Aamir Mukhdoomi · 2006-05-08 · giving the reader a rough idea about the performance of this algorithm. It is, ... 5.3.3 Segmentation by Clustering ... Diﬀerent parts of a mammogram

On The Problem ofSegmenting Fibroglandular

Tissue in Mammograms

Aamir Mukhdoomi

May 5, 2006Master’s Thesis in Computing Science, 20 credits

Supervisor at CS-UmU: Christina OlsenExaminer: Per Lindstrom

Umea UniversityDepartment of Computing Science

SE-901 87 UMEASWEDEN

Abstract

Mammography screening programs have significantly helped to increase the rate of earlydetection of breast cancer in women by radiologists. To increase the efficiency of thesescreening programs, computer aided diagnosis systems are being developed that will beable to facilitate the workload of radiologists by automating different diagnostic tasks.Automated detection and extraction of fibroglandular tissue from mammograms is onesuch task that has been the focus of several researchers in the field of medical imageanalysis.

A novel automatic segmentation algorithm has been developed in this thesis specif-ically to extract glandular tissue in mammograms. This algorithm is divided into twomain phases. In the first phase, image histogram analysis based on the concept of min-imum cross-entropy is used to calculate an optimal threshold that gives a preliminaryglandular segmentation. Due to inherent limitations in segmentation methods based onthresholding, a second phase is designed to improve the quality of segmentation fromphase one. This improvement is comprised of a set of five-step post-processing oper-ations that have been designed based on factors such as location of segmented pixels,their distance from mammilla, etc.

The evaluation of this segmentation algorithm is a complex problem by itself forwhich there are no standardized solutions available at present. Therefore issues regardingthe assessment of the algorithm, developed here, are also discussed with the intention ofgiving the reader a rough idea about the performance of this algorithm. It is, however,concluded in this thesis that the above-mentioned algorithm segments the glandulartissue with similar performance as manual segmentation provided by human experts.

Furthermore, an alternative approach for glandular segmentation, based on fractalanalysis of breast tissue in mammograms, has also been implemented and tested. Pre-liminary results from this approach indicate clearly that fractal dimension by itself is aweak tool to perform texture based segmentation of images like mammograms.

Nevertheless, encouraging results indicate good potential in the presented algorithms,and show possibilities of improvement that can be subject to future work.

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Acknowledgements

I am grateful to all my friends and teachers that have helped me during the recent foursyears of my education. But first and foremost I would like to express my gratitude tomy supervisor Christina Olsen for allowing me to work on this project. I have learneda lot from her, not only about the technical aspect of this thesis, but also about themethods of performing a small scale research project. Her constructive appraisal haskept me motivated and interested in this project the whole time. By suggesting a well-structured plan on how I should distribute the period of six months on different partsof this thesis, she has made it possible for me to work on this project in an organizedmanner. I also appreciate the opportunity she gave me to co-author a paper with herfor the annual symposium of the Swedish Society of Automated Image Analysis.

I would also like to thank my parents, and my family members Aijaz Farooqi andNighat Farooqi for their moral support. I am especially grateful to Aijaz Farooqi for hisexpertise in the methods of working on a research project and writing scientific reports.

Lastly, I thank Anna-Maija Zdunek, who works as a radiologist at the Department ofDiagnostic Radiology at the University of Umea, for answering my questions regardingmammography. Her answers helped me to develop a better perspective for evaluatingthe algorithms developed in this thesis.

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Contents

1 Introduction and Motivation 31.1 Defining Image Processing . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Diversity in Applications of Image Processing . . . . . . . . . . . . . . . 31.3 Biomedical Image Analysis and CAD . . . . . . . . . . . . . . . . . . . . 41.4 Breast Cancer and Mammography . . . . . . . . . . . . . . . . . . . . . 51.5 Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Problem Description 72.1 Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.2 Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Medical Background 93.1 Anatomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.2 Breast Cancer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.3 Mammography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.3.1 Technical Information . . . . . . . . . . . . . . . . . . . . . . . . 103.3.2 Standard Views in Mammography . . . . . . . . . . . . . . . . . 133.3.3 Digital Mammography . . . . . . . . . . . . . . . . . . . . . . . . 13

4 Fundamentals of Digital Image Processing 174.1 Representation of Digital Image . . . . . . . . . . . . . . . . . . . . . . . 17

4.1.1 Range of Intensity Values . . . . . . . . . . . . . . . . . . . . . . 184.2 Histogram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4.2.1 Uses of Histogram . . . . . . . . . . . . . . . . . . . . . . . . . . 204.2.2 Kurtosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.3 Image Segmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.3.1 Thresholding techniques . . . . . . . . . . . . . . . . . . . . . . . 214.3.2 Boundary-based Methods . . . . . . . . . . . . . . . . . . . . . . 234.3.3 Region-based Methods . . . . . . . . . . . . . . . . . . . . . . . . 24

4.4 Morphological Operations . . . . . . . . . . . . . . . . . . . . . . . . . . 244.4.1 Dilation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244.4.2 Erosion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.5 Image Filtering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.5.1 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.6 An Introduction to Fractals in Digital Images . . . . . . . . . . . . . . . 26

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iv CONTENTS

4.6.1 Fractals and Fractal Dimensions . . . . . . . . . . . . . . . . . . 274.6.2 Fractal Dimensions and Image Textures . . . . . . . . . . . . . . 29

5 Implementation 315.1 About the Image Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315.2 Cross-Entropy Based Thresholding . . . . . . . . . . . . . . . . . . . . . 32

5.2.1 Prior Knowledge, Assumptions and Motivation . . . . . . . . . . 325.2.2 Cross-Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.2.3 Minimum Cross-Entropy and the LLBP Approach . . . . . . . . 345.2.4 The Complete Segmentation Algorithm . . . . . . . . . . . . . . 34

5.3 Feature Extraction Using Gabor Filter and Fractal Dimensions . . . . . 435.3.1 Reasons For Filtering The Images . . . . . . . . . . . . . . . . . 435.3.2 The Variation Method . . . . . . . . . . . . . . . . . . . . . . . . 435.3.3 Segmentation by Clustering - The Final Step . . . . . . . . . . . 45

5.4 Evaluation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

6 Discussion and Conclusions 496.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496.2 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 506.3 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

References 51

Appendix 53

A Tables 55

List of Figures

3.1 Side View of female breast, visualizing the pectoralis muscle, the glandular

tissue consisting of lobules and ducts, and the fat tissue [2]. . . . . . . . . . . 103.2 A mammography machine [10]. . . . . . . . . . . . . . . . . . . . . . . . . 113.3 Different positions used to acquire a mammogram [15]. The breast of the patient

is compressed between two plates, in order to create a uniform distribution

of breast tissue. Compression also helps to decrease possibility of unwanted

movement of the breast while taking the X-ray image. . . . . . . . . . . . . 113.4 A diagram of a typical mammography imaging system showing different com-

ponents of a mammography machine [14]. . . . . . . . . . . . . . . . . . . . 123.5 A mammogram showing large glandular tissue. Image mdb003 from MIAS

database [31]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.6 Angles for mammographic international standard views [12]. . . . . . . . . . 143.7 Different parts of a mammogram. It can be seen, by comparing this image to

the mammogram in Figure 3.5, all mammogram images do not have similar

percentage of glandular tissue or muscle tissue or the fat tissue [31, Image

mdb042]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

4.1 Coordinate convention used to represent images in this thesis [13]. . . . . . . 174.2 Image corresponding to matrix A in Equation (4.2). . . . . . . . . . . . . . . 184.3 Illustration of an image histogram. . . . . . . . . . . . . . . . . . . . . . . 194.4 Illustration of effects of histogram equalization [13]. . . . . . . . . . . . . . . 214.5 Illustrating performance of threshold based segmentation on a mammogram.

Image mdb188 from MIAS database [31]. . . . . . . . . . . . . . . . . . . . 224.6 Example of an ideal edge and a blurred edge [13]. . . . . . . . . . . . . . . . 234.7 An example of dilation of set A by set B. . . . . . . . . . . . . . . . . . . . 254.8 An example of erosion of set A by set B. . . . . . . . . . . . . . . . . . . . 254.9 Illustration of 2D convolution of an image by a filter kernel. In the image, filter

response is calculated at pixel C. The filter response then replaces the intensity

value of C in the filtered image. . . . . . . . . . . . . . . . . . . . . . . . . 274.10 Different examples of fractals. . . . . . . . . . . . . . . . . . . . . . . . . . 284.11 Illustration of how to interpret an intensity image as a surface. . . . . . . . . 29

5.1 Illustration of prior knowledge and assumptions about the image data. . . . . 335.2 This figure demonstrates the relation between the image histogram and its

cumulative histogram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

v

vi LIST OF FIGURES

5.3 Illustrates the results from stage 1 of the algorithm explained in Section 5.2.4.

These results pertain to the input image shown in Figure 5.2(a). . . . . . . . 365.4 Illustrates the results from stage 2 of the algorithm explained in Section 5.2.4. 365.5 Illustrates the final result produced at the end of phase I of the segmentation

algorithm described in Section 5.2.4. . . . . . . . . . . . . . . . . . . . . . . 375.6 Demonstrates an unwanted elongated segmentation that usually is a result of

poor muscle segmentation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 385.7 Illustrating results from step 1 to 5 of the phase II described in Section 5.2.4. 415.8 Illustration of final segmentation from algorithm presented in Section 5.2. . . 425.9 An example of how variation is calculated locally in an image. VCP gives the

local variation i.e. difference between the maximum and minimum intensity

value. Size of the window in this image is 3× 3. . . . . . . . . . . . . . . . . 445.10 Image mdb001 from MIAS database [31] after application of assumptions stated

in Section 5.2.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.11 In this figure an example of application of gabor filter in combination with

variation method is illustrated. The input image for all the given examples is

shown in Figure 5.10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465.12 Illustrating an example of inter-expert variations regarding the manual outlin-

ing of the glandular tissue. . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

List of Tables

5.1 Result from an evaluation method proposed by Olsen et al. [24]. The third

column shows the source of segmentation. The last row gives the result for

segmentation from the algorithm developed in this thesis. . . . . . . . . . . . 47

A.1 Number of storage bits for various values of N and k Gonzalez and Woods[13]. Also see Section 4.1.1. . . . . . . . . . . . . . . . . . . . . . . . . . 55

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2 LIST OF TABLES

Chapter 1

Introduction and Motivation

Interest in digital image processing methods stems from two principal applica-tion areas: improvement of pictorial information for human interpretation;and processing of image data for storage, transmission, and representationfor autonomous machine perception [13].

1.1 Defining Image Processing

Any project in digital image processing is usually divided into different stages that areconnected to each other in a sequential fashion, so that the output from one stage isused as input to the next stage. The initial stage involves acquisition of an image indigital format. Images can be produced directly into digital format by use of an appro-priate instrument, for example, images from digital cameras. If an image is available inother formats instead, such as photographic films, then they can be converted to digitalformat. A formal definition of a digital image is given in Chapter 4 of this thesis. Thenext stage of image processing often consists of enhancement procedures to increasethe quality of information present in the image. Enhancement can be done by removalof noise and other unwanted content, and alterations in sharpness and smoothness ofimage. Enhancement might be followed by an analytical stage in which different imagecharacteristics are studied. Image composition and statistical information about theimage are some of the examples of such image characteristics. The final stage mightinvolve inducing some changes to the original image and returning the changed imageas output. The details of procedures in different stages vary from one project to anotherproject depending on the requirements posed on the final result.

Some of the stages mentioned above are referred to as image analysis due to theiranalytical nature, but there is no general agreement among researchers regarding whereto draw the line between image processing and image analysis [13]. To avoid deviatingfrom the main objective of this thesis, it will be prudent not to make an attempt toshow boundaries between image processing and image analysis in this thesis. Howeverassociation of methods, explored in this thesis, to one of the above-mentioned fields isessential for the sake of clarification. Even though these methods are to some extentanalytical in nature, they will be considered to be in the scope of image processing basedon conclusions made by Gonzalez and Woods [13].

1.2 Diversity in Applications of Image Processing

Digital image processing, like several other branches of computer science, plays an im-portant role in our everyday life. One of the earliest applications of digital images canbe traced back to 1920s when submarine cables were used to transmit images [13]. For a

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4 Chapter 1. Introduction and Motivation

long period of time digital image processing was an interest of academic circles becausea common man lacked resources to process an image that would consist of hundredsof thousands of pixels. Advancement in electronic hardware and invention of personalcomputers, capable of performing heavy duty calculations in a matter of seconds, madedigital image processing more easy and accessible to those who were interested in thisfield. Introduction of such machines contributed extensively to speed up research anddevelopment of different aspects of digital image processing, as a result of which, count-less methods and tools of varying complexity were developed. Today one can combinethese methods in different ways and come up with many applications. These appli-cations of digital image processing are not only large in quantity but also diverse innature. An example of diversity is that methods of image processing provides us withan application that can be as simple as an instrument that reads a bar code of a cokecan or as complex as the software that aids in enhancement and analysis of finger printsfor the purpose of solving a crime.

It is not within the scope of this thesis to write about all the areas in which imageprocessing has been made useful. However, the science of diagnosing tumors and cancersbears a special importance to this thesis, as will be evident from Chapter 2 where theprimary goal of this thesis is explained. The following sections discuss in detail howdigital image processing has proven to be vital in the field of medicine in applicationsthat involve diagnosing tumors and cancers, etc.

1.3 Biomedical Image Analysis and CAD

The way human body functions, with all its systems and physiological processes, ishighly intricate. Diseases usually cause disturbances and alterations in normal func-tioning of the human body leading to pathological1 processes and it requires a trainedmedical professional with years of experience to identify nature of the disease and ap-ply the correct treatment. Physicians combine their training and knowledge to studydifferent symptoms and signals to diagnose a disease. Body temperature, heart beat,irregular marks on the skin, etc are some examples of physical symptoms that can beobserved on the outer surface of the body. However, when it comes to diseases liketumors and cancers, physical symptoms are often found deep inside the organs that areplaced well within the body, enclosed in protective layers. Observing these symptomsusually requires use of some form of penetrating radiation that will provide an imageof inner structures of different human organs. This is what is referred to as biomedicalimaging. X-ray, computed tomography (CT) and magnetic resonance image (MRI) aresome examples of the techniques currently used to acquire biomedical images. In somecases, invasive procedures are also required to get additional information.

Humans in general are highly capable in quickly analyzing patterns in images. More-over medical professionals, namely radiologists, are specially trained to detect anomaliesin biomedical images. The current computer-visual system, on the other hand, does nothave capabilities to exactly simulate the intelligence that human visual systems have.Why then should computers be involved in study of biomedical images? What advan-tages do computers have, over humans, which motivate their use in diagnosing diseasesthrough biomedical images? The answers to these questions are provided by Rangayyan[27] and are summarized in the following paragraphs.

Even though humans can quickly learn to detect different visual patterns, they havelimitations when it comes to performing arithmetic operations with large numbers withinseconds. Therefore, it would be highly impractical to let a person process an imagewith millions of pixels and try to remove noise, or extract certain features. Computerscan perform these operations within a considerably short span of time and with smallmargins of error.

1Pathology - The science of the causes and effects of diseases

1.4. Breast Cancer and Mammography 5

Another factor, that has always been an advantage for computers, is that humansare not machines. Humans, unlike machines, are easily affected by fatigue, boredom,environmental factors and psychological factors, which diminishes their efficiency andmakes them susceptible to committing errors. Computers, on the other hand, are con-sistent in their efficiency and can perform repetitive tasks with high accuracy for verylong periods of time.

Analysis by humans is subject to variations from observer to observer. It can be dueto difference in education and experience, among other factors. Results can even vary forthe same observer at different points of time, due to lack of application of knowledge andpractice. Because of such variations, there is a demand for objective analysis that onecan use as a reference or comparison. Computers can easily provide such an objectiveresult with consistent quality. Besides, one can combine knowledge from several expertsand program it into software that, in turn, has a collective ”intelligence”.

These are some of the primary reasons why computers have proven to be vital inthe field of medicine. The use of computers in study and interpretation of biomedicalimages for arriving at a diagnosis is gradually developing into its own area of science,called as computer-aided diagnosis (CAD). It should be noted that CAD systems todayare not only restricted to analysis of images. Methods have been developed that can beused to analyze digital information in forms other than images as well.

1.4 Breast Cancer and Mammography

In Sweden, breast cancer is the most frequently diagnosed type of cancer in women. Atotal of 6300 women are positively diagnosed with breast cancer every year in Sweden.Approximately 1500 of these cases prove to be fatal. The risk of getting breast cancerincreases with age. Approximately half of the diagnosed cases are among women above64 years of age. Less than five percent of the affected women are under 40 [30].

To minimize death rate due to breast cancer, it is imperative that cancer is diagnosedas early as possible. This diagnosis is done by manually analyzing X-ray images offemale breast also called mammograms. Mammography screening has proven to be vitalfor decreasing fatalities from breast cancer. Several studies have shown that death riskdecreases by 20 percent among the group of women involved in mammography screeningprogram [30]. Sweden was one of the first countries in the world to introduce regularmammography screening. This leads to a large number of mammograms that have tobe maintained and analyzed by radiologists all year round. To increase efficiency ofsuch mammography screening programs, use of computers in form of CAD systems isessential. The primary goal of involving computers in mammography is the automaticdetection of patterns that indicate cancers or tumors in a patient’s breast.

At present, several research projects are being carried out, in the field of imageprocessing, with the goal of developing new and improved techniques that will lead to aCAD system capable of automatically analyzing mammograms and helping radiologistsin their diagnosis. This thesis is part of one such research project being conductedby Christina Olsen [23] at the Department of Computing at the University of Umea,with the intent of developing a CAD system that will automatically analyze digitalmammograms and determine if their image quality is adequate according to severalcriterions.

1.5 Thesis Overview

Including this introductory chapter, this thesis comprises of six main chapters. Chapter2 presents the main objective of this thesis and also gives a brief overview of relatedscientific works found in the literature. Medical background on the subject of mam-mography, and a summary of basic concepts in breast anatomy are given in Chapter

6 Chapter 1. Introduction and Motivation

3. Chapter 4 serves to introduce the reader to the fundamental concepts in the field ofdigital image processing. This is followed by the core of this thesis, i.e. Chapter 5. Thischapter is divided into three sections. The first two sections describe in detail the ap-proaches that have been followed to achieve the objectives presented in Chapter 2. Thelast section presents the evaluation of these approaches. Chapter 6 includes a discussionrelated to the results, achieved in Chapter 5, along with the conclusions reached in thisthesis. Suggestions on future works are also presented in this chapter.

Chapter 2

Problem Description

2.1 Objective

The primary goal of this thesis is to examine the task of automatic extraction of regionsdepicting the fibroglandular tissue in a digital mammogram, and to develop an algo-rithm to accomplish this task. In image processing extraction of a region, also calledas segmentation (See Chapter 4), implies the process of determining the exact locationof the image pixels that represent a specific region. Evaluation of an algorithm thatautomatically segments glandular tissue, from a digital mammogram, is important todetermine the quality and accuracy of the results. However, there are certain problemsthat make evaluation of such algorithms a complex task. These problems arise mainlydue to the difficulty in obtaining an objective ground-truth1, a segmentation that can beaccepted as the correct marking of the glandular tissue in a digital mammogram. Such aground-truth, however necessary to determine the correctness of the segmentation pro-vide by the automated segmentation algorithms, is problematic to obtain. Therefore,the problems regarding the ground-truth and evaluation of the segmentation algorithmmust be discussed.

For a deeper understanding of the primary goal, a basic study in the field of medicineis required . This study will comprise of the facts about female breast anatomy, acquisi-tion of X-ray images and mammography techniques. Previously developed methods forautomatic glandular extraction in mammograms will be examined to develop the finalsolution in this thesis. It is a prerequisite that the algorithm developed should give aresult that is as objective as possible. This means that the obtained extraction shouldnot be adjusted to be in agreement with manual outlining of glandular tissue providedby an expert in mammography screening. This requirement of the objectivity is againrelated to the problems in obtaining an acceptable ground-truth, and is discussed indetail later in this report.

2.2 Related Work

Several methods have been proposed previously to perform the segmentation of glandulartissue in digital mammograms. Saha et al. [28] described an automatic and reproduciblemethod to segment dense tissue regions from fat within breasts using scale-based fuzzyconnectivity methods. The accuracy of this method was calculated by comparing theresults to manually segmented regions. In Section 5.4 of Chapter 5 of this thesis, itis described why such a method of evaluation is not desirable if the goal is to createan algorithm that gives an objective segmentation. Sample [29] describes in his Ph.D.

1Ground-truth - In case of segmentation tasks, a ground-truth is the segmentation that fulfills allthe criteria of a completely correct result

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8 Chapter 2. Problem Description

thesis a different method that uses a histogram based analysis to segment glandulartissue. This method has been named histogram difference method and has been shownto give a good result for certain types of glandular tissues. Sample [29] also discussesseveral improvements that can enhance the performance of his method. Bovis and Singh[6] and Karssemeijer [17] have devised segmentation algorithms based on classificationmethods through feed-forward Artificial Neural Network and kNN classifier respectively.A drawback of such methods is that the artificial intelligent tools used in classificationsuch as neural networks have to be trained and updated with time. In addition, verylarge amount of data is required to train these algorithms properly. The amount ofdigital mammographic images available is not sufficient for such purposes, which createsa risk that the classifier tool is trained to work on certain specific type of data. However,if training is carried out properly and a robust set of features is used then such methodscan show very high potential for providing optimal results. Various studies found inthe literature [27], are also based on the texture analysis of tissue in mammograms.Different tools are used for analyzing texture such as fractal dimensions, image filters,etc.

In spite of the presence of numerous methods developed over the past few decades,none has been able to provide a result that can be fully accepted to replace humanexperts and hence there does not exist a standard algorithm for automated extractionof glandular tissue. Moreover, some expert radiologists with extensive experience inmammogram screening share an opinion that computers can not be used to analyzemammograms by themselves due lack of years of experience and knowledge that ispossessed by medical professionals [37]. However, Zdunek [37] is also of the opinionthat computers can be used as decision aids due to certain limitations in humans incomparison with machines, such as those summarized in Section 1.3.

In addition to the methods mentioned above, two more segmentation techniques werefound in the literature. These methods were chosen for further study and examinationin this thesis. The first method is based on a simple concept of analyzing the histogramof an image. From this method an optimal threshold value is calculated that can be usedto create a distinction between two regions of the image that have different intensitycompositions. In this case, the required optimal threshold should create a clear distinc-tion between the fibroglandular tissue and the remaining breast tissue. This method offinding the optimal threshold is called Minimum Cross-Entropy, and was used by Masek[21] to segment different regions in digital mammograms. This method was chosen dueto its simplicity and the fact that the segmentation is independent of the exact locationof the glandular tissue in a mammogram. The results from the implementation of thismethod, as presented by Masek [21], were also encouraging and showed good potentialin this method for the primary objective of this thesis. The second method is based on asegmentation algorithm developed by Kasparis et al. [18]. This method is based on thefact that, based on their textural details, different regions in a digital image inhibit dif-ferent fractal properties that can be expressed in terms of fractal dimensions calculatedlocally in the image. Thus using the fractal dimensions regions with specific texture canbe recognized. Based on the hypothesis that this concept can be used to distinguishglandular tissue from other parts of the breast tissue, in a mammogram, algorithm byKasparis et al. [18] was implemented. This method was basically chosen due to theexceptional results, from segmentation of synthetic images, shown by Kasparis et al.[18]. The terminology used in this paragraph comes from the subject of digital imageprocessing and will be explained in the later chapters of this thesis. The two methodsmentioned here will be explained in more detail in Chapter 5.

Chapter 3

Medical Background

To fully understand the problem posed in this thesis, it is important to attain a basicunderstanding of the anatomy of female breast. In addition, it is equally important tounderstand how a mammogram is created, and how different parts of a female breastare projected in a mammogram.

3.1 Anatomy

The term breast, also known by Latin word mamma in anatomy, refers to the part offemale body which contains glands that produce and secrete milk used to feed infants.Female breasts are dome shaped and usually extend from the level of the second rib tothe sixth rib anteriorly (on the front). Important muscles of the chest wall, such as thepectoralis, are covered by the breast. Pectoralis muscle is attached to the collarbone,breastbone and the cartilage of most of the ribs [23]. The breast itself contains a mass ofglandular, fatty and fibrous tissue, and is connected to the chest wall by fibrous strandscalled Cooper’s ligaments [23].

The pectoralis muscle along with the glandular and the fatty tissue are the threemajor parts that are visible in a mammogram. There are 15 to 20 sections, called lobes,and each of these lobes has many smaller lobules that end in dozens of tiny bulbs thatcan produce milk (Figure 3.1). The lobes, lobules, and bulbs are all linked by thintubes called ducts that transport milk to the mammilla, which is also known as thenipple. The nipple, which is the conic elevation located slightly below the midpoint ofthe breast, contains openings of the milk ducts. The glandular tissue inside the breastis surrounded by layers of fatty tissue that extend throughout the breast.

3.2 Breast Cancer

Cancer in humans is triggered when a single cell or a group of cells deviate from thecontrol that regulates cellular growth. As a result, these cells start to multiply, creatinga mass, tumor, or neoplasms [27]. Tumors can be classified as either benign or malignant.When a tumor is benign, its abnormal growth is confined to a circumscribed, limitedmass of cells. Benign tumors usually run a low risk of spreading to the surroundingtissue. The growth of a malignant tumor, on the other hand, invades the surroundingtissues and may spread to distant parts of the body through bloodstream or lymphsystem [23] - a process also known as metastasis. Even though the benign masses cancause severe complications in a patient, the term ’cancer’ is mostly used to refer tomalignant tumors due to their relatively more severe nature [27].

The earliest form of breast cancer can develop as a non-invasive cancer in the milkducts or lobules, and is referred to as Carcinoma In Situ (CIS) [34]. Ductal CIS is

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10 Chapter 3. Medical Background

Figure 3.1: Side View of female breast, visualizing the pectoralis muscle, the glandular tissueconsisting of lobules and ducts, and the fat tissue [2].

the most common type of breast cancer in women [23]. The cancer in situ can spreadthrough the duct walls and invade the surrounding breast tissue acquiring an invasivenature.

3.3 Mammography

Women are continuously encouraged to examine their breasts for any unusual changes. Ifa cancer is suspected, a complete breast examination by a trained physician is required.Initial examination involves visual inspection and palpation (examine a part of body bytouching). Subsequently, diagnostic images of breasts are taken to get more information.At present, mammography screening is the best way of diagnosing breast disease.

3.3.1 Technical Information

In the earlier explanation of a mammogram in this report, it was described as an X-ray image of a female breast. Figure 3.2 shows a standard mammography instrumentand Figure 3.3 examples of different positions in which a patient has to be relative tothe instrument. In the latter images, one can observe that the breast of the patient iscompressed between two transparent plates. This is referred to as breast compressionand is an important factor in order to achieve high resolution and contrast.

Basic Principle

The basic principle behind mammography is the same that all X-ray imaging is basedupon. X-ray radiations, also called as roentgen radiations after their discoverer, areproduced when an electron beam is focused on a special type of material usually calledas target material. Tungsten and molybdenum are some of the most common examplesof target materials. A spectrum of roentgen radiations emerges from the target and is

3.3. Mammography 11

Figure 3.2: A mammography machine [10].

(a) (b) (c)

Figure 3.3: Different positions used to acquire a mammogram [15]. The breast of the patientis compressed between two plates, in order to create a uniform distribution of breast tissue.Compression also helps to decrease possibility of unwanted movement of the breast while takingthe X-ray image.

directed through that part of the body that has to be examined. The term object willbe used to refer to this body part. On the opposite side of the object, a special filmis placed that is sensitive to roentgen radiations. Once inside the object, each ray ofX-ray photons is attenuated1 by varying degrees dependent on the variation in densityof the contents within the object. For instance, if the object is an arm of a person, then

1Attenuate - To reduce the force, effect, or value of (a signal, electric current, etc.). Sometimes theword ’absorb’ is used instead when speaking of X-ray imaging.


Figure 3.4: A diagram of a typical mammography imaging system showing different compo-nents of a mammography machine [14].

the amount of attenuated radiations due to the bone will be considerably higher thanthe radiations attenuated by the surrounding softer tissue. As a result, the spectrumof X-ray radiations that exits the object is not the same as the one that originallyemerged from the source. This change in the spectrum is measured by exposing theradio-sensitive film to the exiting X-ray spectrum [27]. Thus a 2D projection of a 3Dobject is created on the film. This 2D projection film can be interpreted as a gray-scaleimage. As the attenuation increases within the object, the amount of radiations exitingfrom it decreases, which creates darker patterns on the film. Since the cancer masseshave higher density relative to the normal tissue, it absorbs more radiation resulting ina darker projection. However, the intensities of an X-ray film are inverted, before it isanalyzed by an expert, due to cognitive reasons. This is why bones, tumors, and otherdense masses appear lighter in an X-ray image.

Contrast and Resolution

For easy and correct analysis of a mammogram, high contrast and high resolution are twomajor properties that have to be maintained. Breast tissue is relatively homogeneous,and the difference between attenuation of X-ray radiations by different parts of thebreast tissue is low [27]. This can easily diminish the contrast between tumors andthe surrounding tissue in a mammogram. Therefore mammographic imaging systemuses a special setup of tube target material (molybdenum) that produces particularlyhigh quality X-ray beam (Figure 3.4). Moreover, the film used for mammography hashigher resolution than the ordinary X-ray films [32]. As per the recommendations fromSocialstyrelsen [30], the film resolution should be 15 line pairs per millimeter. Radiationscatter due to high thickness and high density of a tissue can also cause low contrastbetween the tumors and the healthy tissue. Special grids designed for mammographyare placed between the breast and the X-ray sensitive film to further reduce scattering of

3.3. Mammography 13

the radiation [27]. This increases the contrast around the borders of micro-calcificationsand other carcinogenic masses.

Effective breast compression is another factor that improves image quality of a mam-mogram. By physical compression of the breast of a patient, density distribution of thebreast tissue is maintained as uniformly as possible. Compression also eliminates risksfor motion during the X-ray imaging and separates mammary structures, thereby in-creasing visibility of details in the X-ray image [27].

Radio density of the breast tissue is also an important factor taken into considerationfor high quality breast imaging. Radio density refers to the density of the breast tissuethat directly effects the amount of attenuation of X-ray radiations. High radio densityis caused due to abundant presence of fibrotic tissue, glandular tissue, etc. An exampleof a mammogram of a breast with a large portion of dense tissue is shown in Figure3.5. High radio density causes high absorption of X-ray radiations. If this density isnot accounted for, radiations may fail to sufficiently reach the screen, leading to under-exposed film and a poor quality image. Therefore, quality of radiations i.e. the energyspectrum of radiations along with the arrangement of source and film screen should beoptimal in order to get sufficient penetration of the breast and optimal exposure [14].At the same time, over-exposure to roentgen radiations can have adverse effects on thepatient. Socialstyrelsen [30] has formulated specific guidelines that must be followedwhile exposing a patient to X-ray radiations.

Figure 3.5: A mammogram showing large glandular tissue. Image mdb003 from MIASdatabase [31].

3.3.2 Standard Views in Mammography

In Figure 3.3 it was shown that a mammogram can be taken using different angles andpositions of the patient relative to the machine. To ensure the detection of cancerousmasses, radiologists require more than one mammographic image of a patient’s breast[14]. For the same reason, it is also important that different angles are used to achievedifferent views of a patient’s breast. Mediolateral oblique (MLO) view and cranio-caudal (CC) view, used in combination, have become the international standard viewsin mammographic screening [14, 23]. The diagram in figure 3.6 shows the respectiveangles at which X-ray radiations are directed towards a patient’s breast in CC andMLO view. The MLO view in general is regarded as the most important view becauseit is most likely to visualize all the breast tissue including the pectoralis muscle andother tissue adjacent to the chest wall [14].

3.3.3 Digital Mammography

In the previous sections it was briefly explained how an X-ray image of a female breastis created using a screen-film image receptor system. Such mammograms are analog innature, whereas computers work with digital images. Depending on the method they


(a) The cranio-caudal (CC) viewin which the X-ray radiationsare directed downwards fromabove the patient’s breast.

(b) The mediolateral oblique(MLO) view in which radiationsare directed from the side withan oblique angle.

Figure 3.6: Angles for mammographic international standard views [12].

were created, digital mammograms used in the CAD systems can be classified into twotypes - primary and secondary digital mammograms. Primary digital mammograms arethose that are obtained in digital form directly from the patient by replacing screen-filmreceptor with a digital X-ray detector sensor [23]. The photon energy from the radiationspectrum is converted into an electronic signal and stored in a computer as a digitalimage. Digital images acquired in this way have several advantages over the traditionalanalog images as explained by Olsen [23]. Secondary digital mammograms are producedby digitization of analog mammograms. Several factors are involved in this digitizationprocess. It is very important that the quality of the original mammogram is high inorder to create a high quality digitized mammogram.

An Example

Figure 3.7 is an example of a mammogram from MIAS database, taken with MLO view.Region A is the projection of the muscle tissue, which is usually in form of a triangle.Region B, which is a cluster of very light intensity, is the glandular tissue and the greyregion around is the fat tissue depicted by Region C.

3.3. Mammography 15

Figure 3.7: Different parts of a mammogram. It can be seen, by comparing this image to themammogram in Figure 3.5, all mammogram images do not have similar percentage of glandulartissue or muscle tissue or the fat tissue [31, Image mdb042].


Chapter 4

Fundamentals of Digital ImageProcessing

The knowledge of the concepts of digital image processing, presented in this chapter,is required to fully comprehend the methods discussed in this thesis. In addition toexplaining these concepts, this chapter gives a brief introduction to fractal dimensionand its relation to digital image processing.

4.1 Representation of Digital Image

Digital images are usually stored in a matrix form and represented as two-dimensionalfunctions of space. Let f(x, y) be this function representing an image. Intensity functionand impulse function are some of the names used to denote f(.). Parameters x and y arerespective row and column coordinates of a pixel in the image matrix and value of f(.)is the intensity value of that pixel. Note that the terms gray levels and intensity levelsboth imply to the value of a pixel in a grayscale image and are used interchangeably inthis thesis.

Figure 4.1: Coordinate convention used to represent images in this thesis [13].

Coordinate convention that is widely used to show images on a computer screen isshown in Figure 4.1. According to the notation explained above, one can express ageneral digital image mathematically in a matrix form as

17

18 Chapter 4. Fundamentals of Digital Image Processing

f(x, y) =

f(0, 0) f(0, 1) . . . f(0, N − 1)f(1, 0) f(0, 0) . . . f(1, N − 1)

......

...f(M − 1, 0) f(0, 0) . . . f(M − 1, N − 1)

, (4.1)

where it is assumed that the image has M rows and N columns, or in other words, theimage is of M × N dimensions. Each element of the matrix corresponds to an imagepixel, image element or pixel. It should be noted from Figure 4.1 and Equation (4.1)that, unlike Cartesian coordinates, x is the vertical axis and y is the horizontal axis.

4.1.1 Range of Intensity Values

Range of intensity values for an image implies a closed interval [fmin, fmax] where fmin

is the smallest intensity value an image pixel can attain and fmax is the largest. Size ofthe intensity range of images, during an image processing task, is decided in accordancewith the specifications and requirements of the task. It is a common practice to assignbrighter shades to higher intensity values so that the pixel with highest value has whiteshade and the pixel with lowest value has black. An example of an image matrixand its corresponding gray scale image is illustrated in Equation (4.2) and Figure 4.2respectively.

A =

0 0 0 0 0 01 1 1 1 1 12 2 2 2 2 23 3 3 3 3 34 4 4 4 4 45 5 5 5 5 5

(4.2)

Figure 4.2: Image corresponding to matrix A in Equation (4.2).

Image properties mentioned above i.e. image dimensions and format of intensityrange, have certain requirements. Row and column dimensions of an image must bepositive integers. When it comes to intensity range, it might seem that there are nolimitations other than that intensity values can only be real values. This might be truein theory, however, due to the process involved in acquisition of digital images, hardwareconsiderations and other factors, the number of intensity values is typically an integerpower of 2. Assuming that pixel values of an image can have L discrete gray values inthe range [0, L−1], and that the allowed pixel values are equally spaced in this interval,then there is a positive k such that [13]:

L = 2k. (4.3)

4.2. Histogram 19

If the image is of dimension M × N then the number of bits required for storageis given by M × N × k. Table A.1 shows some common values of N and k, and thecorresponding number of bits required. These values are valid for square images i.e.images with M = N .

A special case is when k = 1 which implies only two discrete intensity values allowedfor a pixel. Such an image is also referred to as a binary image or a black & white image.A binary image is usually a result of the image segmentation process. As a commonpractice, white shade is assigned to pixels that are objects of interest and black shadeto rest of the pixels. Hence the name black & white. For further details about imagesegmentation, see Section 4.3.

4.2 Histogram

Histogram is a discrete function that describes occurrence of different gray levels in animage. If the intensity range of an image I(x, y) is [0, L − 1], then a histogram can bedefined as a function h(rk) = nk, where nk is the number of pixels that have kth graylevel rk [13]. Modifying the image matrix A from Equation (4.2) into the matrix B as

B =

0 0 1 1 1 11 1 1 1 1 12 2 2 2 2 23 3 3 3 5 51 1 4 4 4 45 5 5 5 5 5

, (4.4)

the corresponding histogram per its definition will look like Figure 4.3(b). The graylevel 1, for instance, occurs 12 times as shown in the histogram and so on.

(a) Gray scale image cor-responding to image ma-trix B in Equation (4.4).

(b) Image histogram of Image in Figure 4.3(a).

Figure 4.3: Illustration of an image histogram.


4.2.1 Uses of Histogram

A histogram can be used to calculate and derive different properties from an image.Most of the image processing methods based on histogram are statistical in nature andare related to probability distribution of intensity values in the image. In this thesis ithas been observed that a major advantage of performing calculations on a histogram,instead of the image, is time complexity. All the images in this thesis are of dimensions1024 × 1024. Operations on such a large image will take a considerable amount oftime especially if performed without optimization. On the other hand, a histogram willprovide two 1D arrays, each of length L for k-bit image in accordance with Equation(4.3). One of these arrays contains the discrete gray levels and the other array containsthe corresponding values denoting the occurrence of these gray values. Even if theimages are larger than above mentioned size, arrays obtained from histogram are stillof the same size, i.e. length L. Moreover, due to single dimension of both these arraysoperations are relatively simple. Another advantage of working on histograms is thatlocation of pixel intensity values in the image is irrelevant.

Even though, histogram calculations may seem relatively easier and simpler, thereare certain drawbacks to be considered. Before being able to utilize a histogram, one hasto calculate it. Calculation of an image histogram can be a time consuming operationfor very large images. It is also shown in this thesis that methods based alone onhistograms are not adequate for objectives similar to the primary goal of this thesis (forfurther information see Chapter 5 and 6). Last but not the least, a histogram providesan observer with the so called global information about an image. It is not possible toextract any local features from the histogram of the whole image.

Histogram Normalization

Sometimes it is useful to normalize an image histogram by dividing each of its values bythe total number of pixels in the image. This calculation is also referred to as histogramnormalization, and it creates a histogram that is given by p(rk) = nk/n, where n istotal number of pixels in the image. From a statistical point of view, p now containsthe probability distributions of different intensity values.

Histogram Equalization

Histogram equalization is an interesting image enhancement tool. Figure 4.4(a) showsa pollen image with poor contrast and Figure 4.4(b) its histogram that shows a highconcentration of dark intensity pixels. If a visual analysis of this image were required,it would be desirable to enhance the contrast in this image. This can be achievedby applying a transformation function so that the intensity distribution becomes moreuniform in nature as indicated in Figure 4.4(d). Such transformation is called histogramequalization. Resulting image is shown in Figure 4.4(c)

4.2.2 Kurtosis

Kurtosis of a histogram gives a measure of flatness of its peak relative to the peak of anormal distribution, and is defined as

kurtosis =∑N

i=1(Xi − µ)4

(N − 1)σ4, (4.5)

where N is the number of data points, µ is the mean and σ is the standard deviation[29]. For a normal distribution, kurtosis value is 3. As kurtosis value gets higher than3, the peak get narrower near the mean, and declines rather rapidly with a heavy tail.Values lower than 3 indicate a relatively flat peak near the mean.

4.3. Image Segmentation 21

(a)

0 50 100 150 200 250

0

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10000

12000

14000

16000

18000

(b)

(c)

0 50 100 150 200 250

0

1000

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3000

4000

5000

6000

(d)

Figure 4.4: Illustration of effects of histogram equalization [13].

4.3 Image Segmentation

One of the most recurrent prerequisites of an image processing system is the ability toanalyze images and detect regions that have specific characteristics. For instance, thereis a huge demand of methods that can enable computers to extract tumors and othermalformations in human tissue. Such regions, in general, are called regions of interest(ROIs) for obvious reasons.

Segmentation is a process that divides image pixels into smaller structural units thatcorrespond to ROIs. Segmentation of the pixels is usually based on some characteristicsthat are the same for pixels in the same ROI. There are several types of segmentationalgorithms that are available today and the performance of these algorithms is goalspecific. As explained by Rangayyan [27], image segmentation techniques are classifiedinto four main categories:

– thresholding techniques,

– boundary-based methods,

– region-based methods.

4.3.1 Thresholding techniques

The main principle behind thresholding is that image pixels, falling into a predefinedrange of intensity values, are assigned one single intensity value, and the rest of thepixels are assigned a different intensity value. A thresholding function can be formallydefined as


g(x, y) ={

i if f(x, y) ∈ [Ll, Lu]j if f(x, y) /∈ [Ll, Lu] , (4.6)

where g(.) is the thresholded version of image f(.), i and j are the two intensities used todifferentiate between two groups of pixels, and Ll and Lu are lower and upper limits ofthe intensity range used to define the two groups [13]. Definition of the intensity rangecan vary depending on the images and task in question. One can simply define a singleintensity value as a threshold or define several intensity ranges. The resultant image hasonly two intensities. For simplification purposes, it is common to use intensity values 0and 1 for i and j respectively.

The basic principle in thresholding makes it highly suitable for segmenting ROIsthat will always have distinct intensities from the rest of the image. An example is theimage mdb188 from MIAS database shown in Figure 4.5(a). The ROI in this image,corresponding to goals of this thesis, is the brighter glandular tissue. By choosing theintensity value 165 as threshold, the binary image shown in Figure 4.5(b) is acquired sothat the white pixels correspond to the glandular tissue. However encouraging this resultmight seem, there are vital parts of the glandular tissue that have not been included inthe segmentation. This is due to the fact that the type of thresholding explained herehas certain limitations and requires additional image processing to enhance the qualityof the resultant segmentation.

(a) Image mdb188 from MIAS database[31]. Notice that only the fat tissueand the fibroglandular tissue are visi-ble in this image. This is because thepectoralis muscle has been removed toshow the effects of thresholding on glan-dular tissue only.

(b) Thresholded image from Figure4.5(a) with threshold 165.

Figure 4.5: Illustrating performance of threshold based segmentation on a mammogram. Imagemdb188 from MIAS database [31].

The type of thresholding illustrated above is referred to as global thresholding, wherea single threshold intensity is used to segment the image. A common drawback of globalthresholding is that, for many images, a single threshold value does not fully segment allthe ROIs in the image due to poor illumination, noise and other factors. In such cases,adaptive thresholding is required which is more robust. In adaptive thresholding, animage is divided into subimages and then a different threshold is used to segment each

4.3. Image Segmentation 23

subimage. This requires a careful method of creating the subimages and calculatingthreshold for each one of them [27].

4.3.2 Boundary-based Methods

In terms of digital image processing, boundary in an image can be defined as a singlecontinuous edge forming a closed path and thereby enclosing a part of the image thatmight be considered a ROI. A digital edge is formed by a region where transition betweendark and light pixels is sharp. This is illustrated by Figures 4.6(a) that present a specialcase where the transition is rather abrupt. Boundaries made up of such an edge areeasy to detect automatically, and the term ideal edge is used to denote them. However,it is rarely the case when all ROIs are enclosed by an ideal boundary which makesboundary detection more complex. In reality an edge will look more like in Figure4.6(b) that is an image from MIAS database [31]. Optics, sampling and imperfectionsin image acquisition are some of the reasons why edges are blurred [13]. Intensity plotsin Figures 4.6(c) and 4.6(d) show the transition curve of the intensity values. Degreeof blurring of an edge is inversely proportional to the slope of the ramp that representsthe transition [13].

(a) (b)

0 100 200 300 400 500 6000

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0 50 100 150160

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(d)

Figure 4.6: Example of an ideal edge and a blurred edge [13].

The transition at an edge can be detected mathematically by calculating the slope byapplying gradient operators. For 2D images, direction can be specified in which gradienthas to be calculated. Edges detected in this way will be perpendicular to the directionin which image derivative is calculated.

Concepts explained in the previous paragraph are basis for segmentation methodsbased on boundary detection. Initially an image is scanned for very sharp intensity tran-sitions to detect edges with specific characteristics such as orientation, degree of blurring,length, etc. Then edge-linking algorithms are applied to create enclosed boundaries.Edge-linking and boundary detection methods are explained in detail by [13]. Ran-gayyan [27] and Gonzalez and Woods [13] point out that alternative edge detectionmethods, such as Hough-transform-based global transformation and global processing


via graph theoretic techniques, provide more accurate results when it comes to creatingan enclosed boundary from several disjoint sets of edge pixels.

4.3.3 Region-based Methods

Sometimes regions that are to be extracted from an image have a specific texture thatis different from other parts of the image. Thresholding and boundary detection mightnot be able to recognize the specific texture because it is a local property. This is whenregion-based methods can be used. It is assumed that pixels, in a certain region ofinterest with a specific texture, share similar values. A cluster of pixels, sharing similarcharacteristics, is referred to as neighborhood in region-based methods. According toRangayyan [27] there are two types of region-based segmentation: region splitting andmerging and region growing.

In the former type, an image is subdivided into smaller regions until some predefinedconditions are fulfilled. For instance, a condition might be that a region should not besplit if all its pixels have the same intensity or have same fractal dimension (Section4.6). Then the smaller regions are merged according to some pre-specified conditions.This process might be continued until the desired result is achieved.

Region growing methods usually start with a very small group of pixels and growa region by connecting neighborhood pixels that possess certain properties. Differentproperties can lead to different regions. The initial starting point of region growing iscalled a seed pixel. It is important to choose a correct seed pixel to get the desiredresult. Its choice can depend on several task specific conditions and prior assumptions.For instance, if an approximate location of the ROI is known, then the spatial centroidof this region can be used as the seed pixel. Region-based methods are often timeconsuming. It is also difficult to define optimal conditions that will lead to the desiredsegmentation.

4.4 Morphological Operations

Morphological operations in digital image processing are a way of extracting imagecomponents that can be used to express details about a regions shape, its boundaries,its area and so on [13]. Two primitive and most widely used morphological operationsare explained below.

4.4.1 Dilation

Dilation is usually used to smooth boundaries of regions or bridge very small gaps be-tween neighboring regions. According to Gonzalez and Woods [13], the formal definitionof dilation of a set A by another set B is denoted A⊕B and defined as

A⊕B = {z|(B)z ∩A 6= ∅}, (4.7)

where B is the reflection of B. This definition means that dilation of A by B will isdone by reflection B and then shifting B over A by z. Then set of all the displacementsof B such that B and A overlap by at least one element gives the dilation. Set B is alsoreferred to as the dilation mask or structuring element. In Figure 4.4.1, a set A anda set B is shown to illustrate the effects of dilation. Center of the mask B is markedby a small black square. In this case, the reflection B is equivalent to B. Now, if B ismoved within and outside A, then dilation is given by set of all points traversed by thecenter of B until A and B overlapped by at least one element. The result is shown asthe shaded square that is bigger in size than A as indicated by dashed lines.

4.5. Image Filtering 25

�

�

�

�

� � � � � � � �

�

Figure 4.7: An example of dilation of set A by set B.

4.4.2 Erosion

Erosion has an opposite effect of dilation. Following the same notation as in Equation(4.7), a formal definition of erosion is given as [13]

AªB = {z|(B)z ⊆ A}. (4.8)

In other words, erosion of A by B is set of all points traversed by center of B such thatB is totally contained within A at all times. This is illustrated in Figure 4.4.2

�

�

�

�

� � � � � � �

� �

Figure 4.8: An example of erosion of set A by set B.

Erosion can be used for removing small unwanted components, such as thread like struc-tures, from an image by using a structuring element that is bigger than the unwantedregions.

The processes of dilation and erosion can be combined in different ways to makesome interesting changes in an image. Morphological opening and closing are two suchoperations that are defined by specific combinations of dilation and erosion. Gonzalezand Woods [13] provides details about formal definitions of these operations. Openingis generally used to smooth region contours in an image, and it also removes thin pro-trusions. Closing also adds smoothness to image contours, however it generally fusestwo large regions separated by narrow breaks. This effect is opposite to that caused bymorphological opening that breaks the narrow links between two large regions.

4.5 Image Filtering

Representation of a digital image in Section 4.1 was a representation in the spatialdomain. By spatial domain it is meant that the coordinates of the matrix, representingthe image, express the spatial location of image pixels. Digital images can also berepresented in the frequency domain. The representation matrix has the same size inthe frequency domain as in the spatial domain. However, coordinates of the matrixentries are not the spatial locations but the frequency values. These frequencies expressdifferent degrees of intensity changes in an image and how often such changes occur.


In addition to that, frequency domain also shows the orientation of different intensitychanges. To transform an image from spatial domain to frequency domain, the discreteFourier transform (DFT) can be used, which is defined for a discrete 2D function as[13]

F (u, v) =1

MN

M−1∑x=0

N−1∑y=0

f(x, y)e−j2π( uxM + vy

N ), (4.9)

such that F (u, v) gives the DFT of the image f(x, y) of size M ×N where

u = 0, 1, 2, . . . ,M − 1 and (4.10)v = 0, 1, 2, . . . , N − 1. (4.11)

Transformation to frequency domain is used for varying analytical purposes, imagefiltering being one of them. An image can be filtered in both spatial and frequencydomain. A filter kernel is used, which is usually a matrix of smaller size than theimage. Then by the process of convolution, described in Section 4.5.1, this filter kernelis applied to the image introducing some desired changes such as removal of repetitivenoise, reduction of image blur, etc.

4.5.1 Convolution

Discrete convolution of two functions, f(x, y) of size M × N and k(x, y), is formallydenoted by k(x, y) ∗ I(x, y) and is defined by the expression [13]

h(x, y) = k(x, y) ∗ f(x, y) =1

MN

M−1∑m=0

N−1∑n=0

k(m,n)I(x−m, y − n). (4.12)

Convolution in 2D is used for linear image filtering as illustrated in Figure 4.9. Ateach pixel (x, y) of the image f(x, y), the response of the filter kernel k is calculated. Inlinear image filtering, kernel response is calculated as the sum of point products of thefilter matrix and the matrix defined by a window centered at the pixel (x, y) [13]. Thelatter is commonly called as a filter mask and is of the same size as the filter. In thefiltered image h(x, y), kernel response replaces the entry at (x, y) as the new intensityvalue. The filter matrix is rotated by 180 degrees before calculating the filter response.This process is repeated for all the pixels of the image. At image edges and corners,filter mask falls out of the image borders which is why the image is usually padded,either by zeros or by other values depending on the requirements of a task.

4.6 An Introduction to Fractals in Digital Images

The concept of fractals and fractal dimensions has been used in various studies to analyzeand express texture in digital images. In this thesis a method based on fractal dimensionshas been tested to analyze texture in breast tissue in mammograms. To understandthis method, a basic knowledge of the concept of fractal dimension, introduced in thefollowing sections, is necessary. The definition of a fractal dimension, in its original form,is mathematical in nature and is defined for application on mathematical functions,for example a line, a curve, a surface, etc. In Section 4.6.2, it is explained how thismathematical form of fractal dimensions is applied to digital images.

4.6. An Introduction to Fractals in Digital Images 27

Figure 4.9: Illustration of 2D convolution of an image by a filter kernel. In the image, filterresponse is calculated at pixel C. The filter response then replaces the intensity value of C inthe filtered image.

4.6.1 Fractals and Fractal Dimensions

Among various proposed definitions, fractals are often perceived as

a pattern composed of repeated occurrences of a basic unit at multiple scalesof detail in a certain order of generation [27].

”Self-similarity” is a term that is sometimes used in the literature to denote thisnotion of fractals [26]. Fractal patterns in form of such self-replicating forms have beenfound in abundance in nature, such as in snowflakes, plant leafs, branching and spreadingpatterns of the arteries in the heart, etc [27]. Figure 4.10(a) and 4.10(b) show examplesof naturally occurring fractal patterns whereas Figure 4.10(c) and 4.10(d) are examplesof some of the popular synthetic fractal patterns defined by mathematicians. Closeobservation of these images shows that if a small segment is cropped from an image andenlarged, then it will resemble the whole image. This explains the term ”self-similarity”.

Not all patterns that are said to possess fractal characteristics show aspects of self-replication that are as obvious as the examples in Figure 4.10. Clouds, breast tissuein mammograms, etc. are some examples of such complex patterns that possess fractalproperties that are difficult to observe visually. According to Rangayyan [27], fractalnature in such cases is more easily explained in terms of complexity of dimensionalityof the patterns, giving rise to the term fractal dimensions.

This paragraph explains the basic theory behind what a fractal dimension measuresand how it is applicable in texture analysis in digital images. Dimension of a patternimplies how much space is filled by it. A common measure of dimensionality is topologicaldimension (also called lebesgue covering dimension [33]) according to which a straightline is a 1-D pattern, a circular disc is 2-D and a cube or a sphere is 3-D. If we considerpatterns with high complexity, for instance the von Koch coastline curve (Figure 4.10(c)),such an integer valued dimension does not fully express its characteristics. It fills morespace than a straight line but less space than a circular disc. Its topological dimension


(a) A magnification of an actualsnowflake [5].

(b) A plant leaf modeled by fractal pat-terns [4].

(c) The coastline curve by the Swedishmathematician, Helge von Koch [35].

(d) The Sierpinski triangle named afterthe Polish mathematician Waclaw Sier-pinski [35].

Figure 4.10: Different examples of fractals.

is 1, however, fractal dimension value for such a curve will be a real number between 1and 2, which in other words means that fractal dimensions are an alternate and perhapsa more accurate estimate of how the given pattern fills space. Consequently, fractaldimension of a pattern increases with higher irregularity and complexity.

Various methods have been proposed to estimate fractal dimensions based on dif-ferent descriptions of what a fractal dimension is. Some of these common descriptionsare the box dimension, the Minkowski-Bouligand dimension, the Hausdorff-Besicovitchdimension [22].

4.6. An Introduction to Fractals in Digital Images 29

4.6.2 Fractal Dimensions and Image Textures

If we consider a rough 2-D surface, its fractal dimension will be greater than 2. As itsroughness increases, it will attain higher fractal dimension until it reaches 3. A digi-tal image, as we know, can be interpreted as a two dimensional surface whose heightat different coordinates is given by pixel intensity at those coordinates. This is demon-strated in Figure 4.11. The concept of fractal dimensions can thus be applied to a digitalimage to study its roughness and texture. This approach has been applied to analyzemammographic tissues in several scientific works [8, 9].

(a) An example of a syntheti-cally produced grayscale inten-sity image.

0

20

40

60

80

100

120

140

0

20

40

60

80

100

120

140

−0.5

0

0.5

1

1.5

2

2.5

(b) Image from Figure 4.11(a) plotted as a two-dimensional surface.

Figure 4.11: Illustration of how to interpret an intensity image as a surface.


Chapter 5

Implementation

Solutions to the primary objective of this thesis are presented in this chapter. Twodifferent approaches have been assessed and tested, with the intention of modification ifnecessary, to achieve an optimal segmentation of fibroglandular tissue in a mammogram.

The first approach, presented in Section 5.2, is based on a thresholding technique byMasek [21], which uses the concept of cross-entropy to find the optimal threshold inten-sity. This algorithm in its original form was used to decompose a whole mammograminto its different components, with no specific requirements on segmentation quality ofglandular tissue. Therefore some changes have been introduced in this thesis to enhancethis technique specifically for glandular segmentation.

Section 5.3 presents the second approach that is based on the algorithm proposed byKasparis et al. [18]. In this algorithm, regions with specific textures are segmented byfirst enhancing texture details through the application of different Gabor filters. This isfollowed by estimating fractal dimensions (FD) of the pre-filtered image. Fractal dimen-sions of image pixels is then used as a feature in a classification method called K-meansclustering. As demonstrated by Kasparis et al. [18], this algorithm shows an outstandingperformance when applied on synthetic images consisting of varying textures. However,results of this method were not equally encouraging in case of mammograms, which isthe reason why it has not been explored to the same extent as the previous approach.Despite the preliminary results achieved in this thesis, the extent of testing of thismethod is not sufficient enough to make a conclusion about its performance. In rest ofthis thesis, the segmentation algorithm by Kasparis et al. [18] will be denoted as theKASP algorithm.

5.1 About the Image Data

It was mentioned in Chapter 1 that this thesis is part of a bigger research project. In thisproject, a collection of 200 mammograms were randomly chosen from the two databasesthat are commonly used in image analysis projects related to digital mammography, i.e.Mammographic Image Analysis Society’s (MIAS) digital mammography database [31]and the Digital Database for Screening Mammography (DDSM) [1]. These randomlychosen images were manually examined by three radiologists and two groups of severalradiographs [24]. Among other tasks, these experts were asked to mark what theyperceived as anatomical landmarks in each mammogram.

While forming an automated computerized system, it is essential to split the datacollection into a training set and a test set for unbiased and accurate performanceevaluation purposes. Training set is used during the development of an algorithm, andthe test set is used for evaluation of the algorithm. For this thesis, a total of 40 randomlychosen images were used as training data, and 160 of the images were used as the test

31

32 Chapter 5. Implementation

data. It should also be noted that all the images used in this thesis have dimension1024× 1024 and are 8-bit grayscale images.

5.2 Cross-Entropy Based Thresholding

This algorithm is based on certain assumptions made about the images that will beprocessed. This section starts with the explanation of these assumptions along with thedescription of some basic concepts such as cross-entropy, minimum cross-entropy, etc.Then a brief summary of how this algorithm was originally implemented by Masek [21],which is followed by the details of the enhanced version developed in this thesis.

5.2.1 Prior Knowledge, Assumptions and Motivation

It has already been established that the region of interest (ROI) for this thesis is theregion of mammogram that depicts glandular tissue. Following assumptions are madeabout the input image that is to be segmented:

– Position of the mammilla has already been calculated by the available algorithms[23].

– The input image has been run through the algorithms for breast boundary detec-tion, and extraction of the pectoralis muscle and other unwanted regions such aspatient labels and identification markings [23]. As a result, the input image mostlyconsists of the fat tissue and the fibroglandular tissue. Rest of the image consistsof pixels with absolute zero as intensity value. These pixels are also referred to asbackground pixels.

It has also been observed that no pixels belonging to the breast tissue have inten-sity values 0 or 255, i.e. the minimum and maximum intensity limit for 8-bit images.Therefore all the pixels that have intensities 0 or 255 are excluded from the segmenta-tion procedure. A mammogram without application of above mentioned assumptionsis shown in Figure 5.1(a). Figure 5.1(b) illustrates how this mammogram is processedbefore using it as input to this approach.

In order to use thresholding based segmentation, it is important to examine imagesin question to determine if their intensity composition allows segmentation of ROIs bythresholding. Masek [21] emphasizes this fact and points out that manual thresholdingof images can be used to determine if acceptable segmentation is possible. Thereforeseveral mammograms were manually thresholded and their image histograms were stud-ied, using the popular image editing software Adobe Photoshop CS, in this thesis. It wasobserved that there are well defined peaks that correspond to the glandular tissue in thehistogram of each mammogram. The histogram of region corresponding to breast tissueof mammogram in Figure 5.1(b) can be seen in Figure 5.1(c). It was, as well, noticedthat an optimal threshold lies in the vicinity of the peak enclosed in the black rectangle.Such a peak has been found to exist in all the mammograms in varying shapes andsizes. Therefore a robust algorithm is needed that can detect this peak and calculatean acceptable threshold intensity with consistent performance independent of the shapeand size of the peak. Thresholding techniques based on minimum cross-entropy havebeen shown to produce such results by Masek [21].

5.2.2 Cross-Entropy

A very popular measure of discrepancy between two entities is squared distance or insome cases euclidean distance. There is an alternative measure for calculating differencebetween any two given probability distributions - cross-entropy. The concept of cross-entropy was first introduced by Kullback [19], and has been given several names such

5.2. Cross-Entropy Based Thresholding 33

(a) The original image mdb041from MIAS database [31].

(b) Image in Figure 5.1(a)according to the assumptionsstated in Section 5.2.1. This isthe form in which image is sentas input to the segmentation al-gorithm.

(c) Image histogram pertaining to pixels ofbreast tissue in Figure 5.1(b).

Figure 5.1: Illustration of prior knowledge and assumptions about the image data.

as Kullback-Leibler number and directed divergence [21]. It has proven to be veryuseful in many different forms as it provides a robust method to measure how good oneprobability distribution approximates another probability distribution [3].

Let P and Q be two discrete probability distributions, such that P = p1, . . . , pn andQ = q1, . . . , qn, then cross-entropy distance between P and Q is given as:

HCE(Q, P ) =n∑

i=1

qilogqi

pi(5.1)

Cross-entropy is not a true measure of distance in metric space unless Equation (5.1)satisfies following conditions [21]:

1. HCE(Q,P ) ≥ 0

2. HCE(Q,P ) = 0 if and only if Q = P


3. HCE(Q,P ) = HCE(P, Q)

4. HCE(Q,P ) ≤ HCE(Q,R) + HCE(R,P )

It has been established that cross-entropy fulfills only the first two conditions, inform of Equation (5.1) due to lack of symmetry. Even though, calculations in this thesisdo not require validity of the last two conditions, it can be shown that a symmetrizedversion of cross-entropy can be constructed to satisfy condition 3. The symmetrical formof cross-entropy is [21]:

HCE(Q,P ) =n∑

i=1

qilogqi

pi+

n∑

i=1

pilogpi

qi(5.2)

5.2.3 Minimum Cross-Entropy and the LLBP Approach

If a digital image substitutes P in Equation (5.1) and its thresholded version substitutesQ, then the cross-entropy measure explains how good the thresholded image estimatesthe original image. Following this line of thought, the principle of minimum cross-entropy states that in order to find the solution (i.e. thresholded image) that bestestimates the original image, one should minimize the difference (cross-entropy) betweenthem.

Li and Lee [20] and Brink and Pendock [7] have designed image thresholding algo-rithms based on minimum cross-entropy. In both of them, Jayne’s monkey model [16]is used to interpret digital image as probability distributions. In the monkey model, animage is considered to be an empty grid of cells at which a group of monkeys throwa number of balls in a uniformly random fashion. The number of balls hitting a celldepicts its photon count or in other words its intensity level. Thus, a digital image oftwo-dimensional array of pixels (cells) is created. To make this image a true probabilitydistribution, one can normalize the intensity of each pixel.

The two algorithms above, along with two other approaches [36, 25], have beentested and evaluated on synthetic images as well as actual mammograms by Masek [21].Algorithms by Li and Lee [20] and Brink and Pendock [7] are mathematically equivalentexcept for the fact that roles of P and Q in Equation (5.1) are reversed. Masek [21] hascombined them into a single method designated as LLBP approach that has shown togive best result when used to segment components of a mammogram. LLBP approachis the basic idea behind the algorithm presented below. Formally stated, the mainprinciple in LLBP approach is that, given an image with histogram P, one strives tofind a thresholded histogram Q such that distance (cross-entropy) between P and Q isminimized.

5.2.4 The Complete Segmentation Algorithm

Image shown in Figure 5.2(a) will be used as an example of an input image, Iinput toexplain the different steps of this algorithm. Figure 5.2(b) shows the histogram hbreast

of this image. As mentioned in 5.2.1, black background can and should be exemptedfrom any calculations, therefore the histogram is calculated on the breast tissue part,Ibreast, of the image only. This algorithm is divided into two phases. First, an optimalthresholded image is acquired and then image processing operations are applied on thethresholded image to compensate for incorrect segmentations.

Phase I

Let CEM(f(.), t1, t2) denote a function that performs cross-entropy minimizations onthe image as will be explained below. f(.) is the image to be thresholded. Parameterst1 and t2 are explained later in this section.


(a) Image mdb032 from MIAS database[31].

(b) Image histogram of the pix-els belonging to breast tissue ofmammogram in Figure 5.2(a).

(c) Cumulative histogram from his-togram in Figure 5.2(b).

Figure 5.2: This figure demonstrates the relation between the image histogram and its cumu-lative histogram.

The minimization is performed on a range of intensity values. In this range, eachintensity value is chosen as a threshold and cross-entropy is calculated between theoriginal image and the thresholded image until the minimum cross-entropy is reached.The first step of this minimization is to calculate a suitable initial threshold value. Ex-perimentation shows that peak A corresponds to the pixels belonging to the glandulartissue (Figure 5.2(b)). Therefore, an optimal threshold will lie in the close neighbor-hood of mean of this peak. Empirical tests show that a threshold value near the leftend of peak A as an initial value often gives the desired optimal threshold. A ro-bust method to calculate such an initial value has been constructed here that uses thecumulative histogram [13] to detect the ascents and descents of the image histogram.Figure 5.2(c) shows the cumulative histogram of the breast tissue. Results from trainingdata show that this step gives optimal result if done on pixels within intensity range[max(Imin, 20),min(200, Imax)], where Imin and Imin are the minimum and maximumintensity value in the input image respectively. Values 20 and 200 are sent as parameterst1 and t2 respectively.


(a) Segmentation produced bythresholding in stage 1.

(b) Segmentation from Figure5.3(a) marked in the input im-age.

Figure 5.3: Illustrates the results from stage 1 of the algorithm explained in Section 5.2.4.These results pertain to the input image shown in Figure 5.2(a).

(a) Segmentation produced bythresholding in stage 2.

(b) Segmented region from stage2 marked in the input image tostage 2.

Figure 5.4: Illustrates the results from stage 2 of the algorithm explained in Section 5.2.4.

Observe the bends in the cumulative histogram curve in Figure 5.2(c). These bendscorrespond to ascends and descends of peak A in the image histogram. The line segment,joining the end points of cumulative histogram curve, is used to calculate the distancebetween points on the curve and the line segment. In Figure 5.2(c), point A is farthestaway from the line. Such a point has shown to be a good initial value to the cross-entropy minimization loop. Initial value calculated in this manner is also used to createa thresholded image. This thresholded image will be referred to as stage 0 threshold inthis thesis.

Depending on the size and structure of glandular tissue, the form of peak A candiffer. Calculations show that if the tip of this peak is flatter in form compared to thetip of a normal distribution i.e.


(a) The combined segmentationfrom stage 1 and stage 2, ob-tained by union of segmenta-tions in Figure 5.3(a) and 5.4(a).

(b) Segmentation from Figure5.5(a) marked in the input im-age from Figure 5.2(a).

Figure 5.5: Illustrates the final result produced at the end of phase I of the segmentationalgorithm described in Section 5.2.4.

if kurtosis(hbreast) < 3, (5.3)

then above mentioned method gives a good initial value. However, for images that give

kurtosis(hbreast) ≥ 3, (5.4)

such an initial value causes the minimization to converge too fast towards a local minimagiving a too low threshold value. It these cases, the intensity value with highest count inthe image histogram has proven to be a better initial value. This step of using maximumintensity count as initial value instead of using cumulative histogram is called the maxstep.

With a suitable initial value, cross-entropy minimization is performed to give athresholded image. This is referred to as stage 1 thresholding and is illustrated in Figure5.3. Let Ts1 be the threshold value from stage 1 and Is1 denote image pixels from Ibreast

that have intensity higher than Ts1. Then Is1 denotes glandular tissue according tostage 1 thresholding.

In many cases, stage 1 leaves out parts of glandular tissue due to too high thresholdvalue. Therefore thresholding through the function CEM is repeated in stage 2. Theinput image Iinputs2 to stage 2 is such that,

Iinputs2 ∈ Ibreast & Iinputs2 /∈ Is1, (5.5)

which implies that Iinputs2 consists of all the pixels that belong to the breast tissue exceptthose pixels that were segmented in stage 1 thresholding. It has been observed that theimage histogram of input image to stage 2 does not require the max step. Thereforethe initial value is chosen using the cumulative histogram method only. Parameters t1and t2, in this case, should be 20 and max(Iinputs2) respectively. Resulting thresholdedimage from this stage will be referred to as Ioutputs2 . Figure 5.4 visualizes result fromstage 2 thresholding for the image in Figure 5.2(a).

Segmentations from stage 1 and stage 2 are combined to create a single segmentedimage as shown in Figure 5.5. This image is then processed as explained in the nextsection.


(a) Input image pertaining toimage mdb041 from MIASdatabase [31].

(b) The segmentation obtainedfrom input image in Figure5.6(a). The region outlined bythe white rectangle is not part ofthe glandular tissue and there-fore must by removed.

(c) Before apply-ing the slicingalgorithm, thesegmentationfrom Figure5.6(c) is croppedand resized.

(d) Result of applying the slicingalgorithm on Figure 5.6(b).

Figure 5.6: Demonstrates an unwanted elongated segmentation that usually is a result of poormuscle segmentation.

Phase II

In this phase, the final segmentation from phase I is post-processed in order to removeregions that have a high probability of not belonging to the actual glandular tissue.These regions are a result of image noise and/or simply the fact that threshold based


segmentation always has its limitations. To remove unwanted regions and improve thesegmentation, a sequence of image processing operations has been devised, that can bedivided into five steps.

Step 1 The binary image, representing the segmentation from phase I, is labeled suchthat each separate cluster of white pixels is assigned a numerical value/label. Let thisimage be BWinput. The number of pixels, also known as area, of each labeled region isthen calculated to find regions with very large areas. An area limit is then calculated as

Alimit = Alargest/5 (5.6)

where Alargest is area of the largest labeled region. A new binary image, BWlargeinitial

is created that only contains regions from BWinput that have area greater than or equalto Alimit. As mentioned earlier, this step removes the numerous unwanted regions thatmight belong to fat tissue. However, there are certain cases for which further imageprocessing is necessary. An example of such a case is illustrated in Figure 5.6(a). Noticethe elongated upper region, marked by a rectangle, of segmentation of this image shownin Figure 5.6(b). This elongation is usually a result of inadequate extraction of muscletissue and is mostly not a part of the glandular tissue. To remove this unwanted elon-gation, changes in width of the segmented region in BWlargeinitial are calculated. Thepoint at which this change is relatively abrupt is most likely the foot of the elongation.

For this purpose, BWlargeinitial is sliced horizontally into 10 rectangular small imagesof equal heights. In each sliced image, width of the middle row is calculated to representthe width of the whole slice. Width is calculated as the distance between the left mostand the right most white pixel in that row. For each set of two consecutive slices,difference in width is calculated by subtracting width of lower slice from width of theupper slice. If this difference is greater than a certain limit, then the change in widthis too sharp revealing the presence of elongated upper region. For images in this thesis,this limit has been calculated empirically to be 140. However, certain conditions mustbe fulfilled to apply this limit on a segmentation as explained in the next paragraph.

It is important that a limit is calculated such that it has the same relativity to thewidest part of the segmentation. To ensure this, the segmented region is resized so thatthe widest part has width of 500 pixels. Ratio between height and width of the segmentedregion is preserved in all cases. This is followed by the slicing algorithm explained inthe previous paragraph. This resized version of Figure 5.6(b) is shown in 5.6(c). Centerof the slice, that shows width change higher than 140, is used to cut off the segment toexclude the unwanted elongation. This new segmentation is shown in 5.6(d). For imagein Figure 5.5(a), that has been used as an example earlier, corresponding processedsegmentation looks like Figure 5.7(a). Final segmentation created, BWlarge, at the endof this step thus contains only regions with large pixel count. A shortcoming of this stepis that too many small regions might have been excluded resulting is underdeterminedglandular tissue. Therefore following step are necessary to find any extra regions thatmust be included in the final result.

Step 2 For further processing, BWlarge is divided into 4 quadrants like a Cartesiancoordinate system such that the centroid of the largest region acts as the point of origin;see Figure 5.7(b). Since the position of mammilla is already known, its quadrant isdetermined. Let BWextra denote the image BWinput without the regions in BWlarge

i.e. an image with all the smaller regions that were excluded in step 1. Centroid for allthese regions is calculated and a new binary image is created that contains small regionswhose centroid is located in the same quadrant as the mammilla. Motivation behind thisstep is the anatomical fact that glandular tissue gets more concentrated and compactas one moves closer to mammilla from inside of the breast. Therefore an assumptionis made that all the regions that are located in the same quadrant as mammilla must


belong to the glandular tissue. This new binary image, BWmam, is shown in 5.7(c).As one moves farther away from mammilla, the probability of a small region belongingto glandular tissue decreases. Consequently, the regions in quadrants other than themammilla are not included here.

Step 3 In this step, small regions from BWextra, that were not selected in step 2,are processed. For each quadrant a rectangle and a circular sector is calculated. Letxmax and ymax be respective maximum horizontal and vertical distances achieved bysegmented pixels in BWlarge, then xmax and ymax are used as width and height of therectangle. The maximum radial distance from the origin, achieved by segmented pixelsin BWlarge is used as a radius to calculate the sector. All of the small regions, whosecentroid lies within the intersection of the rectangle and the sector, are accepted as partof the glandular tissue. In Figure 5.7(d), the sector and the rectangle for the quadrant1 is illustrated. This step is repeated for all the quadrants except the quadrant of themammilla. Regions extracted in this step result in image BWsmall; see Figure 5.7(e) forillustration.

Step 4 It was mentioned in Section 5.2.4 that a thresholded image is calculated instage 0. In several cases this thresholded image has shown to include parts of glandulartissue that are otherwise not segmented successfully in stage 1 and 2. Therefore, extraregions contributed by stage 0 thresholding are examined in this step for inclusion infinal segmentation.

To begin with, all the regions from stage 0 that lie in quadrant 2 and 3 are rejectedsince these regions are situated too close to chest wall where presence of glandular tissueis relatively low. The remaining regions must be present either in quadrant 1 or 4. Outof these regions, only those are accepted whose centroid lies below the top most pixel ofthe largest segmented region in BWlarge. Corresponding regions extracted in this stepfor Figure 5.2(a) are shown in Figure 5.7(f)

Final step The union of all the binary images, obtained in previous steps, is returnedas the final segmentation. Morphological closing is done to smooth the boundary ofthis segmentation. As a last step, any holes present in this segmentation are also filled.Resultant image is shown in Figure 5.8.


(a) Result at the end of step 1. (b) Division of the image into four quad-rants with centroid of the largest regionas point of origin.

(c) Extraction of small regions that liein the same quadrant as the mammilla.This is the resultant image from step 2.

(d) Defining the intersection of thesector and the rectangle for includingsmall regions in quadrants 1, 2, and 3.This is done in step 3.

(e) Regions extracted in relation withimage in Figure 5.7(d).

(f) Regions extracted in step 4 usingstage 0 thresholding mentioned in Sec-tion 5.2.4.

Figure 5.7: Illustrating results from step 1 to 5 of the phase II described in Section 5.2.4.


(a) The final binary segmentation ob-tained after the final step that involvesmorphological closing followed by fillingof holes.

(b) The final segmentation from in Fig-ure 5.8(a) marked in the original inputimage from Figure 5.2(a).

Figure 5.8: Illustration of final segmentation from algorithm presented in Section 5.2.

5.3. Feature Extraction Using Gabor Filter and Fractal Dimensions 43

5.3 Feature Extraction Using Gabor Filter and Frac-tal Dimensions

The KASP algorithm was devised and applied by Kasparis et al. [18] on syntheticdigital images. In this algorithm, images were pre-processed by filtering them withgabor functions as will be explained later. Images that are the focus of this thesisare mammograms, and before proceeding with the filtering step assumptions stated inSection 5.2.1 are applied to them.

5.3.1 Reasons For Filtering The Images

As mentioned in the introduction to this chapter, a clustering algorithm is used to createthe final segmentation in the KASP algorithm. The basic idea behind any clusteringalgorithm is that a set of different features is defined and each image pixel possesses acertain value for each feature. Number of features to be used and their characteristicsdepends on the goal to be achieved. Accordingly, there will be a vector of feature valuespertaining to each pixel. A measure of similarity is then defined e.g. the euclideandistance to study the discrepancy between these vectors. A clustering algorithm willthen divide these vectors into some predefined number of clusters in such a mannerthat similarity between any two feature vectors from the same cluster will ideally behigher than the similarity between two feature vectors from two different clusters. Theperformance of clustering is thus dependent on the type of features used. Thus a goodset of features will cause pixels, belonging to the same texture, fall into the same cluster.It is stated by Kasparis et al. [18] that FD alone is not a strong feature when it comesto texture segmentation by this approach. They propose the use of several gabor filterson an image to solve this problem with FD values.

Gabor Filter Functions

Gabor functions were originally designed as 1D functions. In the KASP method, a2D extension of these functions was used. This 2D extension is defined as a Gaussianfunction modulated by a complex sinusoid, defined by orientation θ, frequency ω of thesinusoid, and the standard deviations σx and σy of the Gaussian envelope as

Gab(g(x, y), h(x, y)) = exp

{− 1

2

(g(x, y)2

σ2x

+h(x, y)2

σ2y

) }cos(2πωg(x, y)) (5.7)

g(x, y) = x · cos(θ) + y · sin(θ) (5.8)h(x, y) = y · cos(θ)− x · sin(θ) (5.9)

Rangayyan [27] points of that there is no standard definition of 2D Gabor function, andseveral variations can be found in the literature.

5.3.2 The Variation Method

The variation method was first put forward by Dubuc et al. [11] and shown to be analternative and in some cases more accurate method of estimating fractal dimensionsthan the existing methods at that time. In the KASP algorithm, the variation methodwas adopted to calculate fractal dimension of image textures locally.

The variation method of estimating fractal dimensions locally in an image (or a2D surface in general) is based on the study of how the pixel intensities vary in aneighborhood of the each pixel. Consider an image I(x, y) of size N × N . We can


imagine this image as a 2D surface with intensity values depicting the height of eachpixel as we have done before. Let (x1, y1, I(x1, y1)) and (x2, y2, I(x2, y2)) represent twodifferent points on this surface, and I12 is the line joining them. Then the absolutevalue, |S|, of the slope of I12 goes to infinity as (x2, y2) tends to (x1, y1). Let us considera small neighborhood in an image where the texture is very rough i.e. the variationbetween pixel intensities is huge and sharp. In such a neighborhood, the rate at which|S| will go to infinity must be high due to large variations between pixel intensities. Thefractal dimension in the variation method is then defined as this rate at which |S| goesto infinity. Therefore a high value of fractal dimension must indicate a sharp variationof intensities and a very rough texture.

The description of how the variation method is implemented in the KASP algorithmis concise, and perhaps due to limited space, certain details are missing. Implementationof the variation method, presented below, is thus an interpretation by the author of thisthesis.

Implementing Variation Method

This method starts with the calculation of local intensity variation, which is defined asthe difference between highest and lowest intensity values in a small window. This isdone by centering the window on each pixel of the image. Consider the image shown inFigure 5.9. A square in the Figure shows an example of a window centered on a pixeldenoted by CP as in center pixel. The difference VCP , from the Figure, is then the localvariation mapped to CP. If the size of the window is T × T , where T = 2ε + 1, then theformal way of expressing the variation around a pixel is given by

Vε(x, y) = max(I(s, t))−min(I(s, t)), (5.10)

such that distance((x, y), (s, t)) = max(|x − s|, |y − t|) ≤ ε, and I is an input image tothe algorithm. This procedure returns a matrix VI of the same size as I. Each entryin VI holds the value of variation around the corresponding pixel in I. Calculation of avariation matrix is repeated for different window sizes given by ε = 1, 2, 3, . . . , εmax. LetVεi denote the variation matrix from ith ε value, then Vεi(x, y) is the variation mappedto pixel I(x, y).

Figure 5.9: An example of how variation is calculated locally in an image. VCP gives thelocal variation i.e. difference between the maximum and minimum intensity value. Size of thewindow in this image is 3× 3.

5.3. Feature Extraction Using Gabor Filter and Fractal Dimensions 45

For each Vεi, an average matrix Eεi

is calculated as explained here. Let us define awindow W of size L×L centered at Vεi(x, y). The average of Vεi over W is then associatedto the center pixel of W and saved as Eεi

(x, y). Finally, the FD value mapped to thepixel I(x, y) is calculated as the slope of the line that best fits the set Pa and Pb suchthat

Pa =

{log

(L

ε1

), log

(L

ε2

), · · · , log

(L

εmax

) }(5.11)

Pb =

{log

(L3

ε31· Eε1

), log

(L3

ε32· Eε2

), · · · , log

(L3

ε3max

· Eεmax

) }(5.12)

Repeating these steps for all pixels, an FD matrix is created that is of the same sizeas the input image.

5.3.3 Segmentation by Clustering - The Final Step

As mentioned earlier in Section 5.3.1, clustering of the image pixels is based on a setof different features. In the KASP algorithm, features are created by using differentvariants of gabor filtered versions of the image. For each filtered image, an FD matrixis calculated by the variation method. Kasparis et al. [18] demonstrated the algorithmby using four filter orientations and three different center frequencies giving 12 differentfilters. Thus each pixel had 12 different fractal dimension values as features.

Figure 5.11(b), 5.11(d) and 5.11(f) show examples of four different FD matricescreated in this thesis. The corresponding filtered images that were used to calculatethese FD matrices are illustrated in Figures 5.11(a), 5.11(c) and 5.11(e). The inputimage used in this case was image mdb001 from MIAS database [31] and is displayedin Figure 5.10. An ideal result would show a clear difference between the regions ofglandular tissue and rest of the breast tissue, which is not the case here. Therefore ithas not been possible to create the same result as shown by Kasparis et al. [18].

Figure 5.10: Image mdb001 from MIAS database [31] after application of assumptions statedin Section 5.2.1.


(a) Filtered image from gabor filter de-fined by σx = 4, σy = 6, θ = 0, ω = 0.1.

(b) FD matrix created using ε =1, 2, 3, 4, 5 and L = 8.

(c) Filtered image from gabor filter de-fined by σx = 4, σy = 6, θ = 90,ω = 0.1.

(d) FD matrix created using ε =1, 2, 3, 4, 5 and L = 8.

(e) Filtered image from gabor filter de-fined by σx = 4, σy = 6, θ = 45,ω = 0.5.

(f) FD matrix created using ε =1, 2, 3, 4, 5 and L = 8.

Figure 5.11: In this figure an example of application of gabor filter in combination withvariation method is illustrated. The input image for all the given examples is shown in Figure5.10.

5.4. Evaluation 47

5.4 Evaluation

A common method of measuring the performance, of an automated segmentation al-gorithm, is to compare the results with a ground-truth1. In the field of medical imageanalysis, a ground-truth is often obtained from markings manually created by human-experts. Some of the previously suggested methods for glandular segmentation weresummarized in Section 2.2. In most of these methods, the quality of segmentation wastested against ground-truths obtained from experts in radiology [6, 17, 28]. However,the procedure of evaluation was not the same in these cases because there does notexist a standardized evaluation method that is universally accepted by experts fromeither the field of digital image processing or the field of medicine. One of the principalreasons for absence of a standard evaluation method is that the inter-expert variationsamong manually created segmentations are very high. This makes it difficult to createan objective ground-truth.

In Section 5.1, it was mentioned that manual segmentations, from a panel of experts,are available for comparison in this thesis. Figure 5.12(c) shows an example of theseexpert segmentations for the image in Figure 5.12(a). Segmentation from the algorithmdeveloped in this thesis is shown in Figure 5.12(d) and will be referred to as the sys-tem segmentation. Rough visual inspection of these figures, indicates that the systemsegmentation is in the least agreement with the left most expert segmentation fromFigure 5.12(c). This fact can also be illustrated by superimposing this expert segmenta-tion with the system segmentation as shown in Figure 5.12(e). Simultaneously, Figure5.12(f) shows that the system segmentation has relatively larger area common with theright most expert.

From the previous paragraph, it can be concluded that an automated segmenta-tion algorithm cannot be tested against a ground-truth from just one expert, becausethe evaluation would be subjective. One way to create a rather objective ground-truthwould be to combine the results from a group of several experts. Olsen et al. [24] discussproblems involved in using manually created ground-truths and propose a method ofcreating an objective performance measure, based on expert markings, to assess seg-mentation algorithms. Olsen et al. [24] estimate an objective ground-truth from a groupof five expert markings for each image. Then a ranking function is defined that calcu-lates the degree of agreement of a given segmentation with the estimated ground-truth.All the five expert markings along with the system segmentation are individually rankedby this function. A rank sum is then defined which expresses the overall correctness ofexpert markings and the system segmentation for the whole set of test images. Ranksum calculated for the test set of 160 images in this thesis is shown in Table 5.1 below,where ranks are arranged in ascending order.

Rank Sum Source1 510 A1

2 496 A2

3 447 A3

4 414 A4

5 361 A5

6 334 A6 (System)

Table 5.1: Result from an evaluation method proposed by Olsen et al. [24]. The third columnshows the source of segmentation. The last row gives the result for segmentation from thealgorithm developed in this thesis.

The five expert markings used to estimate the ground-truth are denoted as A1...5 and1Ground-truth - In case of segmentation tasks, a ground-truth is the segmentation that fulfills all

the criteria of a completely correct result


A6 denotes the segmentation produced from the algorithm being evaluated. This resultindicates that the extraction of fibroglandular tissue by the system is on average in theleast agreement with the ground-truth. However, the rank sum also indicates that thesystem segmentation lies very close to the expert markings from source A5.

(a) Image mdb293 from MIASdatabase [31].

(b) Image from 5.12(a), withpectoralis muscle cropped beforesending it as input to the seg-mentation algorithm.

(c) Manual segmentations provided by a panel of experts in radiology.

(d) Segmentation givenby the algorithm fromSection 5.2 of Chapter 5.

(e) Image in Figure5.12(d) superimposedon the left most ex-pert marking in Figure5.12(c).

(f) Image in Figure5.12(d) superimposedon the right most ex-pert marking in Figure5.12(c).

Figure 5.12: Illustrating an example of inter-expert variations regarding the manual outliningof the glandular tissue.

Chapter 6

Discussion and Conclusions

6.1 Discussion

In the previous chapter, inter-expert variations were explained as one of the primaryreasons to why it is difficult to make an objective assessment of segmentation algorithms,for fibroglandular tissue in mammograms. Furthermore, an example of evaluation waspresented by showing results from a newly developed method [24]. Since this evaluationwas based on markings from a certain group of experts, it is possible to get differentranks for a different expert panel. Therefore the results presented in Table 5.1 can notbe considered as exact performance evaluation for the algorithm developed in this thesis,but rather an estimation of how good segmentation this algorithm produces on average.

Even though lot of emphasis has been placed on the issues of creating optimal seg-mentation of glandular tissue and designing an objective evaluation method, it is evenmore important to rethink the requirements that are posed on the segmentation results.At present, the requirements are that the glandular tissue extracted by the algorithmshould be as accurate as extraction by an expert in mammographic screening, and atthe same time as objective as possible. Given the evidence of inter-expert variationsprovided in this thesis, this requirement is rather impractical and unrealistic. A morepractical approach would be to aim at developing segmentation algorithms that would beused as an aid by medical professionals so that the final decision is not completely basedon results from the computer. Following this approach, a segmentation algorithm canbe said to perform well if it gives an approximate extraction of glandular tissue. Suchresults have definitely been achieved in this thesis, which is also indicated by Table 5.1that shows that extraction by the system is close to one of the expert segmentations.Moreover, among 160 test images that were segmented for evaluation, several imageswere segmented by the system with highest rank.

An alternate approach that has been partially tested in this thesis is based on fractalproperties of the texture in breast tissue in mammograms. Fractal dimension in generalexpress the roughness of a texture, where roughness is generally a measure of localchanges in pixel intensities. The composition of the breast tissue is such that thereare regions within the fibroglandular disc that are nearly as uniform and homogenousas certain regions belonging to the fat tissue. As a result, locally calculated fractaldimensions of the fat tissue and the fibroglandular tissue can be equivalent. Besides,glandular tissue can possess varying type of textures depending on the density of thetissue. In order to use textural properties as features for segmenting glandular tissue, itis necessary to define and classify breast tissue into different texture types. Because ofall these reasons, fractal dimension alone is a not a sufficiently strong tool for texturebased segmentation of anatomical landmarks in a mammogram.

Another aspect related to the performance of automated segmentation algorithms,especially for mammograms, is that certain limitations in the developed systems arise

49

50 Chapter 6. Discussion and Conclusions

due to the developer’s lack of medical knowledge and practice. An image analyst cannot assess the quality of a system that performs a highly complex task as segmentingX-ray images of a breast tissue. In his book, Rangayyan [27] presents an algorithm forglandular segmentation, named as the EM algorithm. Evaluation of the results fromthe algorithm was done in consultation with a radiologist experienced in mammography.Instead of performing a mathematical evaluation of segmentations against some ground-truth, the radiologist graded the segmentations by visual examination. Approximately80 % of the tested images were considered to be acceptable for CAD purposes. Thisevaluation is also subjected to the perceptions of the concerned expert. Nevertheless,the approach of involving medical professionals in evaluation is by itself an importantaspect of projects in CAD systems.

The goal of any CAD system is to be able to simulate objective and consistentdiagnostic decisions that match in quality with medical professionals who possess years ofexperience and knowledge. Hence, it seems only natural to incorporate their experiencein the process of development of these CAD systems.

6.2 Conclusions

A novel approach for segmentation of glandular tissue has been presented in this thesis.This approach was based on global histogram analysis of fat and glandular regions in amammogram. Based on the arguments in the previous section, it can be concluded thatthis algorithm is able to calculate a good approximation of the location of glandulartissue. Easy implementation and no requirement of training the algorithm are some ofits attractive features as opposed to the methods based on kNN classifiers or ANN.

Another conclusion that can be drawn is that the evaluation of segmentation algo-rithms for mammograms is very complex and requires further research. In fact, forma-tion of an objective evaluation method is as important as construction of the segmen-tation routines that are being evaluated. Moreover, experts in field of mammographyshould be involved in this process. Simultaneously, objectivity of the systems should bemaintained by not adjusting them to a specific individual or group assessment.

6.3 Future Work

The segmentation method presented in Section 5.2 of Chapter 5, shows good potential.Phase II of this method, which is based on morphological processing of the segmentationresult from phase I, requires further refinement. In this phase, the slicing algorithm usesthe same number of slices to process binary segment from all images. For better accuracy,this number has to be automatically determined according to the height of the segmentbeing processed. Additional spatial information such as, the possibility of existence ofglandular tissue at different distances from the nipple, can also be used to create betterresults.

Rangayyan [27] also refers to scientific works in which the intensity values of thepectoralis muscle and the glandular tissue have been found to be related. Therefore, aninteresting study would be to include the pectoralis muscle in the input image to thesegmentation algorithm. If there is a specific relation between the intensity values ofthe muscle region and the glandular region, then this relation can be used in phase II.

In this thesis, gabor filters were used to process the image before applying the vari-ation method. Even though, the initial results do not seem encouraging, it should benoted that all the possible values of gabor filter parameters were not investigated. Inthe literature Rangayyan [27], gabor functions have been used to study tissue texturein mammograms with interesting results. Testing different center frequencies and otherparameters of gabor filters in combination with the variation method can also be asubject of future work.

References

[1] Digital database for screening mammography (ddsm).University of South Florida Digital Mammography,http://marathon.csee.usf.edu/Mammography/Database.html, August 2002.

[2] Breast anatomy. http://www.mentorcorp.com/global/images/breast-anatomy.jpg,March 2006.

[3] Definition of cross-entropy.http://www.cse.unsw.edu.au/ billw/cs9444/crossentropy.html, March 2006.

[4] Fractal model of a leaf.http://www.cad.zju.edu.cn/home/zhaoyb/image/fractal leaf big.GIF, March 2006.

[5] Magnified snow crystals. California State University, LAhttp://www.calstatela.edu/faculty/acolvil/glaciers/snowflake.jpg, March 2006.

[6] K. Bovis and S. Singh. Classification of mammographic breast density using acombined classifier paradigm. PANN Research, Department of Computer Science,University of Exeter, Exeter, UK. In 4th International Workshop on Digital Mam-mography, pages 177–180, 2002.

[7] A. D. Brink and N. E. Pendock. Minimum Cross-Entropy Threshold Selection.Pattern Recognition, 29(1):179–188, 1996.

[8] C. J. Burdett and M. D. Desai. Localized fractal dimension measurement in digitalmammographic images. In B G Haskell and H M Hang, editors, Visual Communi-cations and Image Processing, volume 2094, pages 141–151, October 1993.

[9] C. B. Caldwell, S. J. Stapleton, D. W. Holdsworth, R. A. Jong, W. J. Weiser,G. Cooke, and M. J. Yaffe. Characterisation of mammographic parenchymal patternby fractal dimension. Physics in Medicine and Biology, pages 235–247, February1990.

[10] Community Hospital of Bremen. A Mammogram Machine.http://www.bremenhospital.com/Support/mammo.jpg, March 2006.

[11] B. Dubuc, C. Roques-Carmes, C. Tricot, and S. W. Zucker. Variation method:a technique to estimate the fractal dimension of surfaces. In T R Hsing, editor,Visual Communications and Image Processing II, SPIE Proceedings, volume 845,pages 241–+, january 1987.

[12] F. Georgsson. Algorithms and Techniques for Computer Aided MammographicScreening. PhD thesis, UMINF 01.15 Umea University, Department of Comput-ing Science, 2001.

[13] R. C. Gonzalez and R. E. Woods. Digital Image Processing. Prentice-Hall, Inc.,second edition, 2002. ISBN 0-201-18075-8. International Edition.

51

52 REFERENCES

[14] S. H. Heywang-Kobrunner, I. Schreer, and D. D. Dershaw. Diagnostic Breast Imag-ing. Thieme Medical Publishers, Inc., 1997.

[15] Imaginis Corporation. How Mammography is Performed: Imaging and Position-ing. http://imaginis.com/breasthealth/mammography imaging.asp. Sponsored bySiemens medical, March 2006.

[16] E. T. Jaynes. Monkeys, Kangaroos and N. Maximum Entropy and Bayesian Meth-ods in Applied Statistics, pages 1–25, 1986.

[17] N. Karssemeijer. Automate classification of parenchymal patterns in mammograms.Physics in Medicine and Biology, 43:365–378, February 1998.

[18] T. Kasparis, D. Charalampidis, M. Georgiopoulos, and J. Rolland. Segmentationof textured images based on fractals and image filtering. The Journal of PatternRecognition Society, 34, 2001.

[19] S. Kullback. Information Theory and Statistics. New York: John Wiley, 1959.

[20] C. H. Li and C. K. Lee. Minimum Cross-Entropy Thresholding. Pattern Recogni-tion, 26(4):617–625, 1993.

[21] M. Masek. Hierarchical Segmentation Of Mammograms Based On Pixel Inten-sity. PhD thesis, Centre for Intelligent Information Processing Systems, School ofElectrical, Electronic, and Computer Engineering. University of Western Australia,Crawley, WA, 6009, February 2004.

[22] A. B. Nilsson. Fractal dimensions and projections. Department of MatemathicsUniversity of Umea, November 2005.

[23] C. Olsen. Automatic Assessment Of Mammogram Adequacy. Lic. thesis, Depart-ment of Computing Science, Umea University, ISBN 91-7305-894-7, 2005.

[24] C. Olsen, A. Mukhdoomi, and F. Georgsson. Assessing ground truth basedon subjective markings. In F. Georgsson and N. Borlin, editors, Proceedings of theSymposium of the Swedish Society of Automatic Image Analysis, pages 37–40. 2006.

[25] N. R. Pal. On Minimum Cross-Entropy Thresholding. Pattern Recognition, 29(4):575–580, 1996.

[26] H-O Peitgen and D. Saupe, editors. The Science of Fractal Images. SpringerVerlag, Berlin, Germany / Heidelberg, Germany / London, UK / etc., 1988. ISBN0-387-96608-0 (New York), 3-540-96608-0 (Berlin).

[27] R. M. Rangayyan. Biomedical Image Analysis. Biomedical Engineering Series. CRCPress LLC, 2005. ISBN 0-8493-9695-6.

[28] P. K. Saha, J. K. Udupa, E. F. Conant, D. P. Chakraborty, and D. Sullivan. Breasttissue density quantification via digitized mammograms. IEEE Transactions OnMedical Imaging, 20(792–803):575–580, august 2001.

[29] J. T. Sample. Computer Assisted Screening Of Digital Mammogram Images. Adissertation submitted to the Graduate Faculty of the Louisiana State Universityand Agricultural and Mechanical College in partial fulfillment of the requirementsfor the degree of Doctor of Philosophy, Department of Computer Science, 2005.

[30] Socialstyrelsen. Mammography - Questions and Answers.http://www.sos.se/FULLTEXT/114/2002-114-5/2002-114-5.htm#11, March 2006.

REFERENCES 53

[31] J. Suckling, J. Parker, D. R. Dance, S. Astley, I. Hutt, C. R. M. Boggis, I. Ricketts,E. Stamatakis, N. Cerneaz, S-L Kok, P. Taylor, D. Betal, and J. Savage. The mam-mographic image analysis society digital mammogram database. In A. G. Gale,S. M. Astley, D. R. Dance, and A. Y. Cairns, editors, Digital Mammography, Com-putational Imaging and Vision, pages 375–378. Elsevier Science, The Netherlands,1994.

[32] A. Soderstrom. Using wavelet methods to extract microcalcifications in mammo-grams. Master’s thesis, Department of Computing Science, Umea university, octo-ber 1997.

[33] E. W. Weisstein. Lebesgue covering dimension. From MathWorld – A WolframWeb Resource. http://mathworld.wolfram.com/LebesgueCoveringDimension.html,March 2006.

[34] Wikipedia contributions. Wikipedia, the free encyclopedia.http://en.wikipedia.org/wiki/Carcinoma in situ, March 2006.

[35] Wikipedia contributions. Wikipedia, the free encyclopedia.http://en.wikipedia.org/wiki/Von Koch curve, March 2006.

[36] Y. Xue, Y. Zhang, and X. Lin. Threshold selection using cross-entropy and fuzzydivergence. Proceedings of the SPIE, 3561:152–162, 1998.

[37] A. M. Zdunek. MD, Department of Diagnostic Radiology, Umea University Hospi-tal, March 2006.

54 REFERENCES

Appendix A

Tables

N/k 1(L = 2) 2(L = 4) 3(L = 8) 4(L = 16) 5(L = 32) 6(L = 64) 7(L = 128) 8(L = 256)

32 1024 2048 3072 4096 5120 6144 7168 819264 4096 8192 12288 16384 20480 24576 28672 32768

128 16384 32768 49152 65536 81920 98304 114688 131072256 65536 131072 196608 262144 327680 393216 458752 524288512 262144 524288 786432 1048576 1310720 1572864 1835008 2097152

1024 1048576 2097152 3145728 4194304 5242880 6291456 7340032 83886082048 4194304 8388608 12582912 16777216 20971520 25165824 29369128 335544324096 16777216 33554432 50331648 67108864 83886080 100663296 117440512 1342177288192 67108864 134217728 201326592 268435456 335544320 402653184 469762048 536870912

Table A.1: Number of storage bits for various values of N and k Gonzalez and Woods[13]. Also see Section 4.1.1.

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