An Introduction to Fractals and their Applications in Electrical ...An Introduction to Fractals and their . Applications in Electrical Engineenng by ARNAUD JACQUIN Signal Processing

An Introduction to Fractals and their .

Applications in Electrical Engineenng

by ARNAUD JACQUIN

Signal Processing Research Department, AT&T Bell Laboratories,

600 Mountain Avenue, P.O. Box 636, Murray Hill, NJ 0797440636, U.S.A.

ABSTRACT : We present a very brief history offractals, describe their generation, their characteristics, and their relation with chaos. We point to where and why,fractals and chaotic s_wtems

are commonly found in nature, which implies that they should be good candidates for modeling dijyerent types of real worldsignals. We then list a number ofdioerse research areas in electrical

engineering where ,fractals and,fractal-based techniques have ,found applications. Finally, we present in some detail applications of ,fractals in signal processing, more spec~jically in the

areas qf digital ima~qe modeling, synthesis, and compression.

I. Introduction

The applications of fractals in signal and image processing have been losing their boundaries more and more, as one starts to witness the merging of various fields : image synthesis and computer animation (which traditionally derive from computer graphics), with digital image and video compression (which derive from digital

si~gnalprocessing, specifically information theory). To a large extent, this merging is due to the need to find ever more powerful models for real world digital signals in order to represent, store, and transmit these signals efficiently. We will give examples involving different types of signals, and correspondingly, different types of fractal models, and will try to describe the advantages and shortcomings of each of these models. The reader should be warned that our choice was to present several frameworks and examples which prevented mathematical rigor. This tutorial is mostly aimed at engineers who are interested in an overview of fractals and chaos, rather than in mathematical detail. However, references to articles and books which give the details of the various frameworks are indicated in each section.

The general problem statement which underlies the use of fractals for signal modeling and compression-the main purpose of the last and largest section of this paper-is the following. Given any original object or signal, for example a discrete monochrome image specified by an array of S-bit pixel values, how can a computer construct a fractal object-the encoded object-which is both visually close to the original one, and has a digital representation which requires fewer bits than the original. Note that this latter requirement is not part of the notion of modeling peu se but is essential to the efficiency of the modeling procedure for compression or coding. For each type of object/signal to model, this problem will

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be presented in the following format. First, we describe the objects under study, the class of iterated systems which can generate them, and the generation process. Second, we describe several approaches to solving what is commonly referred to as the inverse problem, which consists of devising a procedure for controlling the generation process in such a way that it produces fractal models of original real world objects. While the first part is mathematically straightforward and merely represents several instances of a very general theory of iterated contractive transformations in a metric space, the inverse problem can only be dealt with on a per- framework basis and does require a significant amount of creativity on the part of the designer of a fractal modeling system.

ZZ. Fractals and Chaos

2.1. What is a jkzctal object.7

Texts on ,fractal geometry abound. We refer the reader with a strong mathematical interest in the topic to (l-7). Although it is difficult to give an all- encompassing definition since fractals can be objects of different types, it is usually agreed that (deterministic) fractals arise from the repeated iteration of a trans- formationt. In other words, fractals are mathematical objects which can in general be written as

A = ,liir ?(A,,),

where A0 denotes an initial object, and

Tn = zozo . ..05 (2)

denotes n iterations of z. They are typically generated by computing and displaying a sequence of iterates

&,A,,Az,..., (3)

where A, = z(A,_,). From Eq. (l), it is clear that fractals satisfy the invariance

equation :

A = z(A), (4)

which confers to them a property which we generically refer to as self-trans-

,formability, and which leads to the well-known properties of fractals to be rugged objects with an infinite amount of detail; objects which can be found again and again in magnified pieces of themselves, however small.

This very simple formulation can lead to objects which have “pathological” mathematical propertiesI, as well as a tremendous visual complexity. It is sufficient

tNote that where mathematicians see “iteration”, engineers might be more likely to see

a “feedback process.” The idea is the same. ISuch as nowhere differentiability for some continuous fractal curves and surfaces, or

infinite length or area. “Pathological” is meant here as opposed to the regularity of the typical curves and surfaces of Cartesian geometry and differential calculus.

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to describe the generation of “mathematical monsters” such as the Cantor dust,

Sierpinski triangle, von Koch curve and snowflake, space-filling Peano curve, Menger sponge, etc. (I, 7), which kept a number of mathematicians at the beginning

of the twentieth century puzzled for a number of years, but which are by now considered to be rather tame examples of fractal objects and are all well-under- stood. Their status of “monsters” caused them to fall into near-oblivion until they were “rediscovered” by the mathematician Benoit Mandelbrot in the 1960s

triggering a shift of attitude towards them. One can safely assume that this shift of attitude was due to the advent of the

computer and graphic display devices which made these objects visual, and often

strikingly so, as opposed to being characterized only by their construction and mathematical properties, or rather lack thereof. This was also due to the realization, initially by Mandelbrot but soon followed by others, that their complexity was typical of objects found in nature (cf. Section 2.2), as opposed to being aberrant. An illustration of this idea can be found in a paper by Mandelbrot entitled “How long is the coast of Britain? [. .I” (8). The argument is the following. If one set

out to measure the coastline of a geographic region such as the west coast of Great Britain with yardsticks of ever decreasing size, the measurements would show that the length associated with each yardstick keeps increasing as the size of the yardstick

decreases, eventually reaching infinity?. The impracticality of describing most natural objects in terms of straight lines-the basis of differential calculus and

engineering-and the hint at the possibility to find accurate fractal models for them both contribute to the fascination exercised by fractals.

2.1.1. Two simple examples qf self-similar fractals : the con Koch curz’e and snoM,-

jake. The fractal curve known as the con Koch curre can be generated in the following way (see Fig. 1) :

1. Take as initial object a line segment of unit length S,, = [0, 11.

2. To produce the first iterate S, distort the segment by introducing a “bump”

in the middle of S, in the form of two sides of an equilateral triangle with side of length f of the length of So.

3. Repeat this introduction of triangular bumps to each line segment until convergence.

One can see that this construction can be equivalently formulated in the framework of Eq. (l), with a transformation r consisting of the union of four similarities which map S, to S,. The construction of the snowflake is very similar and is left as a simple exercise to the reader. The iterative construction of both objects is illustrated in Fig. 1.

?A series of such measurements was studied by L. F. Richardson, who showed that the approximate length L(q) of the west coast of Great Britain, as measured with a yardstick of size q, satisfies an empirical law of the form L(q) cc q-” (9). In (1, 8), Mandelbrot interprets the quantity D = 1 +cc as the ,fractal dimension of a curve. According to the measurements, the west coast of Great Britain has a fractal dimension approximately equal to 1.25. It is easy to see that a curve with a fractal dimension strictly greater than one has infinite length. This feature is characteristic of fractal curves.

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FIG. 1. Construction of the von Koch curve (left) and snowflake (right).

2.2. Fractals and chaos in the real world

2.2.1. Fractal objects in nature. Most objects found in nature such as (i) weeds, ferns, vegetables, and trees, (ii) clouds and mist, (iii) mountain ranges, coastlines, terrain, rocks, aggregates, and ice crystals, and (iv) galaxies, etc. exhibit typical fractal characteristics such as self-transformability, which manifests itself by the fact that “the object looks the same at many different scales.” This observation gives weight to the assumption that the “geometry of nature” is a,fr.actal geometry which arises from iterative processes; a notion which was popularized by Man- delbrot (1). Yet it would be naive to believe that the iterative processes at work in nature are as simple as Eq. (1). Rather, it is reasonable to assume that they are influenced by external forces and perturbations. The growth of two trees from two identical seeds, one in a protected (ideal) environment, the other in an environment plagued with acid rain, poor soil, aridity, strong winds, etc. illustrates this concept. Rather than trying to find perfect deterministic fractals in nature (the naive view) one should realize that real world objects are more likely to be the result of an iterated process of the type

4, = ~(&,)+&I, (5)

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where E, would denote a random perturbation-t. The retrieval of an approximate z (and even perhaps E,) from a real-world object resulting from many iterations of Eq. (5) constitutes the basic principle of,fractal modeling which is addressed in

Section IV. 2.2.2. Chaotic dynamical systems. Fractals are closely related to chaotic systems.

In this section, we introduce the mathematical notion of chaos by describing and analyzing a simple chaotic dynamical system, and briefly describe the relation between fractals and chaos. For an introduction to chaos we refer the reader to (11) and to the texts (5, 7, 12-14) for a comprehensive treatment of chaotic dynamical systems and fractals.

A dynamical system consists of a transformation or map s defined from a metric space of points into itself-we consider an example where the space is the unit interval I = [0, I] endowed with the Euclidean metric. The orbit of a point x,, E I is the sequence of iterates

~O,~~I,~~2,.“, where X, = s(x,_ , ).

The orbit of x0 is said to be periodic if there exists an integer P such thata

VnE N, x,,+~ = x,,,

(6)

(7)

and it is then written x0x, . . . xP_ , . We consider the dyadic map s defined by

i

2x s(x) =

if O<x<i

2x-1 if i6x61, (8)

sometimes referred to as the “stretch-cut-and-paste” transformation because of the analogy with an idealized process of kneading dougha. The analysis of the dynamical system {I, s} can be more easily carried out when representing the real numbers in I by their binary expansions, whereby a point x0 in I is written

to denote that

0*a,a2a,. . . , with uk E {0, l} for all k, (9)

x0 =a,*2-'+a2*2-2+.". (10)

One can easily see that the transformation s is equivalent to the left-shif operator

S which acts on the binary expansions of points in I in the following way :

S(O*a,a,a, . ..) = O*a,a,a,. . . . (11)

TThis type of process can lead to what Mandelbrot refers to as statistical selfsimilarity

(10, 1). $The smallest non-zero integer P which satisfies Eq. (7) is called the period of the orbit. §A piece of dough is modeled by the line segment [0, I]. The transformation s can be seen

as “stretching” the dough to twice its initial length, then “cutting” it into two equal-length pieces, and finally “pasting” the pieces together. The “kneading” comes from the repetition of the above sequence of steps.

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This dynamical system is said to be chaotic because it can be shown? that it exhibits the three characteristics, which we intuitively describe below.

Sensitivity to initial conditions. Let x0 and _vO be two “arbitrarily close” real numbers in 1. There exists a strictly positive threshold T (consider for example T = i) and a (large) number of iterations N such that _Y~~, and Y,~ differ by the threshold T. This property is also known as the “butterfly effect” because of the chaotic weather models developed by Lorenz (16, 17). This is the property which explains the impossibility to successfully perform long-termpredictions with chaotic dynamical systems.

Mixing. The points in an arbitrarily small subinterval of [0, l] of initial values eventually “spread” over the whole interval [0, I].

E.uistence qf’denseperiodicpoints. There exists a set of points with periodic orbits for S which is dense in [0, 11.

Dynamical systems theory is closely related to fractal geometry. One can show that fractalsPattractors of iterated function systems (cf. Section 4.1) in particular-have a naturally associated dynamical system which is chaotic. Fractals are attractors of dynamical systems ; the place where chaotic dynamics occur. For details on the relation between chaos and fractals, we refer the reader to (5, 7, 12- 14).

Chaos in the real lz,orld. The list of the areas of science and engineering where chaotic dynamics are observable is long : astronomy, biology [e.g. models of popu- lation growth (IS)], chemistry (e.g. diffusion-limited aggregation), climatology (16, 17), economics [e.g. stock market prices (l)]. electrical engineering (19), fluid dynamics (e.g. turbulent flows), geology, medicine, physics, physiology and psy- chiatry, seismology (e.g. earthquake prediction). What is often of most interest in each of these areas is the extremely difficult task of trying to accurately model the underlying systems which give rise to the chaotic behavior observedPattempts have been met with varying degrees of success so far.

III. Fractals in Electrical Engineeving

We list below a number of fields of electrical engineering where fractals appear in the context of chaotic phenomena, and/or fractal tools are actively used as analysis tools. Reference (19) includes a special section covering some of these applications.

3.1. Chaos andjiactals in electrical circuits Chua’s circuit (20, 21) is a simple electronic circuit which gives rise to chaotic

behavior and fractals, and has become the paradigm for the study of nonlinear dynamical systems. For a review of the state-of-the-art on chaos in nonlinear electronic circuits, as well as a number of emerging applications from this field, the reader is referred to (22) as well as this special issue.

l_The proofs of these properties can be found in (7, 15). They make extensive use of the following lemma. Let x and J be two real numbers in I = [0, 11. whose binary expansions agree up to the kth position, i.e. x= O.CI,U~ . . . . and y = O~a,~~...a~b~+,h~+~ . . . . then I”-l’l < 2-“.

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3.2. Electromagnetics, waue propagation

A great body of research work exists in the areas of electromagnetics and wave propagation in fractal media, dealing in particular with fractal antennas and arrays, and the scattering from fractal objects and propagation through fractal screens. The reader is referred to (23) for a thorough list of references on these topics.

3.3. Fractal metrics in signal analysis The notion of,fractal dimension has been increasingly used as a tool for signal

analysis in the past few years. Numerous methods and algorithms have been developed for the estimation of the fractal dimension of an object, which measures its degree of irregularity. We refer the reader to (24) for a review of these methods. Used originally mostly in physical sciences such as material science, they are now used in a wide range of fields such as the classification of landforms from digitized topography (25,26), the detection and discrimination of objects (e.g. natural versus man-made) (27, 28), and analysis and characterization of textures in medical images.

IV. Modeling Real World Objects with Fvactals

The artificial, “machine-world” quality of virtually all computer-generated images and animation before 1980 illustrates the impact that fractal modeling tools have had in computer graphics in the past 1.5 years. Because of their inherent ability to look like typical objects found in nature, fractal models have been extensively used for image synthesis to model and render objects such as plants,

trees, terrain, clouds, landscapes, seascapes, etc. (1,7, 29-35). Some of the natural scenes generated from fractal models such as the “fractal forgeries” of Voss (7, 35), Musgrave et al. (7), Peitgen et al. (7,35) have a strikingly natural quality.

A more mathematical approach to the controlled synthesis of fractal patterns and textures has been developed in the context of approximation theory (3638). Moment matching techniques are used to approximate two-dimensional image textures by synthetic fractal ones.

However, the examples of synthetic images given above remain limited by the fact that the fractal techniques used only allow a fairly limited range of objects or textures to be modeled and “forged”. Besides, they do so by generating fractal scenes which, although they do look like they could be real natural scenes, are not fractal versions of an underlying original, and are therefore not candidates for image compression. In the following sections, we will show how other types of fractal forgeries obtained either as attractors of iterated function systems or of recurrent IF%, or, in the framework of Section 4.3, as fixed points of contractive image transformations defined piecewise, are explicitly modeled from original real- world signals.

Why is this interesting to engineers? The modeling of signals such as images is interesting if it can lead to very high compression. With the current state-of-the- art digital communication and coding technology, the very high-quality transmission of simple “head-and-shoulder” scenes for video teleconferencing/telephony

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remains a challenge at bit rates under 64 kbpst. It seems that classical image compression techniques will not be sufficient, and different types of model-based approaches are being considered (3942).

4.1. Iterated,function systems theory

4.1.1. Fractals us attractors sf’ iterated ,function systems. An iterated function system in LV’ is entirely specified by a set of k affine transformations M’,, . , ~1~ in a metric space (KY, d). A transformation M’, which operates on (compact) sets of points in R’, is defined by,

for every B c lRP, w(B) = (12) 1 <r<h

When the k affine transformations IV, are contructive, i.e.

V(x, y) E R”, d(w,(x), w,(y)) < s, - d(x, y), (13)

for some real number s, < l$, it can be shown that it‘ is also a contractive transformation in the metric space ofcompact sets in (LV’, d) endowed with the Hausdorff metric (5) associated with d. The contruction mappingprinciple (43), due to Banach, ensures that w has a unique,fixedpoint A-a set of points in (KY, d), which is called the attractor of the iterated function system. The set A satisfies the invariance equation

A = w(A), (14)

which means that A is exactly made of the union of k affinely transformed copies of itself. It is said to be a glohull~ self-gffine fructul$. The famous 2D Black Spleenwort fern of Barnsley (5, 44) is an example of such an object; it is the self- affine attractor of a set of four two-dimensional affine transformations. It is easy

to show that any iterated sequence of the type

A,, w(A,), \?(A,,), (15)

converges to the attractor A regardless of the choice of the initial set A,. This property provides a simple procedure for theyenerution or synthesis of the attractor.

The above results are due to the work of Hutchinson in the context of self- similar fractals (45). They were later extended by Barnsley (5, 46) to encompass the self-affine case. Barnsley further extended these results by introducing the notion of IFS with condensation sety, whereby the IFS is of the form

tEven though many approaches have been proposed and many systems have been developed, the image quality of these systems is well below that of analog television signals.

$The smallest .s, for which Eq. (13) holds is called the contractivity factor of the transformation IV,.

§If the transformations M‘, are contractive sirdaritie.s, i.e. each W, contracts the p dimensions of RP with the same factor, A is said to be a (globally) self-similar fractal.

11Fractal plants and trees, for example, are objects which arise as attractors of such systems.

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where C is the condensation set associated to the constant set transformation w,(B) = C, and where w is now defined by

w(B) = O<v<, w,(B). (17)

4.1.2. The inverse problem of modeling an object as the attractor of an IFS. Let us now suppose that we start with an original object Aurz9 which seems to be approximately globally self-affine, and therefore a good candidate for an attempt at modeling it as the attractor of an IFS. For the sake of illustration, let this original object be a digitized binary image of a real Black Spleenwort fern for example. The inoerse problem consists of trying to “recover” a set of affine transformations under which the original object is approximately invariant. This can be expressed as finding a set of k (resp. k+ 1) transformations {w,},~,~~ (resp. {w,}~~,~J which satisfy

u r~,(&,,~) (resp. ,g<, r$‘i(A,,,g)) is close to Arry. (18) 1 Ci<k ,,

The union of the transformed copies in Eq. (18) is called a collage of Aor,y (5, 44). The goal of the modeling is to find the “best” possible-in the sense of Eq. (18)-

collage of Aorry with as few affine transformations as possible, in the shortest amount of time. Note that although the notion of “closeness” should be in the sense of the Hausdorff metric, it was interpreted in the sense of “visual closeness” as defined by a human operator in the examples of natural scenes described at the end of this section.

The collage theorem of Barnsley et al. (5,44) states that the fixed point A of the transformation

w = , y<, wi 0-w. Jj, WA (19) \, ,\

an exactly globally self-affine fractal which in practice can be synthesized by iterating w a large number of times on any initial object A,,, is also close to A(+,,

i.e.

w”(A,) z A,],,, when n is large. (20)

Therefore, the information contained in the parameters of the transformation w is sufficient to enable anyone with a knowledge of the iterative reconstruction procedure to synthesize an object which is close to the original. The set of transformations {w,}, s,sk can thus be considered as a,fractal code-lossy in general- for A,,,.

The result of an important stability theorem, which is referred to in (5) as continuous dependence of attractors on transformation parameters can be informally stated as follows. Small errors in the parameters of w-such as the ones that occur by quantizing the parameters of the individual w,‘s for transmission on a digital channel-will induce equally small changes in the attractor. This property is a consequence of the contractivity of the transformation w. It ensures that two different decoders of the attractor of w, which could for example consist of the same reconstruction algorithm implemented on computers with different floating point precision, will produce two visually indistinguishable fractals.

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Synthetic natural scenes. The images in (5) referred to as “Monterey Coast”, “Black Forest”, “Sunflower field”, and “Bolivian Girl”, (5, 31, 47), were all synthesized with this type of fractal modeling technique. More specifically, they were obtained from the following three-step procedure :

1. Real world color images were segmented by a human operator who would strive to identify various self-affine objects.

2. Each object was modeled as the attractor of an IFS, using an interactive software package?.

3. The final images were synthesized by successively reconstructing, rendering, and overlaying the pieces in a specific order-from background to fore- ground-using another software package described in (30).

The fact that this procedure, when viewed as an image coding and decoding schemet, can be seen as “compressing” images with ratios of 1000: I and higher provoked great, and partially unfulfilled, excitement in the signal processing community. The major hurdles with the approach described above are clear : (i) natural scenes are in general not globally self-affine (the way a fern is), and scene segmentation-a difficult problem by itself-is one essential feature, and (ii) the scheme fundamentally involves a human operator whose function cannot easily be automated.

Piece-Iv&e selfltran.~forn?ahility, recurrent IFS. Foreseeing that the above pro-

cedure could only be automated with great difficulty, if it could be automated at all, lead the author to try and bypass the segmentation step by capturing the notion of piececvise self-tran:formahility in the transformations themselves, albeit by con- sidering slightly more general transformations. Instead of segmenting a priori an

object Aorjg into several pieces and modeling each of the pieces separately, one could look for a set of contractive transformations which operate on an object restricted to pieces of its support, i.e. of the general form

w(B) = u w,(B,.,). I <izsk

(21)

and which would leave the object approximately invariant, i.e. such that

This is a considerable relaxation of the self-affinity requirement, which in the context of IFS theory is a global one, and the only reasonable one when the objects to model are not self-affine, such the image of a human face. When working in a space of digital images, the pieces Dj can be for example squares of pixels of size 8 x 8 which form a partition of the image support. This idea can be applied to the modeling of contours of real-world objects as well as to the coding of digital images, two examples that are developed in the next section.

Pin practice, these objects were not as obviously self-affine as ferns. The number of transformations as well as the transformations themselves involved a great deal of both intuition and “trial-and-error”.

ZAlbeit only semi-automated and very lossy.

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FIG. 2. Action of the transformation Ton an interpolating curve.

4.2. Recurrent IFS theory for contour modeling

4.2.1. Fractal curves as attractors of recurrent IF&. This particular theory of recurrent iterated,function systems? (RIFS) was developed in order to be used as a synthesizer of planar, continuous fractal curves which interpolate a set of data points. It was independently developed by Hutchinson (45), and can also be seen as an extension of the theory offractal interpolationjimctions of Barnsley (49).

Let {z,);“=~ be a set of Nf 1 interpolation nodes in Iw2. Let C denote the space of continuous parametrized curves with compact support [0, I]$, which interpolate the set of nodes, i.e.

ZECifZ(0) = zO, Z(1) = z,, and

3(t,, . . .,tN_,)E(WN-‘suchthatZ(t,) =z,foreveryiE{l,...,N-1). (23)

Note that Z(t) is a closedcurve if go = +. Let Z parametrized curve Z(f) to the segment [ti, t,].

,rl,,J(t) denote the restriction of the

To each ordered pair of consecutive range nodes (z,, zi+ I), is associated an ordered pair of domain nodes (z,,,, Q+ , ), also among the set of interpolation points, and a contractive two-dimensional affine transformation We, such that

JV&,,) = Z,? and ~Q,(Q,+,) = zi+[, (24)

which we refer to as a node mapping condition. Let T denote the transformation which operates on a continuous interpolating curve Z in the following way :

(25)

The action of this mapping is illustrated in Fig. 2. It is easy to see that the transformed curve T(Z) is also a continuous interpolating curve. It can also be

tThe term “recurrent” was coined by Barnsley et ul. (5, 48). Its selection will probably not seem obvious to the reader, whom we refer to (48) for a justification of that choice.

$For the sake of simplicity, we use the same notation for a curve 2(*)-a parametric vector function, and its graph-the set of points {Z(t), t E [0, I]}.

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shown (50) that the mapping T is contractive in the space of interpolating curves endowed with an appropriately defined Hausdorff metric and therefore has a unique fixed point A-the attractor of the RIFS {{z,};“=,, T}-which is a continuous fractal curve that interpolates the set of pomts {zi};“=,. The invariance equation

shows that the curve A is the union of N affinely distorted copies of pieces of itself. This type of fractal curve is therefore said to be piecew-ise selfhffine. It can be generated by plotting the iterates of the sequence

ZO, T(Zo), T’G), (27)

where Z, is, for example, the linear interpolation of the nodes. An example of a recurrent IFS structure and its associated attractor are shown in Fig. 3.

4.2.2. Contour modeling. The inverse problem can in this context be viewed as a contour modeling problem, whereby given a set of digital object contours extracted from image data, one would look for a model in the form of the attractor of a recurrent IFS. A version of the collage theorem can be shown to hold. It states that given an original contour and a set of interpolation points on that curve, the attractor of a recurrent IFS which leaves the original curve approximately invariant will be visually close to the original.

Examples of contours extracted from the image “Cumulus” of Fig. 4(a) are shown in Fig. 4(b). These contours were extracted by first applying a median edge- preserving filter and quantizing the image to eight gray levels, producing uniform gray plateaus whose outer boundaries were traced using a left-most looking rule (50). A collage of the cloud boundaries was constructed using interactive software described in (50,51). The final set of interpolation points chosen on the boundaries is shown in Fig. 4(c), with two node mapping conditions indicated in this figure by arrows from domain to range nodes. The collage, made of transformed pieces of the original boundaries is shown in Fig. 4(d). Figure 5 shows a reconstruction sequence of the RIFS structure resulting from the collage procedure. The top-left frame shows the initial graph; a linear interpolation of the nodes. At the fifth iteration, the sequence has converged to the attractor shown in the lower-right frame.

The parameters of the model for the cloud boundaries considered here consisted of approximately 70 nodes, 60 affine transformations, amounting to about 500 bytes of data. Note that this modeling approach still has the undesirable characteristic of requiring human interaction in the selection of nodes and transformation. The author is not aware of any successful attempts to fully automate this procedure, although a similar procedure was implemented by Maze1 et al. for the modeling of speech signals and seismic data (52).

4.3. Fractal block coding of digital images In this section, we look at the modeling and encoding of monochrome S-bit

digital images. Fractal block coding (FBC) is a modeling approach to image coding

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FIG. 3. Example of a recurrent IFS structure and its fractal attractor.

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FIG. 4. Modeling contours extracted from “Cloud” image with recurrent IFSs. (a) “Cloud” image, (b) Extracted contours, (c) Interpolation points, (d) Collage, @1988, SPlE Vol. 1001

Visual Communications and Image Processing ‘88, p. 128.

based, once again, on a theory of iterated contractive transformations defined piecewise. This work was initially published by the author in (51, 53), and is fully described in (51, 54). The two main characteristics of the approach are that (i) it relies on the assumption that image redundancy can be efficiently captured and exploited through piecewise se(f-traruformability on a blockwise basis, and (ii) it approximates an original image by a fractal image obtained from a finite number of iterations of an image transformation defined blockwise called a fractal code. It is truly an image coding technique since it can be used to encode any digital image, as opposed to only images of a specific type, and is implementable as a computer algorithm which requires no human intervention. We now summarize this coding technique.

Let (M, d) denote a metric space of digital images, where d is for example the familiar root-mean-square error metric. The encoding of an original image P,,,~

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FIG. 5. Six iterations converging to the attractor of a RIFS model for “Cloud” contours, $31988, SPIE Vol. 1001 Visual Communications and Image Processing ‘88, p. 129.

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consists in the automatic construction of a contractive image transformation z, defined piecewise, which leaves ,LL+ approximately invariant, i.e. such that

C&L,,,.,, T&,~)) is as “close to zero” as possible.

The transformation z is chosen to be of the general form

(28)

(29)

where

9 = {R,)“S,<,V (30)

denotes a non-overlapping partition of the image support into N range cel/sP usually, but not necessarily, square blocks of possibly different sizes. The symbol p, R, denotes the restriction of the image p to the cell R, ; we call it the image block over Ri and write

p= 1 i+R,, (31) O<,<N

simply to indicate that an image is the union of its restrictions to the cells of the partition. The symbol zi denotes an elemental block transformation from a domain

cell D,t to the range cell Ri. For clarity, 7i is written as the composition of two transformations Si and T, :

T, = T,oS,, (32)

where Si and T, are the so-called geometric and massic parts of T,, respectively. In the case of square blocks, S, contracts domain blocks the size of range block R,,

while T, processes an image block without altering its square support. Designing a fractal block coding system consists in defining a priori the fol-

lowing: (i) a procedure for constructing a “good” image dependent partition, which can consist for example of a quadtree structure or any other type of tiling, and (ii) classes of discrete block transformations and a procedure for searching these classes. The construction of a fractal code 5 for p,,, consists in finding, for each value of the index i, the “best pair” of domain and block transformation (D,, 7’0, i.e. the pair such that the distortion

db, R,, 7; o S,(,u, ,,)) is minimal. (33)

The fractal code consists of the description of the image partition along with the quantized parameters of the block transformations. Its structure is illustrated in

j-These cells have to be larger than the range cells so as to ensure contractivity. A common choice is to consider square blocks of twice the size of the range blocks.

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FIG. 6. Structure of a fractal image block code.

Fig. 6, in which the arrows indicate block transformations from domain block to range block.

Decoding/reconstruction. As always with fractal techniques, the reconstruction of a decoded image is easily achieved through iteration. Here, the fractal block code z is iterated on an initial image, such as a black square (or any other image) until convergence to a stable image-in practice after about ten iterations. This reconstruction procedure is illustrated in Fig. 7 for a fractal code of the familiar digital image “Lena”, with the initial image “peppers”.

Extensions. Over the past five years, a large number of extensions of fractal image block coding have been proposed by various researchers in the signal processing research community (5%79), which shows that the field remains a very active research area. These extensions broadly address the following issues: (i)

the influence of the type of image partition, pool of block transformations, and optimization of the parameters defining these transformations, (ii) the reduction of the complexity of the encoding process-the search for the best matching blocks in the image to encode, (iii) the comparison and possible merging of fractal block coding with more traditional block-based image coding techniques, and (iv) the extension of the theory to the three-dimensional case for fractal block coding of sequences of images. A summary of these extensions can be found in (80).

Unfortunately, the compression ratios achieved with block fractal coding techniques turn out to be quite similar to those obtainable with other more classical image coding techniques, in the order of 30: 17, which seems modest compared with the 1000 : 1 ratios claimed for the pictures described in Section 4.1.2. This is because, in order to make it tractable, the inverse problem had to be broken up into elementary subproblems which can be efficiently dealt with automatically. In particular, the image segmentation with human operator described in Section 4.1.2

?A rather severe breakdown seems to occur fairly systematically at coding rates below 0.2 bpp.

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F :IG. 7. First eight iterations of “peppers-to-Lena” decoding sequence.

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has been replaced by an image partitioning into fairly small blocks, which is of

course reminiscent of other block-based image coding techniques such as block- based DCT (81L83), or vector quantization (VQ) (84,SS). However, by doing so, the possibility to model very large image segments by only very few transformations-as was done in the images of Section 4. l-has been lost.

V. Conclusion

In this tutorial paper, we briefly introduced the notions of fractal objects- objects with an intricate geometry which look the same at many different scales, and chaotic systems-systems for which long term prediction is impossible due to the property of sensitivity to initial conditions. We then described applications of fractals in signal processing, specifically in the modeling and compression of images.

Three basic types of deterministic fractal modeling techniques were described. The first two, iterated function systems (IFS) theory and recurrent IFS theory, could potentially achieve extremely high compression ratios-provided that they could be automated. The third, fractal block coding, is a true image compression technique but leads, to this date, to more modest compression ratios, comparable to those obtained with standard image compression techniques. Even though fractal block coding of images seems to offer a promise of both understanding and exploiting the iterative processes commonly found in nature which give rise to fractal patterns, the actual process of modeling natural scenes with both a very compact (i.e. very low bit rate) fractal model and a great degree of fidelity seems to be eluding us, still.

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