into its red, green, blue (RGB) at the left,and hue, saturation, intensity (HSI) planes at the right. Algebraic calculation allows conversion

from one representation to the other.

HSI (Hue, Saturation and Intensity) space offers many advantages for image processing over RGB. Imagecapture and display devices generally use red, green and blue sensitive detectors and phosphors to mimic the threereceptors in the human eye, which have sensitivity peaks in the long (red), medium and short (blue) portions ofthe visible spectrum. But our brains do not work directly with those signals, and artists have long known that amore useful space for thinking about color corresponds to hue (what most people mean by the "color", an angleon the continuous color wheel that goes from red to yellow, green, blue, magenta and back to red), saturation (theamount of color, the difference for example between pink and red) and intensity (the same brightness informationrecorded by panchromatic "black and white" film). For microscopists, hue corresponds to the color of a stain,saturation to the amount of stain, and intensity to the local density of the section, for instance. Figure 6 shows animage broken down into the RGB and HSI planes.

Figure 7a shows an example of a histogram of a grey scale image. Note that the actual values of pixels do notcover the entire available range, indicating low contrast. Linearly stretching the brightness range with a transferfunction that clips off the unused regions at either end (Figure 7b) improves the visible contrast significantly. Thisparticular histogram also has several peaks (which typically correspond to structures present in the specimen) andvalleys. These allow spreading the peaks to show variations within the structures, using histogram equalization(Figure 7c). The equalization transfer function is calculated from the histogram and therefore is unique to eachimage.


Figure 7. An SEM image with low contrast is shown with its histogram (a). Linearly expandingthe display brightness range (b) increases the visual contrast, and equalization (c) broadens

the peaks to show variations within the uniform regions.

Gamma adjustments to the image contrast compress the display range at either the bright or dark end of the rangeto allow more visibility of small changes at the other end. This is particularly useful when linear CCD cameras areused, as they record images that are visibly different than film or human vision (which have a logarithmicresponse). Figure 8 shows the effect of gamma adjustment. The same nonlinear visual response means thatfeatures are sometimes more easily distinguished in a negative image than in a positive, and this reversal ofcontrast is easily applied in the computer. Solarization, a technique borrowed from the photographic darkroom,


combines contrast reversal at either the dark or bright end of the contrast range with expanded positive contrast forthe remainder, to selectively enhance visibility in shadow or bright areas (Figure 8c).

Figure 8. Gamma adjustments nonlinearly change the display of grey values, in this light microscopeexample (a) increasing the brightness to show detail in shadow areas (b). Solarization (c) reverses the

contrast at the end of the brightness scale to create negative representations, in this example in dark areas.

Each of these techniques can typically be applied to color images as well. It is not usually wise to alter the RGBplanes separately as this will produce radically shifted colors in the image. Transforming the color image data toHSI space allows many kinds of image processing to be performed on the separate hue, saturation and intensityplanes. Many systems apply them to the intensity plane alone (which corresponds to the monochrome image thatwould be recorded by panchromatic film).


It is usually possible to modify a selected portion of an image, rather than the entire area, but this requires manualintervention to select the regions. When the image does not have uniform illumination, for any of the reasonsmentioned above, it makes it difficult to recognize the same type of feature in different locations.

Correcting nonuniform brightness

Figure 9. Portion of a nonuniformly illuminated metallographic microscope image (a) with the background(b) produced by fitting a polynomial to the brightest points in a 9x9 grid and the result of leveling (c).


Leveling the image brightness is typically done by dividing by or subtracting a reference image that represents thevariation. Subtraction is the more common approach, but technically should only be used when the image hasbeen acquired by a logarithmic recording device (film, a densitometer, some tube type video cameras, or a CCDcamera with appropriate circuitry).

There are several approaches to getting the reference image to use. Sometimes it can be acquired directly byremoving the specimen and recording variations in illumination and camera response. More often the specimenitself causes the variation and the reference image must be derived from the original image. There are three basicmethods used:1. If there is some structure present throughout the image (either features or background) that can be assumed to

be the same, the brightness of those points can be used to generate a polynomial curve of background as afunction of location. This is easiest to automate when the structure comprises either the brightest or darkestpixels in each local area. Figure 9 shows an example. This approach works best when the nonuniformbrightness results from lighting conditions and varies as a smooth function of position.

2. If the features can be assumed to be small in at least one direction, a rank operator can be used. This is aneighborhood operator that replaces each pixel in the image with the (brightest or darkest) value within somespecified distance, usually a few pixels. If the features can be replaced by the background in this way, theresulting image is suitable for leveling. Figure 10 shows an example. This technique can handle abruptchanges in lighting that often result from specimen variations but requires that the features be small or narrow.

3. Filtering the image to remove high frequency changes in brightness can be performed either in the spatialdomain of the pixels or by using a Fourier transform to convert the image to frequency space. The spatialdomain approach requires calculating a weighted average of each pixel with its neighbors out to a considerabledistance in all directions, to smooth out local variations and generate a new background image that has only thegradual variations. This method is slow and has the limitations of both methods above and none of theiradvantages, but is sometimes used if neither of the other methods is provided in the software.

Figure 10. Use of a rank operator to produce a background image from Figure 9a by replacing each pixelwith the brightest neighbor within a radius of 4 pixels (a), and the leveled result (b).


Reducing image noise

Noisy images are usually the result of very low signals, which in turn usually arise when the number of photons,electrons or other signals is small. The best solution is to integrate the signal so that the noise fluctuations averageout. This is best done in the camera or detector itself, avoiding the additional noise and time delay associated withreading the signal out and digitizing it, but this depends upon the design of the camera. Video cameras for instanceare designed to continuously read out the entire image every 1/30 or 1/25 of a second, so that averaging can onlytake place in the computer.

0 0 0 0 1 0 0 0 0 0 0 2 4 6 4 2 0 0 0 2 8 21 28 21 8 2 0 0 4 21 53 72 53 21 4 0 1 6 28 72 99 72 28 6 1 0 4 21 53 72 53 21 4 0 0 2 8 21 28 21 8 2 0 0 0 2 4 6 4 2 0 0 0 0 0 0 1 0 0 0 0

Figure 11. A noisy fluorescence microscope image (a) and comparison of the result of smoothing it (b)using a Gaussian kernel of weights (c) with a standard deviation of 1.6 pixels vs. the application of a

median filter (d) that replaces each pixel value with the median of the 9-pixel-wide circular neighborhood.


If the signal strength cannot be increased and temporal averaging of the signal is not possible, the remainingpossibility is to perform some kind of spatial averaging. This approach makes the assumption that the individualpixels in the image are small compared to any important features that are to be detected. Since many images aredigitized with empty magnification, we can certainly expect to do some averaging without losing usefulinformation. The most common method is to use the same weighted averaging mentioned above as a way togenerate a background image for leveling, but with a much smaller neighborhood. A typical procedure is tomultiply each pixel and its neighbors by small integer weights and add them together, divide the result by the sumof the weights, and apply this method to all pixels (using only the original pixel values) to produce a new image.Figure 11 shows this process applied to an image with evident random noise, using the "kernel" of weightsshown. These weights fit a Gaussian profile and the technique is called Gaussian filtering. It is equivalent to a lowpass filter in frequency space (and indeed it is sometimes implemented that way).

Although it is easily understood and programmed, this kind of smoothing filter is not usually a good choice fornoise reduction. In addition to reducing the image noise, it blurs edges and decreases their contrast, and may alsoshift them. Since the human eye relies on edges - abrupt changes in color or brightness - to understand images andrecognize objects, as well as for measurement, these effects are undesirable. A better way to eliminate noise fromimages is to use a median filter (Davies, 1988). The pixel values in the neighborhood (which is typically a fewpixels wide, larger than the dimension of the noise being removed) are ranked in brightness order and the medianone in the list (the one with an equal number of brighter and darker values) is used as the new value for the centralpixel. This process is repeated for all pixels in the image (again using only the original values) and a new imageproduced (Figure 11). As discussed in Chapter 8 on statistics, for any distribution of values in the neighborhoodthe median value is closer to the mode (the most probable value) than is the mean (which the averaging methoddetermines). The median filter therefore does a better job of noise removal, and accomplishes this without shiftingedges or reducing their contrast and sharpness.

Both the averaging and median filtering methods are neighborhood operators, but ranking takes more computationthan addition and so the cost of increasing the size of the neighborhood is greater. For color images the operationsmay either be restricted to the intensity plane of an HSI representation, or in some cases may be applied to the hueand saturation planes as well. The definition of a median in color space is the point whose distances from allneighbors gives the lowest sum of squares.

Rectifying image distortion

Distorted images of a planar surface due to short focal length optics and nonperpendicular viewpoints can berectified if the details of the viewing geometry are known. This can be accomplished either from first principlesand calculation, or by capturing an image of a calibration grid. In the latter case, marking points on the image andthe locations to which they correspond on a corrected image offers a straightforward way to control the process.Figure 12 shows an example. Each pixel address in the corrected image is mapped to a location in the originalimage. Typically this lies between pixel points in that image, and the choice is either to take the value of the nearestpixel, or to interpolate between the neighboring pixels. The latter method produces smoother boundaries withoutsteps or aliasing, but takes longer and can reduce edge contrast.


Figure 12. Foreshortening due to an angled view with a short focal length lens (a). The vignetting is firstcorrected by leveling (b) and then the central portion of the image rectified using 20 control points (c).

Performing this type of correction can be vital for SEM images, due to the fact that specimen surfaces are normallyinclined and the lens working distance is short. Tilted specimens are also common in the TEM, and low powerlight microscopes can produce images with considerable radial distortion. Even with ideal viewing conditions andoptics the image may still be foreshortened in the horizontal or vertical direction due to errors in the digitizationprocess. The electronics used for timing along each scan line have tolerances that allow several percent distortioneven when properly adjusted, and some older frame grabbers made no attempt to grab pixels on an ideal squaregrid. Fortunately most systems are quite stable, and the distortion can be measured once using a calibration gridand a standard correction made to images before they are measured.

When the viewed surface is not planar, or worse still is not a simple smooth shape, the process is much morecomplicated. If the user has independent knowledge of what the surface shape is, then each small portion of it canbe corrected by the same process of marking points and the locations to which they map. Effectively, these points


subdivide the image into a tesselation of triangles (a Voronoi tesselation). Each triangle in the captured image mustbe linearly distorted to correspond to the dimensions of the corrected planar image, and the same mapping andinterpolation is carried out. Sometimes the shape can be determined by obtaining two views of the surface andidentifying the same points in each. The parallax displacements of the points allow calculation of their position inspace. This is stereoscopy, rather than stereology, and lies outside the scope of this text.

Real images may also suffer from blur due to out-of-focus optics or motion of the specimen during exposure.These can be corrected to some degree by characterizing the blur and removing it in frequency space but the detailsare beyond the intended scope of this text, and also beyond the capability of most desktop computer basedimaging systems. The Hubble telescope pictures suffered from blurring due to an incorrect mirror curvature. Asthis defect was exactly known (but only after launch), the calculated correction was able to be applied to producequite sharp images (the remaining problem of collecting only a small fraction of the desired light intensity wascorrected by installing compensating optics). Blurring of images from confocal microscopes due to the lightpassing through the overlying parts of the specimen can be corrected in much the same way, by using the imagesfrom the higher focal planes to calculate the blur and iteratively removing it.

Enhancement

The same basic methods for manipulating pixel values based on the values of the pixels themselves and otherpixels in the neighborhood can also be used for enhancement of images to reveal details not easily viewed in theoriginal. Most of these methods become important when considering automatic computer-based measurement andare therefore discussed in the next chapter. The human visual system is quite a good image processing system inits own right, and can often extract quite subtle detail from images without enhancement, whereas computermeasurement algorithms require more definite distinctions between features and background (Ballard & Brown,1982).

Figure 13. Light micrograph (a) showing the effect of a Laplacian filter (b) to increase the contrast ofedges and sharpen the image appearance.


There are some enhancement methods that take advantage of the particular characteristics of the human visualsystem to improve visibility (Frisby, 1980). The most common of these is sharpening, which increases thevisibility of boundaries. Local inhibition within the retina of the eye makes us respond more to local variations inbrightness than to absolute brightness levels. Increasing the magnitude of local changes and suppressing overallvariations in brightness level makes images look crisper and helps us to recognize and locate edges.

Figure 13 shows a very simple example. The neighborhood operator that was used is based on the Laplacian,which is a second derivative operator. The filter multiplies the central pixel by 5 and subtracts the four neighborpixels above, below and to the sides; this is repeated for every pixel in the image, using the original pixel values.For pixels in smooth areas where the brightness is constant or gradually changing, this produces no change at all.Wherever a sudden discontinuity occurs, such as an edge running in any direction, this increases the localcontrast; pixels on the dark side of the edge are made darker and those on the bright side are made brighter.

The more general approach than the Laplacian that mimics a photographic darkroom technique is the unsharpmask. Subtracting a smoothed version of the image from the original both increases contrast, reduces some localbackground fluctuations, and increases edge sharpness. Figure 14 shows an example in which this method isapplied to the intensity plane of a color image.

Figure 14. Color light micrograph (a) showing the effect of an unsharp mask (b) to increase contrast andsharpen edges. A Gaussian smooth of the intensity plane from the original image with a diameter of six

pixels is subtracted from the original; hue and saturation are not altered.

Other edge enhancement operators that use combinations of first derivatives are illustrated in the next chapter. Soare methods that respond to texture and orientation in images, and methods that convert the grey scale or colorimage to black and white and perform various operations on that image for measurement purposes.


Figure 15. Example of grids overlayed on images on the computer screen: a) a black cycloid grid on acolor light microscope image; b) a colored grid on a monochrome metallographic microscope image.


Overlaying grids onto images

As discussed in Chapter 1 on manual counting, grids provide ways to sample an image. The lines or points in thegrid are probes that interact with the structure revealed by the microscope section image. Counting (and in somecases measuring) the hits the grids make with particular structures generates the raw data for analysis. There areseveral advantages to using the computer to generate grids to overlay onto the image. Compared to placingtransparent grids onto photographic prints, the computer method avoids the need for photography and printing.Compared to the use of reticles in the microscope, it is easier to view the image on a large screen and to have thecomputer keep track of points as they are counted and provide feedback on where counting has taken place. Thecomputer is also able to generate grids of many different types and sizes as needed for particular images andpurposes (Russ, 1995a), and to generate some kinds of grids such as random points and lines which are not easyto obtain otherwise. It can also adjust the grid appearance or color for easy visibility on different images. Perhapsthe greatest hidden advantage is that the computer places the grid onto the image without first looking at what isthere; humans find it hard to resist the temptation to adjust the grid position to simplify the recognition andcounting process, and the result is practically always to bias the resulting measurement data.

Overlaying a grid onto an image so that the user can conveniently use it to mark points requires that the grid showup well and clearly on the image yet not obscure it. One approach that works well in most cases is to display eitherblack or white grids on color images, and color grids on grey scale pictures. The selection of black or white isusually based on whether there are many areas within the color image that are very dark or very light; if so, choosea white or black grid, respectively. Color grids on black and white images are usually easiest to see if they arecolored in red or orange. Blue colors often appear indistinguishable from black, because the eye is not verysensitive at that end of the spectrum. On the other hand, sensitivity to green is so high that the colors may appeartoo bright and make it difficult to see the image (the green lines may also appear wider than they really are). Butallowing the user to choose a color for the grid is easily supported, and solves problems of individual preferenceor color blindness. Figure 15 shows color and grey scale images with grids superimposed.

In generating the grid, there are several different kinds of grids that are particularly suitable for specific purposes.For example, a grid used to count points can consist of a network of lines, with the points understood to lie at theline intersections (or in some cases at the ends of the line segments), or a series of small circles, with the points ofinterest understood to lie at the centers of the circles. The circles have the advantage that they do not obscure thepoint being marked, which can be important for deciding which points to mark or count. The lines have theadvantage that they impose a global order on the points and make it easier to follow from one point to the next, sothat no points are accidentally overlooked. Sometimes both methods are used together. Figure 16 shows severalspecific kinds of grids to generate grids of lines and arrays of points. The routine for generating these grids isshown in Appendix 1. The listings shown there are in a specific Pascal-like macro language used by the publicdomain NIH-Image program, but serves to document the algorithms required. They are easily translated to othersystems and languages. The identical algorithms are included in the set of Photoshop plug-ins mentioned earlier(in that case they were written in C so that they could be compiled to run on both Macintosh and Windowsversions of Photoshop).


Figure 16. Some other types of grids that can be used: a) square array of points; b) staggered array ofpoints, c) randomly placed points; d) circular lines; e) random lines.


The example of a point grid in Figure 16a is a simple array of points marked by crosses and hollow circlesarranged in a square pattern with a spacing of 50 pixels. The similar example in Figure 16b places the points in astaggered array with a higher point density. It is most efficient to select a point density for sampling so thatneighboring points only rarely fall within the same structural unit. When the structure being sampled by the grid isitself random, then a regular array of grid provides random sampling, and is easiest for a human user to traversewithout missing points. But for applications to structures that have their own regularity, such as man-madecomposites, it is necessary for the grid to provide the randomization. The point grid in Figure 16c distributesrandom points across the image to be used in such cases. This is done by generating random pairs of X, Ycoordinates (and rejecting overlapping ones) until 50 points are marked. Counting with a random grid is moredifficult than a regular one because it is easier to miss some points altogether. Changing this number to providereasonable and efficient sampling of the structure is straightforward. A common strategic error in counting ofimages using grids is to use too many points. It is almost always better to count only a few events or occurrencesof structure in each image, and to repeat this for enough different images to generate the desired statisticalprecision. This helps to assure that reasonable sampling of the entire specimen occurs.

The two examples of line grids shown in Figures 16d and 16e supplement those in Figure 15. A square grid canbe used either to count intersections of structure with the lines (PL) or to count the cross points in the grid as apoint grid (PP). If the image is isotropic (in the two dimensional plane of sampling), then any direction isappropriate for measurement and a square grid of lines in easiest for the user to traverse. However, when thestructure itself may have anisotropy it is necessary to sample uniformly in all directions, which the circle gridprovides. If the structure also has any regularity, then the grid must be randomized. The random line gridgenerates a series of random lines on the sample. If the section is itself a random one in the specimen, then therandom line grid samples three-dimensional structures with uniform, isotropic and random probes.

The cycloid shape is appropriate for lines drawn on so-called vertical sections (Baddeley et al., 1986). The spatialdistribution of the lines is uniform in three-dimensional space when the section planes are taken with randomrotation around a fixed “vertical” axis in the specimen. This method is particularly useful in measuring structuresin specimens that are known not to be isotropic. Examples include metals that have been rolled or have surfacetreatments, specimens from plant and animal tissue, etc. The cycloid is easily drawn as a parametric curve asshown in the appendix. The individual arcs are drawn 110 pixels wide and 70 pixels high (110/70 = 1.571, a veryclose approximation to π/2 which is the exact width to height ratio of the cycloid). The length of the arc is twice itsheight, or 140 pixels. Alternating cycloids are drawn inverted and flipped to more uniformly sample the image.

For all of these grids, the number of points marked on the image, or the total length of grid lines drawn, is shownalong with the image area. These values may be used later to perform the stereological calculations.

There are two principal methods used by computer programs to keep track of the user’s marks on the imageemploying a grid. One is to count the events as they occur. Typically these will be made using some kind ofinteractive pointing device, which could be a touch screen, light pen, graphics pad, etc., but in most cases will bea mouse as these are now widely used as the user interface to a graphical operating system (or the track pad, trackball or pointing stick that are employed as mouse substitutes). Counting the clicks as they occur gives a real-timeupdated count of events, but makes editing to correct mistakes a bit more difficult and requires some other form of


user action to select between multiple categories of features that can be counted at once. It also requires some wayto mark the points as they are identified by the user, and for this purpose colored labels are often placed on theimage on the screen.

Figure 17. Example of counting with a grid. Three different types of interface between the white and grey,black and grey and black and white phases have been marked along the circular grid lines in differentcolors. The report of the total length of the grid lines, the area of the image and the number of eachclass of mark allows calculation of the surface area per unit volume of each type of interface and

the contiguity between phases.

The second common method for counting user marks is to just allow placing any kind of convenient mark onto thescreen, (Figure 17) selecting different colors to distinguish different classes of countable events, and then have thecomputer tally them up at the end. Given that most drawing programs have some kind of “undo” capability to editmistakes, the ability to change the size and color of marks, and tools to erase marks or cover them up with one ofa different color, this allows the user quite a lot of freedom to mark up the image using fairly intuitive andstraightforward tools. Of course, since the totaling up is performed at the end there is no continuously updatedresult.

Counting the features present in an image requires a computer algorithm to connect together all of the touchingpixels that may be present. A simplified approach that can be used for the counting of marks is available if it canbe assumed that the cluster of pixels constituting the mark is always convex and contains no holes. It is thissimplified logic that is shown in the Appendix. Each pixel is examined in each line of the image. Pixels that matchthe selected classification color value are compared to the preceding pixel and to the three touching pixels in theline above. If the pixel does not touch any other identical pixels, then it represents the start of a mark and is


counted. Additional pixels that are part of marks already counted are passed over. A more general approach is touse the same logic as employed for feature measurement and labeling in the next chapter, which is capable ofdealing with arbitrarily complex shapes.

The number of marks in each class is reported to the user, who must combine it with knowledge of the grid(length of lines, area of image, number of grid points, etc.) to obtain the desired structural information. Otherparameters such as the placement of the marks (to assess nonuniformities and gradients, for instance) are notusually extracted from the user marks, which means that they do not actually have to be placed on the exact gridpoints but can be put nearby leaving the actual points available for viewing to confirm accuracy.

Basic Stereological Calculations.

The two most common stereological measures that illustrate the use of counting user marks made employing a gridare the determination of volume fraction of each phase present, and the measurement of the surface area per unitvolume. These and other stereological measures are discussed in much more detail in earlier chapters. There maybe several different type of boundaries evident as lines in the image, each of which corresponds to a one kind ofsurface separating the structure in three dimensions. The user can select different colors to measure the variouskinds of boundaries that are of interest.

The volume fraction of a phase is estimated by the number of grid points counted as lying within the phase,divided by the total number of points in the grid. The word "phase" as used in the context of stereology, refers toany recognizable structure that may be present, even voids. If the volume fraction of one structure within anotheris desired, that is the ratio of the number of points in the inner structure divided by the sum of those points andthose which lie within the outer structure. For example, the volume fraction of a cell occupied by the nucleuswould be measured by marking points within the nucleus in one color, and points within the cell but outside thenucleus in a second color. Then the volume fraction of the cell occupied by the nucleus is N1/(N1+N2). The samemethod would be used if there were many cells and nuclei within the image.

The surface area per unit volume is estimated from the number of places where the grid lines cross the boundaryline corresponding to that surface. If the number of points is N, and the total length of the grid lines is L (inwhatever units the image is calibrated in), then the surface area per unit volume is 2•N/L. This measurement has adimension, and so the image calibration is important. The dimension of area per unit volume is (1/units).

All stereological techniques require that the grid be randomly placed with respect to the features in themicrostructure. When the structure itself and the acquisition of the image are random, this criterion is fulfilled by aregular grid of points or lines. There is one important and common case where orientation of the section plane isnot randomized. In the so-called vertical sectioning case, the surface of the sample that is examined isperpendicular to an exterior surface, but multiple sections can be taken in different orientations that include thatsurface normal (actually, the method works whenever some defined direction within the structure can beconsistently defined as the vertical direction, which lies in all section planes imaged). In that case, in order tocount intersections with lines that are uniformly oriented in three-dimensional space, the cycloid grid should beused. This is discussed in more detail elsewhere.


One of the common microstructural measurements used to characterize materials is the so-called "grain size." Thisis actually a misnomer, since it does not actually have anything directly to do with the size of the grains. There aretwo different definitions of grain size included in the ASTM standard. One is derived from the surface area perunit volume of the grain boundaries in the material. If this are measured by counting N intersections of a grid oftotal length L (in millimeters) with the grain boundaries, then the ASTM standard grain size can be calculated asG = – 6.65 • Log (L/N)– 3.3where the logarithm is base ten. Note that depending on the units of the image calibration, it may be necessary toconvert L to millimeters.

The second grain size method counts the number of grains visible in the image. The manual marking procedurecan also be employed to mark each grain, without regard to the grid chosen. For grains that intersect an edge ofthe field of view, you should count those that intersect two edges, say the top and left, and ignore those thatintersect the other two, say the bottom and right. By this method, if the number of grains is N and the image areais A (converted to square millimeters), then the grain size isG = 3.32 • Log (N/A) – 2.95Note that these two methods do not really measure the same characteristic of the microstructure, and will ingeneral only approximately agree for real structures. Usually the grain size number is reported only to the nearestinteger.

More information about the interpretation of the counts, their conversion to microstructural information, and thestatistical confidence limits that are a function of the number of counts can be found in other chapters.Stereological calculations from counting experiments are extremely efficient, and provide an unbiasedcharacterization of microstructural characteristics. It is important to remember the statistical advantage of countingthe features that are actually of interest rather than trying to determine the number by difference. However, forcounting a grid with a fixed total number of points it is best to mark only the minor phase points and obtain thenumber of counts for the major phase by subtracting the number of marked points from the (known) total numberof points in the grid.


Appendix - Computer Assisted Measurements

Manual stereology macros for NIH Image.

Overlay grids on an image with arrays of lines or points (reports the number of points or the length of thelines in image units). Grids provided include three different point arrays and four line arrays, one ofwhich is cycloids and one sine-weighted radial lines for vertical section method. Then use paintbrush set toany of the fixed colors (up to 6) to mark locations to be counted (e.g., where line grids cross featureboundaries). Finally, use macro to count marks in each class, and use results for stereologicalcalculations. For more details, see the paper "Computer-Assisted Manual Stereology" in Journal of ComputerAssisted Microscopy, vol. 7 #1, p. 1, Mar. 1995

© 1995-1998 John C. Russ - may be freely distributed provided that the documentation is included.

Macro 'Point Grid';Var

k,x,y,xoff,pwd,pht,nrow,ncol: integer;area,ppx: real;un: string;

BeginGetPicSize(pwd,pht);nrow:=pht Div 50;ncol:=pwd Div 50;xoff:=(pwd - 50*ncol) Div 2;If (xoff<25) Then xoff:=25;y:=(pht - 50*nrow) Div 2;If (y<25) Then y:=25;SetLineWidth(1);k:=0;Repeat Until >pht

x:= xoff;Repeat Until >pwd

MoveTo (x-5, y);LineTo (x-1, y);MoveTo (x+1, y);LineTo (x+5, y);MoveTo (x, y-5);LineTo (x, y-1);MoveTo (x, y+1);LineTo (x, y+5);k:=k+1; counterx:=x+50;

Until ((x+10)>pwd);y:=y+50;

Until ((y+20)>pht);GetScale(ppx,un);MoveTo (2,pht-6);SetFont('Geneva');SetFontSize(10);Write('Total Points=',k:3);area:=pwd*pht/(ppx*ppx);MoveTo (2,pht-18);Write('Total Area=',area:10:3,'sq.',un);

End;

Macro 'Staggered Grid';Var

i,k,x,y,xoff,yoff,pwd,pht,nrow,ncol: integer;area,ppx: real;un: string;

BeginGetPicSize(pwd,pht);nrow:=pht Div 34;ncol:=pwd Div 50;xoff:=(pwd - 50*ncol) Div 2;If (xoff<25) Then xoff:=25;yoff:=(pht - 34*nrow) Div 2;If (yoff<25) Then yoff:=25;


SetLineWidth(1);k:=0;i:=0;y:=yoff;Repeat Until >height

x:= xoff;If (2*(i Div 2)=i)

Then x:= x + 25;Repeat Until >width

MoveTo (x-5, y);LineTo (x-2, y);MoveTo (x+2, y);LineTo (x+5, y);MoveTo (x, y-5);LineTo (x, y-2);MoveTo (x, y+2);LineTo (x, y+5);MakeOvalRoi(x-2,y-2,5,5);DrawBoundary;KillRoi;k:=k+1; counterx:=x+50;

Until ((x+25)>pwd);y:=y+34;i:=i+1;

Until ((y+25)>pht);GetScale(ppx,un);MoveTo (2,pht-6);SetFont('Geneva');SetFontSize(10);Write('Total Points=',k:3);area:=pwd*pht/(ppx*ppx);MoveTo (2,pht-18);Write('Total Area=',area:10:3,'sq.',un);

End;

Macro 'Cycloids';Var

h,i,j,k,x,y,xoff,yoff,pwd,pht,nrow,ncol,xstep,ystep: integer;len,area,ppx,pi,theta: real;un: string;

Beginpi:=3.14159265;GetPicSize(pwd,pht);nrow:=pht Div 90;ncol:=pwd Div 130;xoff:=(pwd - 130*ncol) Div 2;yoff:=(pht - 90*nrow) Div 2;cycloids are 110 pixels wide x 70 high, length 140SetLineWidth(1);h:=0;For j:=1 To nrow Do

Beginy:=yoff + j*90-10;For i:=1 To ncol Do

Beginx:=xoff+(i-1)*130+10;If (h Mod 4)=0 Then

BeginMoveTo (x,y);For k := 1 To 40 Do

Begintheta:=(pi/40) *k;xstep:=Round(35*(theta-Sin(theta)));ystep:=Round(35*(1.0-Cos(theta)));LineTo (x+xstep,y-ystep);

End;End;


If (h Mod 4)=1 ThenBegin

MoveTo (x,y-70);For k := 1 To 40 DoBegin

theta:=(pi/40) *k;xstep:=Round(35*(theta-Sin(theta)));ystep:=Round(35*(1.0-Cos(theta)));LineTo (x+xstep,y-70+ystep);

End;End;


MoveTo (x+110,y);For k := 1 To 40 Do

Begintheta:=(pi/40) *k;xstep:=Round(35*(theta-Sin(theta)));ystep:=Round(35*(1.0-Cos(theta)));LineTo (x+110-xstep,y-ystep);

End;End;


MoveTo (x+110,y-70);For k := 1 To 40 Do

Begintheta:=(pi/40) *k;xstep:=Round(35*(theta-Sin(theta)));ystep:=Round(35*(1.0-Cos(theta)));LineTo (x+110-xstep,y-70+ystep);

End;End;

h:=h+1;End; for i

End; for jGetScale(ppx,un);len:=h*140/ppx;MoveTo (2,pht-6);SetFont('Geneva');SetFontSize(10);Write('Total Length=',len:10:4,' ',un);area:=pwd*pht/(ppx*ppx);MoveTo (2,pht-18);Write('Total Area=',area:10:3,' sq.',un);

End;

Macro 'Square Lines';Var

i,j,x,y,xoff,yoff,pwd,pht,nrow,ncol: integer;len,area,ppx: eal;un: string;

BeginGetPicSize(pwd,pht);nrow:=pht Div 100;ncol:=pwd Div 100;xoff:=(pwd - 100*ncol) Div 2;yoff:=(pht - 100*nrow) Div 2;If (xoff=0) Then

Beginxoffset:=50;ncol:=ncol-1;

End;If (yoff=0) Then

Beginyoff:=50;nrow:=nrow-1;

End;


SetLineWidth(1);For j:=0 To nrow Do

Beginy:= yoff + j*100;MoveTo (xoff, y);LineTo (pwd-xoff-1, y);

End;For i:=0 To ncol Do

Beginx:= xoff + i*100;MoveTo (x,yoff);LineTo (x,pht-yoff-1);

End;GetScale(ppx,un);len:=(nrow*(ncol+1)+ncol*(nrow+1))*100/ppx;MoveTo (2,pht-6);SetFont('Geneva');SetFontSize(10);Write('Total Length=',len:10:4,' ',un);area:=pwd*pht/(ppx*ppx);MoveTo (2,pht-18);Write('Total Area=',area:10:3,' sq.',un);

End;

Macro 'Circle Grid';var

i,j,x,y,xoff,yoff,pwd,pht,nrow,ncol: integer;len,area,ppx,pi: real;un: string;

BeginGetPicSize(pwd,pht);SetLineWidth(1);pi:=3.14159265;nrow:=pht Div 120;ncol:=pwd Div 120;xoff:=(pwd - 130*ncol) Div 2;yoff:=(pht - 130*nrow) Div 2;For j:=1 To nrow Do

Beginy:= yoff + 15 + (j-1)*130;For i:=1 To ncol Do

Beginx:= xoff + 15 + (i-1)*130;MakeOvalRoi(x,y,101,101);DrawBoundary;KillRoi;

End;End;

GetScale(ppx,un);Len:=nrow*ncol*pi*100/ppx;MoveTo (2,pht-6);SetFont('Geneva');SetFontSize(10);Write('Total Length=',Len:10:4,' ',un);area:=pwd*pht/(ppx*ppx);MoveTo (2,pht-18);Write('Total Area=',area:10:3,' sq.',un);

End;

Macro 'Radial Lines';var

step,i,j,pwd,pht:integer;x,y,theta,temp:real;

Beginstep:=GetNumber('Number of Lines',16,0);SetPalette('GrayScale',6);AddConstant(7);SetForegroundColor(1);


GetPicSize(pwd,pht);setlinewidth(1);For i:=0 To step Do

Begintheta:=i*3.14159265/step;x:=(pwd/2)*Cos(theta);y:=(pht/2)*Sin(theta);MoveTo(pwd/2-x,pht/2+y);LineTo(pwd/2+x,pht/2-y);

End;End;

Function ArcSin(X:real) : real;var

temp:real;Begin

if (X=0) then temp:=0else if (X=1) then temp:=3.14159265/2else if (X=-1) then temp:=-3.14159265/2else temp:=ArcTan(x/sqrt(1-x*x));ArcSin:=temp;

End;

Macro 'Sine-wt. Lines';var

step,i,j,pwd,pht:integer;x,y,theta,temp:real;

Beginstep:=GetNumber('Number of Lines',16,0);SetPalette('GrayScale',6);AddConstant(7);SetForegroundColor(1);GetPicSize(pwd,pht);setlinewidth(1);For i:=0 To step Do

Begintemp:=(-1+2*i/step);theta:=ArcSin(temp);x:=(pwd/2)*Cos(theta);y:=(pht/2)*Sin(theta);MoveTo(pwd/2-x,pht/2+y);LineTo(pwd/2+x,pht/2-y);

End;End;

Macro 'Horiz. Lines';var

space,i,pwd,pht:integer;Begin

space:=GetNumber('Line Spacing',5,0);SetPalette('GrayScale',6);AddConstant(7);SetForegroundColor(1);GetPicSize(pwd,pht);setlinewidth(1);i:= (pht mod space)/2;if (i<2) then i:=(space/2);while (i<pht) do

BeginMoveTo(0,i);LineTo(pwd-1,i);i:=i+space;

End;End;

Macro 'Vert. Lines';var

space,i,pwd,pht:integer;


Beginspace:=GetNumber('Line Spacing',5,0);SetPalette('GrayScale',6);AddConstant(7);SetForegroundColor(1);GetPicSize(pwd,pht);setlinewidth(1);i:= (pwd mod space)/2;if (i<2) then i:=(space/2);while (i<pwd) do

BeginMoveTo(i,0);LineTo(i,pht-1);i:=i+space;

End;End;

Macro 'Random Points';Var

x,y,k,i,pwd,pht,limt: integer;ppx,area: real;un: string;collide: boolean;

BeginGetPicSize(pwd,pht);limt:=50;number of pointsk:=1;Repeat

x:=Random*(pwd-20); 10 pixel margin around bordersy:=Random*(pht-20);collide:=false;If (k>1) Then avoid existing marks

For i:=1 To (k-1) doIf (Abs(x-rUser1[i])<5) and (Abs(y-rUser2[i])<5)

Then collide:=true;If Not collide Then

BeginrUser1[k]:=x;rUser2[k]:=y;MakeOvalRoi(x+6,y+6,7,7);DrawBoundary;KillRoi;k:=k+1;

End;Until (k>limt);GetScale(ppx,un);area:=pwd*pht/(ppx*ppx);SetFont('Geneva');SetFontSize(10);MoveTo (2,pht-18);Write('Total Area=',area:10:3,'sq.',un);MoveTo (2,pht-6);Write('Total Points=',k-1:4);

End;

Macro 'Random Lines';Var

i,j,k,x1,x2,y1,y2,pwd,pht,len,limt: integer;x,y,theta,m,area,ppx,dummy: real;un: string;

BeginGetPicSize(pwd,pht);len:=0;k:=0;limt:=3*(pwd+pht); minimum total length in pixelsRepeat Until length>limt

x:=Random*pwd;y:=Random*pht;


theta:=Random*3.14159265;m:=Sin(theta)/Cos(theta);x1:=0;y1:=y+m*(x1-x);If (y1<0) Then

Beginy1:=0;x1:=x+(y1-y)/m;

End;If (y1>pht) Then

Beginy1:=pht;x1:=x+(y1-y)/m;

End;x2:=pwd;y2:=y+m*(x2-x);If (y2<0) Then

Beginy2:=0;x2:=x+(y2-y)/m;

End;If (y2>pht) Then

Beginy2:=pht;x2:=x+(y2-y)/m;

End;MoveTo(x1,y1);LineTo(x2,y2);len:=len+Sqrt((x2-x1)*(x2-x1)+(y1-y2)*(y1-y2))k:=k+1;

Until (len>limt);GetScale(ppx,un);area:=pwd*pht/(ppx*ppx);SetFont('Geneva');SetFontSize(10);MoveTo (2,pht-18);Write('Total Area=',area:10:3,'sq.',un);len:=len/ppx;MoveTo (2,pht-6);Write('Total Length=',len:10:3,' ',un);

End;

Macro 'Count Marks';note - this routine is VERY slow as a macro because it must access each pixel. The Photoshop drop-in ismuch faster for counting features, and when used by NIH Image will perform exactly as this does and countthe number of marks in each of the six reserved colors.VAR

i,j,k,pwd,pht,valu,nbr,newfeat: integer;Begin

GetPicSize(pwd,pht);For i:= 1 To 6 Do

rUser1[i]:=0;MoveTo(0,0);For i:=1 To pht Do

BeginGetRow(0,i,pwd);newfeat:=0; start of a new image line - nothing pendingFor j:=1 To (pwd-1) Do skip edge pixels

Beginvalu:=LineBuffer[j]; test pixelIf ((valu=0) or (valu>6)) Then

Begin pixel is not a fixed colorIf (newfeat>0) Then End of a line

rUser1[newfeat]:=rUser1[newfeat]+1; newfeat:=0;

End;If ((valu>=1) and (valu<=6)) Then a fixed color point

Begin


nbr:=LineBuffer[j-1]; left sideIf (nbr<>valu) Then test continuation of line

BeginIf (newfeat>0)Then prev touching color

rUser1[newfeat]:=rUser1[newfeat]+1;newfeat:=valu;start of a chord

End;For k:=(j-1) To (j+1) Do check prev line

Beginnbr := GetPixel(k,i-1);If (nbr = valu) Then

Beginnewfeat:=0;touches

End;End;

End;End; for j

LineTo(0,i); progress indicator because getpixel is very slowEnd; for i

ShowMessage('Class#1=',rUser1[1]:3,'\Class#2=',rUser1[2]:3,'\Class#3=',rUser1[3]:3,'\Class#4=',rUser1[4]:3,'\Class#5=',rUser1[5]:3,'\Class#6=',rUser1[6]:3);

can substitute other output procedures as neededEnd;


Chapter 10 - Computer Measurement of Images

The preceding chapter discussed the display of microscope images on a computer screen along with thesuperposition of grids and overlays. Interactive marking by a human able to recognize the various features presentprovides the input for the computer to tally the different counts, from which calculation of stereologically usefuldata can be carried out. The calculations are straightforward, usually requiring no more than a simple spreadsheetand often performed manually.

Some computer-based image processing operations may be useful in preparing the image for display, to aid theoperator in recognizing the structures of interest, but these are primarily limited to correcting image acquisitiondefects such as poor contrast, nonuniform illumination, non-normal viewing angles, etc. Some of the methodsused to perform those processes, such as neighborhood operations that combine or rank pixel values in each smallregion of the image to produce new values that form a new image, can also be used to carry out the moreaggressive image processing operations discussed in this chapter.

The intent of this chapter is to consider ways that the computer can directly perform measurements on the imagesto obtain data that can be used to characterize the structure. This is not really automatic computer-basedmeasurement since there is still a considerable amount of user interaction needed to specify the operations basedon visual recognition of structure. Selection of the appropriate processing and measurement steps requires more,not less human involvement and knowledge than the simple grid counting methods. However, it becomes possibleto measure things that are not otherwise available. For example, humans can count events such as the intersectionsof grid lines with features, but are not good at measuring lengths or angles.

Measurement using grids

The grids of points and lines used in the preceding chapter served only as visual guides to mark points of intereston the image. Sometimes the counting process can be automated, or direct measurements obtained by combiningthe grid with the image. These methods usually rely on first converting the grey scale or color image to a binary orblack and white image. The convention used here is that black pixels are part of the features of interest and whiterepresents the background (about half of the systems on the market use this convention, and the other half theopposite convention that black is background and white represents features; the latter is a hangover from screendisplays that used bright characters on a dark background).

Reducing a grey scale image to black and white is most directly accomplished by thresholding, selecting a range ofgrey (or color) values by setting markers on the histogram and treating any pixel whose value lies within thatrange as foreground, and vice versa. In fact, there are not too many real images in which such a simple procedureactually works, and several methods will be described later in this chapter for processing images so thatthresholding can be used. But to illustrate the use of grids for direct counting we will use the illustrative image inFigure 1, which can be thresholded to delineate the pink-colored "phase." This is a generic name used instereology for any structure of interest, and in many cases does not correspond to a phase in the chemical orphysical sense (a region homogeneous in chemical composition and atomic or crystallographic structure)

Chapter 10 - Computer Measurement Figures - Page # 193

Figure 1. Example of an image (a) in which a structure can be thresholded by brightness or color todelineate it as a phase for measurement, represented as a binary image (b).

Figure 2. Combining the binary phase from Figure 1b with a grid of points (a) using a Boolean ANDoperation (b). This allows automatic counting of the point fraction to estimate the volume fraction.

To determine the area fraction of the selected phase by manual grid counting, the grid would be overlaid on theimage, the user would mark those points that fall on the foreground regions, these would be counted by thecomputer (either by counting mouse clicks, for example, or by counting the marks made on the image in someunique color). Then the number of points marked divided by the number of points in the grid is PP, whose


expected value is equal to the volume fraction of the phase. If the image is a binary image, and the grid isconsidered to be another binary image, then the two can be combined using a Boolean AND operation. Thisexamines the individual pixels in the two images and creates a new image in which a pixel is made black only ifboth of the two original pixels at that location are black.

Figure 3. Processing the binary image from Figure 1b to get just the outline of the phase boundaries (a).Combining these with a grid of lines with known total length (b) using a Boolean AND (c) allows

counting of the points to estimate the surface area per unit volume of the phase.

The grid points that fall onto the black phase are kept, the others are erased. Counting the number of points thatremain is a simple operation, which can be performed directly from the image histogram. Even if the points in the


grid are marked by more complex shapes (to aid in visual recognition), counting the marks is the same process asdiscussed in the previous chapter. As shown in Figure 2, ANDing a point grid (which can be regular or random asappropriate to the sample) with the binary image gives an immediate estimated result for PP.

If a line grid is used to determine the number of points PL at which the lines cross the phase boundary, thisrequires a single additional step. The binary image of the phase of interest is converted to its outline form. This isanother example of a neighborhood operator, but one of a class of operators often used with binary images calledmorphological operators. They examine the pattern of neighboring pixels and turn the central pixel on or offdepending on that pattern. Typical criteria are the number of touching neighbors (used in erosion and dilation) andwhether or not the neighbors that are foreground points themselves are all touching each other (used inskeletonization).

In this particular case, we want to keep the outline of the phase regions, so we keep every pixel that is black(foreground) that touches at least one white (background) pixel, testing the four closest neighbors to the left, right,top and bottom. The result, shown in Figure 3, erases all internal pixels within the regions and keeps just theoutlines. ANDing this image with that of a grid of lines keeps just those points where the lines cross the edges.Counting these points and dividing by the length of lines in the grid (a known constant) gives PL. Note that thepoints are not necessarily single pixels, since the lines may be tangent to the boundaries so that the points ofintersection cover more than one pixel; counting all of the marks (groups of touching pixels) gives the correctanswer.

There is another seemingly more efficient way to perform this same measurement without first getting the outlines,by just ANDing the grid lines with the original features and counting the number of line segments. Since each linesegment has two ends, the number of points must be twice the number of segments. But there is a problem withthis method because some line segments may end at the edge of the image inside the foreground region, whichwould produce an erroneous extra count. Dealing with the edges of images is always a problem, and although itcan be solved (in this example by testing the ends of the lines to see whether they lie on black or white pixels andadjusting the count accordingly) it requires additional logic.

Measurement of structure dimensions, rather than just counting, can also be performed using a grid of lines.Figure 4 shows one simple example. The image shows a cross section of a coating on a metal. Thresholding theimage to delineate just the coating is possible, so no additional image processing is required. This fortunatecircumstance is often possible in metallography by the judicious use of chemical etchants to darken one phase andnot the other. The same approach is used with chemical staining of biological tissues, but as these tend to be morecomplex in structure the selectivity of the stains is not as great.

Once the coating is isolated as the foreground, ANDing a grid of lines oriented in the direction normal to the metalsurface produces a set of line segments that can be measured. The number of pixels in each line, converted to adistance based on the image magnification, is a measure of the thickness of the coating at that point. Thedistribution of line lengths can be analyzed statistically to determine the mean coating thickness and its variation.Of course, creating other templates of lines to measure in various orientations, including radial dimensions, isstraightforward.


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mean = 82.68std.dev. = 3.80

Figure 4. Metallographic cross section of a coating applied to a metal substrate (a). Superimposing a gridof lines normal to the nominal surface (b) using a Boolean AND produces a set of lines whose

lengths can be used to measure the mean value and variation of the coating thickness (d).

The use of a random set of lines to measure the thickness of layers that are not sectioned perpendicular to theirsurfaces (so that the thickness direction of the layer is not directly measurable) is also useful. In this method, thesection plane intersects different layers or portions of layers at various angles. The random lines constitute IUR(isotropic, uniform, random) probes of the structure. The length distribution of straight lines through a layer ofthickness T can range from exactly T up to infinity, depending on the line orientation. But a plot of frequency vs.1/Length (Figure 5) is a simple triangular function, whose mean value is 2/3 the maximum. Since this maximum is1/T, the true layer thickness is just 1.5 times the mean value of the (1/Length) values from the line probes. Theprocess is thus to generate the random line probes, AND them with the image, measure the lengths and calculatethe mean value of 1/Length, multiply this by 1.5 and take the inverse, which is the mean layer thickness. Manystructures in materials (e.g., composites), biology (e.g., membranes) and other applications can be measured inthis way. This is an example of the power of the stereological approach.

Consideration of the geometry of the structure of interest (a layer) leads to the design of a probe (random lines)and a measurement procedure (1/Length of the intersections) that yields the desired result straightforwardly.Measurement of line lengths in the computer can be done in several ways. For straight lines at any angle the


Pythagorean distance between the end points can be calculated. For lines that may not be straight, measurementmethods (and their errors) will be discussed below.

1 / Length1 / T

Figure 5. Distribution of 1/Intercept Length for random lines through a layer of thickness T.

Measuring areas with pixels

Probably the measurement that seems most straightforward with a digitized image consists of counting pixels. Foran array of square pixels that are either black (features) or white (background) the procedure is to count the pixelsand use that as a measurement of area. The basic stereological rule that VV=PP also states that VV=AA. Thefraction of all the pixels in the image that are black must measure the area fraction of the selected phase.

While this is true, it does not allow easy estimation of the accuracy or precision of the measurement. The problemis that pixels are not points (they have finite area, and in most image capture processes represent an average acrossthat area), and the points are too close together to be independent samples of the structure, so that the √N cannotbe used as a measure of counting precision as it could for counting a sparse grid of points.

For the area fraction of the image as a whole, it isn't even necessary to perform thresholding or look at the pixelsin the image. The histogram contains all of the data. Wherever the threshold levels are set to perform theseparation of features from background, the fraction of the total histogram between the thresholds is directly thearea fraction. For the image in Figure 1, this gives an area fraction of 30.51% (19995 of 65536 pixels).

For measurement of the individual features present, we do need to count the actual pixels. Most computermeasurement systems do this by treating features (sometimes called blobs) as all pixels that are in contact, definingcontact as either sharing an edge or sharing an edge or corner (called four-neighbor and eight-neighbor logic).Except for measuring lines, either definition is usually adequate for measurement purposes since we prefer toavoid situations in which features have dimensions as small as a single pixel. The difference between four- andeight-neighbor rules has to do principally with the topological property of connectivity. Whichever rule is used todefine pixels in features, the opposite rule will apply to the background pixels.

There are several sources of error when pixel counting is used to measure areas in binary images. The finite areaof each pixel means that pixels around the periphery of the feature may be included in the binary image or not


depending on the fraction of each pixel that is actually covered. The same feature placed on the pixel grid indifferent positions and rotations will cover a different number of pixels and thus generate different measurementresults. In addition, the thresholding operation to select which pixels to include in the feature is sensitive to noise,which produces grey scale variations in the pixel values. This can particularly affect the edge pixels and result intheir being included or excluded. Capturing and thresholding repeated images of the same feature will not produceperfectly reproducible results.

Notice that these sources of error are related to the edge pixels (the perimeter of the feature) and not to the totalnumber of pixels (the area of the feature). This means that unlike the grid point count method where the number ofpoints can be used to determine the estimated precision, pixel counting produces results with a precision thatdepends on the feature shape. Two features with the same area will be measured with different precisiondepending on the amount of perimeter each has.

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Figure 6. Results of 100 measurements of the area of a 10 unit diameter circle placed at random on asquare pixel grid. True area is 102•(π/4)=78.54. Mean value of distribution above is 77.86, withstandard deviation = 1.639, skew = -1.143, kurtosis = 1.910. This variation demonstrates the

effects of the finite size of pixels for the case of many edge orientations.

Figure 6 illustrates the variability from placing a small circular feature at random on a pixel grid. Each circle has anactual diameter of 10 units (10 pixel widths, for an ideal geometrical area of 78.54 units), but depending onplacement the number of pixels that are at least 50% covered (and hence included in the feature) will vary. Thegraph shows that the results are skewed to smaller measurement values. This effect becomes less important as thefeature size increases, and sets an effective lower limit to the size of objects that can be measured in an image.

Adding noise to the image complicates the thresholding process and hence the measurement. Figure 7 shows aseries of identical circles 10 units in radius with superimposed random noise. Thresholding the image at the idealmidpoint in the histogram (which is known in this case but would not be in general) and applying a morphologicalopening (discussed later in the chapter) produces features with some pixel errors. The area results obtained bycounting pixels are low on the average because of missing internal pixels and indentations along the sides. Insteadof just counting the pixels to obtain the area, a more robust measure is the convex area obtained by fitting a 32-sided polygon around the feature (sometimes called a taut-string perimeter or convex hull). We will see shortly


that there are a variety of ways to define feature measurements that the computer can make which can be selectedto overcome some of the limitations of the pixels in the original image.

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Area ConvexAreaMean 307.23 321.404Std.Dev 17.35 14.20Skew -.96 -1.02Kurtosis 4.17 4.08Median 309 324

Figure 7. Measurement of circles on a noisy background showing thresholding variability. Thegeometrical area of the circles is 314.16 units. The original image with added Gaussian noise (σ=32 grey

levels) was thresholded at a middle grey value of 128, and was processed with a morphological opening toremove isolated pixels before measuring the area and convex area. The results are summarized in the table.


A circle includes pixels along edges that have all possible orientations with respect to the pixel grid. Somedirections are more sensitive to placement than others. For instance a feature consisting of a hollow square(external width of 11 units, internal hole with a width of 7 units so all of the sides are 2 units wide) that is alignedwith its sides parallel to the pixel grid has no variability with placement at all. When it is rotated by 45° so that theedges all cut diagonally across the pixels, the variability is shown in Figure 8. Notice that the mean value is closeto the actual value (good accuracy) but the variation is quite large (poor precision).

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Figure 8. Results of 100 measurements of the area of a randomly placed feature shaped like a hollowdiamond with outer and inner sides of 11 and 7 units (actual geometrical area = 72). Mean value of the

distribution is 71.93, standard deviation = 4.39, skew = -4.674, kurtosis = 2.518. When the same figureis rotated 45 degrees to align with the pixel grid the measured answer is always 72.0 exactly. This

demonstrates the effect of edge orientation on measurement precision.

We can investigate this orientation sensitivity by placing the same hollow square feature on the grid and rotating itin 1° steps (Figure 9a). Since the feature covers various fractions of pixels, a decision of which pixels to countmust be made. If the threshold is set at 50%, the results (Figures 9b and 9c) show that the typical measured valueis low (60 instead of 72 pixels) except when the orientation is within 1° of the actual pixel grid. Setting thethreshold to 25% coverage (Figures 10a and 10b) gives a result that is biased slightly greater than the actual area,but much closer to it. It is also interesting to look at the measured length of the perimeter (another featuremeasurement that is discussed below). For the 50% thresholding (Figure 11) the results are skewed slightly lowand have some evidence of a trend with orientation angle. The trend is much more pronounced for the case ofthresholding at 25% (Figure 12).

In most cases of real images, it is not clear where the threshold actually should be set. Figure 13 shows a portionof a grey scale metallographic microscope image of a three-phase alloy, with its histogram. This image actuallyhas some vignetting and is systematically darker at the corners than in the center. Leveling of the contrast using theautomatic method described in the preceding chapter (by assuming that all of the light colored features shouldactually be the same) sharpens the peaks in the histogram. Measurement of the light colored phase (dendrites)requires setting a threshold; varying its placement over a wide range makes little visual difference in the resultingbinary image, even though the area fraction varies from about 30% to about 40%. Clearly, these different valuescannot all be "correct."


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Figure 9. Effect of rotation on the area measurement of a hollow square, OD=11, ID=7 (actual area = 72units) rotated in 1 degree steps on a square pixel grid (a), by counting those pixels which are at least 50%covered. Note that the true value is only obtained for <2 degree rotation, and a value of about 60 is more

typical. (mean=60.09, σ=3.42, skew=1.83, kurtosis=7.46, median=60)

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Figure 10. Measurement results for the same rotated squares in Figure 9a, but changing the countingcriterion to include all pixels that are at least 25% covered. Mean=74.20, σ=2.25, skew=-1.055,

kurtosis=3.864, median=75


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Figure 11. Results for measured perimeter from the rotated squares in Figure 9a, selecting pixels that are atleast 50% covered. The geometrical answer is 72.0; mean=70.78, σ=1.459, skew=-.70, kurtosis=2.76,

median=71.

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Figure 12. Results for measured perimeter from the rotated squares in Figure 9a, selecting pixels that are atleast 25% covered. The mean=71.25, σ=1.56, skew=-.312, kurtosis=2.79, median=71.3. Although the

overall range of variation is not greatly affected, there is a definite trend of value with angle.

If there is some independent knowledge about the nature of the sample, automatic algorithms can be used to assistwith the critical threshold setting operation. In this case, the dendritic nature of the light regions suggests that theywere formed by nucleation and growth and therefore should have smooth boundaries. A thresholding operationthat seeks the setting giving the smoothest boundaries selects the value shown in Figure 14. Smooth boundariesare one of the criteria that skilled operators also use to judge threshold settings when they are performed manually.Others are the elimination of small features (noise), or minimal variation in area with setting (Russ, 1995b). All ofthis reminds us that the so-called automatic measurement methods using a computer are only as good as the veryhuman input that selects methods on the basis of either prior knowledge or visual judgment.


Figure 13. Upper right corner portion of an original image of a metal alloy (a) showing corner to centershading and the brightness histograms before (b) and after (c) leveling the image contrast.

Figure 14 (continued...)


Figure 14. Section of the metallographic image from Figure 13 (after leveling) with the results ofthresholding it at different grey levels: a) histogram with levels marked, and b) resulting binary images.

(143 = 40%, 215 = 30%, 178 as selected by method discussed in text gives 37.7% area fraction.

Measurement parameters: size

The area, as defined by the number of foreground pixels that touch each other and constitute the digital binaryimage representation of an object, is one rather obvious measure of size. There are others that are accessible tomeasurement using a computer. Selecting the ones that are appropriate from those offered by each particular imageanalysis software package (and figuring out what the highly variable names used to describe them mean) is animportant task for the user.


Figure 15. Measures of area for a feature: Net area, filled area and convex area.

Area can be measured in three principal ways, as indicated in Figure 15.A. The number of touching foreground pixels (it does matter in some cases whether touching is interpreted as 4-

or 8-neighbor connectivity)B. The same as A but with the inclusion of any internal holes in the featureC. The area inside a fitted geometrical shape. The two most common shapes are a fitted ellipse or a polygon; some

more limited systems use a rectangle, either with sides parallel to the image edges or oriented to the feature'smajor axis. The polygon is typically drawn to vertices that are the minimum and maximum pixel addresses inthe feature as the image axes are rotated in steps. This produces the "taut string outline" or "convex hull" thatbridges across indentations in the periphery. The ellipse may be established in several ways, including settingthe major axis to the maximum projected length of the object and defining the minor axis to give the same areaas A or B above, or by calculating the axes to give the same moments as the pixels within the feature.

Which (if any) of these measures makes sense depends strongly on what the image and sample preparation show.For sections through particles, the net area (A) will correspond to the particle mass, the filled area (B) to thedisplaced volume, and the fitted shape (C) to results obtained from sieving methods.

Any of these area measurements can be converted to an "effective radius" or "effective diameter" by using therelationship Area = 4 π Radius2, but of course the underlying shape assumptions are very important. Classicalstereology calculates a distribution of sizes for spherical particles in a volume based on the measured distributionof circle sizes on a section, and in that specific case the effective (circular) diameter is an appropriate descriptor.

Figure 16 shows an example where this is meaningful. The image is a metallographic microscope view of apolished section through an enamel coating. The pores in this coating are spherical due to gas pressure and surfacetension, and the sections are therefore circles. The image is easily thresholded, but the pores have bright internalspots due to light reflection. Filling holes in objects is accomplished by inverting the image (interchanging blackand white pixel values), treating the holes and background as features, discarding the feature (the background)which touches the edges of the image, and then adding the remaining features (the holes) back into the originalimage. This last step is actually done with a Boolean OR that sets each pixel to black if it is black in either the


original image or in the inverted copy. This is an example of the kinds of image processing that are performed onbinary (thresholded) images, several of which are discussed later in this chapter.

Frequency Distribution for Area(mm2)From (≥) To (<) Count NV(spheres)

0.0 30.0 116 5.08230.0 60.0 3 0.14160.0 90.0 2 0.10290.0 120.0 4 0.221

120.0 150.0 0 0.0150.0 180.0 1 0.067180.0 210.0 0 0.0210.0 240.0 1 0.084240.0 270.0 0 0.0270.0 300.0 0 0.0300.0 330.0 0 0.9330.0 360.0 0 0.360.0 390.0 1 0.277390.0 420.0 1 0.211420.0 450.0 0 0

Total 129

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Figure 16. Metallographic microscope image of pores in an enamel coating, thresholded and with holesfilled. The measurement of circle sizes produces a distribution that can be unfolded to produce an estimate

of the size distribution of the spheres.


Measurement of the resulting circles produces a distribution that can be converted to the size distribution of thespheres that generated them by multiplication by a matrix of values. These are calculated as discussed in Chapter11 on shape modeling and in texts such as Underwood, 1970; DeHoff & Rhines, 1968; and Weibel, 1979; for asphere the calculation can be done analytically. If a sphere of given radius is sectioned uniformly, the sizedistribution of the circular sections gives the probability that a circle of any particular size will result. Doing thisfor a series of sphere sizes produces a matrix of values that predicts the number of circles per unit area of imagewith radius in size class i, as a function of the number of spheres per unit volume with radius in size class j

NAi = ∑ α' ij NVj (10.1

The inverse of this matrix, α, is then used to calculate the expected size distribution of spheres from the measured

distribution of circles. The α and α' matrices are published for various standard shapes such as ellipsoids

(DeHoff, 1962), cylinders, disks, and many polyhedra. In most cases the size distributions of the planar sectionsthrough the solids for isotropic, uniform and random sectioning are not easily calculated by analytical geometry(the sphere of course presents the simplest case, which is why it is so tempting to treat objects as being spherical).Monte-Carlo sampling methods can be employed in these instances to estimate the distribution, as discussed in thechapter on Geometric Modeling.

This classical approach is flawed in several ways. The underlying assumption is that all of the three-dimensionalfeatures have the same shape, which is rarely the case. In particular, a systematic change in shape with size willbias the results, and often occurs in real systems. In addition, the mathematics of this type of unfolding is ill-conditioned. Small errors in the distribution of circles (which are inevitable due to the statistics of sampling andcounting) are greatly magnified in the inverse matrix multiplication. This is easy to see by realizing that largecircles can only result from sections of large spheres, while small circles can be produced by either large or smallspheres. Small errors in the number of large spheres calculated from the number of large circles will influence thenumber of small circles those large spheres produce, biasing the resulting calculation of the number of smallspheres and increasing its uncertainty.

For more complicated shapes, the situation is much worse. The most likely size for a random section through asphere is a large circle. For a cube or other polyhedron there is a significant possibility of cutting through a cornerand getting a small feature. The distribution of the areas of sections cut by random planes varies substantially forobjects of different shape. It would in principle be possible to make a better unfolding based on measuring theshape as well as the size of the sections, but in practice this is not done because there are the other fundamentalproblems with unfolding, and new stereological techniques have evolved to deal with measurement of the numberand sizes of particles in a volume that do not depend on knowing their shapes.

There are many possible measures of size available besides area. The length and breadth of a feature can bedefined and measured in quite a few different ways. The major and minor axes of the ellipse fitted by one of themethods mentioned above can be used. Most common as a measure of length is the maximum projected length (ormaximum Feret's diameter, or maximum caliper diameter). Testing all of the points on the periphery of the featureto find the maximum and minimum values as the coordinate system is rotated through a series of angles, as wasused above to determine the convex hull, also identifies the maximum projected length.


It is not necessary to use a large number of angles to find the maximum. If 8 steps of 22.5 degrees are used, theworst case error (when the actual maximum lies midway between two of the orientations tested) is only the cosineof 11.25 degrees, or 0.981, meaning that the measured value is 1.9% low. With 16 steps the worst case angularerror is 5.625 degrees, the cosine is 0.995, and the error corresponds to 1 pixel on a feature 200 pixels wide,which is within the error variation due to edge definition, thresholding, and object placement on the pixel grid.

It is more difficult to determine the breadth in this way. Instead of a cosine error, the maximum error depends onthe sine of the angle and can be much larger. For a very narrow and long feature, the minimum projecteddimension can be hundreds of percent high even with a large number of projections. Other estimates of breadth aresometimes used, including the area divided by the length (which is appropriate for a rectangular feature), theprojected dimension at right angles to the longest direction, the dimension across the center of the feature at rightangles to the longest dimension, and others. It is easy to find specific shapes, even common ones such as asquare, that make each of the estimates seem silly.

Neither the length nor breadth defined in these ways makes sense for features that are curved (fibers, crosssections of lamellae, etc.). A better measure of length is the distance along the midline of the feature, and this canbe determined by reducing the feature to its skeleton as shown in Figure 17. The skeleton is produced by anerosion algorithm that removes pixels from the outside boundary of the feature (that is, any black pixel thattouches a white one) except when the black pixel is part of the midline of the feature. This is determined by thetouching black neighbors. It they do not all touch each other, then the central pixel is part of the skeleton andcannot be removed. The process continues to remove pixels sequentially until there are no further changes in theimage (Pavlidis, 1980).

Figure 17. A feature and its skeleton (green midline). This provides a much more useful estimate of thelength than the maximum caliper or projected length (red arrow).

Skeletons are useful as a way to characterize shape of features, and the branching of the skeleton and counts of itsend points, branch points and loops can give important topological information, but their use here is for length.The length of a line of pixels is difficult to measure accurately. Each individual pixel in the chain is connected to its


neighbors in either the 90 degree (edge touching) or 45 degree (corner touching) direction. The distance to the 90degree neighbors is one pixel and that to the 45 degree pixels is √2=1.414 pixels, so summing the distance alongthe line gives a straightforward estimate of the length.

But the real midline (or any other line in an image) is not well represented by the line of pixels. Only features thatare many pixels wide are represented faithfully, as discussed before. A perfectly straight line oriented at differentangles and mapped onto a pixel array will have different lengths by this method as a function of angle, with thesum of link lengths overestimating the value except in the case of exact 45 or 90 degree orientations. The length ofa very rough line will be underestimated because it has irregularities that are smaller than the pixel dimensions.This problem also arises in the measurement of feature perimeters, which are usually done in the same way. Thevariations in measurement data in Figures 11 and 12 result from this source of error.

For features that have smooth boundaries that are not too highly curved, a fairly uniform width, and of course nobranches, the skeleton length does offer a useful measurement of feature length. Another tool, the EuclideanDistance Map (Danielsson, 1980), can be combined with the skeleton to measure width. The EDM assigns to eachblack pixel in the image a value that is its distance from the nearest white (background) pixel, using a very efficientand fast algorithm. For the pixels along the midline that are selected by the skeleton, these are the centers of aseries of inscribed circles that fit within the feature. Averaging the value of the EDM along the skeleton gives astraightforward measure of the average width of features that are not perfectly straight or constant in width.

Other feature measurements: shape and position

There are four basic approaches to shape measurement, none entirely satisfactory. The most common approachuses ratios of size values that are formally dimensionless, such as

4π•Area/Perimeter2

Length/BreadthArea/Convex Areaetc.

These have the obvious advantage of being easy to calculate. Ideally they are independent of feature size, althoughin practice the better resolution with which pixels can delineate large features means that trends may appear inmeasured data that are not necessarily meaningful. This is particularly true for ratios that involve the perimeterwhich is difficult to measure accurately as discussed above.

The main difficulty with this class of parameters is that they are not unique (many shapes that are recognized asquite different by a human observer can have the same parameter value) and do not correspond to what humansthink of as shape. This latter point is reinforced by the fact that the names given to these parameters are arbitrarilymade up and vary widely from one system to another.

A second approach to shape measurement was mentioned before. The skeleton of a feature captures the importanttopology in ways that can be easily abstracted (Russ & Russ, 1989). The difference that a human notices betweenthe stars in the US, Israeli and Australian flags is the number of points. The number of end points in the skeleton(pixels which have exactly one neighbor) captures this value.


The remaining two methods for shape description concentrate on the irregularities of the feature boundary.Harmonic analysis unrolls the boundary and performs a Fourier analysis of it (Beddow et al., 1977). The firsttwenty or so coefficients in the Fourier series capture all of the details of the boundary to a precision as good asthe pixels that represent the feature, and can be used for statistical classification schemes. This approach hasenormous power to distinguish feature variations, and has been particularly widely used in geological applicationsfor the analysis of sediments. The computation needed to extract the coefficients is significant and few imageanalysis packages offer it, but the primary barrier to the use of harmonic analysis is the difficulty that humanobservers have in recognizing the distinctions that the method finds. This is clearly an example of a case in whichcomputer analysis takes a different direction from human vision, whereas most of the methods discussed here usethe computer to duplicate human judgment but make the measurements more accurate or easier to obtain.

The fractal dimension of boundaries is also used as a shape descriptor. Many real-world objects have surfaces thatare self-similar, exhibiting ever-increasing detail as magnification is increased (Russ, 1994b). There are importantexceptions to this behavior, principally surfaces that have smooth Euclidean shapes arising from surface tension,membranes, or crystallographic effects. But when boundaries are rough they are frequently fractal and themeasured dimension does correlate highly with human judgment of how "rough" the boundary appears.

The principal difficulty with all of these shape measures is that they properly apply only in the two-dimensionalplane of the image, and may not be relevant to the three dimensionsal objects sampled by a section or viewed inprojection. The shapes of sections through even simple solid objects such as a cube vary from three-sided up tosix-sided, and the number of sides is correlated with the size. Given such a set of data it is not easy to realize thatall of the sections come from the same solid object, or what the object is. When variation in the shape and size ofthe three-dimensional object is also present, the utility of measuring shape of the sections may be quite low.

Harmonic analysis and fractal dimension of sections does provide some information about the roughness of thesurfaces of sectioned objects. For instance, in general for a random section through an object the surface fractaldimension is greater than the dimension of the intersection by 1.0, the same as the difference between thetopological dimensions of the line and surface (1 and 2, respectively). This is not true for outlines of projectedimages of particles, which are smoother than the section or actual surface because indentations tend to be hiddenby surrounding peaks.

Measurement of the position of objects also provides stereologically useful information. Since measurements arenot usually performed in systems that have meaningful absolute coordinate systems, these measurements tend tobe relative. An example would be how far features are from each other, or from boundaries. The detection ofpreferred alignment of feature axes or of boundaries is a closely related task. In most cases the most robustmeasure of feature position is the coordinates of the centroid or center of mass of the feature. For a 2D object thisis the point at which it would balance if cut along its boundaries from a stiff sheet of cardboard. Of course, thecentroid of a 2D section through a 3D object does not properly represent the centroid of the 3D object, but is stilluseful for characterizing some properties such as clustering.

Analysis of the distances and directions between pairs of object centers provides information on clustering,orientation, and so forth. The distribution of nearest neighbor distances in a 2D random section providesclustering information on 3D volumes directly; a Poisson distribution indicates a random distribution of objects,


while a mean nearest neighbor distance that is less than or greater than that for the Poisson distribution indicatesclustering of self-avoidance, respectively (Schwarz & Exner, 1983). In making these measurements, it isnecessary to stay far enough from the edges of the image so that the true nearest neighbor is always found.

The axis of orientation of a feature is typically calculated from the same ellipse axes mentioned above based eitheron the longest chord (greatest distance between any two points on the periphery) or on the axis about which thepixels have the smallest second moment. Unfortunately, it is not possible in general to use these measures on 2Dplanes to understand the 3D orientations. Of course, for some specimens such as surfaces (orientation of defects,etc.) the 2D image is quite appropriate and the measurements directly useful.

The precision of measurement of the centroid (first moment) and axis (second moment) are very good, becausethey use all of the pixels within the feature and not just a few along the periphery. Determining locations with sub-pixel accuracy and orientations to better than 1 degree is routinely possible.

Measuring the location of features with respect to boundary lines is less precise. The boundaries are usuallyirregular so that they cannot be specified analytically. This means that mathematical calculation of the distancefrom the centroid point to the boundary is not practical. Instead, the distance from the feature to the boundary canbe determined using the Euclidean distance map. As mentioned above, this procedure assigns to every pixel withinfeatures a value that is the distance to the nearest background pixel. Any pixel that is a local maximum (greaterthan or equal to all neighbors) is considered to be the feature center (in reality it is the center of an inscribed circle).If Euclidean distance map of the entire area, say the grain or cell in which the features reside, is then computed,the values of the pixels identified as "feature centers" in the first operation give the distances to the boundary.

Note that this method does not directly yield distance information for a 3D structure as the measurements are madein the plane of the image. Using the same stereological model introduced above for measuring the thickness oflayers can solve this problem. This procedure uses averaging of 1/distance in order to estimate the mean value ofthe 3D distance from features to the boundary.

Image processing to enable thresholding and measurement

There are some cases in which images can be directly thresholded and the resulting binary images measured.Examples include metals etched to show dark lines along the grain boundaries (Figure 9 in the preceding chapter),and fluorescence images from stained tissue. But in most cases the structures are too complex and there is nounique set of brightness values that represent the desired structure and only that structure. In addition to theprocedures discussed in the preceding chapter to correct defects in the original image, a variety of tools areavailable to enhance the delineation of features to permit thresholding, and to process the thresholded images topermit measurement.

Figure 18 shows an example that looks initially much like the illustration in Figure 16. The sample is a slide withred blood cells. These are easily thresholded and the holes filled (the figure shows the step-by step procedure).Before measurement, those features that touch any edge must be discarded because their size cannot bedetermined. We also discard features smaller than an arbitrary cutoff area, because we know something about thesample and such small features cannot be the blood cells which are of interest.


Dealing with the finite size of images requires some care. Obviously, features that intersect the edge of the imagecannot be measured because there is no way to measure their actual extent. But in general, larger features are morelikely to intersect the edge and hence to be lost from the measurement process which would bias the results towardsmall features. There are two general approaches to correcting for this. The one most commonly used in manualstereology using photographs is to restrict the measurement to an inner region away from the edges, so that anyfeature that has its upper left corner (for example) in the inner region can be measured in its entirety. This is notusually practical for computer measurements as the limited size of the digitized image makes it unwelcome todiscard a wide guard region around the edges.

Figure 18. Microscope image (a) of red blood cells; (b) the result of thresholding; (c) holes identified asfeatures that do not touch the edges of the inverted image; (d) OR combination of the holes with the

original image to produce filled features (with those that are small or touch edges eliminated); (e)watershed segmentation to separate two overlapped features

Instead, it is equivalent to measure all of the features that do not intersect the edge but count them so as tocompensate for the greater likelihood of losing large features. Each measured feature is counted as Wx•Wy/(Wx-Fx)•(Wy-Fy) where Wx and Wy are the image dimensions and Fx and Fy are the projected widths of the featurein the horizontal and vertical directions. For small features this is 1.0 but for large features the effective countbecomes greater than 1.0 and corrects for the probable loss of similar size features that intersect a randomly placedimage boundary. Of course, if the image is not random in placement this is not appropriate. Human operators


must avoid selecting the placement of their measuring frame to reduce or eliminate having features touch edges (orfor any other reason) as it inevitably introduces uncorrectable bias into the measurements.

Two of the features overlap in the image in Figure 18, and a watershed segmentation is used to separate them.This procedure first calculates the Euclidean distance map of the features in the thresholded binary image, and thenby proceeding downhill from each local maximum finds the watershed lines between the features that markpresumed boundaries.

Measuring the area of each feature and using that to calculate an effective circular diameter, as was done for thepores in Figure 16, is not appropriate here. The overlap that is corrected by the watershed leaves each featureslightly too small. Instead, measuring the maximum external dimension and using that as a diameter gives areasonably robust measure of size and can be used to determine the mean size and variance. In this case, thesample is not a section through a set of spheres in a volume, but a projected image of disks that can be assumed tobe lying flat on the slide. Hence no further processing of the size distribution is required.

Figure 19 shows an SEM image of a two-phase ceramic. The grains are distinguished in brightness by atomicnumber variations (the light phase is zirconia and the dark phase alumina), and the boundaries between the grainsby relief produced by polishing and thermally etching the surface. It is not possible to directly threshold the grainboundaries to delineate the grains. The Laplacian edge sharpening procedure from the previous chapter does notsolve this problem. In general second derivative operators are better at locating noise and points in the image,rather than linear edges. A first derivative is very good at locating steps or edges. A typical edge finding kernel isused in the figure. In order to find edges running in any orientation, two derivatives, one for vertical steps and theother for horizontal ones, are calculated. Then the magnitude of the gradient (regardless of direction) is calculatedas the square root of the sum of squares of the two directional derivatives, for every pixel in the image. Thisoperator is the Sobel, one of the most commonly used edge-finding procedures.

Figure 19. SEM image of an alumina-zirconia ceramic, and the grain boundaries delineated by a Sobel operator.


With the boundaries outlined, the same procedures for determing the “grain size” as discussed in the precedingchapter can be used. However, it is easier with the computer to measure slightly different parameters than wouldnormally be determined using the same grid-based schemes used in manual measurement procedures. Instead ofcounting the number of intercepts that a line grid makes with the boundaries to determine PL, we can measure thetotal length of the boundaries after thresholding and skeletonizing, which gives the length of boundary per unitarea of image LA. Both of these quantities are related to the surface area per unit volume SV.

SV = 2 PL = 4/π • LA (10.2

Counting grains per unit area, which is the other approach to estimating a "grain size" parameter, gives a value thatis not a measure of the grain size, but actually of the length of triple line where three grains meet in the samplevolume. But counting the grains in the image automatically is often difficult because even a tiny break in one of thegrain boundaries will cause the computer algorithm doing the counting to join the two grains and get a count that isbiased low. It is usually difficult to assure that all of the grain boundaries are uniformly etched or otherwisedelineated and this results is a significant error. Watershed segmentation is a possible answer to this problem butis a relatively slow operation in the computer, and also tends to introduce bias because it will subdivide grains thatare not convex in shape.

However, the triple points in the 2D image (where three grain boundaries join) are the intersections of the triplelines in the 3D volume with the plane of the image, and these can be counted and are usually well defined byetching. A triple point can be identified in the skeleton as any pixel with more than two adjacent black neighbors.The number per unit area NA is related to the length of triple line per unit volume as

LV = 2 NA (10.3

It is possible using the same computer algorithm applied in preceding examples to count the touching pixels ineach grain and convert this to an area. Plotting the distribution of areas seems at first like a measure of the size ofthe grains but in fact it is not. Just as for the spherical pores in Figure 16, the section areas do not represent thethree dimensional sizes of the grains, and in fact do not even sample the grains in an unbiased way. It is muchmore likely that the random section plane will cut through a large grain than a small one. We dealt with this in thecase of pores by using an α matrix to unfold the sphere size distribution from the circle size distribution. But we

cannot do that for the grains because they do not have a simple spherical shape (indeed, they can not since spheresdo not pack together to fill space).

There are shapes that have been proposed for grain and cell structures that fill space (such as thetetrakaidodecahedron) but this still implicitly makes the assumption that all of the grains have the same shape. Thisis clearly not true, and in fact in general the smaller grains will be quite different in shape that the larger ones, witha smaller number of contact faces and hence more acute angles at the corners. This has been demonstrated byexperiments in which grain structures are literally taken apart to reveal the constituent grains. The dependence ofshape upon size changes the distribution of the intersection areas dramatically and makes unfolding impractical.The parameters that do describe the grain structure (SV and LV) in an unbiased way seem more limited than anactual distribution of the three-dimensional sizes, but they are actually more useful since they really describe thestructure and are not dependent on questionable assumptions.


Figure 20 shows another example of grain boundaries, this one a metallographic microscope image of etchedaluminum. The boundaries are not delineated in this image. Instead, the grains have different grey values resultingfrom chance orientations of the crystallographic structure. It is necessary to use an edge-finding procedure toreplace the brightness differences of the grains with dark lines at the boundaries. The operator shown is thevariance. The brightness values of pixels within each placement of a small circular neighborhood are used tocompute a variance (sum of squares of differences from the mean) and this value is used as the brightness of thecentral pixel in the neighborhood as a new image is formed. The result is a very small variance within each grain,regardless of the absolute grey value, and a large variance shown by the dark lines at the boundaries where achange in brightness occurs.

This image can be thresholded and the boundaries skeletonized as shown in the figure. It can then be measuredeither using by a grid overlay and counting intersections (either manually or automatically), or by measuring totalboundary length to get SV, or counting grains or triple points to get LV. Both give values for ASTM grain size asdiscussed in the previous chapter. Note as mentioned above that the grain count would be biased low by breaks inthe network of lines along the boundaries which allow grains to join in the computer count, but that these breakshave a negligible effect on the determination of LA for the boundary lines, or NA for the triple points.

Figure 20. Metallographic image of etched alumina (a), the application of a variance operator (b) and theskeletonized binary image of the boundaries (c). A manual counting operation to determine SV using a grid

as shown in (d) gives less precision than measuring the line length in (c).


Image processing to extract measurable information

Some information in the original image may be more easily accessed for measurement if processing steps areapplied first to enhance and isolate the important data. An example of this is the use of a gradient or edge detectorto isolate the edges, as discussed above. If the orientation of the edges is of interest, for example to determine thedegree of preferred orientation present in a structure, then some additional steps can be used. Figure 21 shows themajor steps in the procedure.

The Sobel operator determines the magnitude of the gradient of brightness for each pixel in the image bycombining two orthogonal first derivatives to get the square root of the sum of squares as a non-directionalmeasure of gradient magnitude (Figure 21b). Combining the same two derivatives to get the angle whose tangentis their ratio gives the orientation of the gradient vector (Figure 21c). The magnitude and direction of the gradientcan be combined with the original grey scale image as shown in Figure 21d to illustrate the process.

Thresholding the Sobel magnitude values identifies the pixels that lie along the major edges in the structure (Figure21e). Selecting only those pixels and forming a histogram of the angle values for them produces a distribution(Figure 21f) showing the orientation of edges (which are perpendicular to the maximum gradient). Because theangle values have been assigned to the 256 value grey scale or color values available for the image, this graphshows two peaks, 180 degrees (128 grey scale values) apart for each set of edges in the original image.

Figure 21. SEM image of an integrated circuit (a). The Sobel operator gives the magnitude of thebrightness gradient (b) and can also create an image of the direction of the gradient vector (c).These can be combined with the original image in color planes (d). Thresholding the gradient

magnitude (e) and forming a histogram of the pixel values from the direction direction image (f)provides a direct measure of the orientation of edges in the image.


Processing is also useful as a precursor to thresholding structures for measurement when the original image doesnot have unique grey scale values for the structures or phases present. Figure 22 shows a typical example. Thecurds and whey protein are visually distinguishable because the former have a smooth appearance and the latter arehighly textured, but both cover substantially the same range of grey scale values.

Once the human has recognized that the structures have texture as their defining difference, then the use of atexture operator to process the image is appropriate. There are a variety of texture extraction processes available,many developed originally for remote sensing (satellite and aerial photography) to distinguish crops, forest,bodies of water, etc. They generally operate by comparing the brightness differences between pixels as a functionof their separation distance (Peleg et al., 1984). A very simple one is used in Figure 22b; the difference in greyscale value between the brightest and darkest pixel in each 5 pixel wide circular neighborhood. Since the visuallysmooth areas of the image have low pixel-to-pixel brightness variations while the highly textured regions havelarge differences, this range operator produces a new grey scale image in which the smooth areas are dark and thetextured ones bright. Thresholding this gives a representation of the phase structure of the sample, which is thenmeasured conventionally to determine volume fraction, boundary area, mean intercept length, and the otherstereological parameters of interest.

Figure 22. Light micrograph of a thin section of curds and whey. The visually smooth areas (curds)have the same range of brightness values as the textured regions (whey protein). The applicationof a range operator (b) creates a new image that assigns different values to the pixel brightness

based on the difference in values for the neighboring pixels. The derived image can bethresholded to delineate the structures for measurement.

More elaborate texture operators compare brightness differences over a range of distances. Plotting the maximumdifference in brightness between pixels in circular neighborhoods centered around each pixel as a function of thediameter of the neighborhood produces a plot from which a local fractal dimension can be obtained. If this value is


used to generate a new image, it often allows thresholding images in which subtle differences in texture arevisually distinguishable. This is particularly suitable for images of surfaces, such as SEM images of fractures.Human vision seems to have evolved the ability to distinguish surface roughness by the magnitude of the fractaldimension produced by the roughness (and visible in the scattered light or secondary electron images).

Grey scale processing of the images is not the only way to segment images based on a local texture; sometimes itis possible to use the thresholded binary image. Figure 23 shows a light micrograph of etched steel. The lamellarpearlite structure contributes much of the strength of the material and the single phase bright ferrite regions givethe metal ductility. The volume fraction of the two structures is directly related to the carbon content of the steel,but the size distribution of the pearlite regions (measured for instance by a mean lineal intercept) depends uponprior heat treatment.

Figure 23. Metallographic microscope image of etched carbon steel (a), with the dark iron carbide particlesand platelets thresholded(b). Application of a closing (dilation followed by erosion) to merge together theplatelets, followed by an opening (erosion followed by dilation) to remove the isolated carbide particles)produces a binary representation (c) of the pearlite (lamellar) regions. Superimposing the outline on the

original (d) shows the regions selected for measurement.


Thresholding the dark etched iron carbide (Figure 23b) does not identify the structural constituents of interest.Applying dilation and erosion to the binary image first fills in the gaps between the lamellae and then erases theisolated spots in the pearlite (Figure 23c). Superimposing the outlines of these regions back onto the originalimage (Figure 23d) shows that the major structural components of the steel have been isolated and can bemeasured in the usual ways. This is an example of morphological operations (Serra, 1982; Coster & Chermant,1985; Dougherty & Astola, 1994) applied to binary images which add or remove pixels based on the presence ofspecified patterns of neighboring pixels.

Combining multiple images

In many cases, it is necessary to acquire several different images of the same area to fully represent theinformation needed to define the structure. A familiar example of this is X-ray maps acquired with the scanningelectron microscope. Each map shows the spatial distribution of a single element, but it may require several todelineate the phase structures of interest. Figure 24 shows an example using a mineral specimen. The individualX-ray maps for iron and silicon (Figures 24a and b) must be combined to define the phase of interest.

Since the original images are not grey scale, but have been recorded as a series of discrete pulses, they are quitenoisy in appearance. A smoothing operation using a Gaussian filter reduces the noise and produces smooth phaseboundaries that can be thresholded (Figures 24c and d). The size of the smoothing operator should ideally be setto correspond to the range of the electrons in the sample, since this is the size of the region from which X-rays areemitted and hence represents the distance by which a recorded pulse may vary from the actual location of the atomthat produced it. Boolean logic can then be employed to combine the two images, for example in Figure 24e todefine the regions that contain silicon and not iron. Once delineated, which in some cases may require many moreelemental images, the phases can be measured stereologically.

Figure 25 shows a case in which combination of several grey scale images is performed. The images show a pileof small rocks. The situation is similar to many SEM images of particulates. Because of the complex surfacegeometry of this sample, no single picture makes it possible to delineate the individual particles. Shadows aredifferent on the different sides of the particles, so using an edge-finding process does not suffice. Recording aseries of images with the camera and specimen in the same position but with the light source rotated to differentpositions produces the series of images shown in Figure 25a. Subtracting these in pairs to produce the "northminus south", "east minus west" , etc., images accentuates the shadows on corresponding sides of the pictures(Figure 25b), but assures that each side of the particles is well illuminated in at least one pair. Combining thesepictures to keep the brightest value at each pixel location eliminates the shadows and makes the edges uniformlybright around each particle (Figure 25c). Applying an edge-enhancement operator to this image delineates theparticle boundaries.


Figure 24. SEM X-ray maps of a mineralspecimen showing the spatial distribution ofiron (left) and silicon (right). Smoothingoperation using a Gaussian filter andthresholding produces binary maps of theareas with concentrations of the elements.Boolean logic combines these to define theregions that contain silicon and not iron.

Processing this image is actually not the major problem for characterization. There is no robust stereological modelthat allows measurement for such a pile of rocks. When particulates are dispersed through a transparent volumeand viewed in projection, so that some particles are partially or entirely hidden by others, a simple first ordercorrection for the missing (invisible) particles can be made. Normally, the size distribution is shown as ahistogram with number of particles in each size class per unit volume of sample, where the volume is the area of


the image times the thickness of the viewed section. For each large particle, the hidden volume can be estimated asthe area of the particle times half the section thickness (because on the average each particle will be half waythrough the section). Any smaller particles in this region would be missed, so the effect is to reduce the volumefor all smaller particle classes. This increases the number per unit volume value and shifts upwards the histogramfor smaller sizes. The process is repeated for each size class.

Figure 25. Macroscopic image of a pile of rocks: a) montage showing portions of the same image areawith the light source moved to north, east, south and west locations; b) montage showing imagedifferences between opposite pairs; c) montage showing the brightest pixel value at each location

from the difference images, and the result of applying a Sobel edge operator.

For a "pile of rocks" or other similar structures, the only suitable model is based on a "dead leaves" randomgeneration of an image. This is done by placing features taken from a known distribution of size and shape using arandom number generator, and gradually building up an image. This has been used for known shapes such asspheres and fibers. Special attention to the orientation with which individual features are added to the


conglomerate is needed, because in real cases forces of gravity, surface tension, magnetic fields, etc. may or maynot be important.

The generated image is then compared to the real one, and parameters such as the histogram of intercept lengths orthe partial areas of features revealed on the surface determined and compared. If the simulation agrees with theactual images, it offers some evidence that the structure may be the same as the model used for the simulation. Butsince several quite different models may produce images that are very similar and difficult to distinguishstatistically without a great deal of measured data, this is not a particularly robust analytical method. It is usuallypreferable to find other specimen preparation techniques, such as dispersal of the particulates over a surface sothey are not superimposed, or dispersal through a volume which can be sectioned.

Multiple images also arise in other situations, such as polarized light images from a petrographic microscope andmultiple wavelengths in fluorescence microscopy. It may also happen that the multiple images may all be derivedfrom a single original color or grey scale image by different processing, for example to isolate regions that arebright and have a high texture.

Figure 26. Color fluorescence image showing two different stains (a), and only those the red-stained cellsthat contain a green-stained nucleus (b) as selected using a feature-based AND operation.

Boolean logic used to combine multiple images is generally done on a pixel-by-pixel basis. This allows definingpixels of interest depending on whether they are on or off in the various images using AND (both pixels are on),OR (either pixel is on), Exclusive-OR (one but not both pixels are on) and NOT (changing a criterion from on tooff). These conditions can be combined in flexible ways, but do not address applications in which features in oneimage plane are marked by indicators in a second. Figure 26 shows an example. Two different stains have beenused, one to mark cells and the other the chromatin in nuclei. Using the green nuclei as markers, the entire cellscontaining them can be selected for measurement. This method is called a feature-based AND rather than a pixel-


based comparison (Russ, 1993). It keeps all connected pixels in a feature in one image if any of the pixels arematched by ones in the second.

Summary

The kinds of image processing operations can be summarized as follows:a) Ones that operate on a single image based on the brightness values of neighborhood pixels. This includes

arithmetic combinations such as multiplicative kernels as well as ones that rank the pixel values and keep themaximum, minimum or median value. They are used for smoothing, sharpening, edge enhancement and textureextraction.

b) Ones that operate in Fourier space to select specific frequencies and orientations in the image that correspond towanted or unwanted information and allow the latter to be efficiently removed. Measurements of regularspacings can be made in the Fourier image more conveniently than in the spatial domain. There are also otheralternate spaces, such as Hough space, that are convenient for identifying lines, circles or other specific shapesand alignments of features.

c) Ones that operate on two images to combine the grey scale pixel values. This includes arithmetic operations(add, subtract, etc.) as well as comparisons (keep the brighter or darker value). They are used in levelingcontrast, removing unwanted signals and combining multiple views.

d) Ones that operate on single binary images based on the local pattern of neighboring pixels. These are usuallycalled morphological operators (erosion, dilation, etc.). They add or remove pixels from features to smoothshapes, fill in gaps, etc. They can also be used to extract basic information about feature shape (outlines orskeleton)

e) Ones that are based on the Euclidean distance map, in which each pixel within the features in a binary image isgiven a grey scale value equal to its distance from the nearest background pixel. These allow segmentation oftouching convex features (or their rapid counting), and measuring of the distance of features from irregularboundaries.

f) Ones that combine two binary images using Boolean logic. These allow flexible criteria to be used to combinemultiple representations of the same area, either at the pixel or feature level.

Many image analysis systems offer most of these capabilities. It remains a significant challenge for the user toemploy them to best effect. This generally means that the user must first understand what is in the image and howshe or he is able to visually recognize it. Depending on whether the discrimination is based on color, texture,relationship to neighbors, etc., the choice of appropriate processing must be made. Few real problems are solvedby a single step operation; sequences of grey scale and binary image processes are often needed to isolate thedesired structures for measurement.

This demands a considerable effort by the user to understand the available tools and what they can be used for. Inmost cases in which only a few measurements are to be made on complex images, it is more efficient to use thecomputer-assisted manual stereology techniques discussed in the previous chapter. But when hundreds ofmeasurements are needed to permit statistical analysis, and the data are to be accumulated over a long period oftime so that human variability becomes a concern (for either one or multiple operators), the design of anappropriate image processing method to delineate the structures of interest for measurement becomes importantand worthwhile.


Chapter 11 - Geometric Modeling

Stereology is at heart a geometric science, using the principles of geometric probability to estimate measures ofthree-dimensional structures from measurements that can be accessed in lower dimensions using planes, lines andpoints as probes. There are modern texts in geometrical probability (Kendall & Moran, 1986; Matheron, 1975;Santalo, 1976). However, one of the first areas of study in this field (long predating the use of the name"stereology") was the calculation of the probabilities of intersection of various probes with objects of specifiedshape. This has roots back to the Buffon needle problem (18th century) and also involves Bertand's paradox, withits consideration of what a random probe really entails.

Quite a bit of work during the middle part of this century dealt with the relationship between the size distributionof objects in 3D with the size distribution of the linear or planar intercepts that a set of isotropic, uniform andrandom probes would produce (see for example Wicksell, 1925; Cruz-Orive, 1976). These values are needed forthe unfolding of the size distribution of the intercepts to estimate the size distribution of the 3D objects. Asdiscussed elsewhere the trend in recent decades has been away from this approach (Gundersen, 1986) because ofthe ill-conditioned nature of the mathematics used in unfolding (its sensitivity to small variations in the measuredsize distribution producing much larger variations in the calculated one), and because it makes some flawedassumptions about the nature of the specimen. Basically it assumes that the shape of the 3D objects is known (andhas some relatively simple geometric form) and that all of the objects are the same in shape. The most commonshape used for unfolding is a sphere, because of the mathematical simplicity that it yields. However, there are fewreal samples in which all of the objects of interest are actually perfect spheres. Even small deviations from thisshape, especially deviations that are systematic with size, introduce substantial bias into the final results.

Nevertheless it is useful to understand how the process for determining the relationship between object shape andthe size distribution of intercepts works. It is unlikely that a working stereologist will need to develop such a set ofdata individually (and sets of such data have been published for many geometrical shapes such as cylinders,ellipsoids, and polyhedra). But the basic method helps to educate the mind into the relationship between three-dimensional objects and the intercepts which planes and lines make with them, and this is an important step indeveloping an understanding and even an intuition for visualizing the three dimensional objects that are revealed intraditional two dimensional microscopy.

This chapter reviews the tools used for making these calculations and estimations. There are two avenues bywhich to approach the determinations: integration of analytic expressions, and random sampling. The former willinitially be more familiar, as an outgrowth of normal analytic geometry and calculus, but the latter is ultimately bemore powerful and useful for dealing with many of the problems encountered with real three dimensional features.

Methods: Analytic and Sampling

As an introductory example, consider the problem of determining the area of a circle of unit radius. Figure 1shows the familiar analytical approach. The circle is broken into strips of width dx, whose length if expressed as afunction of x is

L = 2 1− x2(11.1

Then the area of the circle is determined by integration, giving

Chapter 11 - Geometric Modeling Page # 225

Area = 2 1 − x2

−1

+1

∫ dx (11.2

which can be directly integrated to give

2sin−1(1)

2−

sin−1(−1)

2

=π2

− (−π2

) = π (11.3

Figure 1. Drawing illustrating the integration of the area of a circle

The other approach to this problem is to draw a circle of unit radius inside a square (of side 2, and area 4). Hangthe paper on a tree, back off some suitable distance, and shoot it with a shotgun. Figure 2 shows an example oftypical results that you might observe. Now count the holes that are inside the circle, and those that are inside thesquare. The fraction that are inside the circle divided by the total number of holes should be π/4.

Figure 2. The "shotgun" method of measuring the circle's area.

This is a sampling method. It carries the implicit assumption of a perfectly random shotgun, in the sense that theprobability of holes (points) appearing within any particular unit area of the target (plane) is equal. This is inagreement with our whole idea of randomness. And, of course, the precision of the result depends on countingstatistics: we will have to shoot quite a lot of holes in the target and count them to arrive at a good answer.


This is easier to do with a computer simulation than a real shotgun. Most computer systems incorporate some typeof random number generator, usually a complex software routine that multiplies, adds and divides numbers toobtain difference values that mimic true random numbers. It is not our purpose here to prescribe tests of how goodthese "pseudo random number" generators are. (Some are quite good, producing millions or hundreds of millionsof values that do not repeat and pass elaborate tests for uniformity and unpredictability; others are barely goodenough for their intended purpose, usually controlling the motion of alien space invaders on the computer screen.)

Assuming the availability of a random number function, and some high-level computer language in which towork, a program to shoot the holes and count them might look like the one in Listing 1 (see Appendix). Thisprogram (and the others in the Appendix) can be translated quite straightforwardly into any dialect of Basic,Fortran, Pascal, C, etc. It uses a function RND that returns a real number value in the range 0 <= value < 1 with auniform random distribution. This means that we are in effect looking not at the entire target, but at one-quarter ofit (the center is at X=0,Y=0). Each generated point is checked, and counted if it is inside the circle. The programprints out the area estimate periodically, in the example every thousand points, and continues until stopped. Atypical run of the program produced the example data shown in Figure 3. Other runs, using a different set ofrandom numbers, would be different, at least in the early stages. But given enough trials, the value approaches thecorrect one.

Figure 3. Typical result of running the program to measure circle area.

Because of the use of random numbers to measure probability, this sampling approach is called the Monte-Carlomethod (after the famous gambling casino). It finds many applications in sampling of processes where theindividual rules and physical principles are well known, but the overall process is complex and the steps are noteasily combined in analytical expressions that can be summed or integrated, such as scattering or electrons,photons or alpha-particles.

In the example here, it is easy to see that the method does eventually approach the "right" answer, but that it takesa lot more work than the straightforward integration of the circle. However, given some other much more complexshape bounded by lines that could individually be described by relationships in X and Y, but for which the integralcannot so readily be evaluated, the Monte-Carlo approach could be a useful alternative way to find the area.Measuring the area of the British Isles for example would simply require hanging up the map and blasting awaywith our shotgun. We will find this approach especially useful for three-dimensional objects.

If the sampling pattern were not random, but instead the program had two loops that varied X and Y through aregular pattern, for instance


FOR Y = 0 TO 1 STEP 0.02FOR X = 0 TO 1 STEP 0.02

.. .then 2500 values would be sampled, with a good result. This systematic sampling is in effect a numericalintegration, if the number of steps is large enough (the step size is small enough). But with systematic sampling,you have the same problems in establishing the limits as for analytic integration. Furthermore, there is no validanswer at all until you have completed the program, and no way to improve the answer by running for a longertime (if you repeat it, you will get exactly the same answer). The random sampling method produces a reasonableanswer (whose accuracy can be estimated statistically) in a short time, and it will continue to improve with runningtime.

Sphere Intercepts

The frequency distribution of the lengths of linear intercepts passing through a sphere is also a problem that can besolved analytically, as indicated in Figure 4. Figure 4a shows many lines passing vertically through the sphere.Because of its symmetry, it is not necessary to rotate the sphere or consider many directions. If the vertical linescorrespond to shots fired at the target (the flat surface beneath it) then the sampling condition is satisfied and weneed only to determine the intercept length of each line.

The length of each intercept line passing vertically through a sphere is uniquely related to its distance from thecenter (Figure 4b). A cross section (Figure 4c) shows that this is 2•(1–r2)1/2), and the number of lines at eachradius is proportional to the area of the circular strip shown on the plane, which is 2π r dr. The result is afrequency curve for the intercept length L whose shape is just the straight line that was shown before.

dη =π2

L dL (11.4

The Monte-Carlo sampling approach to this is similar to that used before for the area of a circle. Because of thesymmetry of the sphere, we can work just in one quadrant. X,Y values are generated by the RND function, andfor each line that hits the sphere, the length is calculated. These lengths are summed in an array of 20 counters thatbecome the frequency vs. length histogram. The program could be written as shown in Listing 2 in the Appendix.

Figure 4 (continued).


Figure 4. Intercept lengths in a sphere (a). The number of lines with the same length is proportional to thearea of a circular segment on the plane (b) and the length is a function of the distance of the line from the

center (c).

As before, a fairly large number of trials are required to obtain a good estimate of the shape of the distributioncurve. Figure 5 shows three results from typical runs of the program, with 100, 1000 and 10,000 trials. The latteris a fairly good fit to the expected straight line. The uncertainty in each bin is directly predictable from the numberof counts, as described in the chapter on statistics.

Figure 5. Probability curves for intercept lengths in a sphere generated by a Monte-Carlo program.

Intercept Lengths in Other Bodies

It is possible to perform both the integration and Monte-Carlo operations very simply for a sphere, because of thehigh degree of symmetry. For other shapes this becomes more difficult. For example, the cube shown in Figure 6


has a radically different distribution of intercept lengths than the sphere. It is very unlikely to find a short interceptfor the sphere (as shown in the frequency histogram) because the line must pass very close to the edge of thesphere. But for a cube, the edges and corners provide numerous opportunities for short intercept lengths.However, if we restricted the test lines to ones vertical in the drawing, no short intercepts would be observed (norwould any long ones; all vertical intercepts would have exactly the same length). Unlike the case for the sphere, itis necessary to consider all possible orientations of the lines with respect to the figure. This will require morerandom numbers.

Figure 6. A cube with illustrative intercept lines.

It is quite straightforward to determine a histogram of frequency versus intercept length in a circle (2-dimensional)or sphere (3-dimensional) using the Monte-Carlo approach, even though both these problems can also be solvedanalytically. For other shapes, the Monte-Carlo method offers many practical advantages, including that it is notdifficult to program in even a small computer. However, care is required so that the intersecting lines are properlyrandomized in space with respect to the feature, and have uniform probability of passing through all regions and inall directions.

In two dimensions, consider the case of determining intercept lines in a square. The square may be specified ashaving a unit side length and corners at (±0.5, ±0.5). Consider the following ways of specifying lines, andwhether they will meet the requirements for random sampling (all of the random numbers indicated by RNDbelow are considered to vary from 0 to 1, with uniform probability density, as produced by most computersubroutines that generate pseudo-random numbers):

1) generate one number Y = RND – 0.5, and use it to locate a horizontal line across the square. This willuniformly sample space (if the square is subdivided into a checkerboard of smaller squares, and the number oflines passing through each is counted, the numbers will be the same within counting statistics). However, all ofthe intercept lengths through the square will have length exactly 1.0 because the directions are not properlyrandomized.


2) generate one number THETA = π RND, and use it to orient a line passing though the origin (center of thesquare). This will uniformly sample orientations, but not positions. The counts in the sampling squares near thecenter will be much higher than those near the periphery. Short intercept lengths will not be generated, and thiswill consequently not produce a true frequency histogram.

3) generate three random numbers: X = RND – 0.5, Y = RND – 0.5, and THETA = π RND. The line is definedas passing through the point X,Y with slope THETA. It is less obvious that this also produces biased data, butit, too, favors the sampling squares near the center at the expense of those near the periphery and produces abiased set of intercept lengths. So do other combinations such as using four random numbers for X1,Y1 andX2,Y2 (two points to define the line).

To reveal the bias in these (and other) possible methods, it is instructive to write a small program to perform thenecessary steps. This is strongly recommended to the practitioner, as the 2-dimensional geometry and simpleshape of the square make things much easier than some of the 3-dimensional problems that may be encountered.Set up an array to count squares (e.g., 10x10) through which the lines pass. Then use various methods togenerate the lines, sum the counts, and also construct a histogram of intercept lengths. It is also useful to keeptrack of the histogram of line lengths within the circumscribed unit circle around the square, since the shape of thatdistribution is known and departures will reflect some fault in the randomness of the line generation routine.

When this is done for the methods described above, arrays like those shown in Figures 7 and 8 indicate impropersampling. Both of these methods, and the others mentioned, also produce quite wrong histograms of interceptlength for both the cube and sphere.

112 110 122 143 147 163 121 114 97 103123 170 163 181 195 175 185 153 138 124134 191 228 242 216 220 188 189 176 157159 181 213 259 260 255 255 247 175 130147 172 201 253 285 287 273 254 200 166156 175 211 259 288 305 286 258 224 151139 183 230 252 283 265 271 270 209 172127 165 230 252 261 230 244 193 189 160123 158 166 190 209 207 208 179 141 125115 126 118 127 145 159 160 132 124 118

Figure 7. Array of counts for 10x10 grid using four random numbers to generate X1 = RND – 0.5,Y1 =RND – 0.5 and X2 = RND – 0.5,Y2 = RND – 0.5 points to define the line. Note the center weighting.

The graph shows the counts as an isometric display.


177 109 59 48 39 37 37 50 63 185197 184 173 167 142 132 138 152 161 168235 232 216 203 213 206 239 279 297 241249 244 282 367 312 301 333 302 266 240257 280 318 383 423 419 341 288 260 225220 250 298 353 422 458 406 313 240 239196 233 280 322 360 356 361 342 274 246197 234 225 245 222 245 399 269 242 222201 233 181 141 128 134 145 209 196 218181 92 51 37 35 33 38 50 101 197

Figure 8. Array of counts for 10x10 grid using two random numbers to generate points along left andright edges of square. In addition to center weighting note the sparse counts along top and bottom edges.

The graph shows the counts as an isometric display.

A proper method for generating randomized lines is:

Generate R=0.5 RND and THETA = π RND. This defines a point within the unit circle, and a vector from theorigin to that point. Pass the line for intercept measurement through the point, perpendicular to the line. (Notethat generating an intercept line from two points on the circumscribed circle using two angles THETA = π RNDdoes not produce random lines, although for a sphere it does.)

An equivalent way to consider the problem is to imagine that instead of the square being fixed, and the linesoriented at all angles, the square is rotated inside the circle (requiring one random number to specify the orientationangle). Then the lines can all be parallel, just as for the earlier method of obtaining intercepts through the circle.This also requires one random number (the position along the axis). With these methods, or others which areequivalent once rotation of coordinates is carried out, proper sampling of the grids is achieved.

The result is a histogram of intercept lengths in the square as shown in Figure 9. This shows a flat shelf for shortlengths, where the line passes through two adjacent sides of the square. The peak corresponds to the length of thesquare's edge, and the probability then drops off rapidly to the maximum intercept (the diagonal, which is equal tothe circle diameter).

The program used to generate the data for the histogram is shown in Listing 3 in the Appendix (it uses the first ofthe randomizing methods described above). In this program, the array CT is defined (to build a 20 point histogramin this example). DG is the radius of the circumscribed circle, or half the square's diagonal; P2 is 2π. The randomfunction is used to generate a point, from which the slope (M) and intercept (B) of the line through this point


perpendicular to the vector from the origin is calculated (the equation of the line is y = Mx+B). The intersection ofthis point with the left side of the square, or if it does not pass through the side, the intersection with the top orbottom is determined as one end of the line, and the process repeated for the other end. The line length is obtainedfrom the Pythagorean theorem, and for lines that intersect the square produces an integer from the ratio of thislength to the circle diameter which is used to build the histogram.

Figure 9. Histogram of intercept lengths in a square, generated by Monte-Carlo program.

Intercept lengths in three dimensions

Progressing to three dimensions, things become more complicated, and there are many more ways to bias thesampling. An extension of the methods described above for the circle and square can be used for a sphere andcube, as follows (refer to Figure 10 for the nomenclature used for a spherical coordinate system).

Figure 10. Spherical coordinates as described in the text.

1) Generate a random vector from the origin. This requires a radius R which is just the circle radius times arandom number, and two angles. Using the nomenclature of the figure above, the θ angle can be generated as

2π•RND, but the Φ angle cannot be just (π/2)•RND. If this were done, the vectors that pointed nearly vertically

would be much denser in space than those that were nearly horizontal, because for each vertical angle therewould be the same number of vectors, and the circumference of the "latitude" circles decreases toward the pole.The proper uniform sampling of orientations requires that Φ be generated as the angle whose sine is a random

number from 0 to 1 (producing angles from 0 to π/2 as desired). Through the point at the end of the vector


defined by these two angles and radius, pass a plane perpendicular to the vector. Then within this plane,generate a random line by the method described above (angle and radius from the initial point to define anotherpoint, and a line through that point perpendicular to the vector from the initial point). This is less cumbersomethan it sounds if proper use of matrix arithmetic is used to deal with the vectors.

2) Locate two random points on the circumscribed sphere. As in the method above, the points are defined by twoangles, θ and Φ, which are generated as θ = 2π RND, Φ = arc sin (RND). Connect these points with a line.

As before, alternate methods where the cube rotates while the line orientation stays fixed in space can be usedinstead (but the same method for determining the tilt angle is required). Conceptually it may be helpful to visualizethe process as one of embedding the cube inside a sphere which is then rotated while the intersection lines passthrough it vertically, as shown in Figure 11. Once the line has been determined, it is straightforward to find theintersections with the cube faces and obtain the intercept length. In the program shown as Listing 4 in theAppendix, this is done by setting up a matrix equation and solving it. Simpler methods can be used for the cube,but this more general approach is desirable when we next turn our attention to less easy shapes.

Figure 11. Linear intercepts in a cube can be determined by embedding the cube in a sphere which is thenoriented randomly and vertical lines passed through it.

This program dimensions an array for the histogram and line and defines the equations for each of the six faces onthe cube (equations such as 1X+0Y+0Z=0.5). Then a random point on the sphere is generated. For convenience,the elevation angle E used here is the complement of the angle F discussed above, and the complicated arctanfunction is used to get the arc sine, which many languages do not provide. The point X1,Y1,Z1 (and the secondpoint X2,Y2,Z2 ) define the intersections of the random line on the sphere.

Coefficients in the A matrix define this line. Each face is checked for intersections with the line, by placing theface coordinates into the matrix. PC is a counter used to find two intersections of the line with faces. Thedeterminant of the matrix (if zero, skip to another face, since the current one is exactly parallel to the line)calculates the coordinates (X,Y,Z) of the intersection of the line with the face plane, which are checked to see if


the intersection lies within the face of the cube. For the first such intersection, the coordinates are saved and thecounter PC incremented. When the second point is found, the distance between them (the intercept length) iscalculated and converted it to an integer for the histogram address, and one count added. The process repeats forthe selected number of intersection lines. Some output of the data as a graph is needed to make the resultsaccessible.

Figure 12. Linear intercepts in other solids: a)tetrahedron; b) many-sided polyhedron; c)cylinder; d) torus; e) arbitrary non-convexshape (a banana).


Several additional features can be added to this type of program to serve as useful checks on whether gross errorsare present in the various parts of the program. First, the sphere intercept length can also be used to build ahistogram (this is just the distance between the points X1,Y1,Z1 and X2,Y2,Z2) which should agree with theexpected frequency histogram for a sphere. Also, by summing the lengths of all the lines in both the sphere andcube, the volume fraction VV and the surface area per unit volume SV of the cube in the sphere can be obtainedand compared to the known correct values. The mean intercept length in the feature λ can also be obtained, which

should equal 4 V/S for the cube. It is also relatively simple to add a counter for each face of the cube, and comparethe number of intersections of lines with each face (if random orientations are correctly used, these values shouldbe the same within counting variability).

Before looking at the results from this program, It is worth pointing out that it may be relatively easily extended toother shapes, even non-regular or concave ones. As shown in Figure 12, for polyhedra the smaller the number offaces the easier it is to produce short intercepts where the lines pass through adjacent faces. The number of faceswill vary, and so some of the dimensions will change, and it may be worthwhile to directly compute the equationsof each face plane from vertex points. The most significant change is in the test to see if the intersection point ofthe line with each face plane lies within the face. For the general polyhedron, the faces are polygons. The simplestway to proceed is to divide the face into triangles formed by two adjacent vertices and the intersection point, andsum the triangle areas. If this value exceeds the known area of the face, then the point lies outside the face(provision for finite numerical accuracy within the computer must be made in the equality test). Many curvedsurfaces can also be described by analytical geometry, but complex objects (e.g. those defined by splines) requiremore difficult techniques to locate the intersection points. For convex objects some lines will create more than oneintercept.

Figure 13. Frequency histograms for intercept lengths through various regular polyhedra,compared to that for a sphere, generated by Monte-Carlo program using 50 intervals

for length, and about 40,000 intercept lines.

Applying this program to a series of regular polyhedra produces the results shown in Figure 13. The results areinteresting in several respects. For instance, note that whereas for the sphere, the most likely intercept length is themaximum (equal to the sphere diameter), large values are quite unlikely for the polyhedra because they requires aperfect vertex-to-vertex alignment (and is impossible for the tetrahedron because even this distance is less than the


sphere diameter). Also, the peaks present in the frequency histogram correspond to the distance between parallelfaces (again, there are none for the tetrahedron), while the sloping but linear shelves correspond to intersectionsthrough adjacent faces.

The mean intercept lengths in the various solids are obtained from the distributions. They are listed below asfractions of the diameter of the circumscribed sphere.

Solid Mean InterceptTetrahedron 0.232Cube 0.393Octahedron 0.399Icosahedron 0.538Sphere 0.667

Intersections of planes with objects

Not all distribution curves are for intercept lengths, of course. It is also possible to model using geometricprobability (or to measure on real samples) the areas of intersection profiles cut by planar sections through anyshape feature. As usual, this is easiest for a sphere. Figure 14a shows that any intersection of a plane with asphere produces a circle. Because of the symmetry, rotation is not needed and a series of parallel planes (Figure14b) can generate the family of circles whose radii are calculable from the position of the plane, analogous to thecase for the line intercept.

a b Figure 14. Intersection of a plane with a sphere in any orientation produces a circle (a).

A family of parallel planes (b) produces a distribution of circle sizes.

For a cube, the intersection can have 3, 4, 5 or 6 sides (Figure 15). It is evident that the cube's corners produce alarge number of small area intersections, and that there are ways to cut areas quite a bit larger than the mostprobable slice (which is nearly equal to the area of a square face of the cube). Figure 16 shows a plot (Hull &Houk, 1958) of the area of profiles cut through spheres and cubes. Generating similar distributions for planarintersections with other shapes is a straightforward if rather lengthy exercise in geometric probability. It alsoreveals that humans are not good intuitive judges of intersections of planes with solids. Figure 17 shows a torus,familiar to many researchers in the form of a bagel. A few of the less obvious cuts which produce sectionsunexpected to casual observers are shown. Considering the difficulty in predicting the shape of section cuts from


known objects, it should not be surprising that the reverse process of imagining the unknown 3D object that mayhave given rise to observed planar sections easily leads to errors.

a b

c d Figure 15. Planes intersection a cube can produce intersections with: a) 3, b) 4; c) 5; d) 6 sides.

For other shapes with even less symmetry, the analytical approach becomes completely impractical. Even Monte-Carlo approaches are time-consuming, because of the problems of calculating the intersection lines and points, andit may take a very large number of trials to get enough counting statistics in the low-probability regions of thefrequency histogram to adequately define it (particularly when the shape of the curve is to be used to deconvolutedistributions from many different feature sizes).


0.000

0.025

0.050

0.075

0.100

0.125

Fre

quen

cy

0.00

0.25

0.50

0.75

1.00

Area/Max Area

Cube

Sphere

Figure 16. Intercept area distributions for a sphere and cube.

Figure 17. Some of the less familiar planar cuts through a bagel (torus), which can produce one or twoconvex or concave intersections.


Bertrand's Paradox

There is a problem which has been mentioned briefly in the previous examples, but which often becomesimportant when dealing with ways to orient complex shapes using random numbers. It is easy to bias (or, inmathematical terminology, to constrain) the orientations so that the random numbers do not produce a randomsampling of the space inside the object. In a trivial example, if we only rotated the cube around one axis, wewould not see all intercept lengths; but if we rotate it uniformly around all three space angles, the proper results areobtained.

A classic example of the problem of (im)proper sampling is stated in a famous paradox presented by Bertrand in1899. The problem to be solved is stated as follows: What is the probability that a random line passing through acircle will have an intercept length greater than the side of the inscribed equilateral triangle in the circle?

Figure 18. Illustrations for Bertrand's Paradox.

Using the three drawings in Figure 18 (and working from left to right), here are three arguments that can bepresented to quickly arrive at an answer.

1. Since the circle has perfect symmetry, and any point on the periphery is the same as any other, we will considerwithout loss of generality, lines that enter the circle at one corner of the triangle, but at any angle. Since the totalrange of the angle theta is from 0 to 180 degrees, and since the triangle subtends an angle of 60 degrees (andany line lying within the shaded region clearly has an intercept length greater than the length of the side of thetriangle), the probability that the intercept line length is greater than the side of the triangle is just 60/180, orone-third.

2. Since each line passing through the circle produces a line segment which must have a center point, which willlie within the circle, we can reduce the problem to determining how many of the line segments have midpointsthat lie within the shaded circle inscribed in the triangle (these lines will have lengths longer than the side of thetriangle, while any line whose midpoint is outside the circle will have a shorter length). Since the area of thesmall circle is just one-fourth the area of the large one, the probability that the intercept line length is greaterthan the side of the triangle is one-fourth.

3. Since the circle is symmetric, we lose no generality in considering only vertical lines. Any vertical line whichpasses through the circle but passes outside of the smaller circle inscribed in the triangle will have a lengthshorter than the side of the triangle, so the probability that the intercept line length is greater than the side of thetriangle is the ratio of diameters of the circles, or one-half.


These three arguments are all, at least on the surface, plausible. But they produce three different answers. Perhapswe should choose one-third, since it is in the middle and hence "safer" than the two extreme answers!

No, the correct answer is one-half. The other two arguments are invalid because they impose constraints on therandom lines. Lines spaced at equal angle steps all passing through one point on the circle's periphery do notuniformly sample it, but are closer together at small and large angles. This under-represents the number of linesthat lie within the shaded triangle, and hence produces too low an answer. Likewise the lines whose midpoints liewithin the shaded inner circle would be correct if only one angular orientation were considered (it would then beequivalent to the third argument), but when all angles are considered it undercounts the lines that pass outside theshaded circle. It can be very tricky to find subtle forms of this kind of bias. They may arise in either the analyticalintegration or the random sampling methods.

The method used to generate random orientation of lines in space for the Monte-Carlo routines described abovemust avoid this problem. The trick is to use θ = 2π RND and Φ = arc sin (RND), using the terminology of Figure

10. This avoids clustering many vectors around the Z axis, and gives uniform sampling.

The Buffon Needle Problem

A good example of the role of angles in geometrical probability is the classical problem known as Buffon's Needle(Buffon, 1777): If a needle of length L is dropped at random onto a grid of parallel lines with spacing S, what isthe probability that it crosses a line? This is another example of a problem easy enough to be solved by eitheranalytical means or random sampling. First, we will look at the procedure of integration.

Figure 19. Geometry for the Buffon needle problem

Figure 19 shows the situation. Two variables, y and theta, are involved; theta is the angle of the needle withrespect to the line direction, and y is the distance from the midpoint of the needle to a line. It is only necessary toconsider the angle range from 0 to π/2, because of symmetry. At any angle, the needle subtends a distance of Lsin Á perpendicular to the lines, so if y is less than half this, there will be an intersection. Hence, the probability ofintersection is, as was stated before

L sin θS

dθ0

π2

∫

dθ0

π2

∫

=

L

S−cos

π2

+ cos 0( )

π2

=2L

πS(11.5

A Monte-Carlo approach to solving this problem is shown as Listing 5 in the Appendix. L and S have beenarbitrarily set to 1, so the result for number / count should just be π/2.


One particular sequence of running this program ten times, for 1000 trials each, gave a mean and standarddeviation of 1.5727 ± 0.0254, which is quite respectable. There is a probably apocryphal story about an easternEuropean mathematician who claimed that he dropped a needle 5 cm. long onto lines 6 cm. apart, and counted 226crossing events in 355 trials. He used this to estimate π, as 2 x 355 / 266 = 3.1415929 (correct through the first 6decimal places).

We can evaluate the (im)probability of the truth in this story. Dropping the needle one more time, regardless of theoutcome, would have seriously degraded the "accuracy" of the result! This tale may serve as a caution to alwaysconsider the statistical precision of results generated by a sampling method such as Monte-Carlo simulation, aswell as the statistics of measured data to which it is to be compared.


Appendix - Geometric Modeling

Listing 1: Random points in a unit square used to measure circle area.Listing 1Integer: Increment = 1000;Integer: Number = 0;Integer: Count = 0;Repeat X = RND;

Y = RND;IF (X*X + Y*Y) < 1

THEN Count = Count + 1;Number = Number + 1;IF Float(Number/Increment) = Integer Number/Increment)

THEN PRINT Number, Float(4*Count/Number);Until Stopped;

Listing 2: Intercept lengths in a sphere.Listing 2Integer Count[1..20] = 0;Integer Number, Bin, i;Float X, Y, rsquared, Icept;Input ("Number of trials= ",Number);for (i = 1; i <= Number; i++) X = RND;

Y = RND;rsquared = X*X + Y*Y;if rsquared < 1 then Icept = sqrt (1 - rsquared);

Bin = Integer (20*Icept) + 1;Count[Bin] = Count[Bin] + 1;

for (Bin = 1; Bin <= 20; Bin++)

PRINT Bin, Count[Bin];

Listing 3: Intercept lengths in a square.Listing 3Integer CT[20] = 0;Integer N, NU, L;Float M, B, TH, R, XL, YL, XR, YR;Const DG = SQR(2)/2;Const P2 = 8*ATN(1);Input ("Number of Lines: ",NU);for (N = 1; N <= NU; N++) R = DG * RND; /* two random numbers */

TH = P2 * RND;M = -1/tan(TH);B = 0.5 + R * sin(TH) - M * (0.5 + R * cos(TH));/* get the coordinates of the ends of the line */XL = 0;YL = B;if ((YL>1) or (YL<0)) then if (YL>1)

then YL=1else YL=0;

XL=(YL-B)/M;XR = 1;YR = M + B;


if ((YR>1) or (YR<0)) then if (YR>1)

then YR=1else YR=0;

XR=(YR-B)/MLEN = sqrt((XR-XL)*(XR-XL)+(YR-YL)*(YR-YL));if (LEN>0) then L = integer(0.5+10*LEN/DG);

CT[L] = CT[L]+1;

Listing 4: Intercept lengths in a cube.Listing 4Float F[6,4] = 1,0,0,0.5

1,0,0,-0.50,1,0,0.50,1,0,-0.50,0,1,0.50,0,1,-0.5; // the six faces of the cube

Float A[3,4]; // used to calculate line/face intersectionInteger LX[50]; // for histogramConst P2 = 8 * atan(1.0); // 2πConst R = sqrt(3.0)/2.0; // radius of circumscribed sphereInput (“Number of Lines:”,NU);for (n = 1; n <= NU; N++) TH = P2 * RND;

E = -1 + 2 * RND;E = atan (E/sqrt(1 - E*E));X1 = R * cos(E) * cos(TH);Y1 = R * cos(E) * sin(TH);Z1 = R * sin(E); // one random point on the sphereTH = P2 * RND;E = -1 + 2 * RND;E = atan (E/sqrt(1 - E*E));X2 = R * cos(E) * cos(TH);Y2 = R * cos(E) * sin(TH);Z2 = R * sin(E); // second random point on sphereA[1,1]=Y1*Z1-Y2*Z1;A[1,2]=Z1*X2-Z2*X1;A[1,3]=X1*Y2-X2*Y1;A[1,4]=0;A[2,1]=Y1*Z2-Y2*Z1;A[2,2]=Z1*X2-Z2*X1+Z2-Z1;A[2,3]=X1*Y2-X2*Y1+Y1-Y2;A[2,4]=Y1*Z2-Y2*Z1; // define the line in A matrixPC=0;for (j=1; j<=6; j++) // check faces for intersection for (k=1; k<=4; k++)

A[3,K]=F[J,K];DE= A(1,1)*(A(2,2)*A(3,3)-A(2,3)*A(3,2))

+A(1,2)*(A(2,3)*A(3,1)-A(2,1)*A(3,3)) +A(1,3)*(A(2,1)*A(3,2)-A(2,2)*A(3,1));

if (DE<>0) then // zero means parallel to the face X= (A(1,4)*(A(2,2)*A(3,3)-A(2,3)*A(3,2))

+A(1,2)*(A(2,3)*A(3,4)-A(2,4)*A(3,3))+A(1,3)*(A(2,4)*A(3,2)-A(2,2)*A(3,4)))/DE;

Y= (A(1,1)*(A(2,4)*A(3,3)-A(2,3)*A(3,4))+A(1,4)*(A(2,3)*A(3,4)-A(2,4)*A(3,3))+A(1,3)*(A(2,1)*A(3,4)-A(2,4)*A(3,1)))/DE;

Z= (A(1,1)*(A(2,2)*A(3,4)-A(2,4)*A(3,2))+A(1,2)*(A(2,4)*A(3,1)-A(2,1)*A(3,4))


+A(1,4)*(A(2,1)*A(3,2)-A(2,2)*A(3,1)))/DE;// intersection point of line with faceif ((abs(X)<=0.5)

and (abs(Y)<=0.5)and (abs(z)<=0.5))

then // see if intersection is inside cube if (PC=0)

then PC=1;X0=X;Y0=Y;Z0=Z;

else LE=SQR((X-X0)*(X-X0)+(Y-Y0)*(Y-Y0)+

(Z-Z0)*(Z-Z0));// distance between points = lengthH=integer(25*LE/R);LX[H]=LX[H]+1; // increment histogram

// if de

// for j // for n

Listing 5: The Buffon needle problem.Listing 5Integer Number, j, Count;Float Y, Theta, Vert;Const HalfPi = 2 * Atan(1); // π/2Input ("How many trials: ",Number);for (j = 1; j<=Number; j++) Y= RND;

Theta = HalfPi * RND;Vert = Sin (Theta) / 2;if ((Y - Vert < 0) OR (Y + Vert > 1))

then Count++;PRINT Number/Count


Chapter 12 - Unfolding Size Distributions

In the preceding chapter, methods were discussed for predicting the distribution of intercepts that a random probewould make with objects in a three-dimensional space. This kind of modeling can be done using either line orplane probes. With line probes the length of the intersection can be measured, and with plane probes the area aswell as the shape of the intersection can in principle be determined. This approach requires making severalassumptions: the size and shape of the objects being sampled must be known, and the probes must be isotropic,uniform and random with respect to the objects.

In actual practice, the experiment is carried out in the opposite direction. A real specimen is probed with lines orplanes, and the data on intercept lengths, areas and perhaps shape information are recorded. Then the goal is tocalculate the (unknown) sizes of the objects that must have been present in order to produce those results. Notethat it is not possible to determine the size of any single object this way, from the intersection that the probe makeswith it. It is only in the statistical aggregate that a distribution of objects can be inferred.

And even then it is far from easy to do so. There are several aspects to the problem, which will be discussedbelow. First, mathematically the inverse problem is ill-conditioned. The small statistical variation in measured data(which arises because a limited number of measurements are made, even if they are an unbiased sample of thespecimen) grows to a much larger variation in the calculated answer because of the ways that these variationspropagate through the calculation.

Second, a critical assumption must be made for the method to be applied at all: we must assume that we knowwhat the shape of the objects is, and usually that they are either all of the same shape or have a very simpledistribution of shapes. The most popular shape assumption is that of a sphere, because its symmetry makes themathematics easy and also relaxes the requirements for isotropic sampling.

But in fact not very many structures are actually spheres, and even a small variation from the assumed shape canintroduce quite a large bias in the calculated results. To make matters worse, many real specimens contain objectsthat have a wide variety of shapes, and the shape variation is often a function of size. This presents majorproblems for unfolding techniques.

In spite of these difficulties, the unfolding approach was the mainstay of classical stereology for decades and isstill used in spite of its limitations in many situations. Certainly there are specimens for which a reasonable shapeassumption can be made - spheres for bubbles, cubes for some crystalline materials (e.g., tungsten carbide),cylinders for fiber composites, and so on. And the use of computer-based measurement systems makes it possibleto collect enough data that the statistical variations are small, limiting the extent of the errors introduced by theinverse solution method.

Linear intercepts in spheres.

The preceding chapter developed the length distribution of lines intersecting spheres. The frequency distributionfor a sphere of diameter D is simply a straight line, and when measured intercept lengths are plotted in aconventional histogram form we would expect to see a result as shown in Figure 1. The presentation of data in

Chapter 12 - Unfolding Size Distributions Page # 246

histogram form is the most common way to deal with measurement data of all types. The bin width δ is typically

chosen to permit each bin to accumulate enough counts for good statistical precision, and enough bins to show theshape of the distribution. In most cases from 10 to 15 bins are linearly spaced in size up to the largest valueobtained in measurement.

Figure 1. Frequency distribution of linear intercepts in a sphere of diameter D,shown as a continuous plot and as a discrete histogram.

One consequence of the number of bins used is the ability to measure spheres (or other objects) whose sizes varyover that range of dimensions. Ten or 15 bins allows determining spheres in that number of sizes. If the size rangeof the spheres is much larger, say 50:1, and information on the smallest is actually needed, then a histogram withat least that many bins is required. This greatly increases the number of measurements that must be made to obtainreasonable counting precision.

When a mixture of sphere sizes is present, the measured distribution represents a summation of the contribution ofthe different sizes each in proportion to the abundance of the corresponding size. As shown in Figure 2, the resultis that the histogram bins corresponding to smaller intercept lengths contain counts from intersections with manydifferent size spheres, while the largest bin contains counts only from the largest spheres.

This offers a straightforward graphical way to unfold the data (Lord & Willis, 1951). As shown in Figure 3,knowing that the largest bin contains counts only from the largest spheres and that those spheres should alsogenerate a proportionate number of shorter intercepts allows subtracting the expected number of shorter intercepts


from each of the other (smaller) bins. This process can then be repeated for the next bin, using the number ofcounts remaining, and so on.

Figure 2. Frequency distribution of linear intercepts from a mixture of sphere sizes,shown as a continuous plot and a discrete histogram.

Figure 3. Unfolding the linear intercept distribution from a mixture of sphere sizes.Lines project the histogram bin heights onto a vertical axis and the increments given

the relative proportion of each sphere size.


Notice that the number of counts in the smaller size bins (and hence the estimated number of smaller size spheres)is obtained by successive subtractions. Subtraction is a particularly poor thing to do with counting data, since thedifference between two numbers has a standard deviation that is the sum of the deviations of the two numberswhose difference was taken. This means that in effect the uncertainty of the estimated number of smaller sizespheres grows rapidly and depends strongly on the number of larger size spheres and the precision with whichthey are determined.

Hence the need to obtain large numbers of interception counts, since the standard deviation of any counting data issimply the square root of the number counted. In other words if you count one hundred events the one-sigmauncertainty is √100 = 10 out of 100 or 10%, and to reduce this to 1% precision would require counting 10,000events.

Plane intersections

Linear intercepts were primarily used when manual measurements were required to obtain data, because theycould be performed by drawing (random) lines on images of sections and measuring the length of theintersections. With modern computer-based instruments, it is usually easier to measure the intersection areas madeby the sampling plane with the objects. It is very difficult to use plane probes that must be isotropic, uniform andrandom (IUR) in a real object because once a single plane section has been taken, other planes that might intersectthat one cannot be generated. However, for specimens which are themselves IUR any plane is as good as another,and examination of plane sections is a very common approach.

Figure 4. Different size circles are produced by sectioning of a sphere.

Every section through a sphere is a circle as shown schematically in Figure 4. Figure 5 shows an image of poresin an enamel coating on steel. The pores may be expected to be spherical as they result from the evolution of gasbubbles in the firing of the enamel, and all of the intersections are observed to be circular. The distribution of thesizes of circles can be calculated analytically. The probability of obtaining a circle of radius d±δd (corresponding

to counts that will fall into one bin of a histogram) from a sphere of diameter D is equal to the vertical thickness ofa slice of the sphere with that diameter. The shape of the distribution is

P(dcircle) = d / (D • (D2 - d2)1/2) (12.1

as shown in Figure 6.


Figure 5. Example of sectioning of spheres: bubbles in a fired enamel coating.

Figure 6. The distribution of planar intercepts through a sphere is proportional to thevertical thickness δz of a slice of the sphere covering a range of circle sizes δr.

As for the linear intercept case, if there are several different sizes of spheres present in the sample a particular sizeof intercept circle could result from the intersection of the plane with several different sphere sizes. The resulting


measurement histogram would show the superposition of data from the different sizes in proportion to theirrelative abundance and to their size (it is more likely for the plane to strike a large sphere than a small one).

This distribution might be unfolded sequentially and graphically as shown before for the linear intercepts, sincethe largest circles could only come from the largest spheres. Then as before, a corresponding number of interceptswould be subtracted from each smaller bin and the process repeated until all sizes are accounted for. This is not avery practical approach because of the tendency for statistical variation to concentrate in the smaller size ranges,and instead a simultaneous solution is generally used.

For the case of spheres, this method was first described by Saltykov (1967) and has been subsequently refined bymany others, with an excellent summary paper by Cruz-Orive (1976) that includes tables of coefficients suitablefor general use. The method is simply to solve a set of simultaneous equations in which the independent variablesare the number of circles of various sizes (measured on the images and recorded as a histogram) using thecoefficient matrix, to obtain the numbers of spheres of each size present in the material.

The matrix is determined by calculating as discussed above, from geometric probability, the frequency distributionfor the number of circles in each size class due to three dimensional objects of each size. This can be written as

NAi = α' ij • NVj (12.2

where the subscript i ranges over all size classes of circles in the measured histogram and the subscript k coversthe corresponding sizes of spheres in the (unknown) distribution of three-dimensional objects. The equations aremore readily solved by using the inverse matrix α, so that

NVj = (1/δ) αij • NAi (12.3

where δ is the size of the bin used in the histogram. A typical α matrix of coefficients is shown in Table 1.

Figure 7. Images of graphite nodules in cast iron.



As an example of the use of this method, Figure 7 shows images of nodular graphite in a cast iron. The sectionsare all at least approximately circular and we will make the (typical) assumption that the nodules are spherical.Figure 8a shows a distribution of circle sizes obtained from many fields of view covering the cross section of theenamel. Converting these NA data using the coefficients in Table 1 produces the plot of sphere sizes shown inFigure 8b.

Figure 8. The distribution of measured circle sizes from the cast iron in Figure 7 (left),and the calculated distribution of sphere sizes that generated it (right).

Other shapes

The "sphere unfolding" method is easily programmed into a spreadsheet, and is often misused when the measuredobjects are not truly spheres. Of course, similar matrices for α' can be computed for other shapes of objects, and

inverted to produce α matrices. In fact, this has been done for an extensive set of convex shapes including

polyhedra, cylinders, etc. (see for example Wasén & Warren, 1990). It is well to remember, however, that theresults are only as useful as the assumption that the three-dimensional shape of the objects is known and is thesame for all objects present.

Another shape model that has been applied to the "nearly spherical" shapes that arise in biological materials inparticular (where membranes tend to produce smooth boundaries as opposed to polyhedra) is that of ellipsoids.These may either be prolate (generated by revolving an ellipse around its major axis) or oblate (generated byrevolving the ellipse around its minor axis), and can be used to approximate a variety of convex shapes.

All of the sections made by a plane probe passing through an ellipsoid produce elliptical profiles (Figure 9). Thesecan be characterized by to dimensions, for instance the lengths of the major and minor axes. DeHoff (1962)presented a method for unfolding the size distribution of the ellipsoids, which are assumed to have a constantshape (axial ratio q) but varying size (major axis length), from the size distribution of the ellipses. This is amodification of the sphere method in which

NVj = (K(q)/δ) • ∑ αij NAi (12.4


δ is the size increment in the histogram, and the additional term K(q) is a shape dependent factor that depends on

the axial ratio of the ellipsoid q and whether the ellipsoids are prolate or oblate; this factor shown in Figure 10 forboth cases.

Figure 9. Ellipses formed by planar sections with prolate (blue) and oblate (green) ellipsoids.

a

bFigure 10. K shape factors for prolate (a) and oblate (b) ellipsoids as a function

of the axial ratio q of the generating ellipse.


Note that it is necessary to decide independently whether the generating objects are oblate or prolate, since eithercan produce the same elliptical profiles on the plane section image. If this is not known a priori, it can sometimesbe determined by examining the sections themselves. If the generating particles are prolate, then the diameters ofthe most equiaxed sections will be close in size to the width of those with the highest aspect ratio, while if theparticles are oblate then the most equiaxed sections will be close to the length of those with the highest aspect ratio(Figure 11). In either case, the axial ratio q is taken as the aspect ratio of the most elongated profiles and used toobtain k from the graph.

Figure 11. Elliptical profiles from prolate and oblate ellipsoids.

It is in principle possible to extend the unfolding method of analysis to convex features that do not all have thesame shape. The profiles observed in the plane section can be measured to determine both a size and shape (e.g.an area and eccentricity) and a two-dimensional histogram of counts constructed (Figure 12). From these data,calculation of both the size and shape of the generating ellipsoids or other objects should be possible. Thecalculation would be of the form

NVij = ∑ αijkl • NAkl (12.5

where the subscripts i,j cover the size and shape of the three-dimensional objects and k,l cover the size and shapeof the intersection profiles. Calculating the four-dimensional a' matrix and inverting it to obtain a is astraightforward extension of the method shown before, although selecting the correct three-dimensional shape andits variation is not trivial. The major difficulty with this approach is a statistical one. The number of countsrecorded in many of the bins in the two-dimensional NA histogram will be sparse, especially for the more extremesizes and shapes. These are important to the solution of the equations and propagate a substantial error into thefinal result.

Shape information from the profiles can be useful even when a single shape of generating object is present. Ohser& Nippe (1997) have shown that measuring the shapes of intersections with cubic particles (which are polygonswith from 3 to 6 sides) can significantly improve the determination of the size distribution of the generating cubes.


Figure 12. An example of a two-dimensional histogram of size and shapeof intersections made by a plane probe.

Simpler methods

The unfolding of linear intercepts is one of the simplest methods available, in use long before modern computersmade the solution of more complicated sets of equations easy. It can be performed using other distributions thanthe linear one that applies to spheres, of course. If the actual shape of the objects present is known, the exactdistribution could be generated as discussed in the preceding chapter and used for the unfolding. A relativelycompact and general approach has been proposed for the case of other convex shapes by several investigators(Weibel & Gomez, 1962; DeHoff, 1964). Using the usual nomenclature of NV (number per unit volume), NA

(number per unit area), VV (volume fraction), PL (intersections per length of line) and PP (fraction of pointscounted), the relationships proposed are

NV = (K/b) • NA3/2 / VV 1/2 (12.6

and

NV = 2 γ • NA PL / PP (12.7

Note than both procedures require determining the total volume fraction of the particles (VV or PP) as well as thenumber of objects per unit area of section plane, and one requires determining the mean linear intercept (theinverse of PL). The parameter K takes into account the variation of the size of features from their mean, and isoften assumed to have a value between 1.0 and 1.1 for many real structures that constitute a single population(e.g., cell organelles). Figure 13 shows K as a function of the coefficient of variation of the sizes of the objects.

Notice that the size distribution of the intercept lengths is not used here. Instead, the fact that the mean value of theintercept length of a line probe through a sphere is just (π/4) times the sphere diameter is used (and adjusted forother shape objects). β and γ are shape factors that depend upon an assumed shape for the three-dimensional

objects. They vary only slightly over a wide range of convex shapes, as indicated in Table 2 and Figure 14. This


method represents a very low cost way to estimate the number of objects present in three dimensions based onstraightforward measurements taken on plane sections. However, it is presently giving way to more exact andunbiased techniques described under the topic of the "new" stereology.

Figure 13. The shape distribution parameter K as a function of therelative standard deviation of the mean diameter.

Table 2. Shape coefficients for particle countingShape γ βSphere 6.0 1.38Prolate Ellipsoid 2:1 7.048 1.58Oblate Ellipsoid 2:1 7.094 1.55Cube 9.0 1.84Octahedron 8.613 1.86Icosahedron 6.912 1.55Dodekahedron 6.221 1.55Tetrakaidekahedron 7.115 1.55

Figure 14. The shape coefficient β for ellipsoids and cylinders of varying axial ratio.


Lamellae

Linear probes are also useful for measurement of other types of structures. One of the advantages of linear probesis the ease with which they can be produced with uniform, random and isotropic orientations (which as notedabove is very difficult for plane probes). Drawing random lines onto a plane section can be used to measureintercept lengths, either manually or using a computer.

Figure 15. A plane section through a layered structure produces an apparent layerthickness that is generally larger than the true perpendicular dimension.

Lamellar structures occur in many situations, including eutectic or other layered structures in materials,sedimentary layers in rocks, membranes in tissue, and so on. Because these structures will not in most cases lie ina single known orientation perpendicular to the section plane, the apparent spacing of the layers will be larger thanthe true perpendicular spacing (Figure 15). There is no guarantee that on any particular section plane the truespacing will be revealed.


Figure 16. Plots of the intercept length distribution for linear intercepts through a layer.

Random (IUR) linear intercepts through a layer of true thickness t produce a distribution of intercept lengths asshown in Figure 16. If the data are replotted as a histogram of 1/λ, the inverse of the measured intercept length, a

much simpler distribution is obtained (Gundersen, 1978). The mean value of this triangular distribution is just(2/3) (1/t) so the true layer thickness can be calculated as

t = 3/2 (1/λµ) (12.8

where λµ is the mean measured intercept length. Furthermore, if the layers are not all the same thickness but have

a range of thickness values, the distribution can be unfolded using the same graphical technique as shown abovefor linear intercepts through spheres.


Chapter 13 - Anisotropy and Gradients

Much of the discussion of stereological measurement in other chapters has emphasized the importance ofisotropic, uniform and random sampling of the structure. The techniques by which isotropic sampling can beaccomplished using vertical sections and cycloids, for example, can be applied even to materials that are notthemselves isotropic. In fact, there are very few real samples for which isotropy or uniformity can be assumed.Biological specimens usually have complex directional relationships between local structure and the organism,geological strata preserve orientation information even when tilted, folded or faulted, plants know which waygravity points and where the sun shines, and materials bear the marks of their solidification and processinghistory, and even (except for a few precious shuttle-grown crystals) gravity.

One of the important ways to study such materials is to quantify the non-uniformity, preferred orientation,clustering and gradients that may be present. Non-uniformity means the variation in measures of structure such asvolume fraction, specific surface area, and particle size, shape or spacing, with position. Anisotropy refers tovariations with respect to orientation. There may also be more complex relationships in which the anisotropyvaries with position, and so forth. Usually a very specific experimental design based on considerable a prioriknowledge of the structure is needed to measure and to describe such combinations of variation.

Grain structures in rolled metals

There are relatively straightforward techniques that can be used to measure the basic parameters that describe theindividual variations. For example, a common measure of preferred orientation that has been applied to metalstructures produced by rolling (in which the metal is reduced in thickness and elongated in the rolling direction) isto examine a vertical surface, and to measure the mean intercept length as a function of orientation. Overlaying atemplate of radial lines on the structure and manual counting of intersections of the lines with grain boundariesgives the basic PL data, which can then be plotted as a rose plot to show the degree of preferred orientationgraphically. The ratio of maximum intercept length (in the rolling direction) to minimum intercept length (in thevertical direction) is simply taken from the inverse of PL.

Figure 1 shows plots of PL obtained by this method. For a set of parallel lines, the distribution of lengths issimply a cosine function (Figure 1a). For a square grid of lines, the distribution becomes more equiaxed but is stillnot isotropic (Figure 1b). For a typical real structure with preferred orientation, the distribution may be morecomplex (Figure 1c) but is always rotationally symmetric. The degree of anisotropy may be measured by thedeparture from a perfect circle.

Of course, if the grain boundary structure is well delineated, manual counting may not be required. Thresholdingthe image to obtain the grains, combining that with the line templates to produce a series of line segments, andthen measuring the length of the lines as a function of angle is one automatic method that can be used to determinethe same information. This use of templates and Boolean combination with binary (thresholded) images has beendiscussed in several other chapters as a substitute for purely manual techniques. Figure 2 illustrates the procedure:the original light micrograph (Figure 2a) of the structure (a low carbon steel whose thickness has been reduced50% by cold rolling) is processed in the computer so that the grains and grain boundaries can be thresholded(Figure 2b). The image of the grains is then ANDed with a set of radial lines to produce line segments (Figure 2c).

Chapter 13 - Anisotropy and Gradients Page # 260

Plotting the average length of the lines as a function of orientation (Figure 2d) reveals the preferred orientation.Either the ratio of maximum to minimum dimension of the plot, or the deviation from a perfect isotropic circle, canbe used to quantify the degree of anisotropy.

a b

c

Figure 1. Rose plots of PL as a function ofdirection for a) a set of horizontal parallellines; b) a grid of orthogonal lines; c) a typicalanisotropic structure.

For more complicated structures such as polymer composites containing many phases this automatic delineation isdifficult to achieve and so manual recognition and counting may be more efficient. Particularly for biologicalspecimens complex image processing is needed to isolate and define structures of interest, and so the manualmethods are often preferred.


a b

c dFigure 2. Measurement using a template of lines: a) original micrograph of 50% cold rold steel;

b) processed image thresholded to delineate grains; c) grains ANDed with a set of uniform radial lines;d) rose plot of the average intercept line length (reciprocal of PL) as a function of angle.

As indicated in Figures 3 and 4, this approach of measuring intercept length or PL as a function of direction doesnot tell the whole story. The grains are also squashed in the lateral direction by rolling and the vertical section thatlies in the rolling direction does not reveal this. It is unusual to sample the three orthogonal surfaces formeasurement, but necessary if the full information on anisotropy is desired. Measurements in three orthogonal


directions using linear probes and the simple determination of intercept length can still be used to calculate ameasure of the degree of anisotropy.

Figure 3. Micrographs taken on three orthogonal planes in a rolled steel, showing the anisotropy of the grains.

Figure 4. Longitudinal and transverse sections through muscle fiber, showing the anisotropy of the structure.

At least two planes are necessary because no single plane, and sometimes not even two planes, can distinguishbetween the possible kinds of anisotropy. Figure 5 shows schematic examples of equiaxed, needle-like and plate-like structures. Each of the structures share some identical faces. In the relatively straightforward cases of rolled ordrawn metals and fiber or laminar composites it is possible a priori to judge which directions are useful as naturalaxes to orient the data. In other applications such as biological tissue this is much more difficult and the axes maychange from place to place or even from specimen to specimen.


a b

c

Figure 5. Examples of some simple kinds ofpreferred orientation: a) equiaxed; b) needles;c) plates.

It is possible to learn much from measurements made in two directions, parallel and perpendicular to the axis ofpreferred orientation, if the preferred orientation in three dimensions is already known either from more completestudies and multiple sections, or from an understanding of the process by which the specimens are created. Evenin a single plane, human vision is quite efficient at determining a principal orientation direction so that twomeasurements (in that direction and in the perpendicular direction) can be made.

In the chapter on modeling, the Buffon needle problem was used as an example of geometric probability. Theresults of that calculation are directly applicable here. The intercept line length per unit area for uniformly randomlines is LA = π/2 PL if the measurement is made using test lines that are parallel to the orientation axis. If the testlines are oriented perpendicular to the orientation axis the value of PL will be greater, and the value of LA (totalline length per unit area of image) is obtained by subtracting PL for the parallel lines. Combining theserelationships gives


LA = PLperpendicular+ 0.571 ⋅ PLparallel

(13.1

for the total line length per unit area. So by performing two counts of PL, one in the principal direction ofelongation and the other perpendicular to it, the total line length per unit area can be determined. The degree ofpreferred orientation is then give by (Underwood, 1970)

Ω12 =PLperp

− PLpar

PLperp+ 0.571 ⋅ PLpar

(13.2

The subscript 12 indicates that we are dealing with lines (1 dimensional) on planes (2 dimensional). The value ofΩ is a dimensionless ratio, often expressed as a percentage.

To determine the preferred orientation of lines is space (e.g., dislocations in metal, fibers in a composite) a similartechnique can be applied. Two perpendicular planes are examined and the number of points where the three-dimensional lines intersect the planes are counted. Then the degree of preferred orientation is

Ω13 =PAperp

− PApar

PAperp+ PApar

(13.3

For surfaces in space, the specific area of the surface SV is estimated by counting PL on sections that are paralleland perpendicular to the orientation axis. For systems that are arrays of parallel, needle-like grains or fibers thatappear equiaxed on a plane perpendicular to the orientation axis the degree of orientation is given by

Ω23 =PLperp

− PLpar

PLperp+ 0.273 ⋅PLpar

(13.4

Arrays of plate-like grains give rise to a more complicated situation, where there are three combinations ofdirections and hence three directions in which to count PL (called perpendicular, parallel and transverse), and threeΩ terms. This is equivalent to the generalized Buffon needle problem in three dimensions. The overall degree of

anisotropy can be estimated as the square root of the sum of squares of the three terms, in cases where a singledescriptive parameter is needed.

Ω23a =PLperp

− PLtrans


+0.571 ⋅ PLtrans

Ω23b =1.571 ⋅ (PLtrans

− PLpar)


+0.571 ⋅ PLtrans

(13.5

Ω23c =PLperp

− 1.571⋅ PLpar+ 0.571 ⋅ PLtrans


+0.571 ⋅ PLtrans


Note that these terms include much less detail than the rose plot, as they compare intercept lengths in only two orthree directions (which are assumed to be the important directions for the structure). All of these coefficients canbe useful to describe the degree of preferred orientation for quantitative comparison in a given class of materials,but they do not really "describe" the nature of the orientation and they depend upon finding the correct naturalcoordinate system for the polished faces.

Boundary orientation

Computer-based image analysis has tools that make it possible to measure the orientation of boundary linesdirectly. The derivative of brightness in the horizontal and vertical directions can be calculated for every pixel inthe image by using local neighborhood operators. For example, multiplying the adjacent pixels in a 3x3neighborhood by the integers

1 0 -12 0 -21 0 -1

produces the derivative in the horizontal direction, and a similar set of numbers rotated by 90 degrees produces thevertical derivative. If these are then combined as

Pixel = 255π

⋅arctan

∂B∂y∂B∂x

(13.6

a new image is produced in which every pixel has a grey scale value that represents an angle, the direction of themaximum local gradient. Using the same local derivatives to get the magnitude of the local brightness gradient as

Pixel = ∂B∂x

2

+ ∂B∂y

2

(13.7

allows selecting just the pixels that actually correspond to edges (this is called the Sobel operator, and is discussedin the chapter on image processing). Then a histogram of the pixel brightness values provides a direct measure ofthe preferred orientation of the boundaries. Because of the limitations of performing this operation on a discretegrid of pixels in a 3x3 neighborhood in which some pixels are farther from the center than others, the results arenot perfect. Measuring the boundaries of circles as shown in Figure 6 shows the directional bias, so thatmeasurements on real structures can be normalized.

This is a relatively fast technique for measurement of real structures. Figure 7 shows the results for the sameimage used above for intercept length measurements (Figure 2a). The histogram of the pixels shows the 45 and 90degree "spikes" mentioned above, but they do not affect the ability to analyze the results. These indicate a strongdegree of preferred orientation in which most of the boundary length is horizontal and little is vertical.


a b

cFigure 6: Measurement of edge orientation: a) image of ideal circles; b) the direction results for the boundaries

(color coded); c) histogram of direction values, showing bias favoring 45 and 90 degree directions.

a bFigure 7. Boundary orientation measurement on the 50% cold rolled steel image in Figure 2a: a) color coding ofgradient direction for boundary pixels; b) histogram of values.


It is important to note that the measurement of intercept length or feature axis (discussed below) is not necessarilythe same thing as measuring boundary orientation. The graph in Figure 2d shows about a 2:1 ratio for the aspectratio of the rose plot, and this is in agreement with the fact that the steel was rolled to half of its original thickness(50% reduction). The histogram in Figure 7b shows a much greater ratio of the peaks at 0 and 180 degrees to thevalleys in between. In part this is due to the fact that the valley covers a wide range of angles (boundaries betweenthe ends of grains are not precisely vertical). It also reflects the fact that grain shapes are not regular.

Figure 8 shows two extreme examples of feature shapes for which boundary orientation produces a very differentresult than a consideration of orientation based on the longest dimension (or the moment axis introduced below).Both objects are visually oriented at about 45 degrees to the left of vertical. The feature in Figure 8a hasboundaries that are primarily oriented at 90 degrees to the main feature direction, but the one in Figure 8b hasboundaries that are primarily oriented in the same direction as the feature. Measuring feature or grain orientationwith intercept lengths or other methods that depend on the interior of the grain produces different results and has adifferent meaning than measuring the orientation of the boundaries. It is important to decide beforehand which hasmeaning in any particular experimental application.

a bFigure 8. Two features whose major axis orientation and boundary orientation have different relationships.

In principle the measurement of boundary orientation can be performed in three dimensions the same as in two. Itis not enough to simply measure the pixels in images on several sections, however. Instead, a true 3D imagingtechnique that generates arrays of cubic voxels must be used. Magnetic resonance imaging and computedtomography are capable of doing this, and so in some situations is confocal light microscopy. Then the same localneighborhood operators, extended to taking derivatives in the three X, Y, and Z directions, can be used tomeasure the orientation of the gradient in three dimensions and used to construct histograms for interpretation.

Gradients and neighbor relationships

Nonuniformity in structures (and hence in images, provided the section planes are properly oriented) can take anendless variety of forms. Human perception is very good at detecting some of them, such as alignments andorientation of features, and quite insensitive to others, such as variations in size. Figures 9 and 10 show a fewexamples of gradients. The first case (Figure 9a) illustrates a variation in area fraction, which can happen in


materials due to compositional differences arising from diffusion, heat treating, welding, etc. If the direction inwhich the gradient exists is known, it can often be measured simply by plotting the average pixel brightness.Figure 9b shows a plot averaged over a band about 40 pixels wide, which in spite of the local bumps and dipsindicates the gradual overall change in area fraction.

a bFigure 9. Concentration gradient in a tool steel due to carburization (a),

and a plot of average brightness in the gradient direction (b).

a bFigure 10. Examples of gradients: a) size; b) orientation.

Gradients such as the variation in size with position (Figure 10a) or orientation with position (Figure 10b) aremuch more difficult to quantify. Generally it requires measuring each feature to determine the appropriate size andshape parameters and the centroid location. The latter may be in some global coordinate system taken from the


microscope stage or the location of specimens in the material being sampled, depending on the scale. Then the datafile containing all of the individual measurements can be subjected to various statistical tests to obtain the desiredquantitative description. When mixed gradients (changes in several parameters of size and shape) or nonlineardirections of variation are present, the situation becomes extremely complex, and requires knowing a great deal inadvance about the specimen.

a b

c

Figure 11. Examples of clustering (a), self-avoidance(b) and a random distribution offeatures (c).

One of the most interesting location-related variables in structures is the distance between neighboring features.Figure 11 shows three example cases, corresponding to a random distribution of features, uniform spacing offeatures, and clustering of features. Many physical situations correspond to one of these. Random distributionsresult when every instance is independent of all others. A section through a random three-dimension structure willshow a random distribution of features on the section plane. Clustering results when structures are attracted to one


another by some force (gravity causes clustering of galaxies, surface tension causes clustering of dust on fluids,human nature causes people to live in cities). Spacing or self-avoidance also arises from physical conditions; forexample the precipitation of particles in metals typically produces self avoidance because forming one particledepletes the surrounding region of the element in the particles and makes it less likely that a second particle willform nearby.

To have a quantitative measure of the tendency of structures to clustering or self avoidance, Exner (Schwarz &Exner, 1983)) has proposed measuring the nearest neighbor distance. For a random structure, this produces aPoisson distribution, with a few particles having quite close neighbors and others much large spacings. In fact,the random distribution is sometimes called a Poisson random process. In a clustered structure, as shown inFigure 12, the nearest neighbor is usually much closer, and conversely in the self-avoiding structure the meannearest neighbor distance is much greater.

a b

c

Figure 12. Histograms of the measured nearestneighbor distances for the features in the images inFigure 11.

For a Poisson distribution, the mean is the only parameter needed to fully characterize the data. If the number offeatures per unit area is known, the mean value that a random distribution will produce is therefore known. Itshould be simply

Mean Distance = 0.5N A

(13.8


which has the proper units of distance. For any Poisson distribution, the standard deviation is simply the squareroot of the mean. If the mean nearest neighbor distance and standard deviation value actually measured from animage is different from the mean value, we can conclude that the specimen is not random and decide whether itshows clustering or self-avoidance. The chapter on statistical analysis presented the t-test, a simple tool that can beused to compare the values and make a determination, with explicit confidence levels. Table 1 shows the valuesfor each of the images in Figure 11.

Table 1Nearest neighbor data from Figures 11 and 12.

Image: Clustered Self-Avoiding RandomMean Nearest Neighbor

Distance (pixels) 7.41 20.81 13.56Standard Deviation 6.16 0.01 6.33

Predicted (equation 7) 12.86 11.40 13.06Standard Deviation 3.59 3.38 3.61

Nearest neighbors can also be used to identify anisotropy. Figure 13 shows an illustration of using the nearestneighbor direction to produce a rose plot of orientation. In the example, most of the nearest neighbors lie in eitherthe northeast or southwest direction, indicating a strong preferred orientation. This method has proven particularlyuseful in finding and correcting the compression of samples produced by microtoming, which can result indistortions of 10-20%. Stretching the image back until the rose plot becomes circular (provided the structure isisotropic) restores the correct dimensions.

a bFigure 13. Illustration of preferred orientation revealed by a rose plot of the number

of nearest neighbors as a function of the orientation angle at which they occur.


Distances and irregular gradients

Measuring the distance of features from each other is generally carried out after the measurements have been takenon each feature. The locations of the centroids of the features can then be compared, pairwise, to find the closestneighbor and the distance or direction used for the statistical comparisons discussed above. The coordinates offeatures are generally in some external coordinate system, based on the pixel location within the image andperhaps the stage micrometer coordinates or the location from which the sample was taken in the originalspecimen. But it is often useful to measure the distances of features from some other structure present in thespecimen. This could be the surface, but in most cases is likely to be an irregular feature such as a grain boundary,some other type of particle, etc. In biological specimens it may be a membrane or cell wall, or another organelle.In a real world it might be the distance from the nearest road.

Calculating this distance in the traditional analytical geometry way would be at least very difficult, since the targetboundary is not likely to have a simple algebraic function to describe its shape. Also, for irregular boundariesthere may be several different perpendicular distances and finding the appropriate shortest one adds morecomplexity.

Fortunately, there is a relatively straightforward and very fast way for the computer to make this measurement byprocessing the original image. It is based on the Euclidean Distance Map (EDM), which assigns a value to eachpixel in the image based on its straight line distance from a defined structure regardless of shape or irregularities.The algorithm for constructing the EDM (Danielsson, 1980) requires only two passes over the image pixels, andno algebra, taking only a fraction of a second on the computer. As shown in Figure 14 this produces an image(color coded for clarity) in which all pixels with the same value lie at the same distance from the boundary.Combining this image with an image of the features of interest assigns the distance values to the features.Measuring each feature's color value then gives the desired information.

In order to use the data in Figure 14e to detect gradients with respect to distance from the boundary, an additionalstep is needed. The shape of the interior region within the boundary is important because there is more space nearthe boundary than far away. This can also be measured with the distance map. A simple histogram of all of thepixels within the region gives the number of pixels at each distance from the boundary. This can be used tonormalize the histogram of feature distances, to reveal gradients that place many features near the boundary orperhaps show depletion there.


a b

c d

e

Figure 14. Illustration of using theEuclidean distance map to measure distancefrom an irregular boundary: a) test imageshowing boundary and features; b) theinterior of the boundary with colorsassigned according to distance (theEuclidean distance map); c) the features; d)colors from figure b assigned to the featuresfrom figure c; e) histogram of distancesobtained by tallying the colors of features infigure d.


Alignment

Human vision is particularly good at finding alignments of features along straight lines (even if they really aren'tbased on any physical reason, which is why we see constellations in the night sky). There is a computer techniquethat can be used to locate these alignments automatically, called the Hough transform (Hough, 1962). Figure 15shows the basic steps involved. Each point in the original image is mapped into Hough space as a sinusoid. The(ρ,θ) coordinates of this space are the polar coordinates of a line in the original spatial domain. Each point on the

sinusoid represents one of the possible lines that could be drawn through the corresponding point in the originalspatial image. Since all of the sinusoids in Hough space are added together, a maximum value results where manyof them cross over. In Figure 15b, color coding has been used to show this peak. The peak corresponds to thecoordinates of the line that passes through the points in the original image, and shown in Figure 15c. This methodcan also be applied to fitting circles and other shapes (Duda & Hart, 1972), but the details go beyond the intendedscope of this text.

a b

c

Figure 15. Illustration of the linear Houghtransform. Each point in the original spatialdomain image (a) generates a sinusoid inHough (ρ,θ) space (b) corresponding toallpossible lines through the point. Thecrossover point in this space identifies a singlestraight line in the spatial domain image (c) thatpasses through the points.


Chapter 14 - Finite Section Thickness

Projected images

The classic applications of stereology and the principal rules which have been derived and applied are specific tothe case of planar sections through three-dimensional solids. This is more-or-less the situation in metallographicmicroscopy, where polished sections cut through the material are examined. It is not so appropriate for many otherfields. Light microscopy in biological and medical applications usually looks through a slice of material, forinstance cut with a microtome, whose thickness is not negligible. Microscopy of particulates also looks through athick layer, in which the matrix is transparent air. Confocal light microscopy images an optical section from withinmaterial, whose thickness is finite and depends on the numerical aperture of the optics, and whose boundaries arenot sharp. Even electron microscopy, which uses much thinner sections than light microscopy, deals with finitesections (the lateral resolution of the images is improved along with the thinner sections so that features muchsmaller than the section thickness are detected). And in scanning electron microscopy although surfaces areimaged to negligible depth (at least with secondary electrons, if not backscattered electrons or X-rays), thesurfaces are rarely flat.

The introduction of finite sections and viewing of images projected through them raises some significant problemsfor stereology. In cases where the section thickness is much less than the dimensions of any features or details ofinterest it may be possible to ignore the differences and treat the images as though they were ideal sections.Indeed, this is the assumption that is commonly made in applying stereological rules to microscope images,although its appropriateness is rarely evaluated.

Viewing images that are projections through a very thick section makes some kinds of measurements quite simple.The outer dimensions of features are seen directly, so that no unfolding is required to ascertain the actual particlesize distribution based on the sizes of sections. There are still some important underlying assumptions, of course.Most basic is that the dispersion is quite sparse so that particles are visible and do not obscure one another. Thespecific case of particles deposited onto a substrate meets this test, since the particles are generally present as asingle layer. The original "thickness" of that layer corresponding to a volume of material from which the particleshave settled onto the substrate is not known, but it is usually possible to control the deposition conditions toprevent the deposit from becoming so dense that particles touch or overlap one another.

A second assumption is that the particles have uniformly random orientations with respect to the viewingdirection. This may or may not be true when viewing through an actual matrix, but is always suspect whenparticles are deposited on a substrate since physical forces can act to reorient the particles when they contact thesurface. These forces range from gravity (for large particles) to electrostatic or magnetic attraction, surface tensionin the fluid as it is removed, and others. The general result is that particles may align with their long axis parallel tothe surface or to each other, and may tend to clump together. This presents an image that is not random and makesthe determination of three-dimensional particle sizes suspect. Special preparation techniques can sometimes beemployed to produce well dispersed samples with unmodified particle orientations. One example is the use of acamphor-napthalene eutectic wax for dispersing cement powders (Thaulow & White, 1971). The wax supportsthe particles and is spread onto a slide as a substrate. It is then evaporated away allowing the particles to settle onto

Chapter 14 - Finite Section Thickness Page # 276

the slide. Obviously this is a specialized method, but similar approaches may be practical for a variety ofapplications.

When particulates are viewed in projection, the external diameter or projected area gives a direct measure of size.This of course is only true for convex features. Any interior details such as surface pits or voids are hidden, andimportant topological characteristics may be hidden as well. Consider for example a distribution of toroids -Cheerios® for instance. As shown in Figure 1, some of the projected images will show the presence of the centralhole but most will not. If it is known a priori that all of the features are the same, the multiple projections make itpossible to determine the three-dimensional shape. In fact this technique has been used to reconstruct the shape ofvirus particles from transmission electron microscope pictures, by combining multiple projections of presumablyidentical particles using tomographic reconstruction. But when a mixture of shapes is present, the assessment ofthe actual shapes and relative populations is difficult or impossible.

Figure 1. Illustration of a projected view of toroids: a) surface image which reveals the shapes;b) projected image in which most images do not indicate the three dimensional shape.

Even if the assumption of random orientation can be met, many projected images do present the viewer with someoverlaps. Figure 2 illustrates this problem. Small particles are more likely to hide behind larger ones, andoverlapping projections of particles reduce the number and alter the size of the features in the projected image. Inthe example, the spatial distribution is not isotropically random. In one direction (Figure 2b) the particles arenearly all separately visible, while in another (Figure 2c) they are heavily overlapped. In most cases, of course, itis not possible to select a particular viewing direction or to compare the results in different directions. Figure 3shows a real example, a section viewed in the TEM containing latex spheres. The apparent clusters of particles arein fact simply superpositions of images along the particular projection direction and the particles are actuallyseparated in three dimensions.

As noted above it is more likely for small particles to be hidden by large ones than the converse. A statisticalcorrection for the systematic undercounting of small features is possible. The size distribution of the visiblefeatures is constructed. Normally such a distribution would report the number per unit volume, where the volumeis the area of the viewed image times the thickness of the section. However, for small features the actual volumebeing viewed is less. The projections of large features block the view of a portion of the area, and this must be


subtracted from the volume measured when counting the smaller features that could hide within that shadow asshown in Figure 4. This correction can be made for each bin in the distribution, increasing the number per unitvolume of the smaller particles.

a

b c Figure 2. Illustration of particles in a thick section, which overlap in a projected image:

a) perspective view; b) projection along the Z axis; c) projection along the Y axis.

There are actually two variants of this correction. When the projected image is a shadow or binary image, thevolume covered by the large feature extends completely through the section thickness so that the sampled volumeis reduced by the projected area times the thickness. When the image shows the object surfaces (Figure 5), smallfeatures lying in front of larger ones can be seen, but particles lying behind larger ones cannot. The hidden volumeis reduced by one half since on the average the large particles lie in the middle of the section.


Figure 3. TEM image of latex spheres, which appear to form clustersbut are actually separate in three dimensions.

Figure 4. Schematic showing the hidden shadow region above and below a large particle. As discussed inthe text, the area of the feature may be reduced if partially protruding objects can be distinguished.

In either case, it is clear that particles are hidden if they are entirely within the shadow area. But if their centers liewithin a band equal in thickness to the radius, they will partially protrude and may be detected depending on the


criteria for visibility. Often, particles whose centers lie outside the shadow but intersect the projection of the largeparticle can be seen and measured.

a b Figure 5. For a surface image, small particles above a larger one are visible and can be measured while

those below are completely or partially hidden: a) perspective view; b) vertical image.

Consequently, it is an application-specific decision whether the area of the large feature is the proper value to usefor the correction, or whether a region smaller or larger by the radius of the smaller feature for which thecorrection is to be made, and whether the hidden volume is equal to this area times the thickness or half thethickness of the section.

Bias in stereological measurements

Finite section thickness introduces two opposite sources of bias into conventional stereological measurements.These have been recognized for a long time (Holmes, 1927), and various correction procedures have been derived(Cruz-Orive, 1983). For the most part, however, these are not used widely. This is because they require knowingthe section thickness, making some assumptions about feature shape and uniformity (also isotropy of orientation),and/or performing more measurement work to obtain results. Determining the thickness of a microtomed sectionaccurately is difficult, and for many materials specimens techniques used to prepare TEM sections such aselectrochemical thinning or ion erosion produce sections whose thickness varies dramatically and imaging isactually performed on wedge-shaped regions around the edges of holes.


Figure 6. Diagram of the cross section of a thick section in which the opaque phaseappears to cover more area because of overprojection.

Figure 7. Diagram of the cross section of a surface with polishing relief, in which the harderphase projects from the surface and appears to cover more area because of overprojection.

For the case in Figure 6, we have a finite section with one transparent and one opaque phase. Note that this isexactly equivalent to the case of a polished metallographic section with finite polishing relief due to one phasebeing harder than the other (Figure 7). In each case, the opaque or harder phase appears to cover a greater fractionof the viewed area than it should, because of overprojection. The apparent area fraction ′ A A is greater than the

actual volume fraction VV by an amount that depends on the thickness of the section t and the amount of surface

area per unit volume SV . (Cahn & Nutting, 1959)

′ A A = VV + 14 SV ⋅ t (14.1

Detemining SV also requires a correction for the section thickness, which depends on the number of features

present and their mean curvature. In terms of practical measurement quantities ′ A A (the apparent area fraction,

which is typically measured by counting a point fraction ′ P P ), ′ B A (the phase boundary length per unit area,

which is typically measured by counting linear intercept points ′ P L ) and the number of features per unit area ′ N A,

the true stereological structure parameters are


VV = ′ P P − t ⋅ 12 ′ P L − ′ N A ⋅ t ⋅ H

t + H

(14.2

SV = 2 ′ P L − 4 ′ N A ⋅ t ⋅ H t + H

(14.3

The parameter t is the section thickness and H is the average mean caliper diameter of the particles or phase

regions in the section. It is usually assumed that H is much less than t , corresponding to the case in which

particles are small compared to the section thickness, but this is not essential. Of course, if H is not small

compared to t it is not likely that H will be seen or measurable in the section images and will have to bedetermined separately from other experiments. The model also assumes that the phase regions or particles areconvex and do not overlap each other (the section thickness is very small, or the dispersion is very sparse and VVis small).

If measurements on two sections of thickness t1 and t2 are performed, a further simplification is possible and we

obtain

VV =t1 ⋅ ′ A A1

− t2 ⋅ ′ A A2

t1 − t2(14.4

SV = 4π ⋅

t2 ⋅ ′ B A1− t1 ⋅ ′ B A2

t1 − t2(14.5

In actual practice, determining the thickness values with sufficient precision to divide by the difference (t1 − t2 ) is

extremely difficult to accomplish.

R. E. Miles (1976) derived a complete set of stereological equations for volume density, surface density, andcurvature based on measurements on the projected image of a section of thickness t which are more general thanthe preceding equations. They apply to the case of an aggregate of opaque, isotropic, random convex particles in atransparent matrix. The required measurements can be obtained using a point count using a grid, and intersectioncount using a grid, and a tangent count using a sweeping test line. Data must be taken on paired sections and theestimated values depend on the differences between the pairs of measurements (requiring a large number of countsfor adequate precision) and the accuracy of the thickness measurements.

The above discussion deals only with the problem of overprojection, features appearing too large in the projectedimage. It is restricted to convex phase regions and cannot deal with particles hiding behind others. An additionaleffect may also be present, usually referred to as the "end cap" problem. Consider the illustration in Figure 8. Thespherical particles are of three types: ones that lie entirely within the section thickness, ones that project out (andare truncated) but whose centers lie inside the section thickness, and ones whose centers lie outside the section.The latter group have only their polar caps in the section. In many cases, the contrast with which the opaque phaseis visible in the projected image is such that these thin end caps may not be seen, or they may literally fall out ofthe section.


Figure 8. Diagram of a cross section of a thick section containing spherical particles that lie entirelywithin the section, ones that protrude out but have their centers inside, and ones with their

centers outside the section but protrude into it producing end caps.

In terms of the number of features seen per unit area of image, the contribution of the first two types of particles is

a function of the section thickness t while that of the third group is a function of the mean particle diameter D ,where NV is the true number density per unit volume.

′ N A = NV ⋅ t + NV ⋅ D (14.6

Making corrections for the presence of end caps or for the fact that some may be missing requires making manyassumptions. Weibel (1980) derives a family of curves to correct VV and SV measurements for invisible end

caps as a function of the diameter at which they disappear. In most cases, it is more useful to consider thethickness at which they disappear since the visibility of the opaque phase is usually related to this dimension. Forthe assumption of spherical particles, the two dimensions are of course related.

The same method can be extended to the sphere unfolding method discussed in Chapter 12. This is typically doneby assuming either that the caps disappear at a constant latitude angle in all size classes, or that they disappear at afixed thickness due to insufficient image contrast. Of course, the assumption of spherical particle shape is alreadya basic part of the unfolding method. If the shape of the distribution can also be assumed (e.g., normal or log-normal), then an iterative calculation of the size distribution of the circles produced by sectioning spheres can beused to determine the mean and variance of the sphere size distibution (Nauman & Cavanaugh, 1998).

Measurements within sections

With all of the difficulties and assumptions involved in applying classical stereology to thick sections, it issurprising to find that some very straightforward measurements are possible. One is the length of linear structures,regardless of their complexity, connectedness or branching. These may be separate discrete lines such as fibers ordislocations, or branching structures such as roots of plants (Figure 9) or neurons in stained tissue. Linearstructures are also formed by the intersection of planar ones, such as the triple line that forms a network in grain


structures where the surfaces of three grains meet. This network is important as a path for diffusion and otherprocesses, but is not often visualized as a three-dimensional linear structure.

Figure 9. A projected image of the roots of a plant. The branching structure can be consideredto occupy a volume in which the transparent matrix is air.

The projected image of a linear structure is of course a set of lines, and we will make the assumption that they areall visible and the magnification and section thickness are appropriate to allow the lines to be resolved and viewed.But of course since the lines are embedded in a three-dimensional volume and can run in any direction, the lengthof the lines may be much greater than the length of the projections. Without making any assumption about theorientation of the lines, it is possible to measure the actual value with little effort.

One of the rules of classical stereology is that a plane probe can be used to intersect lines, and that the expectedvalue of the length of the lines per unit volume LV is equal to 2 • NA where NA is the number of intersections thatthe lines make with the plane. These intersections appear as points, of course, and sometimes they can be useddirectly for classical measurement. For instance, etching a piece of semiconductor to produce pits wheredislocations intersect a polished surface makes it possible to count them. The factor two arises because anisotropic distribution of lines would intersect the surface at many different angles. So the classical measurement ofLV from the value of NA on a section surface is dependent on the assumption of isotropy.

This highlights the usual difficulty with plane probes - it is very difficult to make them isotropic in three-dimensional space, if the specimen itself is not isotropic. Fortunately with thick sections there is a way toaccomplish this. As shown in Figure 10, a line drawn on the projected image represents a surface extendingthrough the thick section. If the line is straight the plane is flat; if the line is curved the surface is a cylindrical one.The area of the surface is the product of the length of the line times the section thickness. Counting points wherethe linear structure of interest crosses the surface can be accomplished by counting the intersections of theprojection of the linear structure with the projection of the surface, which is the drawn line.


Figure 10. The relationship between a line drawn on a projected image and the surfacethat line represents in the original thick section.

Drawing a uniform set of straight lines on the projected image does not produce a set of random planes in threedimensions, so the problem of anisotropic sampling is still present. All such surfaces share a common directionthrough the thickness of the section, and so are far from isotropic. However, the vertical section method discussedin other chapters can be used here as well. Select one direction in the plane of the section as the nominal "vertical"direction, cut several sections that include that direction (experiments show that as few as three sections cut 60degrees apart provide a reasonably sample set), and then draw cycloids on projected images of each section asshown in Figure 11.

Note that the vertical direction is shown on the image and the orientation of the cycloids with respect to thatdirection is not the same as when cycloids are drawn on a planar vertical section. the cycloidal surfaces have moreof their area parallel to the vertical direction than perpendicular to it, since every thick section cut to include thesame vertical section will have the same planes perpendicular to that direction, which would be oversampled byplanes corresponding to isotropic lines drawn on the projection.

Counting the intersections of the linear structure with the cycloids allows an easy and straightforwardmeasurement of the total length of the structure. Sometimes this is performed manually and sometimes by usingimage processing to combine the image of the grid with the linear structure (a Boolean AND) to mark thecrossings for automatic counting. Both methods have been discussed in other chapters.


Figure 11. Transmission electron microscope image of nerve axons with a superimposed grid of cycloids.

In some cases it is not even necessary to cut multiple thick sections. In the case of the root in Figure 9, the entirevolume of the root can be treated as a (very) thick section with a transparent (air) matrix. Rotating the entirestructure to three orientations about 60 degrees apart provides the isotropic sampling around some selected verticaldirection; in this case the plant stem would provide a logical reference. Superimposing a cycloid grid on each ofthe three projected images and counting the intersections can be used to measure the total length of the structure.

As a practical demonstration of this method, it is convenient to take a wire or other structure that can later bemeasured and bend it into a complex three-dimensional shape, place it on a grid (using a transparency andoverhead projector is a good classroom demonstration method), and count the intersections in each of threerotational orientations. As shown in the example of Figure 12, this produces a number of counts that can becombined with the known length of the lines and the area of the grid.

If the volume occupied by the wire is assumed to be a cube whose face is the area of the grid, then the keyrelationships can be written in terms of the length per unit volume of the linear structure LV, the length of the gridline G, the number of counts N (averaged over the three orientations of the specimen), and t the size of the grid(and the assumed section thickness). Equation (14.7) is simply the classical relationship between LV and thenumber of intersections, and equation (14.8) gives the actual length L in terms of the number of counts and thelength of grid line per unit area (a known constant for any test grid).

LV = 2 ⋅ NG ⋅ t

(14.7

L = 2 ⋅ NGA

(14.8


Figure 12. Projected image of a wire tangle superimposed on a cycloid grid, with intersections marked.

In the example of Figure 12, the number of intersections is 23. The total length of grid line is 68.054 cm in an areaof 7.5 x 7.5 cm, so GA = 1.209 cm-1. That produces an estimate of the wire length of 19 cm. The uncertainty in

this estimate comes from the number of counts ( 2323 is about 20%, or 19±4 cm (the actual length of the wire

was 20 cm.). Repeating the measurement with other placements of the grid and other orientations around the samevertical direction would produce more counts and thus improve the statistical precision of the measurement.


Chapter 15 - Three-dimensional imaging

Limitations of stereology

Stereological analysis provides quantitative measures of several kinds of structural information. Whetherdetermined manually or by computer measurement, most of these require only simple counting or measurementoperations and give unbiased parameters (provided that IUR conditions on the point, line and plane probes aremet). Measures of phase volume and surface area are reasonably straightforward. Distributions of feature size andshape, and alignments and neighbor distances, can be determined in many cases. However, the data that areprovided by stereological measurement are sometimes difficult to interpret in terms of the appearance of amicrostructure as it is understood by a microscopist.

While the global metric parameters of a microstructure can be measured with the proper stereological probes andtechniques, and with more difficulty the feature-specific ones can be estimated, topological ones are more difficultto access. These include several fundamental properties that are often of great interest to microscopist, such as thenumber of objects per unit volume. Counting on a single plane of intersection can estimate this for the case ofconvex features of a known size, or with some difficulty for the case of a distribution of such features of a knownshape. Counting with two or more planes (the disector, Sterio, 1984)) can determine the number per unit volumeof convex features of arbitrary size and shape.

But the general case of objects of arbitrary shape, and networks of complex connectivity, cannot be fullydescribed by data obtained by stereological measurement. This is essentially a topological problem, and is relatedto the desire to "see" the actual structure rather than some representative numbers. Humans are much more awareof topological differences in structure than in metric ones.

Serial methods for acquiring 3D image data

The method longest used for acquiring 3D image data is serial sectioning for either the light or electronmicroscope. This is a difficult technique, because the sectioning process often introduces distortion (e.g.,compression in one direction). Also, the individual slices are imaged separately, and the images must be aligned tobuild a 3D representation of the structure (Johnson & Capowski, 1985). This is an extremely critical and difficultstep. The slices are usually much farther apart in the Z direction than the image resolution in the X, Y plane ofeach image (and indeed may be nonuniformly spaced and may not even be parallel, complications that fewreconstruction programs can accommodate).

This alignment problem is best solved by introducing some fiduciary marks into the sample before it is sectioned,for instance by embedding fibers or drilling holes. These provide an unequivocal guide to the alignment, whichattempting to match features from one section to another does not (and can lead to substantial errors). Forsequential polishing methods used for opaque specimens such as metallographic samples, hardness indentationsprovide a similar facility. They also allow a direct measure of the spacing between images, by measuring thechange in size of the indentation before and after polishing.

Chapter 15 - Three-Dimensional Imaging Page # 288

For objects whose matrix is transparent to light, optical sectioning allows imaging planes within the specimennondestructively. Of course, this solves the section alignment problem since the images are automatically inregistration. It also allows collecting essentially continuous information so that there is no gap between successiveimage planes. The conventional light microscope suffers significant loss of contrast and resolution due to the lightfrom a plane deep in the sample passing through the portions of the sample above it. This applies to the case ofreflected light viewing or fluorescence imaging. For transmission imaging the entire thickness of the samplecontributes to the blurring.

Deconvolution of the blurring is possible using an iterative computational procedure that calculates the pointspread function introduced by each layer of the specimen on the affected layer. As shown in Figure 1 this allowssharpening image detail and recovering image contrast. However, it does little to improve the depth of field of theoptics and the resolution in the Z direction remains relatively poor.

Figure 1. Immunofluorescence cell image (a) showing green microtubles, red mitochondrial protein, andblue nucleus. Deconvolution of the point spread due to specimen thickness (b) performed using software

by Vaytek (Fairfield, Iowa), who provided the image.

The confocal microscope solves both problems at once. By rejecting light scattered from points other than thefocal position, contrast, resolution and depth of field are all optimized. Scanning the point across the specimenoptically in the X, Y, plane and shifting the specimen vertically in the Z direction can build up a 3D data set inwhich the depth resolution is 2-3 times poorer than the lateral resolution, but still good enough to delineate manystructures of interest. There are still some difficulties with computer processing and measurement when the voxels(the volume elements which are the 3D analogs to pixels in a plane image) are not cubic, and these are discussedbelow. Confocal microscopy has primarily been used with reflected light and fluorescence microscopy, but inprinciple can also be extended to transmission imaging.

Inversion to obtain 3D data

In confocal microscopy, or even in conventional optical sectioning, light is focused onto one plane within thesample. The light ideally enters through a rather large cone (high numerical aperture objective lens), so that thelight that records each point of the image has passed along many different paths through the portion of thespecimen matrix that is not at the focal point. This should cause the variations in the remainder of the sample toaverage out, and only the information from the point of focus to remain.


Figure 2. The principle of inverse reconstruction. Information from many different views through acomplex structure is used to determine the location of objects within the volume.

This idea of looking at a particular point from many directions lies at the heart of inverse methods used toreconstruct images of microstructure (Figure 2). They are often called tomographic or computed tomographic (CT)methods, and the most familiar of them is the X-ray CAT scan (computed axial tomography, because the mostcommon geometry is a series of views taken from different radial positions around a central axis) used in medicaldiagnosis (Kak & Slaney, 1988). This method actually reconstructs an image of one plane through the subject,and many successive but independent planes are images to produce an actual 3D volume of data. Other projectiongeometries are more commonly used in industrial tomography and in various kinds of microscopy, some of whichdirectly reconstruct the 3D array of voxels (Barnes et al., 1990).

There are two principal methods for performing the reconstruction: filtered back projection (Herman, 1980) inwhich the information from each viewing direction is projected back through the voxel array that represents thespecimen volume, with the summation of multiple views producing the final image, and algebraic methods(Gordon, 1974) that solve a family of simultaneous linear equations that sum the contribution of each voxel toeach of the projections. The first method is fast and particularly suitable for medical imaging, because speed isimportant and the variation in density of geometry of the sample is quite limited, and because image contrast isimportant to reveal local variations for visual examination, but measurement of dimension and density are not


generally required. The second method is more flexible to deal with the unusual and asymmetrical geometriesoften encountered in microscopy applications, and the desire for more quantitatively accurate images from arelatively small number of projections.

Tomographic reconstruction requires some form of penetrating radiation that can travel through the specimenvolume, being absorbed or scattered along the way as it interacts with the structure. There are many situations inwhich this is possible. X-rays are used not only in the medical CAT scan, but also for microscopy. A point sourceof X-rays passing through a small specimen produces a magnified conical projection of the microstructure.Rotating the specimen (preferably about several axes) to produce a small number of views (e.g., 12-20) allowsreconstruction of the microstructure with detail based on variation of the absorption cross section for the X-rays,which is usually a measure of local specimen density. However, using a tunable X-ray source such as asynchrotron, it is possible to measure the distribution of a specific element inside solid samples (Ham, 1993) witha resolution on the order of 1 µm.

Electron tomography (Frank, 1992) in the transmission electron microscope typically collects images as thespecimen is tilted and rotated. Constraints upon the possible range of tilts (due to the stage mechanism andthickness of the sample) cause the reconstruction to have poorer resolution in the Z direction than in X and Y, andit is important to avoid orientations that cause electron diffraction to occur as this is not dealt with in thereconstruction.

Light is used tomographically in some cases, such as underwater imaging methods developed for antisubmarinedetection. It is also used to measure the variation in index of refraction with radius in fibers. Sound waves areused for medical imaging as well as seismic imaging and some acoustic microscopy. In the latter case, frequenciesin the hundreds or thousands of megahertz provide resolution of a few µm. Seismic tomography is complicatedgreatly by the fact that the paths of the sound waves through the specimen are not straight lines, but in fact dependupon the structure so that both the attenuation and path must be part of the reconstruction calculation. However,using pressure and shear waves generated by many earthquakes and detected by a worldwide network ofseismographs, detailed maps of rock density and fault lines within the earth have been computed.

Magnetic resonance imaging (MRI) used in medical diagnosis also has curved paths for the radio waves thatinteract with the hydrogen atom spins in a magnetic field. The use of very high field gradients has allowedexperimental spatial resolutions of a few µm for this technique. Many other signals have been used, includingneutrons (to study the structure of composite materials) and gamma rays (photons with the same energies as X-rays but from isotope sources, which are suitable for use in some industrial applications and for down-boreholeimaging in geological exploration).

In all these cases, the reconstruction methods are broadly similar but the details of the microstructure that interactwith the signal and thus are imaged vary greatly. Some techniques respond to changes in density or composition,others to structural defects, others to physical properties such modulus of elasticity or index of refraction. In theideal case, the image that is reconstructed is a cubic array of voxels. Depending on the application, the size of thevoxels can range from nanometers (electron microscopy) to kilometers (seismic imaging). A significant amount ofcomputing is generally required, and the number of voxels that can be reconstructed is quite small and givesresolution that is poor compared to the number of pixels in a two-dimensional image. Processing is generally


needed to minimize noise and artefacts. This means that tomography is not a "live" viewing technique, and that theresolution of the images is limited

Stereoscopy as a 3D technique

Tomography usually employs a fairly large number of projections taken from as widely distributed a set ofviewpoints as possible. Medical imaging of a single slice through the body may use more than 100 views alongradial directions spaced on a few degrees apart, while some conical projection methods for 3D reconstruction useas few as a dozen projections in directions spread over the full solid angle of a sphere. By contrast, stereoscopyuses only two views that are only slightly different, corresponding to the two points of view of human vision. Theadvantage is that humans possess the 3D reconstruction software to merge these two views into a 3D structuralrepresentation (Marr & Poggio, 1976).

It is important not to confuse stereoscopy with stereology. In both, the root stereo is from the Greek and refers tothree-dimensional structure. Stereology is defined as the study of that structure based on geometric principles andusing two dimensional images. Stereoscopy is the recording or viewing of the structure, and it generally taken tomean the two-eye viewing of structures in a way that reveals them to the brain of the observer. This does notexclude the possibility of making measurements based on those two images (Boyde, 1973).

Figure 3. The principle of stereoscopic measurement. Measurement of the parallax or apparentdisplacement of features in two views separated by a known angle (Θ) allows calculation

of the vertical displacement of the features.

As shown in Figure 3, the parallax between two points viewed in left- and right-eye images gives a direct measureof their vertical separation, provided that the scale of the pictures and the angle between the two views is known.Because the angle is typically small (5-15 degrees) so that the views can be successfully fused by the human


visual system, small uncertainties in the angle strongly affect the precision of the Z calculation. Finite precision inthe lateral measurements that give the parallax measurement, which ultimately depend on the lateral resolution orpixel size in the images, produce uncertainties in the Z measurement about an order of magnitude worse than theX, Y measurements.

Figure 4. SEM image of a three-dimensional clump of yeast cells. Stereoscopy can measure the outerdimensions of the clump but cannot reveal internal details such as the individual cells.

Stereoscopy (and for that matter stereology) do not offer any way to determine the number of features in a clumpas shown in Figure 4. This SEM image of yeast is certainly not densely packed so that the number might beestimated from the outer dimensions, nor are all of the particles the same size nor are those visible on the outsidenecessarily representative of the interior ones. Stereoscopy can provide the height of the clump from two images,but the invisible interior is not accessible.

Viewing stereo images does not require making measurements, of course. It is the relative position of objectswhich humans judge based on whether the parallax increases or decreases, and this is determined by the vergence(motion of the eyes in their sockets) needed to bring points of interest to the fovea in the eyes for detailed viewing.This means that relative position is judged one feature at a time, rather than a distance map being computed for theentire image at once.

When stereo pair images are used for applications such as aerial mapping of ground topography or SEM mappingof relief on integrated circuits, the computer works entirely differently than human vision (Medioni & Nevatia,1985). It attempts to match points between the two images, typically using a cross correlation technique that looksfor similarities in the local variation of grey (or color) values from pixel to pixel. Some systems rely on humanmatching of points in the two images; this can be quite time consuming when thousands of points need to beidentified and marked. Matched points then have their parallax or horizontal offset values converted to elevation,and a range image is produced in which the elevation of each point on the surface is represented by a grey scalevalue.


These techniques are suitable for surface modeling and in fact are widely used for generating topographic maps ofthe earth's surface, but they are much more difficult to apply to volumetric measurement. Matching of pointsbetween the two views is complicated when the left-to-right order in which they appear can change, and thecontrast may vary or points may even disappear because of the presence of other objects in the line of sight.Stereoscopic images of volumes can be used for discrete measurements of z spacings between objects, butproducing a complete voxel array with the objects properly arranged is not possible with only two views - itrequires the full tomographic approach.

Figure 5. Stereoscopic display of a processed voxel array. The left and right images can be viewed to seethe internal structure of a human brain as imaged with MRI, and processed with a 3D gradient operator to

reveal regions where there is a large change in concentration of water.

However, stereoscopic images are very useful and widely employed to communicate voxel-based data back to thehuman user. From a volumetric data set, two stereoscopic projections can be generated quickly and displayed forthe user to view. This is sometimes done by putting the images side by side (Figure 5) and allowing the user torelax his or her eyes to fuse them, or by putting the images into the red and green color planes of a display andusing colored glasses to deliver each image to one eye.

The example display in Figure 5 is a processed image of the human brain. The original data were obtained bymagnetic resonance imaging, which shows the concentration of protons (and hence of water molecules) in tissue.The images were assembled into a voxel array, and processed using a gradient operator as discussed below. Thetwo projections 9 degrees apart were generated for viewing. Other schemes in which LCD lenses on glassesrapidly turn opaque and transparent and the computer display alternately flashes up the left and right images, orlens and mirror systems oscillate to assist the viewer, are also used.

The majority of people can see and correctly interpret stereo images using one or more of these methods, but asignificant minority cannot, for a variety of reasons. Many of the people who cannot see stereo images get thesame depth information by sequential viewing of images from a moving point of view, for example by movingtheir head sideways and judging the relative displacement of objects in the field of view to estimate their relative


distance. The relationship is the same as for the parallax. Displays that rotate the voxel array as a function of timeand show a projection draw on this ability of the human visual system to judge distance. Almost everyonecorrectly interprets such rotating displays.

Visualization

When a three-dimensional data set has been obtained, whether by serial sectioning, tomographic reconstruction orsome other method, the first desire is usually to view the voxel array so as to visualize the volume and thearrangements of objects within it. The example in Figure 5 shows one mode than is often used, namely stereopresentation of two projections. This relies on the ability of the human to fuse those images and understand thespatial relationships which they encode. As mentioned above, the second common technique is to use motion toconvey the same kind of parallax or offset information. This may either be done by providing a continuous displayof rotation of the volume to be studied, or an interactive capability to "turn it over" on the screen and view it fromany angle (usually with a mouse or trackball, or whatever user interface the computer provides).

Producing the projected display through the voxel array can be done in several different ways, to show all of thevoxels present, or just the surfaces (defined as the location where voxel value - density for example - changesabruptly), or just a selected surface cut through the volume. We will examine these in detail below. But in allcases, the amount of calculation is significant and takes some time. So does addressing the voxels that contributeto each point on the displayed image. If the array of data is large, it is likely to be stored on disk rather than inmemory, which adds substantially to the time to create a display. For these reasons, many programs are used tocreate a sequence of images that are then displayed one after the other to create the illusion of motion. This can bevery effective, but of course since it must be calculated beforehand it is not so useful for exploring an unknownsample, but rather is primarily used to communicate results that the primary scientist has already discovered tosomeone else.

Computer graphics allow three principal modes of display (Kriete, 1992): transparent or volumetric, surfacerenderings, and sections. The volumetric display is the only one that actually shows all of the information present.For any particular viewing orientation of the voxel array, a series of lines through the array and perpendicular tothe projected plane image are used to ray-trace the image. This is most often done by placing an extended lightsource behind the sample and summing the absorption of light by the contents of each voxel. This makes the tacitassumption that the voxels values represent density, which for many imaging modalities is not actually the case,but at least it provides a way to visualize the structure. It is also possible to do a ray tracing with other lightsources, to use colored light and selective color absorption for different structures.

The chief difficulty with volumetric displays is that they are basically foreign to human observers. There are fewnatural situations in which this kind of image arises - looking at bits of fruit (which must themselves betransparent) in Jell-O can be imagined as an example. In most cases, we cannot see through objects; instead wesee their surfaces. Even if the matrix is transparent or partially visible, the objects we are interested in are usuallyopaque (consider fish swimming in a pond, for example). For this reason, most volumetric display programs alsoinclude the capability to add incident light to reflect from surfaces, and to vary the color or density assigned tovoxels in a nonlinear way so that surfaces become visible in the generated image.


Figure 6. Several views of magnetic resonance images of a hog heart from a movie sequence of itsrotation. The data set was provided by B. Knosp, R. Frank, M. Marcus, R. Weiss, Univ. of Iowa Image

Analysis Facility, and the images were generated using software from Vital Images (Fairfield, Iowa).

Figure 6 shows an example. The data are magnetic resonance images of a hog heart, and the color and density ofthe heart muscle and the vasculature (which can be distinguished by their different water content) have beenadjusted to make the muscle partially transparent and the blood vessels opaque. Arbitrary colors have beenassigned to the different voxel values to correspond to the different structures. The sequence of images fromwhich the selected frames have been taken shows the heart rotating so that all sides are visible. Note however thatno single frame shows all information about the exterior or interior of the object. The observer must build up amental image of the overall structure from the sequence of viewed images, which happens to be something thatpeople are good at. Movies in which the point of view changes, or the opacity of the various voxel values isaltered, or the light source is moved, or a section plane through the data is shifted, all offer effective tools forcommunicating results. They do not happen to fit into print media, but video and computer projectors arebecoming common at scientific meetings and network distribution and publishing will play an important role intheir dissemination.

Figure 7. The same data set as Figure 6 with the opacity of the voxels with valuescorresponding to muscle tissue increased. This shows the exterior form of the heart.

Changing the nonlinear relationship between the values of voxels in the array and their color and opacity used ingenerating the image allows the transparency of the different structures to be varied so that different portions of thestructure can be examined. Figure 7 shows the same data set as Figure 6 but with the opacity of the heart muscletissue increased so that the interior details are hidden but the external form can be displayed. Again, because of the


number of variables at hand and the time needed to produce the images, this is used primarily to produce finalimages for publishing or communicating results rather than exploring unknown data sets.

The surface rendering produced by changing voxel contributions in volumetric display programs is not as realisticas can be produced by programs that are dedicated to surface displays. By constructing an array of facets (usuallytriangles) between points on the surface defined by the outlines of features in successive planes in the 3D array,and calculating the scattering of light from an assumed source location from the facet as a function of itsorientation, visually compelling displays of surfaces can be constructed.

Figure 8. Views of serial section slices through the cochlea from the inner ear of a bat. The surfacevoxels are shaded to indicate their orientation based on the offset from the voxels above.

The data set was provided by A. Keating, Duke Univ., and the reconstructionperformed using Slicer (Fortner Research., Sterling, VA).

Figure 8 shows data from more than 100 serial section slices through the cochlea from the inner ear of a bat. Theprogram used to generate the surface display simply shades each individual surface voxel according to the offsetfrom the voxels above, so that the surface detail is minimally indicated but not truly rendered. This program alsoallows rotating the object, but only in 90 degree steps, and does not allow varying the location of the light source.The same data set is shown in Figure 9 using a more flexible reconstruction program that draws outlines for eachplane, fills in triangles between the points in adjacent planes, calculates the light scattering from that triangle forany arbitrary placement of the light source, and allows free rotation of the object for viewing (it also generatesmovie sequences of the rotation for later viewing).

Surface rendering provides the best communication of spatial relationships and object shape information to thehuman viewer, because we are adapted to properly interpret such displays by our real-world experiences. Thedisplays hide a great deal of potentially useful information - only one side of each object can be seen at a time,some objects may be obscured by others in front of them, and of course no internal structure can be seen. But thisreduction of information makes available for study that which remains, whereas the volumetric displays contain somuch data that we cannot usually comprehend it all.


Figure 9. The same data set as Figure 8 reconstructed by converting each image plane to a contour outlineand connecting points in each outline to ones in the adjacent outlines forming a series of facets which are

then shaded according to their orientation. The entire array can be freely rotated. Reconstructionperformed using MacStereology (Ranfurly MicroSystems, Oxford UK).

There is a third display mode that is easily interpretable by humans, but (or perhaps because) it is rather limited inthe amount of information provided. The three-dimensional array of voxels is either obtained sequentially, oneplane at a time, or produced by an inverse computation that simultaneously calculates them all. Showing theappearance of any arbitrary plane through this array is a direct analog to the sectioning process that is often used insample preparation and imaging. The amount of computation is minimal, requiring only addressing the appropriatevoxels and perhaps interpolating between neighbors.

As shown in Figure 10a, this allows viewing microstructure revealed on planes in any orientation, not just thoseparallel to the X, Y, Z axes. Sometimes for structures that are highly oriented and nonrandom, it is possible tofind a section plane that reveals important structural information. But such unique planes violate most of theassumptions of stereology (IUR sections) and do not pretend to sample the entire structure.

Arbitrary section displays do not reveal very much about most specimens, except for allowing stereologicalmeasurements to test the isotropy of the structure. In the example of Figure 10, which is an ion microscope imageof a two-phase metal structure, the volume fraction of each phase, the mean intercept lengths, and the surface areaper unit volume of the interface can easily be determined from any plane. Three-dimensional imaging is notrequired for these stereological calculations. The key topological piece of information about the structure is notevident in section images. Reconstructing the 3D surface structure (the figure show two different methods fordoing so) reveals that both phases are connected structures of great complexity. This is the kind of topological


information that requires 3D imaging and analysis to access. But the amount of added information, while key tounderstanding the structure, requires only visualization, not measurement. Measurement is often performed morestraightforwardly with less effort and more accuracy on two-dimensional images than on three-dimensional ones,as discussed below.

Figure 10. Ion microscopic images of a two-phase Fe-Cr alloy (data provided by M. K. Miller, Oak RidgeNational Labs): a) arbitrary sections through the voxel array do not show that the regions for each phase areconnected; b) display of the voxels on the surface of one phase; c) outlines of the phase regions; d) rendered

surface image produced by placing facets between the outlines of (c).

In summary, visualization of three-dimensional data sets typically involves too much data to put all on the screenat once. Selection by discarding parts that are not of interest (e.g., the matrix) or selecting just those parts that aremost easily interpreted (e.g., the surfaces) helps produce interpretable displays. These are often used to createmovies of sequences (e.g., a moving section plane, changing transparency, or rotating point of view) to convey toothers what we have first learned by difficult study. Such pictures are in much demand, but it isn't clear how wellthey convey information to the inexperienced, or how they will be efficiently disseminated in a world stilldominated by print media.

Processing

Image processing in three dimensions is based on the same algorithms as used in two dimensions. Most of thesealgorithms generalize directly from 2 to 3D. Fourier transforms are separable into operations on rows and columnsin a matrix, the Euclidean distance map has the same meaning and procedure, and so forth. The important change


is that neighborhood operations involve a lot of voxels. In a 2D array, each pixel has 8 neighbors touching edgesor corners. In 3D this becomes 26 voxels, touching faces, edges and corners. For most processing, a sphericalneighborhood is an optimal shape. Constructed on an array of cubic voxels, such a 5-voxel wide neighborhoodinvolves 57 voxels (Figure 11). Addressing the array to obtain the values for these neighbors adds significantoverhead to the calculation process.

Figure 11. The 57 voxels that form an approximation to a spherical neighborhoodwith a diameter of 5 voxels in a cubic array.

A cubic array is not the ideal one for mathematical purposes. Just as in a 2D image with square pixels the problemof distinguishing the corner and edge neighbors can be avoided by using a grid of hexagonal pixels for greaterisotropy, so in three dimensions a "face-centered cubic" lattice of voxels is maximally isotropic and symmetrical.But this is not used in practice because the acquisition processes do not lend themselves to it, and computeraddressing and display would be further complicated. Cubic voxels give the best compromise for practical use.But many acquisition methods instead produce voxels that are not cubic, but as noted before have much differentresolution between planes than within the plane. This causes difficulties for processing since the voxels should beweighted or included in the neighborhood in proportion to their distance from the center. The usual approaches todeal with this are to construct tables for the particular voxel spacing present in a given data set, or to re-sample thedata (usually discarding some resolution in one or two directions and interpolating in the others) to construct acubic array.

Because of the size of the arrays and the size of the neighborhoods, processing of three-dimensional images isconsiderably slower and needs more computer horsepower and memory than processing of two-dimensionalimages. The purposes are similar, however. As discussed in earlier chapters, these include correction of defects,visual enhancement, and assistance for segmentation. Figure 5 shows an example, in which the gradient of voxelvalue is calculated by a three-dimensional Sobel operator (combining the magnitude of first derivatives in threeorthogonal directions to obtain the square root of the sum of squares at each pixel) to generate a display ofbrightness values that highlights the location of structures in the brain.


Measurement

Global parameters such as volume fraction and surface area can of course be measured directly from 3D arrays bycounting voxels. In the case of surfaces, the area associated with each one depends on the neighbor configuration.It is not necessarily the case that this type of direct measurement is either more precise nor more accurate thanestimating the results stereologically from 2D images. For one thing, the resolution of the voxel arrays is generallymuch poorer. A 256*256*256 cube requires 16 Megabytes of ram (even if only one byte is required for eachvoxel). A cube 1000 voxels in each direction would require a Gigabyte. However the data are obtained, by a serialsectioning technique or volumetric reconstruction, such a large array is unworkable with the current generation ofinstrumentation and computers. The size of voxels limits the resolution of the image, and so details are lost in thedefinition of structures and surfaces which limits their precise measurement. And accuracy in representing theoverall structure to be characterized generally depends upon adequate sampling. The difficulty in performing 3Dimaging tends to limit the number of samples that are taken. Measurement of global parameters is best performedby taking many 2D sections that are uniformly spread throughout the structure of interest, carrying out only rapidand simple measurements on each.

The measurement of feature-specific values suffers from the same limitation. With resolution limited, the range ofsizes of features that can be measured is quite restricted. The size of the voxels increases the uncertainty in the sizeof each feature, and the result is often no better than can be determined much more efficiently from 2D images.Figure 12 shows an example. The structure is a loosely sintered ceramic with roughly spherical particles, imagedby X-ray microtomography. The figure shows various sections through the structure, and the surfaces of theparticles. Figure 13 shows the measurement results, based on counting the voxels in the 3D array and also bymeasuring the circles in 2D sections and using the unfolding method to determine the sizes of the spheres thatmust have produced them. Even with the same resolution in the 2D images as in the 3D array, the results aresimilar. It would be relatively easy to increase the resolution of 2D section images to obtain a more precisemeasurement of sizes.

Counting voxels to determine volume is analogous to counting pixels in 2D to determine area. Other parametersare not quite so easily constructed. The convex hull in 3D consists of a polygon, ideally with a large number ofsides. In practice, cubes and octahedrons are used. The length (maximum distance between any two points in afeature) is determined by finding a maximum projection as axes are rotated, and since this need be done only infairly crude steps (as discussed in the chapter on measurement of 2D images) this generalizes well to 3D. But theminimum projected dimension requires much more work.

The midline length of an irregular object can be based on the skeleton. Actually there are two different skeletonsdefined in 3D (Halford & Preston, 1984; Lobregt et al., 1980), one consisting of surfaces and a different one oflines; both require quite a bit of work to extract. Line lengths can be summed as a series of chain links of length 1,√2 and √3 (depending on how the voxels lie adjacent to each other), but the poor resolution tends to bias theresulting length value (too long for straight lines, too short for irregular ones). Shape parameters in 3D aretypically very ad-hoc. In principle the use of spherical harmonics, similar to the harmonic shape analysis in 2D,can be used to define the shape. However, the resolution of most images is marginal for this purpose, and thepresence of re-entrant shapes can occur with real features and frustrate this approach.


Figure 12. X-ray microtomographicreconstruction of sintered particles in a ceramic(the data set is artificially stretched by a factor of2 in the vertical direction): a) sections alongarbitrary planes; b) sections along a set ofparallel planes; c) display of the surface voxelsof the spherical particles.

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Figure 13. Comparison of 2D and 3Dmeasurement of size of spherical particles instructure shown in Figure 12: a) size distributionof circles in 2D plane sections; b) estimated sizedistribution of spheres by unfolding the circledata in (a) - note the negative values; c) directlymeasured size distribution of spheres from 3Dvoxel array.


Figure 14. Volumetric reconstruction of Golgi-stained neurons (data provided by Vital Images,Fairfield, Iowa). The two different orientations show the topological appearance of the network,

but the limited resolution of this 256x256x256 array does not actually define a continuousnetwork nor reveal many of the smaller structures.

Figure 14 shows an example of the power and limitation of 3D imaging. The Golgi-stained neurons can bedisplayed volumetrically and the structure rotated to allow viewing of the branching pattern. However, it isprobably not desired and would in any case be very difficult to achieve a quantitative measure of the branching. Infact, the comparatively poor resolution (as compared to 2D images) means that the linear structures are notnecessarily fully connected in the voxel image, and that some of the finer structures are missing altogether. Thisdoes not impede the ability of the viewer to judge the relative complexity of different structures in an qualitativesense, but it does restrict the quantitative uses of the images.

The simple conclusion must be that few metric properties of structures or features are efficiently determined using3D imaging. The resolution of the arrays is not high enough to define the structures well enough, and the amountof computation required for anything beyond voxel counting is too high. Metric properties are best determinedusing plane section images and stereological relationships. On the other hand, topological properties do require 3Dimaging. However, these are not usually measured but simply viewed. Most of the use of 3D imaging is for visualconfirmation of structural understanding that has been obtained and measured in other ways using classicalstereology.


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