NOVEL IMAGE PROCESSING METHODS BASED ON FUZZY LOGIC …

Budapest University of Technology and Economics

Department of Control Engineering and Information Technology

Budapest, Hungary

NOVEL IMAGE PROCESSING METHODS BASED

ON FUZZY LOGIC

Ph. D. Thesis

UJ FUZZY ALAPU KEPFELDOLGOZASI

ELJARASOK KIDOLGOZASA

Ph. D. ertekezes

Laszlo Szilagyi

In Partial Fulfillment of the Requirements for

the Degree of Doctor of Philosophy

Supervisor:

Dr. Zoltan Benyo

Department of Control Engineering and Information Technology

Faculty of Electrical Engineering and Informatics

Budapest University of Technology and Economics

October, 2008

Declaration

Undersigned, Laszlo Szilagyi, hereby state that this Ph. D. Thesis is my own work wherein I

have used only the sources listed in the Bibliography. All parts taken from other works, either

in a word for word citation or rewritten keeping the original contents, have been unambiguously

marked by a reference to the source.

Nyilatkozat

Alulırott Szilagyi Laszlo kijelentem, hogy ezt a doktori ertekezest magam keszıtettem es abban

csak a megadott forrasokat hasznaltam fel. Minden olyan reszt, amelyet szo szerint, vagy azonos

tartalomban, de atfogalmazva mas forrasbol atvettem, egyertelmuen, a forras megadasaval

megjeloltem.

Budapest, 2008. 10. 06.

Szilagyi Laszlo

The reviews of this Ph. D. Thesis and the record of defense will be available later in the Dean

Office of the Faculty of Electrical Engineering and Informatics of the Budapest University of

Technology and Economics.

Az ertekezesrol keszult bıralatok es a jegyzokonyv a kesobbiekben a Budapesti Muszaki es Gaz-

dasagtudomanyi Egyetem Villamosmernoki Karanak Dekani Hivatalaban elerhetoek.

i

Abstract

This thesis incorporates three main topics. First, a hybrid c-means clustering model is proposed

as a generalization of fuzzy, possibilistic, and hard c-means algorithms. Second, novel image

processing methods are introduced, as applications of the hybrid clustering. Finally, a virtual

endoscope model is proposed and implemented, based on the clustering and image segmentation

results.

The hybrid clustering model unifies the classical, fuzzy and possibilistic logics within a single

objective function, introducing an infinite number of different clustering algorithms, which are

obtained by varying the two tradeoff parameters. Based on several tests, some recommendations

have been proposed for the optimal choice of the tradeoff parameters. Further on, an alternative

formulation of the hybrid model is proposed: by dropping the partition matrices and computing

a few sums instead, the memory requirements of the algorithm is drastically reduced, which

makes it suitable for hardware implementation. We have analyzed the properties of the so-called

suppressed FCM algorithm, and found an alternative optimal suppression, achieved as a special

case of the proposed hybrid clustering model. Series of tests have been performed, which revealed

the properties of the hybrid clustering: it is more robust, has quicker convergence and produces

finer partition quality, than earlier c-means clustering methods. The proposed clustering method

is universal, in the sense that it is suitable to classify any kind of vectorial data, for which the

operation of weighted averaging is defined.

Within the scope of image processing, our main goal was to create novel methods for ac-

curate and efficient segmentation of MR brain images. In this order, we have proposed a new

procedure for quick segmentation of MR images contaminated with high-frequency Gaussian and

impulse noises, using histogram-based c-means clustering. Further on, a concept of multi-stage

inhomogeneity compensation was introduced, that was involved into two different segmentation

procedures. We have also proved that the efficiency and accuracy of these c-means based proce-

dures are enhanced by the proposed hybrid clustering model.

Based on our clustering and image segmentation methods, a novel concept of virtual endoscope

was introduced, which was later implemented using 3-D surface models and 3-D computer graphics

techniques.

ii

Kivonat

Jelen ertekezes harom nagy temaval foglalkozik. Egyreszt a fuzzy logikat alkalmazo klaszterezo

algoritmusok elmeletet egeszıti ki egy hibrid osztalyozasi modellel. Masreszt uj kepfeldolgozasi

eljarasokat vezet be, melyekben alkalmazast nyer a hibrid klaszterezesi algoritmus. Harmadreszt

egy virtualis endoszkop modell megalkotasat es gyakorlati megvalosıtasat ismerteti, a javasolt

osztalyozas es kepfeldolgozas alapjan.

A hibrid klaszterezesi modell egyesıti a hagyomanyos, fuzzy es lehetosegfuggvenyek alapjan

osztalyozo algoritmusokat, bevezetve vegtelen sok uj algoritmust, melyet ket sulyozasi parameter

segıtsegevel er el. Szamos adathalmazon vegzett elemzesek alapjan javaslatot tettunk az

osztalyozas idealis parametereinek beallıtasara. Tovabba javasoltunk egy alternatıv modszert

az algoritmus hardveres megvalosıtasara: ez esetben a partıcios matrixok elhagyasaval az

osztalyozashoz szukseges memoria nagysagrendekkel csokkentheto. Elvegeztuk a szakirodalom-

ban fellelheto un. elnyomott fuzzy klaszterezo algoritmus reszletes analitikus elemzeset, ossze-

hasonlıtottuk a javasolt hibrid klaszterezesi eljaras egy sajatos esetevel, majd javasoltunk egy

optimalis elnyomasi modszert a fuzzy klaszterezo algoritmushoz. Sorozatos tesztelesek soran

megallapıtottuk, hogy a javasolt hibrid klaszterezo algoritmus robusztus, valamint gyorsabban

es hatekonyabban osztalyoz, mint elodei. A javasolt osztalyozasi algoritmus univerzalis, elvileg

barmely olyan vektorialis adat osztalyozasara kepes, ahol ertelmezheto a ket vagy tobb vektor

sulyozott atlaga.

A kepfeldolgozasi algoritmusok kereteben az agyi MRI kepek hatekony es precız szeg-

mentalasat tuztuk ki celul. Ennek megvalosıtasara egy uj eljarast javasoltunk a magas frekvencias

zajokat tartalmazo kepek gyors szuresere es fuzzy elven torteno osztalyozasara. Tovabba java-

soltunk egy tobblepcsos kompenzalasi elvet az MRI kepek inhomogenitasanak kompenzalasara,

melynek alapjan ket kulonbozo szegmentalasi eljaras is megvalosult. Bebizonyıtottuk, hogy ez

utobbi modszerek hatekonysagat tovabb lehet fokozni a hibrid klaszterezo algoritmus bevetesevel.

A javasolt klaszterezo es kepfeldolgozo eljarasok alapjan elkeszıtettunk egy sajat kon-

cepcioju agyi virtualis endoszkop modellt, melyet meg is valosıtottunk 3D feluletmodellek es

3D szamıtogepes grafika segıtsegevel.

iii

Contents

1 Introduction 1

2 Contributions to the Development of c-Means Clustering Models 5

2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Related work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2.1 From Zadeh’s fuzzy theory to fuzzy c-means clustering . . . . . . . 9

2.2.2 The suppressed fuzzy c-means algorithm . . . . . . . . . . . . . . . 16

2.2.3 Relaxation of the fuzzy probabilistic constraint . . . . . . . . . . . 18

2.2.4 Optimization via evolutionary computation . . . . . . . . . . . . . . 22

2.2.5 FCM-based and FCM-like competitive clustering algorithms . . . . 24

2.3 An analysis of the suppressed FCM algorithm . . . . . . . . . . . . . . . . 26

2.3.1 Competition by suppression . . . . . . . . . . . . . . . . . . . . . . 26

2.3.2 May one call the s-FCM algorithm optimal? . . . . . . . . . . . . . 30

2.4 A novel hybrid c-means clustering model . . . . . . . . . . . . . . . . . . . 34

2.4.1 The alternating optimization approach . . . . . . . . . . . . . . . . 34

2.4.2 Clustering results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

iv

Contents v

2.4.3 Reducing memory requirements and execution time in case of huge

amounts of data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

2.4.4 The evolutionary computation approach . . . . . . . . . . . . . . . 51

2.5 Suppressed and optimally suppressed FCM, revisited . . . . . . . . . . . . 57

2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3 Medical Image Segmentation Using FCM-Based Algorithms 61

3.1 Magnetic resonance imaging . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.2 Image segmentation techniques . . . . . . . . . . . . . . . . . . . . . . . . 66

3.3 Quick segmentation of MR brain images . . . . . . . . . . . . . . . . . . . 67

3.3.1 Spatial constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.3.2 Proposed method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.3.3 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . 75

3.4 Segmentation of MR brain images in the presence of intensity inhomogeneity 80

3.4.1 FCM-based bias and gain field estimation . . . . . . . . . . . . . . 81

3.4.2 Smoothening filter . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.4.3 Multi-stage bias and gain field estimation . . . . . . . . . . . . . . . 84

3.4.4 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.4.5 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.5 Application of the hybrid clustering model for inhomogeneity compensation

and image segmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

3.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4 Virtual Endoscopy 92

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

vi Contents

4.2 Existing applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.3 Existing methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.4 Proposed methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.4.1 Specification of an own virtual endoscopy system . . . . . . . . . . 99

4.4.2 Input data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

4.4.3 2-D image processing tasks . . . . . . . . . . . . . . . . . . . . . . . 100

4.4.4 3-D surface reconstruction for virtual endoscopy . . . . . . . . . . . 100

4.4.5 The enhanced marching cube algorithm . . . . . . . . . . . . . . . . 101

4.4.6 Interactive visualization and measurements . . . . . . . . . . . . . . 103

4.5 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5 Contributions 106

5.1 First Thesis Group: Contributions to the Development of c-Means Cluster-

ing Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.2 Second Thesis Group: Medical Image Segmentation Using FCM-Based Al-

gorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.3 Third Thesis Group: Virtual Endoscopy . . . . . . . . . . . . . . . . . . . 108

6 List of Publications and Independent Citations i

6.1 Publications related to the thesis . . . . . . . . . . . . . . . . . . . . . . . i

6.2 Other publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Bibliography xvi

Chapter 1

Introduction

Nowadays the modern medical diagnosis is unthinkable without using medical imaging.

The outstanding development of atomic physics in the 20th century made it possible for

the physician (doctor) to look inside the patient’s body and to locate its structures and

possible malformations. As time went by, the growing knowledge concerning the biological

effects of physical phenomena revealed the necessity of such imaging techniques, which

• do not need actual penetration of the body

• do not cause pain or discomfort to the patient

• have minimized side effects (e.g. radiation).

These conditions are mostly fulfilled by magnetic resonance imaging (MRI) and ultra-

sound imaging.

The extremely fast evolution of personal computers, mostly from the point of view

of computation speed and storage space, enabled us to perform algorithms having hard

computational load and processing huge amounts of data. For example, the processing of

a set of MRI slices or cross sections, belongs to this category. Modern medical imaging

software packages require such algorithms, which can perform image segmentation and

1

2 Chapter 1. Introduction

interpretation efficiently, accurately, and automatically, without relevant amount of human

interaction.

Medical image processing contains several processing steps: beginning with the pro-

cedures assisting the image formation, and continuing with posterior image enhancement

(filtering, contrast enhancement, inhomogeneity compensation), image registration (inte-

gration of several 2-D images into a 3-D volume, by detecting their relative position with

each other) and segmentation (distinguishing the objects from each other or from back-

ground within 2-D images or 3-D volumes), and the interpretation of the recognized and

localized objects.

With this work I intend to contribute to the development of image segmentation tech-

niques and the supporting theories, by introducing new algorithms and procedures. Of

course, the final product of my individual work cannot compete with imaging systems

developed by multinational companies investing thousands of man-years of effort, but the

efficiency and accuracy of image processing algorithms that stand behind such products

still can be and still need to be improved. My basic research activity concentrates on

novelties in clustering theory, and as an application of them, I will propose some efficient

and accurate medical image processing methods.

Shortly after the introduction of fuzzy logic, by Zadeh [196] in 1965, several applica-

tion fields emerged and they continued to grow and multiply with time: control systems

[84, 108, 172], intelligent cooperative robot systems [14, 80], speech recognition [114], com-

puter vision and image processing [60], portfolio management systems [173], medical di-

agnosis systems [152]. Besides all these, fuzzy logic opened the way to the development

of sophisticated data classification and clustering methods [149, 20]. I intend to place my

work within this streamline, by contributing to the development of data clustering based

on fuzzy logic, and giving it some applications in medical image processing and virtual

endoscopy.

3

The remainder of this work is structured as follows. Chapter 2 presents my contributions

in the development of c-means clustering theory. Chapter 3 introduces and evaluates some

novel image processing methods based on c-means clustering models. Chapter 4 gives

an account of my contributions to the development of a virtual endoscope, supported by

the results of the previous chapters. Chapter 5 summarizes the contributions of my PhD

research activity.

Acknowledgements

This thesis is the result of my PhD research work conducted at Budapest University of Tech-

nology and Economics, Department of Control Engineering and Information Technology.

Some parts of the work were finished at Sapientia Hungarian University of Transylvania,

Department of Electrical Engineering.

I would like to thank Professor Dr. Zoltan Benyo, my supervisor at the Department of

Control Engineering and Information Technology, Budapest University of Technology and

Economics, for his professional guidance and unrestrained help during the research work.

Without his lead and support, my work would not have been possible.

I would like to thank Professors Dr. Peter Arato and Dr. Laszlo Szirmay-Kalos, former

and current heads of Department of Control Engineering and Information Technology, and

all of my colleagues from the Department and from the Biomedical Engineering Laboratory,

for their encouragements and working conditions provided during my work.

Also, I would like to express my sincere gratitude to Professor Dr. Laszlo David,

rector of Sapientia - Hungarian Science University of Transylvania, Mr. Tibor Bıro (former

physics teacher and current friend), and Professor Dr. Francis Rodes (Universite Bordeaux

I, France). They are the ones, besides my parents, who had the most valuable influence

upon my mind during my undergraduate and PhD studies.

I would like to acknowledge Dr. Attila Frigy (County Medical Clinic of Marosvasarhely),

Dr. Sandor Miklos Szilagyi, David Iclanzan, Tamas Vajda, and all my colleagues from

4 Chapter 1. Introduction

the Department of Electrical Engineering, Sapientia - Hungarian Science University of

Transylvania, for the given help and support during the research work.

I would like to thank for the research support received from

• Hungarian National Research Program (OTKA) (Grants # T069055, T042990,

T029830, U47793),

• Domus Hungarica Scientiarum et Artium Foundation,

• Pro Progressio Foundation,

• Communitas Foundation of Transylvania,

• Hungarian National Office for Research and Technology,

• Sapientia Institute for Research Programmes,

and the Hungarian Ministry of Education for the scholarship provided during my PhD

studies.

Finally, I would like to express my gratitude to my parents, for their unrestrained

support and infinite patience.

Chapter 2

Contributions to the Development of

c-Means Clustering Models

2.1 Definitions

The main objective in pattern recognition is to separate a set of objects O = {o1, o2, . . . , on}

into a given number of groups. Each object can be described by its physical or behavioral

properties. These properties, encoded into numerical and logical data, may serve as fea-

tures that allow us to distinguish different types of objects. Although all properties can

be used as features, it is not a good choice to do so, because that would cause an un-

necessary computational load to the classifier algorithms, without having relevant benefits

to the quality of the result. Therefore the feature vector, which is an ordered collection

of features, usually consists of those selected properties of the objects, which are sup-

posed to provide good separation characteristics. The set of feature vectors is denoted by

X = {x1, x2, . . . ,xn}, where vector xk consists of the feature variable values of object ok,

k = 1 . . . n.

For example, if we wish to distinguish a polar bear from a rattlesnake, the number

of eyes is not a relevant property, because both have two eyes. However, if we consider

5

6 Chapter 2. Contributions to the Development of c-Means Clustering Models

the number of legs, this property clearly distinguishes the two individuals. This is why,

when the set of objects we need to group consists of polar bears and rattlesnakes only, the

number of legs is not only a property, but also a good choice for a feature variable.

Most pattern recognition problems have the following main procedural steps: pattern

generation, pattern selection, classifier design, and system evaluation. Feature generation

comprises all the methods that convert physical properties into feature variables. Feature

selection has the main goal to select those generated feature variables, which have relevant

separation abilities. Classifier design represents the choice of a classifier algorithm that suits

the formulated problem and its efficient implementation. System evaluation represents the

validation of the classification process.

Pattern recognition problems, from the point of view of the employed classification

algorithm, can be separated into two main branches: supervised and unsupervised clas-

sification. Supervised classification means that we have a set of labeled feature vectors

called learning data, based on which the classifier algorithm can develop its separation cri-

teria, which will be later applied to separate the test data set. Unsupervised classification,

also known as clustering, means that the input feature vectors need to be separated into

groups based on similarities characterized with so-called proximity measures. This work

is restricted to the study of clustering algorithms and their applications in image process-

ing. Proximity measures (PM) are functions that quantify the similarity or dissimilarity

between two feature vectors, a feature vector and a cluster, or two clusters. In this order, a

PM can be a similarity measure (SM) if it gives larger values for more similar vectors, or a

dissimilarity measure (DM) if it is proportional with the distance or dissimilarity between

vectors.

Classifier design in case of unsupervised algorithms addresses two main issues:

1. Clustering criterion refers to the type or shape of the expected clusters: besides

the most frequent agglomerative clusters, in some applications linear [63], circular

[43, 58], or elliptical [11, 44, 90, 92] etc. clusters are encountered.

2.1. Definitions 7

2. Clustering algorithm refers to the choice of a specific algorithmic approach employed

to obtain the clusters.

Whatever clustering method is being performed, each cluster is represented by one

(or sometimes more than one) representative element, which is also known as prototype or

centroid (this latter only in case of agglomerative clusters), and is denoted by vi, i = 1 . . . c.

In case of agglomerative clusters, these prototypes are vectors having the same dimensions

as the input feature vectors. Of course, in case of clusters of a specific shape (e.g. elliptical

clusters), the prototype is defined as an ordered collection of parameters that describe the

given shape.

Having performed the clustering, there are two more steps to accomplish: the validation

or evaluation of clustering results, and the interpretation of the results. Several cluster

validity indices have been introduced to validate the clustering results [23, 129, 178, 187].

Their usage depends on the chosen clustering algorithm. The interpretation of the results

mainly depends on the scientific field where input data come from. For example, if the pixel

intensity values of a medical image are clustered, then different clusters will be interpreted

as different tissue types. Basically all clustering methods aim at finding the ideal positions

of the cluster prototypes, so that the input feature vectors are situated optimally among

or around them. There are several clustering principles, which can be applied at the

formulation of clustering algorithms. In the followings we will make an attempt to list

them:

• Clustering algorithms may use one feature vector at a time or all feature vectors at

the same time when updating its cluster prototypes. The former scheme is called

feature mode algorithm, while the latter is referred to as batch mode algorithm.

• The term sequential clustering refers to that feature mode clustering approach, which

looks at the input feature vectors only once and immediately decides the cluster

to which the vector belongs, based on the nearest neighbor rule or some similar


criterion [176]. These algorithms are not efficient; that’s why some post-processing

adjustments like relabeling have been proposed.

• The term hierarchical clustering refers to the approach, which performs the formation

of clusters gradually. In this order, the agglomerative hierarchical clustering starts

from singleton clusters and gradually unifies couples of clusters selected by some

optimality criterion (e.g. highest similarity). Divisive hierarchical clustering initially

considers all input feature vectors in the same cluster, and then iteratively decides

which cluster to cut into two and how to separate the elements of the divided cluster.

The optimal cut can be suitably established based on the entropy of the clusters.

Hierarchical algorithms usually work in batch mode, and are extremely slow in their

classical formulation.

• One of the most commonly used clustering approaches is based on the optimization

of an objective or cost function. The cost function generally contains squares of

distances between cluster prototypes and input vectors. Most of these algorithms use

a previously set number of clusters (usually denoted by c), and that’s why they (with

some exceptions discussed later) are collectively called c-means clustering methods.

There are several branches and versions of c-means clustering, distinguished by the

logic applied to assign the degrees of membership to which given vectors belong to

different clusters, and the extra cost terms introduced to enhance some behavioral

properties of the basic version of the applied algorithmic scheme. Most c-means

approaches work in batch mode.

• Competitive clustering is also an objective function driven clustering approach, fun-

damentally different from c-means methods. Input vectors are considered one by one,

and the cluster prototypes compete for each vector: it is established, which cluster

prototype is situated the closest to the given feature vector, and the winner makes a

step towards the feature vector, according to the previously set and monotonically de-

creasing learning rate. Enhanced competitive algorithms also update the non-winner

2.2. Related work 9

prototypes, but those use a considerably smaller learning rate.

• Several further clustering schemes exist that stem from different ideas. For example:

clustering based on graph theory [47], binary morphology clustering algorithms [112],

simulated [83] and deterministic annealing [146], etc.

The current chapter is dedicated to the enhancement and efficient combination of c-

means and competitive algorithmic schemes.

2.2 Related work

2.2.1 From Zadeh’s fuzzy theory to fuzzy c-means clustering

Data clustering was one of the first research fields, where the fuzzy set theory received its

applications. Only a few years after Zadeh [196] established the notion of fuzzy sets in

1965, Ruspini [149] introduced the fuzzy partition into clustering theory. With this first

step, the hard c-means clustering algorithm (HCM) [105, 164] received a way towards its

fuzzy extension. The first formulation of fuzzy clustering was given by Dunn [46], and

later it was extended by Bezdek [20] to more general cases in 1981, by establishing the

well-known fuzzy c-means algorithm.

The hard c-means clustering problem, often referred to as k-means clustering or Lloyd

algorithm, optimizes the following cost function:

JHCM =n∑

k=1

c∑i=1

hik||xk − vi||2A , (2.1)

where hik = 1 if vector xk is assigned to cluster having the prototype vi and hik = 0

otherwise, while || · ||A stands for the generalized norm defined as ||s||A =√

sT As, where

A is a positive definite square matrix. Each vector xk is assigned to exactly one cluster at

a time, which can be formulated asc∑

i=1

hik = 1 ∀k = 1 . . . n. This constraint is necessary in


order to avoid the trivial minimum of the cost function JHCM obtained when all hik values

are set to zero.

The optimization of this cost function is achieved using the following alternating opti-

mization (AO) scheme:

1. Initialize vi, i = 1 . . . c with randomly chosen input vectors that differ from each

other. This is likely to be possible, unless we have an ill-posed problem.

2. For each feature vector xk, find cluster prototype vi for which ||xk−vi||A is minimal.

Set hik = 1, and hjk = 0 for any j ∈ {1, 2, . . . , c} \ {i}. Ties are resolved arbitrarily.

3. Update the cluster prototypes according to the following formula:

vi =

n∑k=1

hikxk

n∑k=1

hik

. (2.2)

In other words, set each cluster prototype equal to the mean of the vectors belonging

to the cluster. Singularity may occur in this formula if a cluster has no elements, but

this is hardly possible with suitably chosen initial prototypes [5].

4. Repeat steps 2-3 until cluster prototypes converge.

The equation (2.2) is obtained from the zero crossing of the derivative of JHCM with

respect to vi.

Hard clustering is reported to have a quick convergence and to provide poor partition

quality [20, 49]. Some reasons for this latter could be:

• There is no guarantee that the convergence of the prototypes is reached in a minimum

of the cost function [176]. The algorithm stops as soon as there is no change within

the partitions in the latest performed iteration.

• The initialization of the cluster prototypes: the initialization technique mentioned

above assures each cluster to have at least one vector in any iteration. However, the

2.2. Related work 11

initial vectors assigned to each clusters will hardly be able to move to another cluster

in any further iteration.

A recent work that emphasizes the importance of intuitive initialization techniques, and

also provides such rules is [5] by Arthur and Vassilvitskii.

The fuzzy c-means (FCM) was introduced by Bezdek in 1981 [20]; it represented a

solution to most of the above mentioned problems of HCM. The main difference in the

formulation of FCM and HCM is the partition logic. Defining the partitions according

to the fuzzy philosophy allows us to assign a fuzzy membership function to each couple

(xk, vi), signifying the degree to which vector xk belongs to cluster Ci represented by

prototype vi. Let us denote these fuzzy membership variables by uik, where i = 1 . . . c

and k = 1 . . . n. In fact, these degrees of memberships can be interpreted as probabilities,

which means that the constraintc∑

i=1

uik = 1 for any k = 1 . . . n persists.

As defined by Bezdek [20], FCM clustering minimizes the following objective function:

JFCM =n∑

k=1

c∑i=1

umik||xk − vi||2A =

n∑k=1

c∑i=1

umikd

2ik , (2.3)

where m is the so-called fuzzyfication exponent or parameter, and dik represents the dissim-

ilarity (distance) between vector xk and cluster prototype vi. The fuzzyfication exponent

influences the behavior of the algorithm. A sufficient condition of the convergence is to set

m > 1. As we will see in the followings, the implementation of the algorithm is easiest for

m = 2. That is why this is the most popular value used in the literature.

The minimization of the objective function (2.3) is reached using the well-known AO

technique: by alternately applying the optimization of uik with vi fixed, and optimizing vi

with uik fixed, until cluster prototypes stabilize. A detailed description of this optimization

theory is given in [130].

The cost function is quadratic and each term has non-negative coefficient, so we can

get some first order necessary conditions of the optimum from the zero crossings of the


cost function’s partial derivatives with respect to uik and vi. Differentiating JFCM with

respect to uik and equaling it to zero leads to the trivial solution uik = 0 for any i =

1 . . . c and k = 1 . . . n, which is unacceptable as it gives no partitioning and contradicts

the probability constraint. This kind of problems are solved using Lagrange multipliers.

Instead of differentiating JFCM, we consider the following functional

LFCM =n∑

k=1

c∑i=1

umik||xk − vi||2A +

n∑k=1

λk

(1−

c∑i=1

uik

), (2.4)

where the second term is obviously zero. Differentiating LFCM with respect to uik leads

to the zero crossing condition mum−1ik d2

ik = λk. Taking into consideration the probability

constraintc∑

i=1

uik = 1, we can eliminate λk and obtain the solution:

u?ik =

d−2/(m−1)ik

c∑j=1

d−2/(m−1)jk

∀ i = 1 . . . c, ∀ k = 1 . . . n . (2.5)

The obtained formula requires the following remarks:

• Given an input vector xk, its degrees of memberships regarding the clusters only

depend on the distances or dissimilarities: the more distant a cluster is, the lower

membership it receives.

• There is a singularity case in the formula (2.5): when a cluster prototype, say vw

coincides with the input vector xk, we have dwk = 0. In this case we cannot apply

the computed formula, so we set uwk = 1 and ujk = 0 for any j 6= w.

• The fuzzification parameter m defines the penalty applied to distant clusters. Lower

fuzzyfication exponent means stronger favoring of closer prototypes.

• When m→∞, all degrees of membership uik → 1/c.

• When m→ 1+, all degrees of membership uik → hik.

The update formula for the cluster prototypes is obtained from the zero crossing con-

dition of the partial derivative of LFCM or JFCM with respect to vi: differentiating gives


2aiiumik(xk − vi) = 0, which yields

v?i =

n∑k=1

umikxk

n∑k=1

umik

∀ i = 1 . . . c . (2.6)

Thus in each iteration, the cluster prototypes are computed as weighted averages of the in-

put feature vectors, where the weights are provided by the m’th power of the corresponding

degrees of memberships.

The AO solution of the FCM problem can be summarized as follows:

1. Initialize cluster prototypes with values differing from each other. More intuitive

initialization is recommended [5] but not absolutely necessary.

2. Update degrees of membership using equation (2.5).

3. Update cluster prototypes using equation (2.6).

4. Repeat steps 2-3 until cluster prototypes stabilize. This is checked by comparing the

sum of norms of the variations of cluster prototype vectors from cycle to cycle with

a previously set constant ε.

It is possible to invert the order in which we apply the formulae (2.5) and (2.6); in this

case the membership values need to be initialized and the stopping criterion needs to check

the stabilization of these memberships. As this latter step would be time consuming, this

is not a frequently applied version.

The defuzzyfication of the obtained partition is performed by assigning each input

vector to the cluster whose prototype is closest, that is, with respect to which it has the

highest degree of membership.

The convergence of the FCM algorithm has a well established theory: details of this

topic can be found in [20, 57, 65].


Figure 2.1: Effect of an outlier (drawn in black) on the cluster prototypes (represented by

the larger circles). Wherever we have an input vector on the gray line, it will have equal

degrees of membership (0.5) with respect to both clusters

Cluster validation is also a widely researched topic. Several cluster validity indices have

been proposed by Bezdek [20], Windham [187], Bezdek and Pal [23, 129], Tolias et al. [178].

Some of these will be applied in later chapters.

It is generally accepted by the literature that FCM provides significantly better par-

titions than HCM does, but it needs considerably more iterations to stabilize. Another

disadvantage of FCM (and also of HCM) is its sensitivity to outliers. A single outlier input

vector can shift the cluster prototypes out of the cluster’s real range. This is demonstrated

by Figure 2.1.

In spite of these obvious defects, the fuzzy c-means clustering has become one of the

most popular clustering algorithm, in the sense that it has applications in almost all tech-

nical and nature sciences.

However, there is also a wide range of the literature, which aims to correct its short-

comings. First let us consider the time complexity. In the early times, when personal

computers had reduced computing power, Cannon et al. [34] proposed an approximative

FCM algorithm implemented with operations on integer numbers only. Although the com-

puters speeded up thousands of times since then, their solution can be still useful if we

need to implement a clustering in a microcontroller. Another approximative solution was

given by Kamel and Selim [73].


A different approach to combat time complexity is parallelization. Lazaro et al. [95]

proposed a parallel hardware implementation to its algorithms and gave it an application in

signal processing. Kolen and Hutcheson [87] reorganized the FCM algorithm to reduce the

necessary memory storage by eliminating the partition matrix. They found their solution

significantly quicker than FCM especially in case of large amount of input data.

Another way to reaching higher speed of the clustering is data reduction. This approach

was followed by Eschrich et al. [48], who introduced a data reduction scheme by aggregating

similar input vectors into a weighted example. Their solution can speed up FCM by an

order of magnitude. Cheng et al. [37] proposed a data reduction scheme based on random

sampling, which yields a quick approximative FCM clustering. Fan et al. [49] attempted to

speed up the convergence of the FCM by combining it with HCM’s behavior. This solution

will be discussed in details later.

The FCM clustering has been also criticized for its sensitivity to outliers. The easiest

approaches of this problem reformed the distance function applied in FCM. In this order,

Wu and Yang [189] proposed an alternative FCM algorithm based on an exponential dis-

tance function. A similar solution was given by Zhang and Chen [200] using a formulation

with kernels. Another reformulation of FCM uses similarities instead of distances: this

approach was followed by Yang and Wu [193] and Pedrycz et al. [133]. The other branch

of research, aiming at FCM-based and FCM-like algorithms less sensitive to outliers, revo-

lutionizes the algorithm by changing its constraints regarding the degrees of memberships.

These solutions will be discussed in the next sections.

Other improved FCM versions, which deserve to be remarked are:

• Noordam et al. [125] published an FCM version, which is insensitive to cluster sizes.

• Frigui and Krishnapuram [52, 53] presented a modified FCM algorithm, which adap-

tively recognizes the ideal number of clusters in the input data, by setting up a

competition among clusters and eliminating clusters of low cardinality.


• Some further FCM algorithms have been created for special types of data: D’Urso

et al. [181] introduced by weighted model for fuzzy data, while Murata et al. [119]

presented an FCM with tolerance for low precision input data.

• Belacel et al. [18] introduced a heuristic version of c-means clustering and called it

fuzzy J-means.

• Fuzzy c-means clustering with supervision was introduced by Pedrycz and Vukovich

[134].

2.2.2 The suppressed fuzzy c-means algorithm

One of the improved FCM algorithms, which will be extensively analyzed within this

chapter, is the so-called suppressed FCM algorithm (s-FCM) introduced by Fan et al.

[49]. Having the intention of combining the higher convergence speed of HCM with the

more accurate partitioning properties of FCM, they added an extra computation step into

the FCM iteration, which created a competition among clusters: after performing the

computation of fuzzy memberships according to Eq. (2.5), the highest membership of

each input vector is increased in the detriment of the lower ones. In other words, low

membership values are suppressed according to the previously defined suppression rate α,

and the largest membership is raised by swallowing all the suppressed parts of the low

memberships.

This new step performs the following task: for each datum xk, after having obtained

its new optimal fuzzy membership values uik, we search for the highest one uwkk, and

declare cluster with index wk ∈ {1, 2, . . . , c} the winner. The fuzzy memberships are then

modified such a way that all non-winner values are decreased via multiplying by a so-

called suppression rate α, (0 ≤ α ≤ 1), and the winner membership is increased so that

the probability constraint relation is fulfilled by the modified memberships. Therefore, the


suppression formula of s-FCM is:

µwkk = 1− α∑j 6=wk

ujk = 1− α + αuwkk if i = wk , (2.7)

µik = αuik ∀i ∈ {1, 2, . . . , c} \ {wk} , (2.8)

where µik, i = 1 . . . c, k = 1 . . . n, represent the fuzzy memberships obtained with the

modification introduced by the s-FCM algorithm. Cluster prototypes are then updated

according to the intuitive formula:

v?i =

n∑k=1

µmikxk

n∑k=1

µmik

∀ i = 1 . . . c . (2.9)

Fan et al. did not give a recipe for choosing a suppression rate that is optimal in any

sense, or suitable for any given purpose. They set the suppression rate to the middle of

the interval (α = 0.5), and found s-FCM insensitive to the fuzzification parameter m [49].

The authors also remarked that setting the suppression rate α = 0 makes s-FCM and

HCM identical, while α = 1 reduces the algorithm to the conventional FCM. The s-FCM

algorithm was found successful based on some numerical analysis, but unfortunately, the

authors left several issues wide open:

1. They failed to establish whether s-FCM is optimal in any sense, that is, whether it

minimizes any kind of objective function.

2. The extra step of s-FCM was inspired by the basis of competitive learning [49], but

the authors failed to give any evidence of its competitive behavior.

3. The authors found s-FCM clustering insensitive to the fuzzyfication parameter m,

based on a few experiments. However, since the competitivity of FCM is controlled

using m [20, 63], this cannot be the established so easily.

4. The authors failed to provide any strategy to choose the suppression rate α. This was

already pointed out by Hung et al. [66, 67], who formulated a criterion for α based


on considerations regarding cluster validity. Their criterion, however, has nothing to

do with the nature of the s-FCM algorithm.

Recently, another way of “suppression” was introduced by Xie et al. [191], using a dif-

ferent formulation. Similarly to the support vector machine (SVM) [182], which guarantees

a margin on both sides of its decision hyperplane, this clustering model assures a margin

between the fuzzy memberships of the winner and all non-winner clusters.

In this chapter we will clarify most of these open questions, and will give some further

applications of s-FCM in the field of medical image processing in the next chapter.

2.2.3 Relaxation of the fuzzy probabilistic constraint

Ten years after the establishment of the fuzzy c-means algorithm [20], the clustering re-

search community realized that the probabilistic constraint, namelyc∑

i=1

uik = 1 ∀k = 1 . . . n,

was the main obstacle in making c-means clustering accurate and efficient.

A set of relaxation schemes have followed and are introduced also nowadays. The most

simple solution is to introduce an extra class into FCM called noise class, which should

incorporate all outlier input vectors. This noise class is not represented by a prototype.

Instead of this, it is supposed that the distance between the noise class and any input

vector is equal to a constant denoted by d0, which is computed from the input vectors.

Thus, for any vector xk, we obtain the degrees of memberships with respect to normal

clusters:

uik =d−2/(m−1)ik

d−2/(m−1)0 +

c∑j=1

d−2/(m−1)jk

, (2.10)

while the membership to the noise class is

u0k =d−2/(m−1)0

d−2/(m−1)0 +

c∑j=1

d−2/(m−1)jk

. (2.11)


The constant distance d0 is computed as the average distance between input vectors:

d0 =2

n(n− 1)

n∑k=1

n∑l=k+1

||xk − xl|| . (2.12)

Obviously, the probabilistic constraint does not hold anymore, the sum of degrees of mem-

bership is less than 1. The order among clusters remains the same, the highest FCM

membership belongs to the same cluster. The value of u0k gives us a probability value that

the input vector belongs to the noise class. Finally, having the cluster prototypes updated

according to (2.6), the sensitivity to outliers vanishes.

In 1993, Krishnapuram and Keller [91] introduced the foundations of possibilistic clus-

tering. In their view, the relation between a vector and a cluster is not described by the

degree of membership signifying the probability that the vector belongs to the cluster.

Instead of this probability, they talk about typicality, which shows how much an element

is typical to or compatible with a class. Just like the probability, the typicality is also

described by a real number within the interval [0, 1], but the probability constraint is not

valid for typicality values. In the followings, we will denote by tik the compatibility of

vector xk with cluster Ci.

The constraints affecting typicality values are:

1. Any typicality value is within the interval tik ∈ [0, 1].

2. Any cluster must have at least one element with nonzero typicality and no cluster

can have all elements with maximum typicality: 0 <n∑

k=1

tik < n ∀i = 1 . . . n .

3. Any element must have nonzero typicality to at least one cluster: ∀k = 1 . . . n ∃i ∈

{1, 2, . . . c}: tik > 0.

The possibilistic c-means (PCM) clustering minimizes the following objective function:

JPCM =n∑

k=1

c∑i=1

[tpik||xk − vi||2 + ηi(1− tik)

p]

, (2.13)


where ηi are suitably chosen positive numbers, responsible for the variance of clusters. Just

like in case of FCM, the exponent p is also required to satisfy p > 1. The minimization of

the PCM cost function doesn’t need Lagrange multipliers. The update formula of typicality

values is obtained from the zero crossing of the partial derivative with respect to tik:

t?ik =

[1 +

(d2

ik

ηi

)1/(p−1)]−1

. (2.14)

Cluster prototypes are updated similarly to FCM, using the typicality values instead

of fuzzy memberships. The authors also gave a scheme for choosing the right values for

constants ηi. These constants influence the magnitudes of the typicality values: generally

larger ηi values result in larger typicality values. The authors proposed to compute ηi from

the result of a previously performed FCM algorithm, with the following formula:

ηi = κ ·

n∑k=1

umikd

2ik

n∑k=1

umik

. (2.15)

Most of the literature uses κ = 1.

The first version of possibilistic c-means clustering had an instant solution for the

sensitivity of FCM to outliers, but several new problems emerged, the most important of

which is that cluster prototypes seemingly tend to attract each other. The root of the

problem in fact is the cost function, which is formulated such a way that typicality values

to different clusters and consequently the cluster prototypes, too, are totally independent

from each other. Under such circumstances, there is no obstacle for two or more cluster

prototypes to converge to the same position. This is the main cause of the virtual attraction

[13, 15].

In order to avoid the coincidence of possibilistic cluster prototypes, Pal et al. [131]

proposed a mixed solution, which combines the FCM algorithm with a possibilistic term.

Their clustering model minimizes the following objective function:

Jmix =n∑

k=1

c∑i=1

(umik + tpik)d

2ik , (2.16)


subject to constraints m > 1, p > 1, 0 ≤ uik, tik ≤ 1, and

c∑i=1

uik = 1 ∀k = 1 . . . n andn∑

k=1

tik = 1 ∀i = 1 . . . c . (2.17)

The AO scheme of this method uses besides Eq. (2.5) the following update formulae:

t?ik =d−2/(p−1)ik

n∑l=1

d−2/(p−1)il

∀ i = 1 . . . c, ∀ k = 1 . . . n , (2.18)

v?i =

n∑k=1

(umik + tpik)xk

n∑k=1

(umik + tpik)

∀ i = 1 . . . c . (2.19)

The main advantage of this method is that both components manage to significantly

compensate the disadvantage of the other, so the resulting algorithm is less sensitive to

outliers than FCM was, and cluster coincidence is not likely to occur. However, it still has

a relevant disadvantage: in case of large data sets, as n→∞, we get tik → 0. This means

the more input vectors we have, the closer to FCM our algorithm gets. This problem could

be avoided by settingn∑

k=1

tik = nc, but in this case there is no guarantee for the tik ≤ 1

condition.

Another remarkable effort to avoid the coincident clusters was made by Timm et al.

[175]. They added an extra term to the objective function shown in Eq. (2.13), which

set up a repulsive force between all couples of cluster prototypes, the strength of which

decreases with distance. Their method succeeded in avoiding coincident clusters, but failed

to correctly treat cases when two clusters are really close to each other.

Recently, Pal et al. [132] published an article on a hybridized possibilistic-fuzzy c-

means (PFCM) clustering model. Its cost function is a combination of the previous two

approaches:

JPFCM =n∑

k=1

c∑i=1

(a · umik + b · tpik)d

2ik +

c∑i=1

ηi

n∑k=1

(1− tik)p , (2.20)

with a and b positive real numbers that define the tradeoff between the strength of FCM

and PCM components. The update formulae for the probabilistic fuzzy memberships


is identical with Eq. (2.5), while typicality values and cluster prototypes are updated

according to:

t?ik =

[1 +

(b

ηi

d2ik

)1/(p−1)]−1

, ∀ i = 1 . . . c, k = 1 . . . n , (2.21)

v?i =

n∑k=1

(a · umik + b · tpik)xk

n∑k=1

(a · umik + b · tpik)

∀ i = 1 . . . c . (2.22)

After a long series of tests using standard data sets, the authors found PFCM a reliable

robust clustering scheme, which has the following main properties:

1. If exponents grow without bounds, that is m → ∞ and p → ∞, the algorithm acts

the same way as FCM at m→∞: cluster prototypes approach the mean of the input

vectors and consequently also each other.

2. In order to reduce the effect of outliers, it is advised to set b > a.

3. It is also possible to reduce the effect of outliers by setting m > p.

4. However, if m becomes too large, the algorithm will resemble PCM, which is not

desired.

These methods showed the right way towards a robust and accurate clustering method,

which seems possible to obtain if we try to put together the advantages of each clustering

principle and set them to eliminate or compensate each other’s deficiencies.

In the following sections, we will study some cases of further combinations of existing

clustering principles.

2.2.4 Optimization via evolutionary computation

Classical c-means clustering methods have their easy-to-understand and easy-to-implement

alternating optimization scheme. Probably that’s why they are so popular in all scientific


fields. However, if one produces several generalizations within the frame of FCM clustering,

it is possible to reach an objective function, which is difficult or impossible to minimize

this way. For such cases, it is recommended to use evolutionary computation (EC).

In 1995, Hathaway and Bezdek [63] introduced some reformulated versions of the cost

functions for all three classical c-means clustering methods, with the intention of reducing

the search space during prototype optimization. They eliminated the variables hik, uik,

and tik, which represent the probability or typicality of input vector xk with respect to

cluster Ci. They obtained the following reformulated objective functions:

RHCM =n∑

k=1

min{d21k, d

22k, . . . , d

2ck} , (2.23)

RFCM =n∑

k=1

(c∑

i=1

d2/(1−m)ik

)1−m

, (2.24)

RPCM =n∑

k=1

c∑i=1

[d

2/(1−p)ik + η

1/(1−p)i

]1−p

. (2.25)

The authors proved the equivalence of these objective functions with their original.

Having reduced the number of sought parameters, they opened the way towards the opti-

mization of c-means objective functions via evolutionary computation.

The generalized c-means clustering scheme by Yu [195], which was introduced as a

further development and generalization of the methods given by Karayiannis et al. [75]

and Hathaway et al. [64], indicates a decision criterion on the topic of choosing a suitable

optimization technique: under what circumstances is it possible to apply an iterative AO

scheme, and when is it necessary to turn to neural networks, evolutionary computation, or

simulated annealing.

In the remainder of this chapter, starting from section 2.3, we will investigate several

aspects of combining the three classical clustering methods, and based on this study, we

will attempt to formulate various recommendations.


2.2.5 FCM-based and FCM-like competitive clustering algo-

rithms

Competitive clustering stems from the work of Kohonen [85], who introduced the notion

of self organizing maps into pattern recognition. Competitive algorithms usually work in

pattern mode and update their cluster prototypes according to the result of a competition:

whenever an input vector is presented to the algorithm, the cluster prototype showing the

highest similarity is assigned winner of the competition. The winner prototype is updated

by making a step towards the considered input vector. The length of the step depends on

the current learning rate η, whose value is decreasing in time according to a previously

defined rule.

Enhanced competitive clustering models update more than one cluster prototypes in

every iteration. This is necessary because, for example, if a cluster prototype is initialized

with an outlier value, it will never win a competition and will never be updated during the

basic competitive learning. So it is recommended to update all non-winner prototypes in

every cycle, by making a small step towards the input vector [176]. Self organizing maps

update those nodes which are situated in the neighborhood of the winner prototype [85].

Some further developed, enhanced competitive learning schemes are listed below:

1. The soft competitive scheme (SCS) introduced by Yair et al. [192] updates all cluster

prototypes in every cycle. It distributes the learning rates among clusters similarly

to the update formula of degrees of membership in FCM, but uses an exponential

distance function, which is beneficial for the formation of fine partitions. However, as

the iteration count grows beyond a limit, the algorithm turns into a winner-takes-all

competitive scheme, which helps prototypes converge quickly.

2. The fuzzy learning vector quantization (FLVQ) algorithm introduced by Tsao et al.

[180] has two versions, the so-called ascending (↑FLVQ) and descending (↓FLVQ)

schemes, depending on the direction the exponent m evolves within the interval


m ∈ [1.1, 7.0]. Similarly to SCS, FLVQ also applies an FCM-like fuzzy membership

update formula for the distribution of learning rates. The main advantage over SCS

is the quicker convergence.

3. The generalized learning vector quantization (GLVQ) algorithm introduced by Pal

et al. [128], in spite of its mistaken formulation remarked by Gonzalez et al. [56] and

repaired by Karayiannis et al. [74], has the main merit of showing the road towards

competitive algorithms based on generalized means.

4. Karayiannis and Bezdek [75] introduced some batch competitive algorithms based

on the generalized mean of real numbers. They found that some properly chosen

generalized mean Dp(xk, V ) of the distances of datum xk from cluster prototypes vj,

j = 1 . . . c, given by the formula

Dp(xk, V ) =

[1

c

c∑j=1

(d2jk)

p

]1/p

, (2.26)

where V represents the set of cluster prototypes, summed up for a set of input

vectors X = {x1, x2, . . . ,xn}, differs from the reformulated function of FCM in a

single multiplicative constant:

n∑k=1

Dp(xk, V ) = κn∑

k=1

[c∑

j=1

(d2jk)

p

]1/p

= κ · JFCM . (2.27)

The properly chosen generalized mean refers to the choice of p: p = 1/(1−m). The

constant κ is given as: κ = c−1/p = cm−1. The authors found the objective function

given by Eq. (2.27) suitable for competitive clustering for any p < 1, corresponding

to m ∈ (−∞, 0) ∪ (1,∞) [75].

The classical c-means clustering models are not called competitive algorithms as there is

a clear distinction between their objective functions [128]. However, the crisp logic applied

in HCM for membership assignment can be thought of as a winner-takes-all competition.

In case of any input, the cluster prototype situated the closest will win the competition and


will be rewarded with a unitary membership, while all non-winners receive zero. Similarly

thinking, by reducing the fuzzy exponent within the FCM framework, we increase the

competition among clusters. This sort of competition will be studied later in this chapter.

2.3 An analysis of the suppressed FCM algorithm

As it was presented in the previous section, suppressed FCM was admittedly introduced

by Fan et al. [49] on the basis of competitive learning, having the main goal to combine the

quick convergence of HCM with the accurate partitioning properties of FCM. Besides giving

the extra formulae (2.7) and (2.8), which compute the suppressed degrees of membership

µik after having computed FCM’s memberships in each iteration, the authors failed to

clarify some main issues like:

• What kind of competition does s-FCM create within the FCM framework?

• From a rigorous mathematical point of view, may one call s-FCM optimal? If so,

what does s-FCM minimize?

In the followings we will attempt to answer these questions and give some further, well

supported recommendations.

2.3.1 Competition by suppression

We will start the investigation from Eq. (2.5). When the new degrees of membership of

datum xk are computed, everything depends on the distances dik, i = 1 . . . c. If we change

the scale of the metric such a way that distances are lengthened or shortened proportionally,

the obtained memberships remain the same. In other words, the ratio of two membership

values, say uik/ujk does not depend on the above mentioned linear scale.

Conversely, when memberships to non-winner clusters are proportionally suppressed

via multiplying them by α, it can be interpreted as their distances from vector xk remain

2.3. An analysis of the suppressed FCM algorithm 27

unchanged, but as the winner cluster receives a higher degree of membership, prototype

vwkis counted as it were closer to xk than it really is. Again, in other words, when new

cluster prototypes vi are computed using the weighted averaging formula in eq. (2.6), using

the suppressed fuzzy memberships µik instead of FCM memberships uik, the vectors xk

whose competition the currently computed prototype has won, are taken into consideration

as they were at a reduced distance d′wkk < dwkk, giving those winner vectors a higher impact

than in FCM.

This reduced distance can be characterized with a quasi learning rate defined similarly

to the conventional learning rate of competitive algorithms:

ηs−FCM = 1−d′wkk

dwkk

, (2.28)

whose value will be computed in the followings. Let us introduce the notations: δik =

d−2/(m−1)ik ∀i = 1 . . . c, ∀k = 1 . . . n, δ′wkk = d

−2/(m−1)wkk , and let us suppose δ′wkk = γδwkk,

where we expect γ ≥ 1. Using these new notations, we can rewrite (2.5) for both winner

and non-winner clusters. For the winner cluster we have:

µwkk =δ′wkk

δ′wkk +c∑

j=1,j 6=wk

δjk

=γδwkk

γδwkk +c∑

j=1,j 6=wk

δjk

=γδwkk

(γ − 1)δwkk +c∑

j=1

δjk

, (2.29)

while the non-winner clusters receive the memberships:

µik =δik

δ′wkk +c∑

j=1,j 6=wk

δjk

=δik

γδwkk +c∑

j=1,j 6=wk

δjk

=δik


j=1

δjk

. (2.30)

On the other hand, we have the definitions of µwkk and µik using the uik memberships

and the suppression rate α. Now we can compare the suppressed memberships defined in

(2.7) with the computed (2.29), and (2.8) with (2.30), respectively. From the former two

we get:

(1− α) + αδwkkc∑

j=1

δjk

=γδwkk


j=1

δjk

, (2.31)


which implies

(1− α)(γ − 1)δwkk + (1− α)c∑

j=1

δjk + α(γ − 1)δ2wkk

c∑j=1

δjk

+ αδwkk = γδwkk . (2.32)

From here we intend to compute γ, so we go on this way:

δwkk(γ − 1)

(1− α) + αδwkkc∑

j=1

δjk

− 1

= (1− α)

[δwkk −

c∑j=1

δjk

], (2.33)

which then becomes

α(γ − 1)

δwkkc∑

j=1

δjk

− 1

= (1− α)

1−

c∑j=1

δjk

δwkk

. (2.34)

Taking in consideration the relation uwkk = δwkk/c∑

j=1

δjk, we get

(γ − 1)(uwkk − 1) =(1− α)(uwkk − 1)

αuwkk

. (2.35)

In FCM, the degree of membership assigned to the winner cluster uwkk = 1 only if the

input vector xk and the cluster prototype vwkcoincide. In this trivial case there is no need

to compute the suppression as there is nothing to suppress. Excluding this trivial case, we

may simplify the previous equation, so we get:

γ = 1 +1− α

αuwkk

. (2.36)

On the other hand, if we start from (2.8) and (2.30) we obtain:

δik


j=1

δjk

= αδik

c∑j=1

δjk

. (2.37)

As δik is never zero, this equation can be restructured as follows:

c∑j=1

δjk = α(γ − 1)δwkk + αc∑

j=1

δjk , (2.38)


or

γ = 1 +

(1− α)c∑

j=1

δjk

αδwkk

= 1 +1− α

αuwkk

, (2.39)

which, according to Eqs. (2.7) and (2.8), can be further transcribed to any of these forms:

γ = 1 +1− α

αuwkk

=µwkk

αuwkk

=µwkk

µwkk − (1− α). (2.40)

So we obtained the same γ value both ways. Although this is not yet the learning rate,

we should discuss about the possible singularities:

• The degree of membership assigned by FCM to the winner class, uwkk, cannot be zero,

because then all other uik values would be zero, and that contradicts the probability

constraint of FCM.

• The suppression rate α can be zero, but that would reduce s-FCM to HCM, which

is a trivial case with strict winner-takes-all competition.

• As the suppression rate α and the winner cluster’s fuzzy membership are both in the

interval (0, 1], we indeed have γ ≥ 1. Equality holds when α = 1, that is, there is no

suppression.

We can conclude that Eq. (2.40) is valid if 0 < α ≤ 1. Under these circumstances, the

learning rate of the s-FCM is:

ηs−FCM = 1−d′wkk

dwkk

= 1− γ(1−m)/2 = 1−(

1 +1− α

αuwkk

)(1−m)/2

. (2.41)

It was expected that the fuzzyfication parameter m and the suppression rate α influence

the learning rate. In addition, another factor is present, namely the fuzzy membership

value of the winner cluster, uwkk. Some graphical representations of the learning rate vs.

suppression rate are shown in Fig. 2.2. So far we can conclude that s-FCM has a quasi

competitive behavior with a variable learning rate.


Figure 2.2: Effect of suppression rate on the learning rate: with uw = 0.8 and different

values of m (left), and with m = 2 and different values of winner membership uw (right)

2.3.2 May one call the s-FCM algorithm optimal?

Fan et al. [49] mentioned that setting the suppression rate α = 0 reduces s-FCM to HCM,

while α = 1 takes us back to FCM. What happens between these two bounds, is a good

question. In the followings, we consider 0 < α < 1, as the two extreme cases are trivial

anyway and need no investigation.

In the everyday practice, the concept of optimality is often interpreted as a definite

qualitative attribute, not necessarily having a rigorous mathematical support. In this

work, however, we are treating this concept from the rigorous mathematical point of view,

as Pal et al. did in [130].

Under such circumstances, we cannot call s-FCM optimal unless we find an objective

function, whose AO minimization gives the optimization formulae shown below, which are

all necessary conditions:

µik = α · d−2

m−1

ikc∑

j=1

d−2

m−1

jk

∀k = 1 . . . n ∀i 6= wk , (2.42)


µwkk = 1− α + α ·d−2

m−1

wkkc∑

j=1

d−2

m−1

jk

∀k = 1 . . . n, wk = arg mini{dik} , (2.43)

vi =

n∑k=1

µmikxk

n∑k=1

µmik

∀i = 1 . . . c . (2.44)

Unfortunately, this kind of analytical function is not likely to exist. There exists at

least one function, namely

Jnot s−FCM =n∑

k=1

c∑i=1

[µik − (1− α)hik]md2

ik , (2.45)

which satisfies the first two conditions for any α ∈ (0, 1), but the corresponding cluster

prototype update formula generates prototypes which coincide with the ones of FCM.

Instead of hunting for an unlikely-to-exist objective function of s-FCM, let us propose a

novel approach for an optimal suppression and name it optimally suppressed fuzzy c-means

algorithm (Os-FCM):

JOs−FCM = α · JFCM + (1− α) · JHCM =n∑

k=1

c∑i=1

[αumik + (1− α)hik]d

2ik , (2.46)

where α is a parameter that is intended to mix fuzzy and hard c-means clustering, similarly

to s-FCM, but creating a pure mixture of FCM and HCM. Obviously, there are two values

of α, where s-FCM and Os-FCM coincide: 0 and 1, corresponding to HCM and FCM,

respectively. In case of any other α, s-FCM and Os-FCM differ. The AO iteration formu-

lae of Os-FCM are easy to obtain via the well-known technique of Lagrange multipliers.

Despite the presence of parameter α in the proposed cost function, the update criteria we

obtain for uik and hik are the same as in case of FCM and HCM algorithms, respectively.

The update rule for cluster prototypes differs from Eq. 2.6, as it becomes:

vi =

n∑k=1

[αumik + (1− α)hik] xk

n∑k=1

[αumik + (1− α)hik]

. (2.47)


In the followings, we will compare the s-FCM algorithm with our newly proposed op-

timal clustering model, from two different points of view:

1. We will compute the quasi-competitive learning rate of Os-FCM and compare it with

Eq. (2.41) and Fig. 2.2.

2. We will analyze the behavior of both algorithms by employing them to cluster the

IRIS data [2] with several different settings.

According to Eq. 2.47, the prototype who wins the competition of vector xk receives

a weight 1 − α + αumik, while non-winners get αum

ik, where uik coincides with the fuzzy

memberships provided by FCM using Eq. (2.5). These weights correspond to the µmik

values in Eq. (2.9), which on their turn, can be expressed using the terms defined at the

introduction of the quasi competitive learning rate. Using the notations defined at the

computation of the learning rate of s-FCM, and based on Eq. (2.47), we can write the

following equation in these new circumstances:

α

δwkkc∑

j=1

δjk

m

+ (1− α) =

γδwkk


j=1

δjk

m

, (2.48)

which represents the weighting coefficient received by a vector xk whose competition was

won by cluster prototype vi. According to Eq. (2.5), we can transcribe the following

equation as:

αumwkk + (1− α) =

(γuwkk

1 + (γ − 1)uwkk

)m

, (2.49)

which is an equation that is difficult (if possible) to solve analytically. Let us ease the

circumstances by setting m = 2, which then yields:

γ2u2wkk = [1− α + αu2

wkk][1 + (γ − 1)uwkk]2 , (2.50)

which leads to the following second order equation in γ:

αu2wkk(1 + uwkk)γ

2 − 2uwkk[1− α + αu2wkk]γ − (1− uwkk)[1− α + αu2

wkk] = 0 . (2.51)


Figure 2.3: Effect of optimal suppression rate on the learning rate: with m = 2 and

different values of winner membership uw

This equation has two solutions: one of them is negative, which is not acceptable for us.

The other solution has the following formula:

γ =1− α + αu2

wkk +√

1− α + αu2wkk

αuwkk(1 + uwkk). (2.52)

Let us verify the extreme values: if α → 0, then γ → ∞, which is what we expected.

Also, if we set α → 1, we get γ →u2

wkk+uwkk

uwkk(1+uwkk)= 1, which means zero learning rate,

corresponding to FCM.

The learning rate is given by:

ηOs−FCM = 1− γ1−m

2 = 1−√√√√ αuwkk(1 + uwkk)

1− α + αu2wkk +

√1− α + αu2

wkk

. (2.53)

The graphical representation of this function is shown in Fig. 2.3. The curves on in this

graph are quite similar but not identical with the ones shown in Fig. 2.2(right). This is

not yet a proof for similar behavior of the two algorithms, this is what was possible to

show the analytical way.


The comparison of s-FCM and Os-FCM should continue with a numerical analysis.

However, now we continue on with a more general hybrid c-means clustering model, and

we will return later and discuss Os-FCM as a special case of the hybrid model.

2.4 A novel hybrid c-means clustering model

2.4.1 The alternating optimization approach

In this section we propose a hybrid c-means clustering model, which consists of a mixture

of hard, fuzzy, and possibilistic criteria. The objective function of such a model is:

Jhybrid = βαJFCM + (1− β)JPCM + β(1− α)JHCM

=n∑

k=1

c∑i=1

[βαumik + (1− β)tpik + β(1− α)hik] d

2ik

+(1− β)c∑

i=1

ηi

n∑k=1

(1− tik)p,

(2.54)

• where hik, uik, and tik represent the degrees of membership of input vector xk with

respect to cluster Ci, assigned by the hard, fuzzy, and possibilistic criteria, respec-

tively, constrained by hik ∈ {0, 1},c∑

i=1

hik = 1 ∀k = 1 . . . n; uik ∈ [0, 1],c∑

i=1

uik = 1

∀k = 1 . . . n; tik ∈ [0, 1], 0 <c∑

i=1

tik < c ∀k = 1 . . . n, and 0 <n∑

k=1

tik < n ∀i = 1 . . . c.

• m and p are the exponents of the fuzzy and possibilistic terms, restricted by m > 1

and p > 1

• dik = ||xk − vi|| is the distance between input vector xk and cluster prototype vi,

• ηi are the variance parameters of the possibilistic term, first introduced in (2.13),

• α and β are tradeoff parameters, which control the relative strength of the three terms

within the hybrid cost function. In this order, β controls the presence of possibilistic

clustering, while α is responsible of the FCM-HCM mixture, similarly to the case of

suppressed FCM.

2.4. A novel hybrid c-means clustering model 35

Figure 2.4: Parametrization of the proposed hybrid clustering model, its domain of defin-

ition with special cases at the boundary and corners: β = 0 corresponds to PCM, β = 1

and α = 0 is HCM, β = 1 and α = 1 reduces to FCM, α = 1 and β ∈ (0, 1) is the PFCM

model of Pal et al. [132], and finally β = 1 and α ∈ (0, 1) is not exactly the s-FCM of Fan

et al. [49], but the recently introduced Os-FCM

• Setting α = 1 makes the hybrid equivalent with PFCM, with linear relations among

their parameters, while β = 1 reduces Jhybrid to JOs−FCM.

In order to obtain the AO iterative solution scheme, we need to apply the Lagrange

multiplier method. We will need to compute the zero crossing of the derivative of the

following functional:

Lhybrid = Jhybrid +n∑

k=1

λk

(1−

c∑i=1

uik

). (2.55)


From the partial derivative of the above functional with respect to uik we get:

∂Lhybrid

∂uik

= 0 ⇒ βαmum−1ik d2

ik − λk = 0 ⇒ uik =

(λk

βαmd2ik

) 1m−1

. (2.56)

Taking the probability constraint into consideration, we obtain the AO formula for uik:

c∑j=1

ujk = 1 ⇒(

λk

βαm

) 1m−1

=

(c∑

j=1

d−2

m−1

jk

)−1

⇒ uik =d−2

m−1

ikc∑

j=1

d−2

m−1

jk

, (2.57)

which, not at all surprisingly, coincides with (2.5). The derivative with respect to typicality

value tik provides:

∂Jhybrid

∂tik= 0 ⇒ (1− β)p[tp−1

ik d2ik − ηi(1− tik)

p−1] = 0 ⇒(

d2ik

ηi

) 1p−1

=1− tik

tik, (2.58)

which yields a formula that coincides with (2.14)

tik =

(1 +

(d2

ik

ηi

) 1p−1

)−1

. (2.59)

With respect to hik, the cost function is not differentiable, so the hard membership degrees

will be attributed according to the crisp logic

hik =

1 if i = wk

0 if i 6= wk

, (2.60)

where wk = arg min{dik : i = 1 . . . c}, that is, the index of the cluster that wins the

competition for vector xk.

Finally, we need to differentiate the cost function with respect to vi:

∂Jhybrid

∂vi

= 0 ⇒n∑

k=1

[βαumik + (1− β)tpik + β(1− α)hik] (xk − vi) = 0 , (2.61)

so we obtain the following AO formula for cluster prototypes:

vi =

n∑k=1

[βαumik + (1− β)tpik + β(1− α)hik] xk

n∑k=1

[βαumik + (1− β)tpik + β(1− α)hik]

. (2.62)


Algorithm

Let us summarize the optimization algorithm of the proposed hybrid c-means clustering

scheme:

1. Initialize cluster prototypes vi with randomly chosen input vectors differing from

each other, or employ some more intuitive initialization scheme.

2. Choose the values of exponents m and p, and tradeoff parameters α and β.

3. Compute new fuzzy membership values uik with Eq. (2.57).

4. Compute new typicality values tik with Eq. (2.59).

5. Compute new hard membership values hik with Eq. (2.60).

6. Update cluster prototypes with Eq. (2.62).

7. Repeat steps 3-6 until cluster prototypes stabilize.

We can observe that steps 3-5 do not depend on each other, so they can be computed

in any order, or even they can be performed by parallel tasks. The order shown here was

chosen arbitrarily.

Remarks

By analyzing the AO formulae of the proposed hybrid c-means clustering model, for any

i = 1 . . . c and for any k = 1 . . . n we can remark the followings:

1. limm→1+

uik = hik, signifying the elimination of the FCM component from the hybrid,

thus having similar effects to setting α = 0.

2. limm→∞

uik = 1/c, which means that the FCM component turns into a term that tries

to equalize memberships, leading to a slower convergence and worse partitions.


3. limp→∞

tik = 0.5, having the same effect as the above one.

4. limp→1+

tik has a more complicated form:

limp→1+

tik =

1 if d2

ik < ηi

0.5 if d2ik = ηi

0 if d2ik > ηi

. (2.63)

5. If d2ik = ηi then tik = 0.5, whatever is the value of p.

6. limm→∞,p→∞,α→1

vi = 1n

n∑k=1

xk.

2.4.2 Clustering results

The proposed hybrid c-means clustering model has undergone a thorough test, using the

IRIS data set [2], which is one of the most popular test data sets applied by the clustering

algorithm developer community. IRIS data represent a collection of 150 feature vectors

containing 4 different measures of individual iris flowers. Each feature vector contains a

label that indicates the species of iris to which the given individual belongs. The data

set contains three classes, each class having exactly 50 elements. IRIS data vectors are

available, for example, as an appendix of [24].

Supervised and unsupervised classification techniques can all be tested using the IRIS

data set. In our case, when testing unsupervised c-means clustering, we are allowed to use

the labels only to verify the accuracy of the partitioning created by the algorithm. Classical

c-means approaches have a reported classification score of 133-134 correctly clustered IRIS

data vectors. As a first part of the evaluation of the proposed hybrid c-means algorithm,

we will analyze the influence of the system parameters upon the classification score.

As it was previously stated, α and β are the tradeoff parameters of the proposed hybrid

method, which determine the degrees to which FCM, PCM, and HCM components are

present. In case of β = 0, the value of α is completely irrelevant: that is why the α-β plane


Figure 2.5: Average classification score in case of ideal ηi values

can be best represented as a sector of a circle, having the site of β = 0 in the center. Such

diagrams are shown in Figs. 2.5, 2.6, and 2.7. These diagrams represent the distribution

of the average number of the correctly partitioned IRIS data vectors with respect to α

and β, computed over several hundred of tests performed with various cluster prototype

initializations.

Besides α and β, the parameters ηi are also expected to influence the behavior of the

algorithm. If we follow the rule proposed by Krishnapuram and Keller [91], presented in

Eq. (2.15), we obtain (in case of κ = 1): η1 = 0.3427, η2 = 0.5824, and η3 = 0.6894.

In the followings, we will refer to these values as ideal ηi values. These values yield the


Figure 2.6: Average classification score in case of ηi = 1 ∀i = 1 . . . c

classification score diagram shown in Fig. 2.5. Setting larger values to this possibilistic

parameter leads to similar classification scores, represented in Figs. 2.6 and 2.7.

By examining Figs. 2.5–2.7, and the simulation results that stand behind them, we can

remark the followings:

The classification score slightly varies with the initial cluster prototype, but generally

the algorithm is stable. The only unstable case is when β = 0, and thus the hybrid c-

means is reduced to PCM, which has the well-known tendency of leading to clusters whose

prototypes attract each other. This property is also visible in Fig. 2.10, where the validity


Figure 2.7: Average classification score in case of ηi = 2.5 ∀i = 1 . . . c

of clusters is analyzed. The classification score mainly depends on β and ηi, not on α. This

is visible from the shape of different colored regions, which mostly resemble some arcs of

concentric circles, or the leaves of a cabbage.

Each of these three diagrams have a region with maximum classification score, 139 or

140. The position of this region depends on ηi. At ideally chosen ηi values, the best regions

are situated on the arcs corresponding to β = 0.075 and β = 0.1. As the parameters ηi

grow, this maximum scoring region drifts outwards to larger β’s: setting ηi = 1 ∀i = 1 . . . c

leads to β ∈ [0.175, 0.225], while ηi = 2.5 yields β ∈ [0.425, 0.475].


Figure 2.8: The number of necessary iterations to reach a convergence of 10−6 precision,

plotted against hybridization parameters α and β, at ideal ηi values

Figure 2.9: Same as Fig. 2.8, at different resolution of α and β


Figure 2.10: Cluster validity index plotted against hybridization parameters α and β, at

ideally chosen ηi values

According to our experiments, the maximum classification score slowly reduces as we

increase ηi. As shown in Figs. 2.5–2.7, ideal ηi’s can give 140 correct decisions, with

ηi ∈ [1.0, 2.5] we obtain only 139. Tests revealed that at certain high values of ηi may

cause this score fall to 138 or 137.

Tradeoff parameter α, responsible for the FCM/HCM ratio within the hybrid c-means

model, has only a slight influence on the classification score. In other words: if we go along

the whole arc determined by a fixed β, by gradually changing α from 0 to 1, the number

of correct decisions will hardly vary more than one unit. In most of the cases, the better

accuracy is obtained at the α = 1 end of the arc, which supports the generally accepted

opinion that FCM makes more accurate partitions than HCM. This relation between FCM

and HCM persists when mixed with the same amount of PCM.

The classification scores are quite stable: they hardly depend on the initially chosen

cluster prototypes.

The next important issue to be treated is the number of necessary iterations to reach a


given level of convergence. The easiest way to test the convergence is to set up a criterion

that evaluates the change in the cluster prototypes during one cycle: ifc∑

i=1

||vi−v(old)i || < ε,

then the convergence is reached. The variable ε represents a previously set small positive

number.

A series of tests have been performed to establish the average number of necessary

cycles at given α and β parameter values. The hybrid c-means clustering algorithm was

performed several times for each setting, with different cluster prototype initializations.

The averaged iteration counts plotted against α and β are represented in Figs. 2.8 and

2.9. Figure 2.8 shows the whole domain (α, β) ∈ [0, 1] × [0, 1] at a coarse resolution. A

relevant peak is clearly visible at α = 1 and β ∈ [0.1, 0.2], indicating that a pure PCM-

FCM mixture is likely to need significantly more iterations than any other hybrid mixture.

In order to find a more exact position of the peak, further tests have been performed at a

finer resolution of the vicinity of the peak. Test results are plotted in Fig. 2.9.

As it was expected, in these figures it is clearly visible that mixing FCM with HCM

by reducing α from its maximum value also reduces the number of necessary computation

cycles. This idea was also pointed out by Fan et al. in [49], but their mixture was different

from our proposed hybrid model.

Besides the relevant peak already mentioned, there is also a visible tendency of the

algorithm to require more iterations as the amount of PCM grows within the mixture, that

is, as β gets closer to 0.

Another important issue is cluster validity. Here we will employ one of the most simple

and most easy-to-understand cluster validity index (CVI):

CVI = min{||vi − vj|| : 1 ≤ i < j ≤ c} , (2.64)

which is the minimum distance between cluster prototypes. Obviously, we will call a

solution stable or valid if its CVI is high, and unstable or invalid if its CVI is low.

The proposed hybrid c-means clustering algorithm was thoroughly tested in order to


Figure 2.11: Graphical representation of the most stable solution having 140 correctly

classified feature vectors, corresponding to 6.67 % misclassification rate


Figure 2.12: Graphical representation of the most stable solution having 139 correctly

classified feature vectors, corresponding to 7.33 % misclassification rate


establish a relation between the tradeoff parameters and the cluster validity. The obtained

results are graphically represented in Fig. 2.10. In this figure we can observe that the PCM

algorithm, and mixtures situated close to the PCM have low CVI, giving a low credibility

to these cases. Those cases mentioned above, which need an extremely high number of AO

cycles, represented by the peak in Figs. 2.8 and 2.9, correspond to the maximum slope of

the CVI function, indicating the highest lability point of the α− β domain. Consequently,

these spots should be avoided at the selection of tradeoff parameters.

In the followings, let us take a look at the partitions we can obtain using the hybrid c-

means clustering model. Figures 2.11 and 2.12 are detailed representations of two different

scenarios, both selected as having the highest CVI at their high classification score. Each

figure contains six 2D projections of the 150 four-dimensional feature vectors of the IRIS

data. Colors signify the ground truth, that is, blue stands for the first cluster (called

“setosa”), green is the color of the second cluster (“versicolor”), and finally red represents

the third cluster (“virginica”) [2]. The partitions given by the proposed clustering model

are represented with shapes: circles, squares and triangles stand for the first, second, and

third class, respectively. In case of a perfect clustering, all circles should be blue, all squares

green, and all triangles red. However, we will see, this is hardly possible because of the

overlapping clusters. The obtained cluster prototype positions are indicated by the center

of large blue circle, large green square, and large red triangle.

Figure 2.11 represents the results of the setting α = 0.2, β = 0.075. The final cluster

prototypes are

v1 =

5.00496

3.40291

1.49196

0.25213

, v2 =

6.03476

2.84170

4.54372

1.48235

, v3 =

6.37390

2.92896

5.10796

1.80651

, (2.65)

while the cluster validity index takes an acceptable value of CVI = 0.73896. The classifi-

cation score is 140 correct decisions out of 150.


We can get almost this precision with a considerably higher stability: classification

score of 139, at α = 0.55, β = 0.175, and the quality index turns out to be CVI = 1.10510.

The obtained cluster prototypes are

v1 =

5.00533

3.40582

1.48983

0.25223

, v2 =

5.97193

2.81659

4.46294

1.44137

, v3 =

6.50017

2.97473

5.29671

1.91257

. (2.66)

The partitions generated by these cluster prototypes are visualized in Fig. 2.12.

As a short remark on the topic of sensitivity of the proposed method to outliers: FCM

and HCM components present in the hybrid model can cause this kind of effect. However,

as the recommended value for β stays within the bounds [0.1, 0.3], the tradeoff coefficients

of these components are reduced, so their effect upon the cluster prototypes is also consid-

erably weaker as in case of pure FCM or HCM. This theoretical observation was totally

confirmed by the performed test series. In case that we wish to further reduce the sen-

sitivity to outliers, it is recommendable to apply the technique of the extra noise class

presented in Eqs. (2.10)-(2.12).

Further tests have been performed involving the WINE data set [7], which contains 178

vectors of 13 dimensions. We have established that the hybrid model behaves similarly in

such a multidimensional environment.

Choosing parameter values

Based on the clustering results presented above, we can formulate a set of recommendations

regarding the parameters of the hybrid clustering model:

• The parameters that regulate the strength of possibilistic terms, namely ηi, i = 1 . . . c,

should be adjusted according to Eq. (2.15), using κ ∈ [1, 2]. Larger values would


deteriorate the best achievable partition quality. Smaller values would make the area

of best partitions drift into places where the cluster validity index is too low.

• According to Fig. 2.5, the best partitions are obtained at β ∈ [0.075, 0.2].

• The value of α, which defines the FCM/HCM ratio within the mixture, should be

chosen far enough from FCM to ensure a quick convergence, and close enough to FCM

to obtain good partitioning. These criteria determined us to recommend choosing

α ∈ [0.25, 0.75].

These parameter settings ensure at least 139 correct decisions at the clustering of the

IRIS data, regardless on the initial cluster prototypes. This is a relevant improvement

in comparison with the most commonly reported scores of 133-136, obtained with similar

clustering techniques.

2.4.3 Reducing memory requirements and execution time in case

of huge amounts of data

There are two kinds of limitations, which may turn out to be annoying as the number of

input vectors grows excessively. These are: time complexity and memory requirements.

The time complexity of each iteration cycle is directly proportional with the amount of

input feature vectors. The number of execution cycles may slightly vary with the number of

input vectors, but this change is not so relevant. There exist some approximative solutions

to reduce time complexity by aggregating similar input vectors. They were presented in

the beginning of this chapter, in section 2.2.1.

Reducing the memory requirements of the proposed hybrid c-means clustering is a

possible task. The algorithm can be formulated such a way that probabilistic fuzzy mem-

berships uik, typicality values tik and crisp memberships hik are all computed in every

iteration cycle, but their values are not stored individually. In other words, instead of


storing the partition matrices, we will compute the sums that are required by the cluster

prototype update formula. Let us transcribe the Eq. (2.62):

vi =βαSiu + (1− β)Sit + β(1− α)Sih

βαQiu + (1− β)Qit + β(1− α)Qih

, (2.67)

where

Siu =n∑

k=1

umikxk, and Qiu =

n∑k=1

umik , (2.68)

Sit =n∑

k=1

tpikxk, and Qit =n∑

k=1

tpik , (2.69)

Sih =n∑

k=1

hikxk, and Qih =n∑

k=1

hik . (2.70)

According to these notations, we can reformulate the inner cycle of the hybrid c-means

clustering model:

• Initialize Siu = Sit = Sih = 0 and Qiu = Qit = Qih = 0

• For each k = 1 . . . n do

Compute fuzzy probabilistic memberships uik, i = 1 . . . c using Eq. (2.57)

Compute possibilistic memberships (typicalities) tik, i = 1 . . . c using Eq. (2.59)

Compute crisp degrees of membership hik, i = 1 . . . c using Eq. (2.60)

For ∀i = 1 . . . c, update Siu ← Siu + umikxk and Qiu ← Qiu + um

ik

For ∀i = 1 . . . c, update Sit ← Sit + tpikxk and Qit ← Qit + tpik

For ∀i = 1 . . . c, update Sih ← Sih + hikxk and Qih ← Qih + hik

• Update cluster prototypes by applying Eq. (2.67) for each i = 1 . . . c.

Using this formulation of the problem, the number of stored floating point variables

changes from c(3n + z) to c(4z + 3), where n in the number of input vectors, z is the

dimension of the input vectors, and c is the number of clusters. As the number of input


data grows to the order of thousands or even millions (e.g. pixels of an image), this is a

very convenient change in the necessary storage space. For example, at the segmentation of

a single channel T1-weighted magnetic resonance image of a brain, consisting of 256× 256

pixels, to the classical three classes, the necessary memory storage space is reduced 28000

times. As the number of performed arithmetical operations is practically the same, there

cannot be speed changes of orders of magnitude. However, as Kolen and Hutcheson showed

it in [87], as the number of load and store operation is reduced, it is possible to reduce the

execution time with 30− 70 %.

Algorithms having this reduced memory requirements are also suitable for hardware

implementation [3].

2.4.4 The evolutionary computation approach

In this section we propose to study the genetic algorithm approach of the hybrid clustering

model defined in Eq. (2.54). Based on the reformulated objective functions of classical

c-means clustering models, given in [63], the generalized hybrid model can be formulated

as the following mixture criterion:

Jhybrid = βαRFCM + (1− β)RPCM + β(1− α)RHCM , (2.71)

where α, β ∈ [0, 1] represent balancing parameters that control the tradeoff among the

three terms. Obviously, setting β = 0 returns us to PCM, β = α = 1 means FCM, while

β = 1 and α = 0 yield HCM clustering. Further parameters are fuzzy exponents m and p

(applied to the FCM and PCM terms) and ηi (for PCM only). According to the equivalence

proofs given by Hathaway et al. [63], this cost function is equivalent with the one given in

Eq. (2.54).

In the followings, in order to study the properties of the proposed clustering method,

we will turn to evolutionary computation techniques. The optimization is performed in

two steps: first a global optimizer genetic algorithm [55, 69] is applied which reaches a sub-


Figure 2.13: Graphical representation of the average number of correct decisions out of

150, vs. α and β tradeoff parameters

optimal solution after 50-100 generations. This is followed by a local Nelder-Mead simplex

algorithm [94], which finds the best solution within the neighborhood of the suboptimum.

The properties of the proposed generalized clustering method were tested using the IRIS

data. We tested the algorithm with various parameter settings: α and β independently

varied from 0 to 1, with steps of 0.1. The algorithm was performed 10 times with each

parameter setting. Figures 2.13 and 2.14 show the average and maximum number of

correctly clustered vectors for all parametrized cases.

Looking at the graphs in Figs. 2.13 and 2.14, we can formulate the followings. Two

interesting cases need to be remarked. Most of the accurate clustering techniques report

133-135 correct decisions out of 150. Our graph in Fig. 2.14 shows two independent spots

with better decision rate. We can have 140 correct decision at β = 1, that is, when the

hybrid model is reduced to a mixture of HCM and FCM. This suggests that under certain

circumstances, mixing FCM and HCM can lead to a superior clustering algorithm. One


Figure 2.14: Graphical representation of the maximum number of correct decisions out of

150, vs. α and β tradeoff parameters

set of prototypes, which was found to give this partition, is:

v1 =

4.65310

3.07784

1.50021

0.20540

, v2 =

5.82625

2.70022

3.98397

1.31061

, v3 =

6.72685

3.09917

5.59478

2.43399

. (2.72)

Besides their high classification score, the above cluster prototypes show a fine stability:

their cluster validity index is CVI = 2.19703, which is well above the best achieved value

of the AO solution.

The whole IRIS data set and its partition obtained with the vectors given above, are

represented in two-dimensional projections in Fig. 2.15.

The other key thing that deserves a remark is the fact that pure PCM (β = 0) is quite

unstable, which means in most cases it leads to poor quality partitions (as indicated in

[13]), and is always characterized by a low CVI, but sometimes it finds a set of prototypes


Figure 2.15: Partition with 140/150 success rate signifying 6.67 % misclassifications.

Shapes indicate the ground truth labeling; misclassified vectors appear in different color.

Cluster prototypes are indicated by the large circle, triangle and square


Figure 2.16: Partition with 149/150 success rate signifying 0.67 % misclassifications.


offering 149 correct decisions.

One such case, having CVI = 0.27547, is depicted in eq. (2.73), and Fig. 2.16:

v1 =

5.14843

3.51076

1.51996

0.13272

, v2 =

6.02770

2.85190

4.62071

1.67480

, v3 =

5.94317

2.76658

4.81742

1.82569

. (2.73)

Cluster prototypes with this high clustering score are difficult to find with genetic

algorithms, and impossible with AO.

These results of poor stability or validity, provided by pure PCM are easily corrected

by giving β a small, but non-zero value, for example 0.1. Figures 2.13 and 2.14 suggest

this choice to be the best at this resolution, whatever α might be. In other words, for any

α ∈ [0, 1], the best choice is to set a low value β (e.g. β = 0.1), and we get a stable and

accurate solution with 136-137 correct decisions. Similar optimal β values were obtained

with the AO technique, too, but that way we could achieve 139-140 correct decisions with

a very high stability.

In this section we proposed a mixture c-means clustering model, and involved a genetic

algorithm for its optimization. Using the IRIS data to test and evaluate the proposed

approach, we have concluded the followings:

• Mixtures of HCM and FCM, under certain circumstances, can outperform pure HCM

or pure FCM.

• Pure PCM is very unstable, but mixing 80−90 % of PCM with some HCM and FCM

shows excellent in both partition quality and stability.

• The more FCM and less HCM we have in a mixture, the better the accuracy will be.

However, this variation is slow.

2.5. Suppressed and optimally suppressed FCM, revisited 57

Although under certain circumstances the genetic algorithm was able to provide solu-

tions with very reduced misclassification rate, the AO version is more suitable for clustering

problems because

• it requires less computations at least by an order of magnitude,

• it appears to be incomparably more reliable, its results are much easier to reproduce.

2.5 Suppressed and optimally suppressed FCM, re-

visited

After having our hybrid c-means model analyzed, it is easier to judge the difference between

the behavior of these two algorithm. If we set the parameter β = 1, we exclude the

possibilistic component from the hybrid model and we obtain the optimally suppressed

FCM clustering algorithm.

The classification results, which we can obtain with the Os-FCM are shown in Figs.

2.5-2.7 on the external (longest) arc of the pie diagrams. These three diagrams show the

same behavior of Os-FCM, which is not at all surprising, as the η term is also excluded

from the cost function, so it cannot have any influence in our case.

Now let us compare the main behavioral parameters of the two algorithm. Figure 2.17

shows the clustering score of s-FCM and Os-FCM, averaged along several hundreds of

tests performed with different initializations. Figure 2.18 presents the average number of

iterations of s-FCM and Os-FCM, necessary to reach a given level of convergence. The

data represented here are also computed from hundreds of tests.

From the shape of the two graphs it seems obvious that the two methods are related

somehow. Not only the approximative values are similar for any α, but also the shape of

the graphs. The graph in Fig. 2.17, and also the formulation of the two clustering models

would support the idea, that s-FCM and Os-FCM should exhibit the highest similarity if


Figure 2.17: Average clustering score of s-FCM and Os-FCM, plotted against α

Figure 2.18: Average number of necessary iterations for s-FCM and Os-FCM to reach an

ε = 10−8 convergence, plotted against α

2.6. Conclusions 59

their suppression rates satisfy (α(s−FCM))m = α(Os−FCM). But even in this case, these two

algorithms are not identical, as it will be shown in the followings.

Figure 2.17 indicates that the result of clustering and the misclassification rate does

depend on the initialization: the clustering score can be either 133 or 134 in all cases.

The shape of the graphs in Fig. 2.17 suggests that there might be a direct relation

between α(s−FCM) and α(Os−FCM). If s-FCM and Os-FCM were equivalent, then according

to Eqs. (2.7), (2.8), (2.6) and (2.47), we should have ∀uik ∈ [0, 1],∀m > 1(α(s−FCM)uik)

m = (α(Os−FCM)uik)m

(1− α(s−FCM) + α(s−FCM)uik)m = (α(Os−FCM)uik)

m + 1− α(Os−FCM)

. (2.74)

In order to have any chance for equivalence between s-FCM(α(s−FCM)) and Os-

FCM(α(Os−FCM)), this equation system should be compatible. But this isn’t the case:

the only case of compatibility is m→ 1 and α(s−FCM) = α(Os−FCM). However, m→ 1 is the

case of HCM, where there is nothing to suppress.

As a diagnosis of the suppressed FCM algorithm we can say: we cannot take for granted

the optimality or non-optimality of s-FCM, but we can assert that it behaves very similarly

to an optimal clustering model (Os-FCM).

2.6 Conclusions

In this chapter, as an extension to the theory of c-means clustering, I have introduced a

hybrid c-means clustering model that contains variable amounts of fuzzy c-means (FCM),

possibilistic c-means (PCM), and hard c-means (HCM) clustering, weighted by two tradeoff

parameters, α and β. Tests involving various input data sets revealed that the proposed

algorithm is robust and fast, and it provides high quality partitions, within most part of

the weighting parameter domain [L4, L23].

I have evaluated the behavior and the performance of the proposed hybrid clustering


model within the whole α-β domain involving relevant parameters like misclassification

rate and expected number of iterations. Based on this study, I have established the rules

of choosing suitable weighting parameters α and β [L4, L23].

I have formulated a reduced memory usage version of the hybrid c-means clustering

model. Memory reduction is achieved via not storing partition matrices, and computing

those sums instead, which contribute to the computation of cluster prototypes. Memory

usage can be reduced by three or four orders of magnitude, while in case of high amount

of data the algorithm will also perform better against the clock [L23].

I have performed an analytical and numerical evaluation of the suppressed FCM cluster-

ing algorithm (s-FCM) and I have established the formula of its quasi-competitive behavior.

Based on the functional similarity of s-FCM with a special case of the proposed hybrid

c-means clustering model, namely the β = 1, I have proposed an optimally suppressed

FCM algorithm [L3, L5] and revealed its slight advantages over s-FCM.

Chapter 3

Medical Image Segmentation Using

FCM-Based Algorithms

Whenever we take a picture with our camera, an image is created. If we are professional or

lucky enough, the image will look mostly the way we wanted, but there are some possible

circumstances that may represent difficulties to several photographers (e.g. bad weather,

darkness, shaded areas, etc.). Images, which were not created under ideal circumstances,

need to be enhanced in order to satisfy our expectations. Most cameras have their in-

tegrated image enhancement algorithms, which even cannot be switched off, causing an

image enhancement to every picture taken. If we show our photographed and enhanced

image to several people, they will probably see different things in it. For example, the

Eiffel tower means an architectural masterpiece for some people, and a big piece of old iron

for others. This is the result of different interpretation.

If we consider medical images, several issues will resemble the case of digital photog-

raphy, but there will be others quite different ones. A medical image processing problem

may (but doesn’t always) contain the following steps:

1. Image formation, which is performed by the image acquisition device.

61

62 Chapter 3. Medical Image Segmentation Using FCM-Based Algorithms

2. Image enhancement has the main goal of making objects visible, by eliminating noise,

improving edges and contrast, equalizing histograms.

3. Image segmentation refers to finding and distinguishing objects in the image, sepa-

rating objects from background or separating relevant objects from irrelevant ones.

Segmentation can also refer to separating a region of interest (ROI) from the other

parts of the image. For example: brain and non-brain in an MRI image [4, 8].

4. Sometimes in medical imaging, during an investigation, not only one 2-D picture is

taken, but several parallel ones or different views of the same volume. In such cases,

the 2-D segmentation is followed by the reconstruction of the 3-D surface of objects.

5. Sometimes the segmentation is performed in 3-D. In this case there is one more

operation that is performed simultaneously, called image registration, which has the

main goal to define the actual placement of the 2-D slices or different views.

6. Image interpretation is absolutely a problem specific question. In a brain MR slice

it is likely to look for white and gray matter, while in an X-ray lung scan we are

probably interested in the contour of the lung. On the other hand, different gray

levels of an MR image are closely related to the tissue type that is being scanned,

while in ultrasound imaging the intensity values of individual pixels have hardly any

meaning.

This list may be completed with a further step called visualization, which is not anymore

image processing: it is an issue of computer graphics and machine vision.

Computerized medical image processing has the main goal to perform steps 2-6 of the

list presented above as automatically as possible.

In this chapter we will mostly concentrate on improving image enhancement and image

segmentation methods, based on the clustering techniques created in the previous chapter.

The proposed methods are presented in Sections 3.3 and 3.4.

3.1. Magnetic resonance imaging 63

In the next chapter we will present full implementation details of a virtual endoscope.

This device has the main goal to visualize the internal structure of the human body without

actual penetration, based on parallel 2-D slices taken with magnetic resonance imaging

technology.

3.1 Magnetic resonance imaging

Magnetic resonance imaging (MRI) is one of the way to look inside the human body,

that is, to take pictures of its cross sections without cutting. Among all other computed

tomography (CT) methods, MRI has several advantages:

1. Neither the constant magnetic field, nor the radio frequency modulated one, which

are present in the MRI device, represent a danger to humans. Other CT methods,

for example the positron emission tomography, uses X-rays to create the image.

2. The image contrast of a CT scan only depends on the electron density of the tissues

considered. The MRI signal is determined by the proton density of the tissue, T1 and

T2 relaxation times, the type of sequence used, and the selected acquisition parame-

ters. These parameters give the opportunity to enhance the image contrast between

two tissues by cleverly choosing the type of sequence and acquisition parameters, and

thus optimize the differentiation between tissue structures.

3. Some MRI devices give us the opportunity to produce multi-spectral images, that is,

two or more images of the same cross section with different parameter settings.

In order to prevent possible misinterpretations, most MR images are acquired such

a way that the tissue contrast of various images is determined mainly on a single tissue

parameter. In this context, T1, T2, and PD-weighted images are produced.

Human MR scans are subject to different kinds of noise:


1. High frequency noises that are present in MR images are frequently modeled or

treated as impulse or salt-and-pepper noise [1], Gaussian noise [6, 135, 145, 148, 186],

or mixture of these two [30][L7, L8]. However, lately it has been proved that these

high frequency components of the noises contaminating MR images are following

a Rician distribution [9]. In spite of this latter mentioned fact, the Gaussian and

impulse-Gaussian mixture approaches can represent good approximations under some

circumstances, and are much easier ways to implement.

2. Low frequency noises in magnetic resonance imaging are generally referred to as

intensity inhomogeneity or intensity non-uniformity (INU) artefact, and manifest

as a smooth intensity variation across the image [185]. This phenomenon causes

regions of the image belonging to the same tissue class have different intensities. Low

magnitude INU is hardly noticeable for the human eye, but it can induce confusion for

segmentation algorithms with high sensitivity. But the magnitude of the INU artefact

can be very high, and thus the segmentation requires sophisticated compensation

techniques to perform accurately. Considering the sources of inhomogeneity, we can

distinguish two kinds of artefacts:

a. INU related to the MRI device has efficient prospective compensation and

calibration methods, for example based on usage of a uniform phantom to produce

prior information [10].

b. INU related to the patient’s shape, position and orientation. This form of

inhomogeneity is much more difficult to handle [185], but there are several reported

approaches to compensating them. Homomorphic filtering represents a popular com-

pensation method [29, 72], built upon the theoretical assumption that the frequency

spectra of the image structures and of the INU artefact are not overlapping each

other. These methods are quite fast, but their accuracy is limited because the initial

assumption does not hold. Surface fitting methods usually fit a polynomial or spline-

based parametric surface to the image features [185], and use this surface to model

3.1. Magnetic resonance imaging 65

a gain field that multiplies the noise-free image. The parameters of the surface are

then estimated using pixel intensities [183] or image gradient measures [86]. There

are further methods that incorporate the INU compensation into image segmenta-

tion techniques: in this group we can distinguish maximum likelihood and maximum

a posteriori estimation techniques [97, 98, 198] and c-means clustering based tech-

niques [1, 162]. Liew et al. [100] introduced a compensation model that unifies the

FCM-based segmentation approach with spline surface fitting. Several and various

histogram based approaches exist: the nonparametric nonuniformity normalization

technique proposed in [161] maximizes the high frequency components of the tis-

sue intensity distribution. Further histogram-based techniques rely on information

minimization [101], and histogram matching [158]. Inhomogeneity correction was

successfully integrated into image registration techniques, too. Two such methods

are reported in [104, 166]. A recently written, more detailed review of INU compen-

sation methods is given in [185].

3. The partial volume artefact (PVA), also known as partial volume effect (PVE), repre-

sents a phenomenon that is present in MR medical images due to its coarse resolution.

Even if MRI reportedly has higher resolution than other medical imaging techniques,

is still has only 1-3 pixels per millimeter. Under such circumstances it is unavoidable

to have pixels that are shared between two or even among three or more tissue types.

Any accurate segmentation is required to provide partial volume estimation for such

pixels. The most frequently used partial volume models are the mixel model [38, 147]

and the finite Gaussian mixture model [135, 153, 186, 200]. The most popular estima-

tion approach is based on maximum a posteriori probability estimation and Markov

random fields [6, 77, 198]. However, fuzzy c-means clustering is also reported to give

acceptable partial volume estimations [99].

In this chapter we will attempt to provide improvements to c-means clustering based

correction techniques that handle all three artefact types.


3.2 Image segmentation techniques

The segmentation of an image represents the separation of its pixels into non-overlapping,

consistent regions, which appear to be homogeneous with respect to some criteria concern-

ing gray level intensity and/or texture.

The segmentation of T1-weighted brain MR images generally aim at separating white

matter (WM) from gray matter(GM) and cerebro-spinal fluid (CSF). These three main

tissue classes can generally be separated by their gray level intensity if the contaminating

noises are well compensated or corrected.

T2-weighted brain MR images are often partitioned into 5 or 6 classes [39]: besides

GM, WM, and CSF, they may also distinguish fat, bone, and air.

Image processing textbooks [163] classify the classical image segmentation techniques

into three main groups:

1. Segmentation based on global information: generally performed using the histogram

of the image, and thresholding operations with optimized threshold values. Segmen-

tation using FCM clustering belongs to this group as long as no spatial constraints

are introduced [22].

2. Segmentation based on edge detection methods: starting from gradient based edges

and the classical Canny filter [31], and continuing on with contour models [127, 156].

Details on this latter issue will be presented in the next chapter, section 4.3., as these

methods coincide with the surface reconstruction techniques discussed there.

3. Region based segmentation techniques include watershed methods and split-and-

merge algorithms. This latter produces homogeneous regions by first splitting the

image into as small regions as necessary to have them homogeneous (a single pixel

is always homogeneous), and then merging as many neighbor regions as possible

without losing the homogeneity. This method is extremely time consuming, the only

3.3. Quick segmentation of MR brain images 67

Figure 3.1: A piece of gray matter detected using the split-and-merge segmentation tech-

nique.

reason it deserves being remarked is the quality of regions obtained. Fig. 3.1 shows

an area of gray matter detected with this method.

The rest of this chapter is organized as follows: the second section presents a histogram

based fuzzy c-means clustering technique for quick and robust segmentation of MR images

contaminated with high frequency noise. The third section reports a multi-stage FCM

based inhomogeneity correction and segmentation technique for MR brain images. Each

of these sections contain a summary of closely related works, a set of contributed meth-

ods presented in full details, and evaluation of results. The fourth section relates on the

experiments carried out in order to test the efficiency of the hybrid clustering model in

inhomogeneity compensation and image segmentation. The chapter finally presents the

conclusions and summarizes the contributions presented in the chapter.

3.3 Quick segmentation of MR brain images

The fuzzy c-means (FCM) algorithm is one of the most widely used methods for data

clustering, and probably also for brain image segmentation [21]. However, in this latter

case, considering each pixel of the image an input vector, standard FCM is not efficient by


itself, as it fails to deal with that significant property of images that neighbor pixels are

strongly correlated. Ignoring this specificity leads to strong noise sensitivity and several

other imaging artifacts.

Recently, several solutions were given to improve the performance of segmentation.

Most of them involve using local spatial information: the own gray level of a pixel is not

the only information that contributes to its assignment to the chosen cluster. Its neigh-

bors also have their influence while getting a label. Pham and Prince [137] modified the

FCM objective function by including a spatial penalty, enabling the iterative algorithm to

estimate spatially smooth membership functions. Ahmed et al. [1] introduced a neighbor-

hood averaging additive term into the objective function of FCM, calling the algorithm

bias corrected FCM (BCFCM). This approach has its own merits in bias field estimation,

but it gives the algorithm a serious computational load by computing the neighborhood

term in every iteration step. Moreover, the zero gradient condition at the estimation of

the bias term produces a significant amount of misclassifications [162]. Chuang et al. [39]

proposed averaging the fuzzy membership function values over a predefined neighborhood

and reassigning them according to a tradeoff between the original and averaged member-

ship values. This approach can produce accurate clustering if the tradeoff is well adjusted

empirically, but it is enormously time consuming.

Aiming at reducing the execution time, Szilagyi et al. [L19], and Chen and Zhang

[36] proposed to evaluate the neighborhoods of each pixel as a pre-filtering step, and per-

form FCM afterwards. The averaging and median filters, followed by FCM clustering,

are referred to as FCM S1 and FCM S2, respectively [36]. Once having the neighbors

evaluated, and thus having extracted a scalar feature value for each pixel, FCM can be

performed on the basis of the gray level histogram, clustering the gray levels instead of

the pixels, causing a significant reduction of the computational load, as the number of

gray levels is generally smaller by orders of magnitude [L18]. This latter quick approach,

combined with an averaging pre-filter, is referred to as enhanced FCM (EnFCM) [30].


All BCFCM, FCM S1, and EnFCM suffer from the presence of a parameter denoted by

α, which controls the strength of the averaging effect, balances between the original and

averaged image, and whose ideal value unfortunately can be found only experimentally.

Another disadvantage emerges from the fact that averaging and median filters, besides

eliminating salt-and-pepper and Gaussian noises, also blur relevant edges. Due to these

shortcomings, Cai et al. [30] introduced a new local similarity measure, combining spatial

and gray level distances, and applied it as an alternative pre-filtering to EnFCM, calling

this approach fast generalized FCM (FGFCM). This approach is able to extract local in-

formation causing less blur than the averaging or median filters, but failed to eliminate the

experimentally adjusted parameter, denoted here by λg, which controls the effect of gray

level differences.

Another remarkable approach, proposed by Pham [139], modifies the objective function

of FCM by the means of an edge field, in order to exclude the filters that produce edge

blurring. This method is also significantly time consuming, because the estimation of the

edge field has no direct analytical solution.

In this section we propose a novel method for MR brain image segmentation that

simultaneously targets high accuracy in image segmentation, low noise sensitivity, and

high processing speed.

3.3.1 Spatial constraints

FCM clustering has invaluable merits in making optimal clusters, but in image processing

it has severe deficiencies. The most important one is the fact that it fails to take into

consideration the position of pixels, which is also relevant information while performing

image segmentation. This drawback led to the introduction of spatial constraints into fuzzy

clustering [L8, L9].

Ahmed et al. [1] proposed a modification to the objective function of FCM, in order to


allow the labeling of a pixel to be influenced by its immediate neighbors. This neighboring

effect acts like a regularizer that biases the solution to a piecewise homogeneous labeling

[1]. The objective function of BCFCM is:

JBCFCM =n∑

k=1

c∑i=1

[um

ik(xk − vi)2 +

α

nk

∑r∈Nk

umik(xr − vi)

2

], (3.1)

where xr represents the gray level of pixels situated in the neighborhood Nk of pixel k,

and nk is the cardinality of Nk. The parameter α controls the intensity of the neighboring

effect, and unfortunately its optimal value can be found only experimentally. Having

the neighborhood averaging terms computed in every computation cycle, this iterative

algorithm performs extremely slowly.

Chen and Zhang [36] reduced the time complexity of BCFCM, by previously computing

the neighboring averaging term or replacing it by a median filtered term, calling these

algorithms FCM S1 and FCM S2, respectively. These algorithms outperformed BCFCM,

at least from the point of view of time complexity.

Szilagyi et al. [L19, L20] proposed a regrouping of the processing steps of BCFCM.

In their approach, an averaging filter is applied first, similarly to the neighboring effect of

Ahmed et al. [1]:

ξk =1

1 + α

(xk +

α

nk

∑r∈Nk

xr

), (3.2)

followed by an accelerated version of FCM clustering. The acceleration is based on the

idea that the number of gray levels is generally much smaller than the number of pixels.

In this order, the histogram of the filtered image is computed, and not the pixels, but the

gray levels are clustered [L19], by minimizing the following objective function:

JEnFCM =

q∑l=1

c∑i=1

hlumil (l − vi)

2 , (3.3)

where hl denotes the number of pixels with gray level equaling l, and q is the number of

gray levels. The optimization formulae in this case will be:

u?il =

(vi − l)−2/(m−1)

c∑j=1

(vj − l)−2/(m−1)

∀ i = 1 . . . c, ∀ l = 1 . . . q , (3.4)


v?i =

q∑l=1

hlumil l

q∑l=1

hlumil

∀ i = 1 . . . c . (3.5)

EnFCM drastically reduces the computation complexity of BCFCM and its relatives.

If the averaging pre-filter is replaced by a median filter, the segmentation accuracy also

improves significantly [30].

Based on the disadvantages of the aforementioned methods, but inspired of their merits,

Cai et al. [30] introduced a local (spatial and gray) similarity measure that they used to

compute weighting coefficients for an averaging pre-filter. The filtered image is then subject

to EnFCM-like histogram-based fast clustering. The similarity between pixels k and r is

given by the following formula:

Skr =

s(s)kr · s

(g)kr if r ∈ Nk \ {k}

0 if r = k

, (3.6)

where s(s)kr and s

(g)kr are the spatial and gray level components, respectively. The spatial

term s(s)kr is defined as the L∞-norm of the distance between pixels k and r. The gray level

term is computed as

s(g)kr = exp[−(xk − xr)

2/(λgσ2k)] , (3.7)

where σk denotes the average quadratic gray level distance between pixel k and its neigh-

bors. Segmentation results are reported to be more accurate than in any previously pre-

sented case [30].

3.3.2 Proposed method

Probably the most relevant problem of all techniques presented above, BCFCM, EnFCM,

FCM S1, and FGFCM, is the fact that they depend on at least one parameter, whose value

has to be adjusted experimentally.


The zero value in the second row of Eq. (3.6) implies that in FGFCM, the filtered

gray level of any pixel is computed as a weighted average of its neighbor pixel intensities.

Having renounced to the original intensity of the current pixel, even if it is a reliable,

noise-free value, unavoidably produces some extra blur into the filtered image. Accurate

segmentation requires this kind of effects to be minimized [139, 140].

Context dependent filtering

In this section we propose a set of modifications to EnFCM/FGFCM, in order to improve

the accuracy of segmentation, without renouncing to the speed of histogram-based clus-

tering. In other words, we need to define a complex filter that can extract relevant feature

information from the image while applied as a pre-filtering step, so that the filtered image

can be clustered fast afterwards based on its histogram. The proposed method consists of

the following steps:

A. As we are looking for the filtered value of pixel k, we need to define a small square

or diamond-shape neighborhood Nk around it. Square windows of size 3×3 and 5×5 were

used throughout this study, but other window sizes and shapes are also possible.

B. We search for the minimum, maximum, and median gray value within the neighbor-

hood Nk, and we denote them by mink, maxk and medk, respectively.

C. We replace the gray level of the maximum and minimum valued pixel with the

median value (if there are more than one maxima or minima, replace them all), unless

they are situated in the middle pixel k. In this latter case, pixel k remains unchanged, just

labeled as unreliable value.

D. Compute the average quadratic gray level difference of the pixels within the neigh-

borhood Nk, using the formula

σk =

√1

nk − 1

∑r∈Nk\{k}

(xr − xk)2 . (3.8)


E. The filter coefficients will be defined as:

Ckr =

c(s)kr · c

(g)kr if r ∈ Nk \ {k}

1 if r = k ∧ xk 6∈ {maxk, mink}

0 if r = k ∧ xk ∈ {maxk, mink}

. (3.9)

The central pixel k will have coefficient 0 if its value was found unreliable, otherwise it has

unitary coefficient. All other neighbor pixels will have coefficients Ckr ∈ [0, 1], depending

on their space distance and gray level difference from the central pixel. In case of both

terms, higher distance values will push the coefficients towards 0.

F. The spatial component c(s)kr is a negative exponential of the Euclidean distance be-

tween the two pixels k and r: c(s)kr = exp(−L2(k, r)). The gray level term is defined as

follows:

c(g)kr =

12

[1 + cos

(π xr−xk

4σk

)]|xr − xk| ≤ 4σk

0, |xr − xk| > 4σk,

. (3.10)

The above function has a bell-like shape within the interval [−4σk, 4σk], and is constant

zero outside the interval. A graphical representation of this function is given in Fig. 3.2.

G. The extracted feature value for pixel k, representing its filtered intensity value, is

obtained as a weighted average of its neighbors:

ξk =

∑r∈Nk

Ckrxr∑r∈Nk

Ckr

. (3.11)

Algorithm

We can summarize the proposed method as follows:

1. Pre-filtering step: for each pixel k of the input image, compute the filtered gray level

value ξk, using Eqs. (3.8), (3.9), (3.10), (3.11).

2. Compute the histogram of the pre-filtered image, obtain the values hl, l = 1 . . . q.


Figure 3.2: The gray level difference based components of the averaging weights are com-

puted using this bell-shaped function, defined in Eq. (3.10)

3. Initialize vi with valid gray level values, differing from each other.

4. Compute new uil fuzzy membership values, using Eq. (3.4).

5. Compute new vi prototype values for the clusters, using Eq. (3.5).

6. If there is relevant change in the vi values, go back to step 4. This is tested by

comparing any norm of the difference between the new and the old vector v with a preset

small constant ε.

The algorithm converges quickly, however, the number of necessary iterations depends

on ε and on the initial cluster prototype values.

Handling partial volume effect

Whatever resolution an MR scanner may have, the scanned images will contain such pixels

where more than one tissue classes are present. This phenomenon is referred to as partial

volume effect (PVE). Although it is not granted, it is reasonable to assume that within a

given pixel, PVE only occurs over two classes [135]. Pixels involved in PVE are generally

modeled using the mixel model [147], which states that the gray level intensity of pixel k


is given by:

xk = αk · vµ + (1− αk) · vν + ηk, (3.12)

where ηk represents the noise of pixel k that will be ignored after context dependent

filtering, while vµ and vν are the centroids of the two involved classes, assuming vµ ≤ xk ≤

vν .

Fuzzy membership values given by FCM-based clustering techniques are reported to

give a good estimate of the partial volumes [198]. Let us inspect now on theoretical basis,

under what circumstances will the fuzzy memberships satisfy Eq. (3.12). In this order, we

would like to have

uµk

uνk

=αk

1− αk

. (3.13)

By applying Eqs. (2.5) and (3.12), we obtain

uµk

uνk

=

(xk − vµ

vν − xk

) −2m−1

=

((1− αk)(vµ + vν)

αk(vµ + vν)

) −2m−1

=

(αk

1− αk

) 2m−1

, (3.14)

which equals the desired value shown in Eq. (3.13) if and only if m = 3.

Consequently, if FCM is required to give estimation of partial volume ratios, the usage

of fuzzification exponent m = 3 is recommendable [L7, L8, L13].

3.3.3 Results and discussion

In this section we test and compare the accuracy of four algorithms: BCFCM, EnFCM,

FGFCM, and the proposed method, on several synthetic and real images. All the following

experiments used 3× 3 or 5× 5 window size for all kinds of filtering.

The proposed filtering technique uses a convolution mask whose coefficients are context

dependent, and thus computed for the neighborhood of each pixel. Fig. 3.3 presents the

obtained coefficients for two particular cases. Fig. 3.3(a) shows the case, when the central

pixel is not significantly noisy, but some pixels in the neighborhood might be noisy or might

belong to a different cluster. Under such circumstances, the three pixels on the left side


Figure 3.3: Filter mask coefficients in case of a reliable pixel intensity value (a), and a

noisy one (b). The upper number in each cell represents the intensity value, while the

lower number shows the obtained weight. The arrows indicate that the coefficients of

extreme intensities are contributed to the median valued pixel.

Figure 3.4: Segmentation results on phantom images: (a) original, (b) segmented with

traditional FCM, (c) segmented using BCFCM, (d) segmented using FGFCM, (e) filtered

using the proposed pre-filtering, (f) result of the proposed segmentation.


Figure 3.5: A comparison of the numbers of misclassifications at rising noise level (from

left to right).

having distant gray level compared to the value of the central pixel, receive small weights

and this way they hardly contribute to the filtered value. Fig. 3.3(b) presents the case of

an isolated noisy pixel situated in the middle of a relatively homogeneous window. Even

though all computed coefficients are low, the noise is eliminated, resulting a convenient

filtered value 76. The arrow-indicated migration of weights from the local maximum and

minimum towards the median valued pixel, caused by step C of the filtering method, is

relevant in the second case and useful in the first.

The noise removal performances were compared using a 256× 256-pixel synthetic test

image taken from IBSR [71] (see Fig. 3.4(a)). The rest of Fig. 3.4 also shows the degree

to which these methods were affected by a high-magnitude mixed noise. Visually, the

proposed method achieves best results, slightly over FGFCM, and significantly over all

others.

Fig. 3.5 shows the evolution of misclassifications obtained using three of the presented


methods, while segmenting the phantom shown in Fig. 3.4(a), corrupted by an increasing

amount of mixed noise (Gaussian noise, salt-and-pepper impulse noise, and mixtures of

these). Moreover, not only an extra amount of noise is added to the image step by step,

but also the original cluster centroids (the base intensities of the clusters) are moved closer

and closer to each other. This complex effect is obtained using a variably weighted sum

of three different noisy versions of the same image (all available at IBSR). Fig. 3.5 reveals

that the proposed filter performs best at removing all these kinds of noises. Consequently,

the proposed method is suitable for segmenting images corrupted with unknown noises,

and in all cases it performs at least as well as his ancestors.

We applied the presented filtering and segmentation techniques to several T1-weighted

real MR images. A detailed view, containing numerous segmentations, is presented in

Fig. 3.6. The original slice (a) is taken from IBSR. We produced several noisy versions

of this slice, by artificially adding salt-and-pepper impulse noise and/or Gaussian noise,

at different intensities. Some of these noisy versions are visible in Fig. 3.6 (d), (g), (j),

(m). The filtered versions of the five above mentioned slices are presented in the middle

column of Fig. 3.6. The segmentation results are shown in Fig. 3.6 (c), (f), (i), (l),

(o), accordingly. From the segmented images we can conclude, that the proposed filtering

technique is efficient enough to make proper segmentation of any likely-to-be-real MRI

images in clinical practice, at least from the point of view of Gaussian and impulse noises.

Table 3.1 takes into account the behavior of three mentioned segmentation techniques,

in case of different noise types and intensities, computed by averaging the misclassifications

on 12 different T1-weighted real MR brain slices. The proposed algorithm has lowest

misclassification rates in most of the cases.

We applied the proposed segmentation method to several complete head MR scans in

IBSR. The dimensions of the image stacks were 256× 256× 64 voxels. The average total

processing time for one stack was around 10 seconds on a 2.4 GHz Pentium 4.


Figure 3.6: Filtering and segmentation results on real T1-weighted MR brain images,

corrupted with different kinds and levels of artificial noise. Each row contains an original

or noise-corrupted brain slice on the left side, the filtered version (using the proposed

method) in the middle, and the segmented version on the right side. Row (a)-(c) comes

from record number 1320 2 43 of IBSR [71], row (d)-(f) is corrupted with 10% Gaussian

noise, while rows (g)-(i), (j)-(l), and (m)-(o) contain mixed noise of 3% impulse + 5%

Gaussian, 3% impulse + 10% Gaussian, and 5% impulse + 5% Gaussian, respectively.


Table 3.1: Misclassification rates in case of real brain MR image segmentation

Noise type EnFCM FGFCM Proposed

Original, no extra noise 0.767% 0.685% 0.685%

Gaussian 4% 1.324% 1.131% 1.080%

Gaussian 12% 4.701% 2.983% 2.654%

Impulse 3% 1.383% 0.864% 0.823%

Impulse 5% 1.916% 1.227% 0.942%

Impulse 10% 3.782% 1.268% 1.002%

Impulse 5% + Gaussian 4% 2.560% 1.480% 1.374%

Impulse 5% + Gaussian 12% 6.650% 4.219% 4.150%

3.4 Segmentation of MR brain images in the presence

of intensity inhomogeneity

Magnetic resonance imaging (MRI) is a very popular medical imaging technique, mainly

because of its high resolution and contrast, which represent great advantages above other

diagnostic imaging modalities. Besides all these good properties, MRI also suffers from

three considerable obstacles: noises (mixture of Gaussian and impulse noises), partial vol-

ume artefacts (pixels containing at least two types of tissues), and intensity inhomogeneity

[162]. This latter one, also known as intensity non-uniformity (or INU artefact), mani-

fests as a spatially slowly varying function that makes pixels belonging to the same tissue

be observed having different intensities. In order to produce a correct segmentation or

registration of MR images, the INU artefact needs to be modelled and compensated.

Although several INU compensation approaches exist [97, 136, 184, 198], one of the

most widely used methods is the adaptation of the fuzzy c-means clustering algorithm to

3.4. Segmentation of MR brain images in the presence of intensity inhomogeneity 81

iteratively approximate the INU as a smooth varying bias or gain field. In this order,

Pham and Prince introduced a modified objective function producing bias field estimation

and containing extra terms that force this artefact vary smoothly [137, 138]. They also

provided a multigrid technique to speed up the computationally heavy algorithm, but even

this way, the algorithm performs slowly. A probabilistic formulation leading to the same

objective function was given in [99]. Liew and Hong created a log bias field estimation

technique that models the INU with smoothing B-spline surfaces [100].

Further FCM-based bias field estimation techniques were introduced recently by Ahmed

et al. [1] and Siyal and Yu [162]. The modification introduced by Ahmed et al. allows the

labeling of a voxel to be influenced by its immediate neighbors. This approach has reduced

some of the complexity of its ancestors, but the zero gradient condition that was used for

bias field estimation leads to several misclassifications [162]. The other approach provided

a mean spread filtering method to smoothen the estimated bias field in every cycle of the

FCM algorithm. This approach reduces the amount of necessary computations, but the

result of the segmentation is not deterministic due to the nature of the smoothing filter.

In this section we propose a multi-stage FCM-based technique for bias- or gain field

estimation of the intensity inhomogeneity. Furthermore, we introduce two filtering tech-

niques to improve the segmentation accuracy. The proposed methods are tested using real

MR images and artificial phantoms.

3.4.1 FCM-based bias and gain field estimation

The above presented algorithm clusters the set of data {xk}, which was recorded among

ideal circumstances, containing no noise. However, in the real case, the observed data {yk}

differs from the actual one {xk}: there are impulse and Gaussian noises, which were treated

in the previous subsection, and there is the intensity non-uniformity (INU) artefact, which

will be handled here.


Literature recommends two different data variation models for intensity inhomogeneity:

the bias and the gain field model. If we consider the INU as a bias field, for each pixel k

we will have yk = xk + bk, where bk represents the bias value at pixel k. In case of gain

field modelling, there will be a gain value gk for each pixel k, such that yk = gkxk. In case

of both models, the variation of the intensity between neighbor pixels has to be slow. This

is assured by the smoothing filter presented in the next section.

In case of modelling INU as a bias field, the objective function becomes:

JFCM−b =n∑

k=1

c∑i=1

umik||yk − bk − vi||2 . (3.15)

Using the Lagrange multiplier technique, taking the derivatives of JFCM−b, with re-

spect to uik, vi and bk, respectively, and equaling them to zero, we obtain the following

optimization formulas:

u?ik =

||yk − bk − vi||−2/(m−1)

c∑j=1

||yk − bk − vj||−2/(m−1)

∀ i = 1 . . . c, ∀ k = 1 . . . n , (3.16)

v?i =

n∑k=1

umik(yk − bk)

n∑k=1

umik

∀ i = 1 . . . c , (3.17)

and

b?k = yk −

c∑i=1

umikvi

c∑i=1

umik

∀ k = 1 . . . n . (3.18)

If we approximate the INU artefact as a gain field, the objective function should be:

JFCM−g =n∑

k=1

c∑i=1

umik||yk/gk − vi||2 . (3.19)

Because the derivatives of this function are hard to handle, we slightly modify this

objective function the following way:

JFCM−g =n∑

k=1

c∑i=1

umik||yk − gkvi||2 . (3.20)


This is an affordable modification, which distorts the objective function such a way that

it gives slightly higher impact to lighter pixels (as their gain field value will probably be

over unity). Taking the derivatives of JFCM−g, with respect to uik, vi and gk, respectively,

and equaling them to zero, we obtain the following optimization formulas:

u?ik =

||yk − gkvi||−2/(m−1)

c∑j=1

||yk − gkvj||−2/(m−1)

∀ i = 1 . . . c, ∀ k = 1 . . . n , (3.21)

v?i =

n∑k=1

umikgkyk

n∑k=1

umikg

2k

∀ i = 1 . . . c , (3.22)

and

g?k = yk ·

c∑i=1

umikvi

c∑i=1

umikv

2i

∀ k = 1 . . . n . (3.23)

Similarly to the conventional FCM, these optimization formulas are applied alterna-

tively in each iteration.

3.4.2 Smoothening filter

The intensity inhomogeneity artefact varies slowly along the image. This property is ig-

nored by both the bias or gain field estimation approaches presented above. To avoid this

problem, a filtering technique is applied in each computation cycle, to smoothen the bias

or gain field. This filtering introduces an extra step into each optimization cycle, after

proceeding with eq. (3.18) or (3.23).

Several, not only FCM-based INU compensation approaches apply large sized, 11-31

pixels wide averaging filters performed once or several times in each cycle [186, 201]. These

filters efficiently hide tissue details, which may appear in the estimated bk or gk values, at

the price of transferring bias or gain components to distantly situated pixels. Using larger

averaging windows amplifies this latter undesired effect. In order to reduce the transfer of


bias data to distant pixels, we need to check the necessity of averaging at all locations, and

decide to proceed or skip the averaging accordingly. Averaging is declared necessary or not,

based on the maximum intensity difference encountered within a small neighborhood of

the pixel. The computation of the maximum difference is accomplished by a morphological

gradient operation using a 3× 3 square or slightly larger cross-shaped structuring element.

Wherever the morphological gradient value exceeds the previously set threshold value θ,

the averaged bias or gain value will be used; otherwise the estimated value is validated.

The proposed filter can be easily implemented and efficiently performed by batch-type

image processing operations.

3.4.3 Multi-stage bias and gain field estimation

Bias or gain field estimation using the previous FCM-based approaches [1, 162, 201] can

only handle the INU artefact to a limited amplitude. For any pixel, the FCM algorithm as-

signs the highest fuzzy membership to the closest cluster. Consequently, when the INU am-

plitude is comparable with the distance between clusters, these pixels will be attracted by

the wrong cluster, and the bias or gain field will be estimated accordingly. The smoothen-

ing of the bias and gain field may repair this kind of misclassifications, but the larger these

wrongly labelled spots are, the harder will be to eliminate them via smoothing.

In order to deal with high-amplitude INU artefacts, we propose performing the bias

or gain field estimation in multiple stages. When the FCM-based algorithm given by

eqs. (3.16)-(3.18) or (3.21)-(3.23) has converged, we modify the input (observed) image

according to the estimated bias or gain field:

yk = y(old)k − bk or yk = y

(old)k /gk , (3.24)

and then restart the algorithm from the beginning, using the modified input image [L6,

L12].


3.4.4 Algorithm

The presented algorithm can be summarized as follows:

1. Remove the Gaussian and impulse noises from the MR image using the context

dependent pre-filtering technique.

2. Initialize cluster prototypes vi, i = 1 . . . c, with random values differing from each

other.

3. Initialize the bias (gain) field values with 0-mean (1-mean) random numbers having

reduced variance, or simply set bk = 0 (gk = 1) for all pixels.

4. Compute new fuzzy membership function values uik, i = 1 . . . c, k = 1 . . . n, using

(3.16) or (3.21).

5. Compute new cluster prototype values vi, i = 1 . . . c, using (3.17) or (3.22).

6. Perform new bias or gain field estimation for each pixel k using (3.18) or (3.23).

7. Smoothen the bias or gain field using the proposed smoothing filter.

8. Repeat steps 4-7 until there is no relevant change in the cluster prototypes. This

is tested by comparing any norm of the difference between the new and the old vector v

with a preset small constant ε.

9. Modify the input image according to the estimated bias or gain field using (3.24),

and repeat steps 2-8 until the INU artefact is compensated. The algorithm usually requires

a single repetition.

3.4.5 Results and discussion

We applied the presented filtering and segmentation techniques to several T1-weighted real

MR images, artificially contaminated with different kinds of noises.

The results of bias and gain field estimation performed on a phantom image are shown


Figure 3.7: Inhomogeneity correction using a phantom: (a) original, (b) FCM segmentation

result without correction, (c) estimated bias field, (d) segmentation result with bias field

estimation, (e) estimated gain field, (f) segmentation result with gain field estimation

in Fig. 3.7. Conventional FCM is unable to compensate the INU artefact, but with the

use of smoothened bias or gain field, this phenomenon is efficiently overcome.

In case of low-amplitude inhomogeneity, a single stage of bias or gain field estimation

is sufficient. Figure 3.8. shows the accuracy and efficiency of the proposed segmentation

technique, using a real T1-weighted MR brain slice. The presence of the smoothening filter

supports the accurate segmentation, while the pre-filter has a regularizer effect on the final

result.

Figure 3.9. shows the intermediary and final results of a segmentation process, per-

formed on a heavily INU-contaminated MR image. The inhomogeneity correction succeeds

after two stages. Figure 3.9(i) shows the behavior of the proposed smoothing technique:

white pixels indicate places which required averaging in a given computational cycle, while

black ones signify those places where averaging was unnecessary.


Figure 3.8: Inhomogeneity correction demonstrated on an artificially contaminated real MR

image: (a) original, (b) FCM segmentation without correction, (c) result of FCM-based

segmentation with no pre-filtering, (d) result of FCM-based segmentation with context

sensitive pre-filtering

Table 3.2: Misclassification percentages with various smoothening filters, in case of heavily

INU-contaminated MR images

Structuring Window size 11 size 11 Window size 19 size 19

element execution once 3 times execution once 3 times

Averaging 7.357% 5.766% 4.368% 7.141%

3× 3, square 6.281% 6.201% 2.852% 3.379%

5× 5, cross 6.711% 5.674% 3.873% 3.938%

7× 7, cross 6.254% 5.518% 3.470% 5.029%

11× 11, cross 6.351% 4.432% 3.271% 6.437%


Figure 3.9: Segmentation of a heavily inhomogeneous real MR image: (a) original, (b)

segmentation without compensation, (c) bias field estimated in the first stage, (d) com-

pensated MR image after first stage, (e) FCM-based segmentation after first stage, still

unusable, (f) bias field estimated in the second stage, (g) final compensated image, (h)

segmented image, (i) a smoothening mask computed by the proposed filter

3.5. Application of the hybrid clustering model for inhomogeneity compensation and image segmentation 89

Table 3.2. shows the misclassification percentages of the proposed INU compensation

and MR image segmentation method, depending on the size of the averaging window ex-

pressed in pixels, the structuring element of the morphological criterion of the proposed

filter, and the number of smoothening iterations performed in each cycle of the modified

FCM algorithm. The experimental data reveal that the proposed filtering technique im-

proves the segmentation quality assured by the averaging filter. The best segmentation

was obtained using a 3 × 3 square shaped structuring element used by the morphological

criterion, combined with averaging using a window size of 19× 19 pixels.

Using several repetitive stages during INU compensation may reduce the intensity dif-

ference between tissue classes, which leads to misclassifications. That is why the estimation

is limited to two steps, performing several stages is not recommendable.

3.5 Application of the hybrid clustering model for in-

homogeneity compensation and image segmenta-

tion

In the previous chapter, a hybrid clustering model was proposed for partitioning vectorial

data. Thorough numerical tests have been performed using 4 and 13 dimensional data,

in order to evaluate its performance and to establish the strategy for the choice of its

parameters α and β.

In this section we will replace the FCM algorithm with the hybrid clustering model

within the inhomogeneity estimation and image segmentation algorithm, presented in Sec-

tion 3.4. Under these circumstances, the objective function indicated in Eq. (3.15) be-

comes:

Jhybrid−b =n∑

k=1

c∑i=1

ξik||yk − bk − vi||2 + (1− β)c∑

i=1

ηi

n∑k=1

(1− tik)p , (3.25)

where ξik = βαumik + β(1− α)hik + (1− β)tmik. All notations are used in the context of Eq.


Table 3.3: Misclassification percentages obtained with the hybrid clustering model

Structuring Window size 11 size 11 Window size 19 size 19

element execution once 3 times execution once 3 times

Averaging 6.011% 4.772% 3.614% 5.909%

3× 3, square 5.193% 5.137% 2.377% 2.801%

5× 5, cross 5.562% 4.707% 3.197% 3.264%

7× 7, cross 5.190% 4.583% 2.865% 4.174%

11× 11, cross 5.291% 3.668% 2.724% 5.223%

(2.54). The minimization of this objective function is straightforward. The computation of

ξik in every iteration is performed through its components. Optimal weighting parameter

values were chosen using the strategy established in Chapter 2.

Test results confirmed the initial expectations, namely:

1. If we compare the misclassification percentages obtained with FCM and the hybrid

model under the same circumstances, values contained in Tables 3.2 and 3.3, respec-

tively, we can conclude that the hybrid model produces partitions of better quality

than FCM. The misclassification rate was reduced by 17.8% in average.

2. The average number of necessary cycles in case of the hybrid model is 45% less than

in case of FCM. This means a reduction of the execution time by 38%.

3.6 Conclusions

A modified FCM algorithm has been proposed for automatic segmentation of MR brain

images. The algorithm was presented as a combination of a context dependent pre-filtering

technique and an accelerated FCM clustering performed over the histogram of the filtered

3.6. Conclusions 91

image. The pre-filter uses both spatial and gray level criteria, in order to efficiently elimi-

nate Gaussian and impulse noises without significantly blurring the real edges.

Several test series were carried out using synthetic brain phantoms and real MR images.

These investigations revealed that the proposed technique accurately segments the different

tissue classes under serious noise contamination. Test results revealed that the proposed

approach outperformed other recently reported methods many aspects, especially in the

accuracy of segmentation and processing time.

Further works in this topic will target more precise treatment of partial volume artifacts,

and adaptive determination of the optimal number of clusters.

For partial volume estimation, an optimal value of the fuzzy exponent m has been

determined via numerical computations.

On the other hand, a novel smoothing filter has been proposed to assist bias or gain

field estimation embedded into the conventional FCM algorithm scheme. The proposed

method proved to segment accurately and efficiently MR images in the presence of severe

intensity non-uniformity. Although the proposed method segments 2-D MR brain slices, it

gives a relevant contribution to the accurate volumetric segmentation of the brain, because

the segmented images and the obtained fuzzy memberships can serve as excellent input

data to any level set method that constructs 3-D cortical surfaces. Further works aim at

developing a context sensitive pre-filter for the elimination of INU artefacts, too, so that

the segmentation can be performed using a histogram-based quick FCM algorithm [L10,

L11, L16].

By testing the hybrid clustering model in inhomogeneity correction and image segmen-

tation, it has been established again that the hybrid approach can outperform the fuzzy

c-means clustering algorithm.

Chapter 4

Virtual Endoscopy

4.1 Introduction

Conventional endoscopy (CE) is a medical imaging modality, which requires a penetration

inside the human body. Usually a small device is introduced into the cavity or hollow

organ that needs investigation, where it can create some images that enable the physician

to establish the diagnosis. This intrusion usually causes pain, or at least discomforts the

patient, who consequently needs some kind of sedation or anaesthesia.

Recent advances in computer technology, imaging techniques, and computer graphics

tools, enabled the researcher community to develop a more patient-friendly way of diag-

nosing inner anomalies. A virtual endoscope creates 3-D inner views of the human body

based on image data collected with computed tomography (CT) techniques. There are two

types of commonly used data sources:

• Data provided by helical (or spiral) CT scans are quite popular because they give

more information in the craniocaudal axis, and are less sensitive to respiratory motion

[188].

• Equidistant parallel slices produced by MRI scans make the image registration

92

4.1. Introduction 93

process easier and quicker.

Virtual endoscopy (VE) can be applied for several medical reasons [188]:

• to enhance diagnosis, as specified above;

• to provide computer aided intervention, involving pre-operative planning, operative

technique, and post-operative monitoring [51, 144];

• for education purposes [93, 157][L21].

At this point we should make an account of all reported advantages and disadvantages

of virtual endoscopy. The most important advantages are listed below:

• VE is non-invasive or definitely less invasive than CE;

• No sedation is required in VE;

• VE can create the image of an entire organ (e.g. colon), while CE mainly focuses on

investigated areas;

• in bronchoscopy examinations, CE cannot pass areas of stenosis or occlusion, but VE

can visualize these distal areas;

• CE can have procedural risks, which are avoided in VE: e.g. classical colonoscopy

is reportedly associated with 1/1000 risk of bowel perforation and 1/5000 risk of

mortality [188];

• VE can be performed faster;

• Easier procedure for the physician.

Another relevant advantage, which I did not find in the literature, is the fact that VE

can reach areas of the human body that are even unthinkable with CE. VE doesn’t need

cavities or hollow organs to produce inner 3-D views, which makes it possible to visualize

94 Chapter 4. Virtual Endoscopy

the cortical surfaces of the brain, viewed from the inside of the white or gray matter [L10,

L11, L14].

However, there are also some disadvantages, like:

• VE has a higher cost, than CE, but this may reverse in time as technology advances;

• Patients are exposed to radiation, if X-ray technology is used for image data acqui-

sition. This can be avoided using magnetic resonance imaging or ultrasound;

• As no actual penetration is performed, VE definitely cannot take biopsy specimens

(e.g. polyps in colonoscopy);

• VE has lower image resolution, which yields higher misinterpretation rate for small

objects;

• VE cannot show texture and color details on surfaces.

In spite of these few disadvantages of virtual endoscopy, dozens of applications were

created in the last decade, which aim at imaging different parts of the human body. These

applications will be listed in the next section.

4.2 Existing applications

Virtual endoscopy has seemingly covered all the applications of its classical ancestor, and

even developed further ones. Most of them are associated with a new name like colonoscopy,

angiography or pancreatoscopy. They will be enumerated in the followings:

1. Virtual colonoscopy is responsible for the internal visualization of the colon and rec-

tum, and mostly serves the purpose of polyp and tumor spotting. It has an increased

sensitivity compared to CE in case of polyps larger than 10mm [117, 179, 188]. Vir-

tual colonoscopy, together with lower gastrointestinal endoscopy [113], require 3-D

surface reconstruction techniques specialized on tubular objects [59];

4.2. Existing applications 95

2. Detection of malignant duodenal lesions and malformations using virtual duo-

denoscopy [155], that is, the VE examination of the duoden;

3. In case of stomach imaging [115], VE are reported successful in showing subtle al-

teration in the gastric mucosal folds. Gastric cancer, polyps, ulcers, erosions, and

gastritis are generally clearly visualized. Both from the point of view of sensitivity

and specificity, VE outperforms the real gastric fiberoscopy [70];

4. Sata et al. [154] created a successful VE application for the pancreas, to diagnose

intraductal papillary mucinous neoplasm;

5. Two applications with several similar specific issues are bronchoscopy [62, 88, 110,

116, 143] for the investigation of airways in the lung, and angiography for blood ves-

sels [82, 120, 121, 142]. The similarities stem from their structure, as both systems

consist of lots of tubes with several bifurcations, and their common malformations:

stenosis and occlusion. Image processing techniques generally applied in these appli-

cations can be classified as skeletal or non-skeletal ones, both having pro’s and con’s,

presented in details in [17, 169, 170].

6. In case of heart imaging, a 4-D application is required (3 space + 1 time dimensions),

because of the quasi-periodic movement of the heart. Such an application may ex-

amine the inner cavities and structures of the heart [40, 144, 167], or the heart wall

motion [28][L35], mostly based on ultrasound or MRI data.

7. VE applications for prostate imaging include diagnosis systems [197], intervention

aid systems [12, 89], and therapy systems [50].

8. There are several urological applications involving the imaging of the lower urinary

tract [32, 165], the upper urinary tract and its tumors [16], and the bladder [19];

9. Han et al. [61] presented a VE application for visualizing nose cavities, and a com-

parison with fiberoptic endoscopy. They found VE capable of visualizing anatomic


structures and pathological masses larger than 3mm, but unable to examine the sur-

face of the mucosa. In spite of this deficiency, the authors predicted VE to become

the basic tool of the future in computer-aided nasal surgery;

10. Further VE applications include the monitoring of the liver [26], breast cancer diag-

nosing systems [68], virtual arthroscopy for joints [151], VE of the inner and outer

ear and the auditory canal [27, 122].

4.3 Existing methods

A successful implementation of a virtual endoscope software system requires the followings:

1. Good quality image data collected via CT, which are treated with a set of image

enhancement filtering methods;

2. Some 3-D surface reconstruction technique, the arsenal of which mostly overlaps with

the 3-D segmentation methods;

3. A set of computer graphics procedures that interactively visualize the the extracted

surfaces in three dimensions.

Image enhancement techniques were treated in full details in Chapter 3.

Modern 3-D surface reconstruction and image segmentation techniques include the

followings:

1. Probably the most easy-to-comprehend method for 3-D surface reconstruction is the

so-called marching cube method, which divides the whole object volume with an

equidistant cubic mesh, and extracts the elements of the surface within each of these

small cubes. It was originally proposed by Lorensen and Cline [102], and later fur-

ther developed by Nielson et al. [123, 124] due to its initial mistakes and ambiguities.

4.3. Existing methods 97

Some consider the marching cube an obsolete tool, but even nowadays, recent geo-

metric contour model (see below) applications use it to extract the actual shape of

the evolved surface.

2. Contour models stem from the Mumford-Shah functional [118], which is an equi-

librium equation of internal and external forces that determine the motion of the

contour. The evolution of contour models can be followed two different ways [156]:

a. Parametric contour models introduced by Kass et al. [76](also known as

snakes), define a set of control points along a closed contour, and the movement of

the contour is modeled by the motion of these control points. Internal forces depend

on the relative position of close neighbor control points. External forces generally

depend on image gradient, probably the most efficient field of external forces is the

gradient vector flow (GVF) or generalized GVF by Xu and Prince [190].

b. The other approach, called geometric contour model [126, 127], evolves the

contour as a zero level set of a scalar field computed in the intersection points of

an equidistant grid. Geometric contour models act like a propagating front having

the goal to approximate the zero level set as accurate as possible. The accuracy will

mainly depend on the gradient values around the zero level, because sharper edges

are easier to localize. A stopping force is needed so that the propagating surface

stops at the appropriate place. Some of the most important stopping forces applied

in geometric contour model propagation are

b1. Gradient-driven stopping force was introduced by Caselles et al. [33] and

Malladi et al. [106]. Their solution had a significant weakness with the pulling back

feature, that is, when a front crossed the aim boundary, it could not return.

b2. To remedy the above problem, Yezzi et al. [194] and Kichenassami et al.

[79] introduced their additional stopping force term due to edge strength.

b3. To improve the boundary leak characteristics, Siddiqui et al. [160] added

another extra stopping term due to area minimization.


b4. The most evolved advance in the domain is the usage of curvature depen-

dent stopping forces introduced by Lorigo et al. [103] and Malladi et al. [107].

c. Different level set based approaches are followed in [78, 81, 141, 171, 174].

d. Valuable reviews of the level set approaches are found in [25, 109, 168].

e. A revolution in the concept of geometric contour models was brought by Chan

and Vese [35]. Their model doesn’t act like a front. Instead, the interior and the

exterior set of the contour is computed, while the contour at any moment can be

extracted using an adequate marching cube method.

f. Geometric contour models outperform the parametric ones, both in speed and

accuracy.

3. Based on the active contour models, two further developments have been introduced.

The active shape models (ASM) [41] and active appearance models (AAM) [42, 111]

are supervised learning methods for detecting objects of a given shape in images.

The difference between these two is the usage of texture data specific to AAM’s.

Computer graphics and navigation techniques also consist widely researched domains.

Some of the recent advances are listed below:

1. The introduction of panoramic views [54, 177];

2. Use of ray casting [159] and ray templates [96] for efficient navigation;

3. Use of virtual fly-over navigation instead of well established fly-through method, in

case of tubular shapes (e.g. colonoscopy)[150];

DiMaio et al. [45] gives a detailed description of current challenges in virtual endoscopy

and image guided surgery aid and therapy systems.

4.4. Proposed methodology 99

4.4 Proposed methodology

4.4.1 Specification of an own virtual endoscopy system

In this section, a short specification will be given for an own virtual endoscopy system

(VES) having the main goal to visualize the inner structures of the brain. The main

specification rules [L10, L11] are listed below:

1. The input data of VES consists of a set of parallel cross sections of the brain, created

with MRI technology.

2. The input images need to be investigated, whether they contain relevant amount of

inhomogeneity. This decision will strongly influence the processing time.

3. Based on the response of the previous step, we proceed the quick histogram-based

segmentation or the segmentation with inhomogeneity correction to all MRI slices.

4. The partition matrices given by the FCM or the hybrid clustering algorithm mainly

define the regions within the object volume. Surface extraction is performed in order

to detect the boundary surfaces among regions.

5. The reconstructed surfaces are triangulated in the next step using an own corrected

version of the marching cube algorithm.

6. Computer graphics techniques are applies in order to produce 3-D views of the region

boundaries.

7. Interactive navigation is provided using OpenGL technology.

8. The VES should be able to quantify important physical measures like distances be-

tween any two points in the object space, areas and volumes of connected regions

(white matter, gray matter, tumor, etc.).


4.4.2 Input data

Input data, consisting of several sets of MR brain slices, were taken from two sources:

1. Internet Brain Segmentation Repository [71], which is a database created especially

for reliable testing of MR image segmentation methods. This database consists of

dozens of complete sets of slices, which were evaluated by physicians, to provide

the ground truth for tests. The distances between neighbor pixels in slices, and the

inter-slice distance is also given in most cases.

2. Two sets of MRI slices were obtained from the County Medical Clinic of Targu Mures,

Romania.

Both of these data sources provide data available for qualitative test. For quantitative

evaluation, only the first source is suitable.

4.4.3 2-D image processing tasks

The image processing tasks necessitated here are the ones presented in Chapter 3. Their

main goal is to enhance the image quality by performing different filters, and then they

produce the segmentation of each slice using an FCM-based or hybrid clustering. Details

were presented in Chapter 3.

4.4.4 3-D surface reconstruction for virtual endoscopy

3-D surface reconstruction has the main goal to generate a closed surface that stands at the

boundary of a given region. For example this surface can be the cortical surface between

the white matter and gray matter or the outer surface of the gray matter.

Let us consider we are standing somewhere inside the white matter. We would like to

detect the surface of the current connected region. This surface can be approximated with


the 0.5 level set of the fuzzy membership functions with respect to the class of the current

region. That is because any voxel which has its fuzzy membership greater than 0.5, is

detected to be inside the region.

Let us define a region indicator field for each class: GM, WM and CSF, using the

formula: Rik = 1−2uik ∀k = 1 . . . n, where uik is the fuzzy membership function indicating

the degree to which voxel k belongs to class i. Figure 4.2 shows the region indicator field

for white matter, taken from five consecutive slices of a brain MRI record.

For surface reconstruction purposes, I have applied the geometric contour model without

edges, introduced by Chan and Vese [35].

4.4.5 The enhanced marching cube algorithm

The marching cube algorithm is involved in the virtual endoscopy system in order to

triangulate the reconstructed region boundary surfaces.

The original marching cube algorithm, introduced by Lorensen and Cline in [102],

divides the whole investigated volume into unitary sized cubes, having at its corners 2× 2

adjacent pixels of two neighbor slices. Based on the region indicator values of these 8

voxels, it determines whether the zero level set intersects this cube and if so, it also locates

and triangulates the intersection. As any of the 8 voxels can be either inside or outside the

3-D region we wish to detect, there are 28 = 256 different cases.

According to the first formulation of the algorithm, symmetry assigns these cases to 14

different topologies. Unfortunately, the first version of the marching cube algorithm makes

mistakes, yielding holes in the surface. A corrected version of the algorithm emerged soon

[123, 124], which makes perfect triangulated closed surfaces.

The proposed version of the enhanced marching cube algorithm is obtained by the

relaxation of the symmetry between regions: pixels belonging to the inner region, situated

at distance 1 or√

2 from each other are considered connected, while in the outer regions


Figure 4.1: The 17 different topologies of the enhanced marching cube algorithm

Table 4.1: The 17 topologies to which the 256 cases of the enhanced marching cube algo-

rithm belong

Topology T1 T2 T3 T4 T5 T6 T7 T8 T9

Cases 2 16 24 12 12 8 48 8 8

Triangles 0 1 2 4 2 2 3 5 3

Connected N/A yes yes yes no no yes no no

Topology T10 T11 T12 T13 T14 T15 T16 T17

Cases 24 24 6 8 24 24 2 6

Triangles 5 3 2 4 4 4 4 4

Connected yes no yes yes no yes no no


pixels are connected only if they are at distance 1 and belong to the same region.

Having assumed the above rule, we obtain 17 different topologies. One example for

each topology is shown in Fig. 4.1. Table 4.1 shows some detailed information on each

topology: the number of cases that belong there, the number of triangles obtained within

the cube during the triangulation process, and the fact whether the triangles within the

cube are connected to each other.

If we previously define the current region, having assumed the differentiation rules

between voxels situated inside and outside the region, as presented above, this enhanced

marching cube algorithm always produces closed surfaces.

4.4.6 Interactive visualization and measurements

The interactive visualization uses OpenGL technology. The marching cube algorithm pro-

duces a huge number (2-3 millions) of triangles that need to be visualized. At the initializa-

tion step, the OpenGL system builds up an object set from the triangles. Once the objects

are put together, OpenGL is able to provide 3-D view of the objects depending on the

current position and orientation of the camera. The fine quality of the surfaces are assured

by a technique of bending the triangles, according to some normal vectors approximated

using the local gradient vector of the region indicator scalar field.

The physician can interact with the visualization software in order to navigate through

the investigated volume. The camera can be taken to any position and orientation. Using

a modern PC with Pentium4 processor and a high quality video card, the changes of the

view seems continuous.

Distance measurement can be performed between any two annotated points. The outer

area and the volume of any connected object can be commanded while the camera is inside

the object. These latter computations use approximations based on the surface elements

provided by the marching cube algorithm.


Figure 4.2: Region indicator for white matter, 5 consecutive slices

Figure 4.3: Endoscopic view of a cortical surface

4.5 Results

Figure 4.3 shows the endoscopic view of a cortical surface obtained from the fuzzy partitions

of MR brain slices. The triangulated surface became smooth via a graphical transformation,

automatically produced by the OpenGL based application that implements the proposed

method.

The visual quality of the image shown in Fig. 4.3 indicates that the macrostructure

of the human brain, the cortical surface, can be reconstructed from a set of ordinary

MRI brain slices. In order to visualize smaller details of the human body with acceptable

accuracy, higher-resolution images will be needed.

4.6. Conclusions 105

4.6 Conclusions

In this chapter, I have formulated an own concept of a virtual endoscope model, and I have

created its 3-D surface reconstruction algorithm based on 2-D cross sections segmented

with the FCM or the hybrid model [L10, L11, L14]. Furthermore, I have implemented the

virtual endoscope software system, which provides 3-D views according to an interactive

navigation, and is capable to approximate important physical measures [L10].

Further works will have the main goal to create other endoscopy applications. This

includes the development of such image processing procedures, which can accept different

sorts of computed tomography data, and the creation of surface models for different human

organs.

Chapter 5

Contributions

5.1 First Thesis Group: Contributions to the Devel-

opment of c-Means Clustering Models

1. As an extension to the theory of fuzzy c-means clustering (FCM) and possibilistic

c-means clustering (PCM), I have proposed a hybrid clustering model that besides

FCM and PCM also contains hard clustering (HCM). These three components are

weighted using two tradeoff parameters α and β. By performing several tests I found

the proposed algorithm robust and fast, and it provides high quality partitions within

most part of the weighting parameter domain [L4, L23].

2. I have evaluated the behavior and the performance of the proposed hybrid clustering

model within the whole α-β domain involving relevant parameters like misclassifica-

tion rate and expected number of iterations. Based on this study, I have established

the rules of choosing suitable weighting parameters α and β [L4, L23].

3. I have formulated a reduced memory usage version of the hybrid c-means clustering

model. Memory reduction is achieved via not storing partition matrices, and comput-

ing those sums instead, which contribute to the computation of cluster prototypes.

106

5.2. Second Thesis Group: Medical Image Segmentation Using FCM-Based Algorithms107

Memory usage can be reduced by three or four orders of magnitude, while in case of

high amount of data the algorithm will also perform better against the clock [L23].

4. I have performed a thorough analysis of the suppressed FCM clustering algorithm (s-

FCM) and I have established the formula of its quasi-competitive behavior. Based on

the similarity of s-FCM with a special case of the proposed hybrid c-means clustering

model, I have proposed an optimally suppressed FCM algorithm [L3, L5].

5.2 Second Thesis Group: Medical Image Segmenta-

tion Using FCM-Based Algorithms

1. I have proposed a fast and robust histogram based automatic classification method

for brain MR image filtering and segmentation. The MR image is previously filtered

using an adaptive context sensitive low pass filtering technique that uses spatial and

gray level constraints when locally establishing its weights. This procedure success-

fully eliminates impulse and Gaussian noises, and reduces the execution time of the

classification by 1-2 orders of magnitude, with respect to other FCM-based methods

[L2, L7-L11, L19].

2. Using analytical computations, I have established an optimal value of the fuzzy expo-

nent m, so that the degrees of membership provided by the FCM algorithm accurately

estimate the structure of pixels contaminated with partial volume effect [L7, L13].

3. I have proposed two similar formulations of a multiple stage compensation and seg-

mentation algorithm for MR images with inhomogeneous intensity. These algorithms

can accurately handle cases with extremely low signal-to-noise ratio [L6, L12].

4. I have applied the hybrid c-means clustering model for MR brain image segmentation.

The hybrid model performs quicker and produces better partitions than its ancestors

[L23].

108 Chapter 5. Contributions

5.3 Third Thesis Group: Virtual Endoscopy

1. I have formulated an own concept of a virtual endoscope model, and I have created

its 3-D surface reconstruction algorithm based on 2-D cross sections segmented with

the FCM or the hybrid model [L10, L11, L14].

2. I have implemented the virtual endoscope software system, which provides 3-D views

according to an interactive navigation, and is capable to approximate important

physical measures [L10].

Chapter 6

List of Publications and Independent

Citations

6.1 Publications related to the thesis

Books and book chapters

[L1] Benyo Z, Palancz B, Szilagyi L: Insight into computer science with Maple. Scientia

Publishing House, Kolozsvar, 416 pages (2005), ISBN: 973-7953-56-8.

[L2] Szilagyi L, Szilagyi SM, Benyo Z: Fast and Robust Fuzzy C-Means Algorithms for

Automated Brain MR Image Segmentation. In: Wickramasinghe N, Geisler E (eds.):

Encyclopaedia of Healthcare Information Systems, IDEA Group Publishing: Hershey -

New York 2:578-586 (2008), ISBN: 978-1599048895.

Refereed journal papers

[L3] Szilagyi L, Szilagyi SM, Benyo Z: Analytical and numerical evaluation of the sup-

pressed fuzzy c-means algorithm. Lecture Notes in Computer Science 5285:146–157 (2008),

ISSN: 0302-9743, IF: 0.402

[L4] Szilagyi L, Iclanzan D, Szilagyi SM, Dumitrescu D: A novel generalized approach

i

ii Chapter 6. List of Publications and Independent Citations

to c-means clustering. Lecture Notes in Computer Science 5197:235–242 (2008), ISSN:

0302-9743, IF: 0.402

[L5] Szilagyi L, Szilagyi SM, Benyo Z: A thorough analysis of the suppressed fuzzy c-means

algorithm. Lecture Notes in Computer Science 5197:203–210 (2008), ISSN: 0302-9743, IF:

0.402

[L6] Szilagyi L, Szilagyi SM, David L, Benyo Z: Multi-stage FCM-based intensity inhomo-

geneity correction for MR brain image segmentation. Lecture Notes in Computer Science

5164:527–636 (2008), ISSN: 0302-9743, IF: 0.402

[L7] Szilagyi L, Szilagyi SM, Benyo Z: A modified fuzzy c-means algorithm for MR brain

image segmentation. Lecture Notes in Computer Science 4633:866–877 (2007), ISSN:

0302-9743, IF: 0.402

[L8] Szilagyi L, Szilagyi SM, Benyo Z: Efficient feature extraction for fast segmentation

of MR brain images. Lecture Notes in Computer Science 4522:611–620 (2007), ISSN:

0302-9743, IF: 0.402

[L9] Szilagyi L, Szilagyi SM, Benyo Z: A Modified Fuzzy C-Means Classifier for Fast

Segmentation of MR Brain Images. Series of Advances in Soft Computing, Springer Verlag

41:119–127 (2007), ISSN: 1615-3871.

[L10] Szilagyi L: Medical Image Processing Methods for the Development of a Virtual

Endoscope. Periodica Polytechnica Ser. Electrical Engineering 50(1-2):59–68 (2006),

ISSN: 0324-6000.

[L11] Szilagyi L: Virtual Brain Endoscopy Based on Magnetic Resonance Images. Sci-

entific Bulletin of the Politechnica University of Timisoara, Transactions on Automatic

Control and Computer Science 49(63):47–50 (2004), ISSN: 1224-600X. 1

1Referenced by: [R01] Benyo Z: Education and research in biomedical engineering of the BudapestUniversity of Technology and Economics. Acta Physiologica Hungarica 93:13–21 (2006).

6.1. Publications related to the thesis iii

Papers in refereed international conference proceedings

[L12] Szilagyi L, David L, Szilagyi SM, Benyo B, Benyo Z: Improved Intensity Inho-

mogeneity Correction Techniques in MR Brain Image Segmentation. 17th IFAC World

Congress, Seoul 9625–9630 (2008), ISBN 978-1-1234-7890-2.

[L13] Szilagyi L, Szilagyi SM, Benyo B, Benyo Z: A Novel Clustering Method for Quick

Partial Volume Estimation in MR Brain Images. 17th IFAC World Congress, Seoul 9619–

9624 (2008), ISBN 978-1-1234-7890-2.

[L14] Szilagyi L, Benyo B, Szilagyi SM, Benyo Z: Medical image segmentation techniques

for virtual endoscopy. 6th IFAC Symposium on Modelling and Control in Biomedical Sys-

tems (MCBMS’06) Reims (France). In: Feng DD, Dubios O, Zaytoon J, Carson ER:

Modelling and Control in Biomedical Systems, Elsevier IFAC Publications, Oxford UK,

243–248 (2006) ISBN 0-0804-4530-6.

[L15] Szilagyi L, Szilagyi SM, Fordos G, Benyo Z: Quick ECG analysis for on-line Holter

monitoring systems. 28th Annual International Conference of IEEE Engineering in Medi-

cine and Biology Society, New York 1678–1681 (2006), ISBN 1-4244-0033-3.

[L16] Szilagyi L, Szilagyi SM, Benyo Z: Automated medical image processing methods

for virtual endoscopy. World Congress on Medical Physics and Biomedical Engineering

(WC2006), Seoul. IFMBE Proceedings 14:2267–2270 (2006), ISSN 1727-1983.

[L17] Szilagyi L, Szilagyi SM, Benyo Z: Medical image segmentation for virtual endoscopy.

16th IFAC World Congress, Prague 243–247 (2005), ISBN: 008045108X.

[L18] Szilagyi L, Benyo Z, Szilagyi SM: Brain image segmentation for virtual endoscopy.

26th Annual International Conference of IEEE Engineering in Medicine and Biology So-

ciety, San Francisco 1730–1732 (2004), ISBN: 0-7803-8439-3.2

2Referenced by: [R02] Song J, Zhao Q, Wang Y, Tian J: Gain field correction fast fuzzy c-meansalgorithm for segmenting magnetic resonance images. Lecture Notes in Computer Science 4099:1242–1247(2006). [R03] Zhao Q, Song J, Wu J: Improved fuzzy c-means segmentation algorithm for images withintensity inhomogeneity. Series of Advances in Soft Computing, Springer Verlag 41:150–159 (2007).

iv Chapter 6. List of Publications and Independent Citations

[L19] Szilagyi L, Benyo Z, Szilagyi SM, Adam HS: MR brain image segmentation using

an enhanced fuzzy c-means algorithm. 25th Annual International Conference of IEEE

Engineering in Medicine and Biology Society, Cancun (Mexico) 724–726 (2003), ISBN:

0-7803-7789-3.3

[L20] Szilagyi L, Benyo Z: Magnetic resonance brain image segmentation using an en-

hanced fuzzy c-means algorithm. World Congress on Medical Physics and Biomedical

Engineering (WC2003), Sydney. IFMBE Proceedings 4(4406):1-5 (2003), ISBN: 1-8770-

4014-2.

[L21] Benyo Z, Szilagyi L, Benyo B, Varady P, Palancz B, Szlavecz A, Bongar Sz, Fordos

G: Biomedical engineering education and research activity in Hungary. 24th Annual In-

ternational Conference of IEEE Engineering in Medicine and Biology Society, Houston

2658–2659 (2002), ISBN 0-7803-7612-9.

3Referenced by: [R04] Yuan K, Wu L, Cheng Q, Bao S, Chen C, Zhang H: A novel fuzzy c-meansalgorithm and its application. Int’l Journal of Pattern Recognition and Artificial Intelligence 19:1059–1066(2005). [R05] Moussaoui A, Benmahammed K, Ferahta N, Chen V: A new MR brain image segmentationusing an optimal semi-supervised fuzzy c-means and pdf estimation. Electronic Letters on ComputerVision and Image Analysis 5:1–11 (2005). [R06] Cai WL, Chen SC, Zhang DQ: Fast and robust fuzzy c-means clustering algorithms incorporating local information for image segmentation. Pattern Recognition40:825–838 (2007). [R07] Moussaoui A, Frahta N: Algorithmes neuro-flous de segmentation d’imagesIRM. 4th Int’l Conference on Computer Integrated Manufacturing CIP’07 pp. 1–6 (2007). [R08] Liao L,Lin TS: A fast spatial constrained fuzzy kernel clustering algorithm for MRI brain image segmentation.Int’l Conference on Wavelet Analysis and Pattern Recognition ICWAPR’07 1:82–87 (2007). [R09] PanW, Fu J, Wei J, Hao CY: Weighed-FCM image segmentation algorithm combined with Gibbs randomfield (Chinese). Electronic Measurement Technology 30:190–192 (2007). [R10] Pan W, Fu J, Wang XF,Hao CY: An automatic classification weighted fuzzy c-means image segmentation algorithm (Chinese).Periodical of Ocean University of China 37:485–488 (2007). [R11] Pourreza H, Ghazikhani M: Evaluatingfuzzy c-means with spatial constraints algorithms for vessel detection in retinal image (Persian). IKT’07– Information and Knowledge Technology Mashad (Iran) pp. 1–5 (2007). [R12] Tian JW, Huang YX,Yu YL: A fast FCM cluster multi-threshold image segmentation algorithm based on entropy constraint(Chinese). Pattern Recognition and Artificial Intelligence 21:221–226 (2008). [R13] Li XH, Zhang T, QuZ: Image segmentation using fuzzy clustering with spatial constraints based on Markov random field viaBayesian theory. IEICE Transactions on Fundamentals of Electronics, Communications and ComputerSciences E91-A(3):723–729 (2008). [R14] Liao L, Lin TS, Li B: MRI brain image segmentation and biasfield correction based on fast spatially constrained kernel clustering approach. Pattern Recognition Letters29:1580–1588 (2008).

6.2. Other publications v

[L22] Szilagyi L, Benyo Z, Szilagyi SM: A new method for epileptic waveform recognition

using wavelet decomposition and artificial neural networks. 24th Annual International

Conference of IEEE Engineering in Medicine and Biology Society, Houston 2025–2026

(2002), ISBN 0-7803-7612-9. 4

Papers in refereed Hungarian conference proceedings

[L23] Szilagyi L, Szilagyi SM, Benyo Z: Inhomogen intenzitasu agyi MRI felvetelek szeg-

mentalasa egy uj hibrid klaszterezo algoritmussal. BUDAMED’08 Orvostechnikai Konfer-

encia, Budapest, accepted paper (2008).

6.2 Other publications

Book chapters

[L24] Szilagyi SM, Szilagyi L, Benyo Z: Echocardiographic Image Sequence Compression

Based on Spatial Active Appearance Model. In: Wickramasinghe N, Geisler E (eds.):

Encyclopaedia of Healthcare Information Systems, IDEA Group Publishing: Hershey -

New York 2:472–479 (2008), ISBN: 978-1599048895.

[L25] Szilagyi SM, Szilagyi L, Luca CT, Cozma D, Ivanica G, Benyo Z: Modification of the

4Referenced by: [R15] Firpi H, Goodman E, Echauz J: Epileptic seizure detection by means of ge-netically programmed artificial features. Genetic and Evolutionary Computation Conference GECCO’05461–466 (2005). [R16] Firpi H, Goodman E, Echauz J: On prediction of epileptic seizures by comput-ing multiple genetic programming artificial features. Lecture Notes in Computer Science 3447:321–330(2005). [R17] Firpi H, Goodman E, Echauz J: Genetic programming artificial features with applicationsto epileptic seizure prediction. 27th Annual Int’l Conference of IEEE EMBS 4510–4513 (2005). [R18]Urrestarazu E, Iriarte J: Analısis matematicos en el estudio de senales electroencefalograficas. Revistade Neurologia. 41:423–434 (2005). [R19] Xia MF, Liu JB: Waveform identification technology in intelli-gent fault diagnosis (Chinese). Electro-Mechanical Engineering 22:49–51 (2006). [R20] Ataee P, AvanakiAN, Shariatpanahi HF, Khoee SM: Ranking features of wavelet-decomposed EEG based on significance inepileptic seizure detection. 14th European Signal Processing Conference EUSIPCO’06 paper#1568982271,1–4 (2006). [R21] Parreira FJ: Deteccao de crisis epilepticas a partir de sinais electroencefalograficos. PhDthesis, Universidade Federal de Uberlandia, Brasil (2006). [R22] Firpi H, Goodman E, Echauz J: Epilep-tic seizure detection using genetically programmed artificial features. IEEE Transactions on BiomedicalEngineering 54:212–224 (2007).

vi Chapter 6. List of Publications and Independent Citations

Accessory Pathway Localization Method to Improve the Performance of WPW Syndrome

Interventions. In: Wickramasinghe N, Geisler E (eds.): Encyclopaedia of Healthcare Infor-

mation Systems, IDEA Group Publishing: Hershey - New York 2:921–930 (2008), ISBN:

978-1599048895.

[L26] Szilagyi SM, Szilagyi L, Frigy A, Gorog LK, Benyo Z: Spatial Heart Simulation and

Adaptive Wave Propagation. In: Wickramasinghe N, Geisler E (eds.): Encyclopaedia of

Healthcare Information Systems, IDEA Group Publishing: Hershey - New York 3:1253–

1260 (2008), ISBN: 978-1599048895.

[L27] Szilagyi SM, Szilagyi L, Benyo Z: Spatial Heart Simulation and Analysis Using Uni-

fied Neural Network. In: Wickramasinghe N, Geisler E (eds.): Encyclopaedia of Healthcare

Information Systems, IDEA Group Publishing: Hershey - New York 3:1261–1268 (2008),

ISBN: 978-1599048895.

[L28] Szilagyi SM, Szilagyi L, Benyo Z: Volumetric Analysis and Modeling of the Heart

Using Active Appearance Model. In: Wickramasinghe N, Geisler E (eds.): Encyclopaedia

of Healthcare Information Systems, IDEA Group Publishing: Hershey - New York 3:1374–

1382 (2008), ISBN: 978-1599048895.

Refereed journal papers

[L29] Medves L, Szilagyi L, Szilagyi SM: A modified Markov clustering approach for

protein sequence clustering. Lecture Notes in Computer Science 5265:110–120 (2008),

ISSN: 0302-9743, IF: 0.402

[L30] Szilagyi SM, Szilagyi L, Gorog LK, Luca CT, Cozma D, Ivanica G, Benyo Z: An

enhanced accessory pathway localization method for efficient treatment of Wolff-Parkinson-

White syndrome. Lecture Notes in Computer Science 5197:269–276 (2008), ISSN: 0302-

9743, IF: 0.402

[L31] Szilagyi SM, Szilagyi L, Benyo Z: Spatial visualization of the heart in case of ectopic

beats and fibrillation. Lecture Notes in Computer Science 4872:548–561 (2007), ISSN:

6.2. Other publications vii

0302-9743, IF: 0.402

[L32] Szilagyi SM, Szilagyi L, Benyo Z: Adaptive ECG compression using support vector

machine. Lecture Notes in Computer Science 4756:594–603 (2007), ISSN: 0302-9743, IF:

0.402

[L33] Szilagyi SM, Szilagyi L, Frigy A, Gorog LK, Benyo Z: Unified neural network based

pathologic event reconstruction using spatial heart model. Lecture Notes in Computer

Science 4756:851–860 (2007), ISSN: 0302-9743, IF: 0.402

[L34] Szilagyi SM, Szilagyi L, Benyo Z: Echocardiographic image sequence compression

based on spatial active appearance model. Lecture Notes in Computer Science 4756:841–

850 (2007), ISSN: 0302-9743, IF: 0.402

[L35] Szilagyi SM, Szilagyi L, Benyo Z: Volumetric analysis of the heart using echocar-

diography. Lecture Notes in Computer Science 4466:81–90 (2007), ISSN: 0302-9743, IF:

0.402

[L36] Szilagyi SM, Szilagyi L, Benyo Z: Spatial Heart Simulation and Analysis Using

Unified Neural Network. Series of Advances in Soft Computing, Springer Verlag 41:346-

354 (2007), ISSN: 1615-3871.

[L37] Szilagyi SM, Szilagyi L, Benyo Z: Support Vector Machine-Based ECG Compression.

Series of Advances in Soft Computing, Springer Verlag 41:737-745 (2007), ISSN: 1615-3871.

[L38] Szilagyi SM, Szilagyi L, Benyo Z: Unified Neural Network Based Adaptive ECG

Signal Analysis and Compression. Scientific Bulletin of the Politechnica University of

Timisoara, Transactions on Automatic Control and Computer Science 51(65):27–36

(2006), ISSN 1224-600X.

[L39] Szilagyi M, Szilagyi L: Opinions on the mathematical activity of Janos Bolyai. Acta

Physica Hungarica New Series, Heavy Ion Physics 11:99–108 (2000), ISSN: 1219-7580, IF:

0.270

viii Chapter 6. List of Publications and Independent Citations

Other journal papers

[L40] Kovacs L, Benyo B, Torok L, Reiss A, Szilagyi L, Fordos G: Jarmuvezetok elettani

jeleinek merese, tarolasa es tovabbıtasa. A Jovo Jarmuve, 06(1-2):65–66 (2006).

[L41] Szilagyi SM, Frigy A, Gorog LK, Szilagyi L, Benyo Z: A pitvar-kamrai jarulekos

nyalabok Arruda-fele lokalizacios modszerenek erzekenysegi analızise. ORKI Orvos- es

Korhaztechnika 42(6):164–167 (2004), ISSN: 1585-7360.

[L42] Szilagyi L: EEG jelek kiertekelese, epilepszias jelalakok lokalizalasa wavelet transz-

formacio es neuralis halozatok alkalmazasaval. ORKI Orvos- es Korhaztechnika 41(1):12–

13 (2003), ISSN: 1585-7360.

[L43] Benyo Z, Benyo B, Szilagyi SM, Varady P, Szilagyi L: Research Activity of the

Biomedical Engineering Laboratory at TU Budapest. Research News, 8–13 (1999).

[L44] Frigy A, Szilagyi L, Incze S: A szıv vegetatıv statuszanak noninvazıv felmerese.

Orvostudomanyi Ertesıto 71:29–31 (1998), ISSN: 1453-0953.

Papers in refereed international conference proceedings

[L45] Szilagyi SM, Szilagyi L, Benyo Z: Sensibility Analysis of the Arruda Localization

Method and Modifications in Left Ventricle Analysis. 28th Annual International Confer-

ence of IEEE Engineering in Medicine and Biology Society, New York 3998–4001 (2006),

ISBN 1-4244-0033-3.

[L46] Szilagyi L, Szilagyi SM, Frigy A, David L, Benyo Z: Quick ECG segmentation, arti-

fact detection, and risk estimation methods for on-line Holter monitoring systems. World

Congress on Medical Physics and Biomedical Engineering (WC2006), Seoul. IFMBE Pro-

ceedings 14:914–917 (2006), ISSN 1727-1983.

[L47] Szilagyi SM, Szilagyi L, Benyo Z: Inverse 3D heart model for ECG signal simulation

and analysis. World Congress on Medical Physics and Biomedical Engineering (WC2006),

Seoul. IFMBE Proceedings 14:27–31 (2006), ISSN 1727-1983.

6.2. Other publications ix

[L48] Szilagyi SM, Szilagyi L, Gorog LK, Mathe Zs, Benyo Z: Modifications in Arruda’s

localization method in left ventricle analysis. World Congress on Medical Physics and

Biomedical Engineering (WC2006), Seoul. IFMBE Proceedings 14:117–120 (2006), ISSN

1727-1983.

[L49] Szilagyi L, Szilagyi SM, D. Iclanzan D, Benyo Z: Quick ECG signal processing

methods for on-line Holter monitoring systems. CONTI’2006 Int’l Conference on Technical

Informatics, Timisoara, Romania 1:221–226 (2006), ISBN 973-625-320-1.

[L50] Szilagyi SM, Szilagyi L, Iclanzan D, Benyo Z: Adaptive ECG signal analysis for

enhanced state recognition and diagnosis. CONTI’2006 Int’l Conference on Technical

Informatics, Timisoara, Romania 1:209–214 (2006), ISBN 973-625-320-1.

[L51] Iclanzan D, Szilagyi SM, Szilagyi L, Benyo Z: Advanced heuristic methods for ECG

parameter estimation. CONTI’2006 Int’l Conference on Technical Informatics, Timisoara,

Romania 1:215–220 (2006), ISBN 973-625-320-1.

[L52] Szilagyi SM, Szilagyi L, Frigy A, Gorog LK, Laszlo SE, Benyo Z: 3D heart simu-

lation and recognition of various events. 27th Annual International Conference of IEEE

Engineering in Medicine and Biology Society, Shanghai 4038–4041 (2005), ISBN 0-7803-

8741-4.

[L53] Szilagyi L, Szilagyi SM, Frigy A, Laszlo SE, Gorog LK, Benyo Z: Quick QRS complex

detection for on-line ECG and Holter systems. 27th Annual International Conference of

IEEE Engineering in Medicine and Biology Society, Shanghai 3906–3908 (2005), ISBN

0-7803-8741-4. 5

[L54] Szilagyi SM, Szilagyi L, Benyo Z: Recognition of various events from 3-D heart

model. 16th IFAC World Congress, Prague 107–112 (2005), ISBN 008045108X.

[L55] Szilagyi SM, Szilagyi L, Benyo Z: Risk estimation techniques in case of WPW

syndrome. 16th IFAC World Congress, Prague 184–189 (2005), ISBN 008045108X.

5Referenced by: [R23] Singh SS: Effectiveness of a handheld remote ECG monitor. PhD Thesis,University of North Carolina, Chapel Hill (2006).

x Chapter 6. List of Publications and Independent Citations

[L56] Szilagyi SM, Benyo Z, David L, Szilagyi L: Adaptive wavelet-transform-based ECG

waveforms detection. 25th Annual International Conference of IEEE Engineering in Medi-

cine and Biology Society, Cancun (Mexico) 2412–2415 (2003), ISBN: 0-7803-7789-3. 6

[L57] Benyo B, Benyo Z, Palancz B, Kovacs L, Szilagyi L: A fully symbolic design and

modelling of nonlinear glucose control with Control System Professional Suite (CSPS) of

Mathematica. World Congress on Medical Physics and Biomedical Engineering (WC2003),

Sydney. IFMBE Proceedings 4(2813):1–4 (2003), ISBN: 1-8770-4014-2. 7

[L58] Szilagyi L, Benyo Z: Epileptic waveform recognition using wavelet decomposition

and artificial neural networks. 5th IFAC Symposium on Modelling and Control in Biomed-

ical Systems (MCBMS’03) Melbourne. In: Feng DD, Carson ER: Modelling and Control

in Biomedical Systems, Elsevier IFAC Publications, Oxford UK, 301–303 (2003), ISBN:

0-0804-4159-9. 8

[L59] Szilagyi SM, Benyo Z, Szilagyi L: Comparison of malfunction diagnosis sensibility for

direct and inverse ECG signal processing methods. 24th Annual International Conference

of IEEE Engineering in Medicine and Biology Society, Houston 244–245 (2002), ISBN

0-7803-7612-9.

6Referenced by: [R24] Kim TS, Min CH: ECG based patient recognition model for smart healthcaresystems. Lecture Notes in Computer Science 3398:159–166 (2005). [R25] Ghosh D, Midya BL, Koley C,Purkait P: Wavelet aided SVM analysis of ECG signals for cardiac abnormality detection. Annual Int’lConference IEEE INDICON’05 9–13 (2005). [R26] Wang LC, Chen YQ, Pan M: Development of QRSdetection technique. Space Medicine and Medical Engineering 19:231–234 (2006). [R27] Midya BL, KoleyC: Pattern classification of ECG signals using wavelet aided self-organizing feature map. In: Garg D,Singh A: Soft Computing Allied Publishers: Mumbai, 267–274 (2006). [R28] Boutaa M, Bereksi-ReguigF, Debbal SMA: ECG signal processing using multiresolution analysis. Journal of Medical Engineering& Technology 6:1–13 (2008). [R29] Rizzi M, D’Aloia M, Castagnolo B: ECG-QRS detection methodadopting wavelet parallel filter banks. 7th WSEAS Int’l Conference on Wavelet Analysis & MultirateSystems 158–163 (2007).

7Referenced by: [R30] Makroglou A, Li J, Kuang Y: Mathematical models and software tools for theglucose-insulin regulatory system and diabetes: an overview. Elsevier Applied Numerical Mathematics56:559–573 (2006).

8Referenced by: [R31] Benyo B: Analysis of temporal patterns of physiological parameters. In: BeggR, Kamruzzaman J, Sarker R (eds.): Neural networks in healthcare. Potential and challenges, Idea GroupPublishing:Hershey - London - Melbourne - Singapore 284–316 (2006).

6.2. Other publications xi

[L60] Szilagyi L, Benyo Z, Szilagyi SM, Szlavecz A, Nagy L: On-line QRS complex detec-

tion using wavelet filtering. 23rd Annual International Conference of IEEE Engineering

in Medicine and Biology Society, Istanbul 1872–1874 (2001), ISBN: 0-7803-7211-5. 9

[L61] Szilagyi SM, Szilagyi L: Efficient ECG signal compression using adaptive heart

model. 23rd Annual International Conference of IEEE Engineering in Medicine and Biol-

ogy Society, Istanbul 2125–2128 (2001), ISBN: 0-7803-7211-5. 10

[L62] Nagy L, Szilagyi L: Catheter calibration using template matching line interpolation

algorithm. 23rd Annual International Conference of IEEE Engineering in Medicine and

Biology Society, Istanbul 387–389 (2001), ISBN: 0-7803-7211-5.

[L63] Szilagyi SM, Szilagyi L- Wavelet transform and neural-network-based adaptive fil-

tering for QRS detection. 22nd Annual International Conference of IEEE Engineering in

Medicine and Biology Society, Chicago 1267–1270 (2000), ISBN: 0-7803-6465-1.

[L64] Varady P, Nagy L, Szilagyi L: On-line detection of sleep apnea during critical care

monitoring. 22nd Annual International Conference of IEEE Engineering in Medicine and

Biology Society, Chicago 1299–1301 (2000), ISBN: 0-7803-6465-1. 11

[L65] Szilagyi L: Wavelet-transform-based QRS complex detection in on-line Holter sys-

tems. 21st Annual International Conference of IEEE Engineering in Medicine and Biology

9Referenced by: [R32] Alexandridi A, Panagopoulos I, Manis G, Papakonstantinou G: R-peak detectionwith alternative Haar wavelet filter. Int’l Symposium on Signal Processing & Information TechnologyISSPIT’03 219–222 (2003). [R33] Darrington J: Towards real time QRS detection: a fast method usingminimal pre-processing. Biomedical Signal Processing and Control 1:169–176 (2006). [R34] Duraj A:Algorytmy rozpoznawania zespo lu QRS w sygna lach elektrokardiograficznych pochodzacych od pacjentowzwszczepionym uk ladem stymulujacym. PhD thesis, Universytet Zielonogorski, Wydzia l Elektrotechniki,Informatyki i Telekomunikacji, Zielona Gora, Polska (2007).

10Referenced by: [R35] Simske SJ, Blakley DR, Zhang T: System for compression of physiologicalsignals. US Patent 7310648 (2007).

11Referenced by: [R36] Wen KT: The study of intelligent auto-screening and analysis techniques in clin-ical information. MSc Thesis, Chung Yuan Christian University (2002). [R37] Lee YK, Bister M, SallehYM: A generic algorithm for detecting obstructive sleep apnea hypopnea events based on oxygen satura-tion. IFMBE Proceedings 15:338–341 (2007). [R38] Barbosa Rendon D, Rojas Ojeda JL, Crespo FoixLF, Sanchez Morillo D, Fernandez MA: Mapping the human body for vibrations using an accelerometer.29th Annual Int’l Conference of IEEE EMBS 1671–1674 (2007).

xii Chapter 6. List of Publications and Independent Citations

Society, Atlanta 271 (1999), ISBN: 0-7803-5674-8. 12

[L66] Szilagyi L: Application of the Kalman filter in cardiac arrhythmia detection. 20th

Annual International Conference of IEEE Engineering in Medicine and Biology Society,

Hong Kong 98–100 (1998), ISBN: 0-7803-5167-3. 13

[L67] Szilagyi SM, Szilagyi L, David L: Comparison between neural-network-based adap-

tive filtering and wavelet transform for ECG characteristic points detection. 19th Annual

International Conference of IEEE Engineering in Medicine and Biology Society, Chicago

272–274 (1997). 14

12Referenced by: [R39] Varady P, Benyo Z, Micsik T, Moser Gy: A hybrid on-line ECG segmenting sys-tem for long-term monitoring. Acta Physiologica Hungarica 87:217–240 (2000). [R40] Wen LF, Meng ZH,Zhang YH: New developments of QRS complex detection methods. Tsinghua Report 1–9 (2000). [R41]Wen LF, Meng ZH, Zhang YH, Bai J: New developments of QRS complex detection methods. ForeignMedicine: Biomedical Engineering 24:193–197 (2001). [R42] Alexandridi A, Panagopoulos I, Manis G,Papakonstantinou G: R-peak detection with alternative Haar wavelet filter. Int’l Symposium on SignalProcessing & Information Technology ISSPIT’03 219–222 (2003). [R43] Duraj A: Algorytmy rozpoznawa-nia zespo lu QRS w sygna lach elektrokardiograficznych pochodzacych od pacjentow zwszczepionym uk lademstymulujacym. PhD thesis, Universytet Zielonogorski, Wydzia l Elektrotechniki, Informatyki i Telekomu-nikacji, Zielona Gora, Polska (2007).

13Referenced by: [R44] Perrot D, Dabonneville M, Hou KM, Ponsonnaille J: Plate-forme de validationdediee a la telemedicine: application a la detection d’arythmie dans les signaux ECG. In: Beuscart R,Zweigenbaum P, Venot A, Degoulet P (Eds.): Telemedicine et e-sante. Springer-Verlag. pp. 149–156(2002). [R45] Dayan G: Arrhythmia classification with SOM. MSc Thesis, The Graduate School of Naturaland Applied Sciences, Izmir, Turkey (2006).

14Referenced by: [R46] Wen LF, Meng ZH, Zhang YH: New developments of QRS complex detectionmethods. Tsinghua Report 1–9 (2000). [R47] Wen LF, Meng ZH, Zhang YH, Bai J: New developmentsof QRS complex detection methods. Foreign Medicine: Biomedical Engineering 24:193–197 (2001). [R48]Stadler RW, Shannon N: Axis shift analysis of electrocardiogram signal parameters especially applicable formultivector analysis by implantable medical devices, and use of same. US Patent 6324421 (2001). [R49]Stadler RW, Shannon N: Axis shift analysis of electrocardiogram signal parameters especially applicablefor multivector analysis by implantable medical devices, and use of same. US Patent 6397100 (2002).[R50] Botter EA, Nascimento CL Jr, Yoneyama T: Redes neurais auto-organizaveis para classificacaode sinais eletrocardiograficos atriais, Integracao XI(40):51–56 (2005). [R51] Matsuyama A, Jonkman M:The application of wavelet and feature vectors to ECG signals. Int’l Conference IEEE TENCON’05 ar-ticle #4085178 1–4 (2005). [R52] Li XJ, Chen YQ: New progress in QRS detection algorithm based onfrequency transform. Biomedical Engineering Foreign Medical Sciences 28:281–286 (2005). [R53] Mat-suyama A, Jonkman M: The application of wavelet and feature vectors to ECG signals. AustralasianPhysical and Engineering Sciences in Medicine 29:13–17 (2006). [R54] Benyo Z: Education and researchin biomedical engineering of the Budapest University of Technology and Economics. Acta PhysiologicaHungarica 93:13–21 (2006). [R55] Tian XL, Yan CH, Yu YQ, Wang TX: R-wave detection of ECG sig-nal by using wavelet transform. Journal of Biomedical Engineering 23:257–261 (2006). [R56] Benyo B:

6.2. Other publications xiii

[L68] Szilagyi SM, Szilagyi L, David L: ECG signal compression using adaptive predic-

tion. 19th Annual International Conference of IEEE Engineering in Medicine and Biology

Society, Chicago 101–104 (1997).

Papers in other conference proceedings

[L69] Szilagyi L: EEG jelek kiertekelese, epilepszias jelalakok lokalizalasa wavelet transz-

formacio es neuralis halozatok alkalmazasaval. BUDAMED’02 Orvostechnikai Konferencia,

Budapest, 54–55 (2002), ISBN 963-8231-92-0.

[L70] Szilagyi L: Application of dummy patient in on-line monitoring systems. BU-

DAMED’99 Orvostechnikai Konferencia, Budapest, 117–119 (1999).

[L71] Szilagyi SM, Szilagyi L: Artifact separation and classification from ECG recordings.

Proceedings of the Conference on the Latest Results in Information Technology, Budapest

85–90 (1998), ISBN: 963-421-548-3.

[L72] Szilagyi L: Neural network based QRS complex and arrhythmia detection in on-

line Holter systems. Proceedings of the Conference on the Latest Results in Information

Technology, Budapest 66–72 (1998), ISBN: 963-421-548-3.

[L73] Szilagyi L: Note on application of wavelet transform in industrial processes. Pro-

Analysis of temporal patterns of physiological parameters. In: Begg R, Kamruzzaman J, Sarker R (eds.):Neural networks in healthcare. Potential and challenges, Idea Group Publishing:Hershey - London - Mel-bourne - Singapore 284–316 (2006). [R57] Tsipouras MG, Exarchos TP, Fotiadis DI, Kotsia A, Naka KK,Michalis LK: A decision support system for the diagnosis of coronary artery disease. IEEE Symposiumon Computer-Based Medical Systems article #1647582 279–284 (2006). [R58] Benyo B: Computer-aidedanalysis of physiological systems. Acta Polytechnica Hungarica 4:55–68 (2007). [R59] Matsuyama A,Jonkman M, de Boer F: Improved ECG signal analysis using wavelet and feature extraction. Methods ofInformation in Medicine 46:227–230 (2007). [R60] Stadler RW, Nelson SD, Stylos L, Sheldon T: Methodand apparatus for analyzing electrocardiogram signals. Patent No. 1164931 (2007). [R61] MatsuyamaA, Jonkman M: Using Fisher’s linear discriminant to classify feature vectors of ECG signals. Int’l Con-ference on Information and Communications Technolgy ICICT’07 1–10 (2007). [R62] Sotos JM, AmauJMB, Aranda AMT, Melendez CS: Removal of muscular and artefacts noise from the ECG by a neuralnetwork. IEEE Int’l Conference on Industrial Informatics INDIN’07 2:687–692 (2007). [R63] TsipourasMG, Exarchos TP, Fotiadis DI, Kotsia A, Vakalis KV, Naka KK, Michalis LK: Automated diagnosis ofcoronary artery disease based on data mining and fuzzy modeling. IEEE Transactions on InformationTechnology in Biomedicine 12:447–458 (2008).

xiv Chapter 6. List of Publications and Independent Citations

ceedings of the Conference on the Latest Results in Information Technology, Budapest

47–49 (1997), ISBN: 963-421-545-9.

[L74] Szilagyi SM, Szilagyi L: Adaptive estimator for ECG signal compression. Proceed-

ings of the Conference on the Latest Results in Information Technology, Budapest 50–53

(1997), ISBN: 963-421-545-9.

[L75] Szilagyi L: Cardiac Arrhythmia Detection Using the Kalman Filter. XXI. Neumann

Kollokvium es Kiallıtas Veszprem, A szamıtastechnika orvosi es biologiai alkalmazasai, pp.

59–61 (1998).

[L76] Szilagyi L: Valos ideju osztott Holter tavmero rendszer, BME Vegzos Konferencia,

Budapest, pp. 321–326 (1998).

[L77] Szilagyi L, Szilagyi SM: Parameterbecslo modszerek alkalmazasa szıvaritmiak felis-

meresere, III. Fiatal Muszakiak Tudomanyos Ulesszaka, Kolozsvar, pp. 61–64 (1998).

[L78] Szilagyi L, Szilagyi SM: Az EKG jel tomorıtese genetikai algoritmus alkalmazasaval,

II. Fiatal Muszakiak Tudomanyos Ulesszaka, Kolozsvar, pp. 149–152 (1997).

[L79] Szilagyi SM, Szilagyi L, Moldovan IZ: Uj lehetosegek az orvostudomanyban az EKG

jelek feldolgozasa teren, I. Fiatal Muszakiak Tudomanyos Ulesszaka, Kolozsvar, pp. 1–4

(1996).

Abstracts

[L80] Szilagyi L, Szilagyi SM, Benyo Z: Level set methods in 3-D medical imaging,

MACS06 - 6th Joint Conference on Mathematics and Computer Science, Pecs, pp. 87

(2006)

[L81] Szilagyi SM, Szilagyi L, Z. Benyo Z: Parallelism in inverse 3-D heart modeling,

MACS06 - 6th Joint Conference on Mathematics and Computer Science, Pecs, pp. 88

(2006)

[L82] Szilagyi L: Kepfeldolgozasi modszerek egy virtualis endoszkop megvalosıtasahoz, 10

6.2. Other publications xv

eves az Orvosbiologiai Mernokkepzes Magyarorszagon Konferencia, Budapest (2005)

[L83] Szilagyi L: Informatikai modszerek alkalmazasa az orvosi kepfeldolgozasban egy

Virtualis Endoszkop megvalosıtasahoz, MPV Informatika Konferencia, Budapest (2004).

[L84] Benyo Z, Kovacs L, Varallyay Gy Jr, Szilagyi L: OTKA T042990: Biologiai jelek

informaciojanak diagnosztikai celu kutatasa rendszerelmeleti megkozelıtessel, XXIII. Cen-

tenariumi Neumann Kollokvium es Kiallıtas, Veszprem, Hungary (2003).

[L85] Szilagyi L: Mupaciens alkalmazasa valosideju monitorozo rendszerekben, A Magyar

elettani Tarsasag LXIV. Vandorgyulese, Budapest, pp. 136 (1999)

[L86] Varady P, Szilagyi L: Ein Neues Konzept fur Medizinische Information Systems,

Bajor-Magyar Mernokinformatikai Konferencia, Balatonfured (1999).

[L87] Incze S, Frigy A, Szilagyi F, Szilagyi L, Cotoi S: Az akaratlagos belegzesi ap-

noe erteke a szıv paraszimpatikus statusanak noninvazıv felmereseben, A Magyar Kar-

diologusok Tarsasaga Tudomanyos Kongresszusa, Balatonfured, pp. 58 (1998).

[L88] Szilagyi L: Szamıtogep a kardiologiaban, Erdelyi Magyar Muszaki Tudomanyos

Tarsasag Szakmai Nyılt Napja, Marosvasarhely (1996).

[L89] Szilagyi SM, Szilagyi L, Frigy A, Incze A: Holter telemetry in the study of Heart

Rate Variability, Romanian Heart Journal 2(6):143 (1996).

[L90] Szilagyi SM, Moldovan IZ, Szilagyi L: EKG jelek feldolgozasanak hardver es szoftver

kerdesei, Orvosbiologiai Mernokkepzes Kutatasok es Oktatas nemzetkozi tudomanyos kon-

ferencia, Budapest, pp. 24 (1996).

[L91] Szilagyi SM, Moldovan IZ, Szilagyi L: Uj lehetosegek az orvostudomanyban az EKG

jelek feldolgozasa teren, Orvosbiologiai Mernokkepzes Kutatasok es Oktatas nemzetkozi

tudomanyos konferencia, Budapest, pp. 23 (1996).

Bibliography

[1] Ahmed MN, Yamany SM, Mohamed N, Farag AA, Moriarty T: A modified fuzzy

c-means algorithm for bias field estimation and segmentation of MRI data. IEEE

Transactions on Medical Imaging 21:193–199 (2002)

[2] Anderson E: The IRISes of the Gaspe peninsula. Bulletin of the American IRIS

Society 59:2–5 (1935)

[3] Arato P, Visegrady T, Jankovits I: High level synthesis of pipelined datapaths. Wiley

& Sons (2001)

[4] Ardekani S, Kangarloo H, Sinha U: Region based fuzzy clustering for automated brain

segmentation. 24th Int’l Conference of IEEE Engineering in Medicine and Biology

Society - 2nd Joint Conference of IEEE EMBS and BMES Houston 1041–1042

(2002)

[5] Arthur D, Vassilvitskii S: k-means++: The advantages of careful seeding. Symposium

on Discrete Algorithms 1027–1035 (2007)

[6] Ashburner J, Friston KJ: Unified segmentation. NeuroImage 26:839–851 (2005)

[7] Asuncion A, Newman DJ: UCI Machine Learning Repository. Irvine, CA:

University of California, School of Information and Computer Science.

http://www.ics.uci.edu/ mlearn/MLRepository.html (2007)

[8] Atkins MS, Mackiewich BT: Fully automatic segmentation of the brain in MRI. IEEE


xvi

Bibliography xvii

[9] Awate SP, Whitaker RT: Nonparametric neighborhood statistics for MRI denoising.

Lecture Notes in Computer Science 3565:677–688 (2005)

[10] Axel L, Costanini J, Listerud J: Inhomogeneity correction in surface-coil MR imaging.

American Journal of Roentgenology 148:418–420 (1987)

[11] De Backer S, Scheunders P: A competitive elliptical clustering algorithm. Pattern

Recognition Letters 20:1141–1147 (1999)

[12] Balogh E, Deguet A, Susil RC, Krieger A, Wiswanathan A, Menard C, Coleman

JA, Fichtinger G: Visualization, planning, and monitoring software for MRI-guided

prostate intervention robot. Lecture Notes in Computer Science 3217:73–80 (2004)

[13] Baraldi A, Blonda P: A survey of fuzzy clustering algorithms for pattern recognition -

Part I-II. IEEE Transactions on System, Man, and Cybernetics - Part B: Cybernetics

29:778–801 (1999)

[14] Baranyi P, Nagy I, Korondi P, Hashimoto H: General guiding model for mobile robots

and its complexity reduced neuro-fuzzy approximation. 9th IEEE Int’l Conference

on Fuzzy Systems San Antonio 1029–1032 (2000)

[15] Barni M, Capellini V, Mecocci A: Comments on a possibilistic approach to clustering.

IEEE Transactions on Fuzzy Systems 4:393–396 (1996)

[16] Battista G, Sassi C, Schiavina R, Franceschelli A, Baglivo E, Martorana G, Canini

R: Computerized tomography virtual endoscopy in evaluation of upper urinary tract

tumors: initial experience. Abdominal Imaging in press (2008), doi 10.1007/s00261-

008-9387-5

[17] Bauer C, Bischof H: Extracting curve skeletons from gray value images for virtual

endoscopy. Lecture Notes in Computer Science 5128:393–402 (2008)

[18] Belacel N, Hansen P, Mladenovic N: Fuzzy J-means: a new heuristic for fuzzy clus-

tering. Pattern Recognition 35:2193–2200 (2002)

xviii Bibliography

[19] Bernhardt TM, Rapp-Bernhardt U: Virtual endoscopy of the bladder based on CT

and MRI data. Abdominal Imaging 26:325–332 (2001)

[20] Bezdek JC: Pattern recognition with fuzzy objective function algorithms. Plenum:

New York, NY (1981)

[21] Bezdek JC, Pal SK: Fuzzy models for pattern recognition. IEEE Press, Piscataway,

NJ (1991)

[22] Bezdek JC, Hall L, Clarke L: Review of MR image segmentation using pattern recog-

nition. Medical Physics 20:1033–1048 (1993)

[23] Bezdek JC, Pal NR: Some new indexes of cluster validity. IEEE Transactions on

System, Man, and Cybernetics - Part B: Cybernetics 28:301–315 (1998)

[24] Bezdek JC, Keller J, Krishnapuram R, Pal NR: Fuzzy models and algorithms for

pattern recognition and image processing. Springer (1999)

[25] Bookstein FL: Shape and the information in medical images: a decade of the mor-

phometric synthesis. Computer Vision and Image Understanding 66:97–118 (1997)

[26] Boctor EM, Fichtinger G, Yeung A, Awad M, Taylor RH, Choti MA: Robotic strain

imaging for monitoring thermal ablation in liver. Lecture Notes in Computer Science

3217:81–88 (2004)

[27] Boor S, Maurer J, Mann W, Stoeter P: Virtual endoscopy of the inner ear and the

auditory canal. Neuroradiology 42:543–547 (2000)

[28] Bosch JG, Mitchell SC, Lelieveldt BPF, Nijland F, Kamp O, Sonka M, Reiber JHC:

Automatic segmentation of echocardiographic sequences by active appearance motion

models. IEEE Transactions on Medical Imaging 21:1374–1383 (2002)

Bibliography xix

[29] Brinkmann BH, Manduca A, Robb RA: Optimized homomorphic unsharp masking

for MR grayscale inhomogeneity correction. IEEE Transactions on Medical Imaging

17:161–171 (1998)

[30] Cai W, Chen S, Zhang DQ: Fast and robust fuzzy c-means algorithms incorporating

local information for image segmentation. Pattern Recognition 40:825–838 (2007)

[31] Canny J: A computational approach to edge detection. IEEE Transactions on Pattern

Recognition and Machine Intelligence 8:679-714 (1986)

[32] Carrascosa P, Capunay C, Mariano B, Lopez EM, Carrascosa J, Borghi M, Sueldo C,

Papier S: Virtual hysteroscopy by multidetector computed tomography. Abdominal

Imaging 33:381–387 (2008)

[33] Caselles V, Catte F, Coll T, Dibos F: A geometric model for active contours in image

processing. Numerische Mathematik 66:1–31 (1993)

[34] Cannon RL, Dave JV, Bezdek JC: Efficient implementation of the fuzzy c-means

clustering algorithms. IEEE Transactions on Pattern Recognition and Machine In-

telligence 8:248–255 (1986)

[35] Chan TF, Vese LA: Active contours without edges. IEEE Transactions on Image

Processing 10:266–277 (2001)

[36] Chen S, Zhang DQ: Robust image segmentation using FCM with spatial constraints

based on new kernel-induced distance measure. IEEE Transactions on System, Man,

and Cybernetics - Part B: Cybernetics 34:1907–1916 (2004)

[37] Cheng TW, Goldgof DB, Hall LO: Fast fuzzy clustering. Fuzzy Sets and Systems

93:49–56 (1998)

[38] Choi HS, Haynor DR, Kim Y: Partial volume tissue classification of multichannel

magnetic resonance images - a mixel model. IEEE Transactions on Medical Imaging

10:395–407 (1991)

xx Bibliography

[39] Chuang KS, Tzeng HL, Chen S, Wu J, Chen TJ: Fuzzy c-means clustering with spa-

tial information for image segmentation. Computerized Medical Imaging and Graphics

30:9–15 (2006)

[40] Cirillo S, Bonamini R, Gaita F, Tosetti I, De Giuseppe M, Longo M, Bianchi F,

Vivalda L, Regge D: Magnetic resonance angiography virtual endoscopy in the as-

sessment of pulmonary veins before radiofrequency ablation procedures for atrial

fibrillation. European Radiology 14:2053–2060 (2004)

[41] Cootes TF, Taylor CJ, Cooper DH, Graham J: Active shape models - their training

and application. Computer Vision and Image Understanding 61:38–59 (1995)

[42] Cootes TF, Edwards GJ, Taylor CJ: Active appearance models. IEEE Transactions

on Pattern Recognition and Machine Intelligence 23:681–685 (2001)

[43] Dave RN: Fuzzy shell clustering and applications to circle detection in digital images.

Int’l Journal on General Systems 16:343–355 (1990)

[44] Dave RN, Bhaswan DK: Adaptive fuzzy c-shells clustering and detection of ellipses.

IEEE Transactions on Neural Networks 3:643–662 (1992)

[45] DiMaio S, Kapur T, Cleary K, Aylward S, Kazanzides P, Vosburgh K, Ellis R, Duncan

J, Farahani K, Lemke H, Peters T, Lorensen W, Gobbi D, Haller J, Clarke L, Pizer

S, Taylor R, Galloway R Jr, Fichtinger G, Hata N, Lawson K, Tempany C, Kikinis R,

Jolesz F: Challenges in image-guided therapy system design. Neuroimage 37:S144–

S151 (2007)

[46] Dunn JC: A fuzzy relative of the ISODATA process and its use in detecting compact

well separated clusters. Journal of Cybernetics 3:32–57 (1974)

[47] Enright AJ, van Dongen S, Ouzounis CA: An efficient algorithm for largescale de-

tection of protein families. Nucleic Acids Research 30:1575–1584 (2002)

Bibliography xxi

[48] Eschrich S, Ke J, Hall LO, Goldgof DB: Fast accurate fuzzy clustering through data

reduction. IEEE Transactions on Fuzzy Systems 11:262–270 (2003)

[49] Fan JL, Zhen WZ, Xie WX: Suppressed fuzzy c-means clustering algorithm. Pattern

Recognition Letters 24:1607–1612 (2003)

[50] Fichtinger G, Fiene JP, Kennedy CW, Kronreif G, Iordachita II, Song DY, Burdette

EC, Kanzanzides P: Robotic assistance for ultrasound-guided prostate brachyther-

apy. Medical Image Analysis in press (2008), doi:10.1016/j.media.2008.06.002

[51] Freudenstein D, Bartz D, Merkle M, Ernemann U, Skalej M, Duffner F: A new

virtual planning system for neuroendoscopic interventions. Klinische Neuroradiologie

10:153–160 (2000)

[52] Frigui H, Krishnapuram R: Clustering by competitive agglomeration algorithm. Pat-

tern Recognition 30:1109–1119 (1997)

[53] Frigui H, Krishnapuram R: A robust competitive clustering with applications in com-

puter vision. IEEE Transactions on Pattern Recognition and Machine Intelligence

21:450–465 (1999)

[54] Geiger B, Chefd’hotel C, Sudarsky S: Panoramic views of virtual endoscopy. Lecture

Notes in Computer Science 3749:662–669 (2005)

[55] Goldberg DE: Genetic algorithms in search, optimization and machine learning.

Addison-Wesley, Reading (1989)

[56] Gonzalez AI, Grana M, D’Anjou A: An analysis of the GLVQ algorithm. IEEE Trans-

actions on Neural Networks 6:1012–1016 (1995)

[57] Groll L, Jakel J: A new convergence proof of fuzzy c-means. IEEE Transactions on

Fuzzy Systems 13:717–720 (2005)

xxii Bibliography

[58] Gustafson EE, Kessel WC: Fuzzy clustering with a fuzzy covariance matrix. IEEE

Conference on Decision and Control, San Diego 761–776 (1979)

[59] Haker S, Angenent S, Tannenbaum A, Kikinis R: Non-distorting flattening for virtual

colonoscopy. Lecture Notes in Computer Science 1935:358–366 (2000)

[60] Han JM, Koczy LT, Poston T: Fuzzy Hough transform. Pattern Recognition Letters

15:649–658 (1994)

[61] Han P, Pirsig W, Ilgen F, Gorlich J, Sokiranski R: Virtual endoscopy of the nasal

cavity in comparison with fiberoptic endoscopy. European Arch Otorhinolaryngology

257:578–583 (2000)

[62] Haponik EF, Aquino SL, Vining DJ: Virtual bronchoscopy. Clinics in Chest Medicine

20:201–217 (1999)

[63] Hathaway RJ, Bezdek JC: Optimization of clustering by reformulation. IEEE Trans-

actions on Fuzzy Systems 3:241–245 (1995)

[64] Hathaway RJ, Bezdek JC, Hu Y: Generalized fuzzy c-means clustering strategies

using Lp norm distances. IEEE Transactions on Fuzzy Systems 8:576–582 (2000)

[65] Hoppner F, Klawonn F: A contribution to convergence theory of fuzzy c-means and

derivatives. IEEE Transactions on Fuzzy Systems 11:682–694 (2003)

[66] Hung WL, Yang MS, Chen DH: Parameter selection for suppressed fuzzy c-means

with an application to MRI segmentation. Pattern Recognition Letters 27:424–438

(2006)

[67] Hung WL, Chang YC: A modified fuzzy c-means algorithm for differentiation in MRI

of ophtalmology. Lecture Notes in Computer Science 3885:340–350 (2006)

[68] Ichihara S, Ando M, Maksimenko A, Yuasa T, Sugiyama H, Hashimoto E, Yamasaki

K, Mori K, Arai Y, Endo T: 3-D reconstruction and virtual ductoscopy of high-

Bibliography xxiii

grade ductal carcinoma in situ of the breast with casting type calcifications using

refraction-based X-ray CT. Windows Arch 452:41–47 (2008)

[69] Iclanzan D: The creativity potential within evolutionary algorithms. Lecture Notes

in Computer Science 4648:845–854 (2007)

[70] Inamoto K, Kouzai K, Ueeda T, Marukawa T: CT virtual endoscopy of the stomach:

comparison study with gastric fiberscopy. Abdominal Imaging 30:473–479 (2005)

[71] Internet Brain Segmentation Repository, http://www.cma.mgh.harvard.edu/ibsr

[72] Johnston B, Atkins MS, Mackiewich B, Anderson M: Segmentation of multiple scle-

rosis lesions in intensity corrected multispectral MRI. IEEE Transactions on Medical

Imaging 15:154–169 (1996)

[73] Kamel MS, Selim SZ: New algorithms for solving the fuzzy clustring problem. Pattern

Recognition 27:421–428 (1994)

[74] Karayiannis NB, Bezdek JC, Pal NR, Hathaway RJ, Pai PI: Repairs to GLVQ: a

new family of competitive learning scheme. IEEE Transactions on Neural Networks

7:1062–1071 (1996)

[75] Karayiannis NB, Bezdek JC: An integrated approach to fuzzy learning vector quanti-

zation and fuzzy c-means clustering. IEEE Transactions on Fuzzy Systems 5:622–628

(1997)

[76] Kass M, Witkin A, Terzopoulos D: Snakes: active contour models. Int’l Journal on

Computer Vision 1:321–331 (1987)

[77] Kato Z, Zerubia J, Berthod M: Unsupervised parallel image classification using

Markovian models. Pattern Recognition 32:591–604 (1999)

[78] Kelemen A, Szekely G, Gerig G: Elastic model-based segmentation of 3-D neurora-

diological data sets. IEEE Transactions on Medical Imaging 18:828–839 (1999)

xxiv Bibliography

[79] Kichenassami S, Kumar A, Olver P, Tannenbaum A, Yezzi A Jr: Conformal curvature

flows: from phase transitions to active vision. Archive for Rational Mechanics and

Analysis 134:275–301 (1996)

[80] Kim H, Kim S, Park M, Noh H: Cooperative mobile robots using fuzzy algorithm.

IEEE/RSJ Int’l Conference on Intelligent Robots and Systems 2:796–802 (1992)

[81] Kim JS, Singh V, Lee JK, Lerch J, Ad-Dab’bagh Y, MacDonald D, Lee JM, Kim SI,

Evans AC: Automated 3-D extraction and evaluation of the inner and outer cortical

surfaces using a Laplacian map and partial volume effect classification. NeuroImage

27:210–221 (2005)

[82] Kirchberg KJ, Wimmer A, Lorenz CH: Modeling the human aorta for MR-driven

real-time virtual endoscopy. Lecture Notes in Computer Science 4190:470–477 (2006)

[83] Kirkpatrick S, Gelatt CD Jr, Vecchi MP: Optimization by simulated annealing. Sci-

ence 220:671–680 (1983)

[84] Koczy LT: Algorithmic aspects of fuzzy control. Int’l Journal of Approximative Rea-

soning 12:159–219 (1995)

[85] Kohonen T: The self-organizing map. Proceedings of the IEEE 78:1474–1480 (1990)

[86] Koivula A, Alakuijala J, Tervonen O: Image feature based automatic correction of

low-frequency spatial intensity variations in MR images. Magnetic Resonance Imag-

ing 15:1167–1175 (1997)

[87] Kolen JF, Hutcheson T: Reducing the time complexity of the fuzzy c-means algo-

rithm. IEEE Transactions on Fuzzy Systems 10:263–267 (2002)

[88] Konen E, Katz M, Rozenman J, Ben-Shlush A, Itzchak Y, Szeinberg A: Virtual bron-

choscopy in children: early clinical experience. American Journal of Roentgenology

171:1699–1702 (1998)

Bibliography xxv

[89] Krieger A, Csoma Cs, Iordachita II, Guion P, Singh AK, Fichtinger G, Whitcomb

LL: Design and preliminary accuracy studies of an MRI-guided transrectal prostate

intervention system. Lecture Notes in Computer Science 4792:59–67 (2007)

[90] Krishnapuram R, Nasraoui O, Frigui H: The fuzzy c spherical shells algorithm: a

new approach. IEEE Transactions on Neural Networks 3:663–671 (1992)

[91] Krishnapuram R, Keller JM: A possibilistic approach to clustering. IEEE Transac-

tions on Fuzzy Systems 1:98–110 (1993)

[92] Krishnapuram R, Frigui H, Nasraoui O: Fuzzy and possibilistic shell clustering algo-

rithms and their application to boundary detection and surface approximation I-II.

IEEE Transactions on Fuzzy Systems 3:29–60 (1995)

[93] Kwon KJ, Shin BS, Chung MS: Computer-aided training system of educational vir-

tual dissection using Visible Korean Human. Lecture Notes in Computer Science

3942:1284–1287 (2006)

[94] Lagarias J, Reeds J, Wright M, Wright P: Convergence properties of the Nelder-Mead

simplex algorithm in low dimensions. SIAM Journal on Optimization 9:112–147

(1998)

[95] Lazaro J, Arias J, Martın JL, Cuadrado C, Astarloa A: Implementation of a modified

fuzzy c-means clustering algorithm for real-time applications. Microprocessors and

Microsystems 29:375–380 (2005)

[96] Lee HJ, Lim HS, Shin BS: Unfolding of virtual endoscopy using ray-template. Lecture


[97] van Leemput K, Maes F, Vandermeulen D, Suetens P: Automated model-based bias

field correction of MR images of the brain. IEEE Transactions on Medical Imaging

18:885–896 (1999)

xxvi Bibliography

[98] van Leemput K, Maes F, Vandermeulen D, Suetens P: A unifying framework for

partial volume segmentation of brain MR images. IEEE Transactions on Medical

Imaging 22:105–119 (2003)

[99] Li X, Li LH, Lu HB, Liang ZG: Partial volume segmentation of brain magnetic

resonace images based on maximum a posteriori probability. Medical Physics 32:2337–

2345 (2005)

[100] Liew AWC, Hong Y: An adaptive spatial fuzzy clustering algorithm for 3-D MR

image segmentation. IEEE Transactions on Medical Imaging 22:1063–1075 (2003)

[101] Likar B, Viergever A, Pernus F: Retrospective correction of MR intensity inho-

mogeneity by information minimization. IEEE Transactions on Medical Imaging

20:1398–1410 (2001)

[102] Lorensen WE, Cline HE: Marching cubes: a high resolution 3D surface construction

algorithm. Computer Graphics 21:163–169 (1987)

[103] Lorigo LM, Grimson W, Eric L, Faugeras O, Keriven R, Kikinis R, Navabi A,

Westin CF: Codimension-two geodesic active contours for the segmentation of tubular

structures. IEEE Conference on Computer Vision and Pattern Recognition 444–451

(2000)

[104] Luo J, Zhu Y, Clarysse P, Magnin I: Correction of bias field in MR images using

singularity function analysis. IEEE Transactions on Medical Imaging 24:1067–1085

(2005)

[105] MacQueen JB: Some methods for classification and analysis of multivariate observa-

tions. 5th Berkeley Symposium on Mathematical Statistics and Probability 1:281–297

(1967)

[106] Malladi R, Sethian JA, Vemuri BC: Shape modelling with front propagation. IEEE

Transactions on Pattern Recognition and Machine Intelligence 17:158–175 (1995)

Bibliography xxvii

[107] Malladi R, Sethian JA: An O(N log N) algorithm for shape modeling. Applied Math-

ematics Proceedings of the National Academy of Sciences 93(18):9389–9392 (1996)

[108] Mamdani EH, Assilian S: An experiment in linguistic synthesis with a fuzzy logic

controller. Int’l Journal of Man Machine Studies 7:1–13 (1975)

[109] McInerney T, Terzopoulos D: Deformable models in medical image analysis: a survey.

Medical Image Analysis 1:91–108 (1996)

[110] Men S, Cenk Ecevit M, Topcu I, Kabakci N, Erdag TK, Sutay S: Diagnostic con-

tribution of virtual endoscopy in diseases of the upper airways. Journal of Digital

Imaging 20:67–71 (2007)

[111] Mitchell SC, Bosch JG, Lelieveldt BPF, van der Geest RJ, Reiber JHC, Sonka M:

3-D active appearance models: segmentation of cardiac MR and ultrasound images.

IEEE Transactions on Medical Imaging 21:1167–1178 (2002)

[112] Morales E, Shih FY: Wavelet coefficients clusterings using morphological operators

and pruned quadtrees. Pattern Recognition 33:1611–1620 (2000)

[113] Moorthy K, Munz Y, Orchard TR, Gould S, Rockall T, Darzi A: An innovative

method for the assessment of skills in lower gastrointestinal endoscopy. Surgical En-

doscopy 18:1613–1619 (2004)

[114] de Mori R: Computer models of speech using fuzzy algorithms. Plenum: New York,

NY (1983)

[115] Mori K, Oka H, Kitasaka T, Suenaga Y, Toriwaki J: Virtual unfolding of the stom-

ach based on volumetric image deformation. Lecture Notes in Computer Science

3217:389–396 (2004)

[116] Mori K, Ema S, Kitasaka T, Mekada Y, Ide I, Murase H, Suenaga Y, Takabatake H,

Mori M, Natori H: Automated nomenclature of bronchial branches extracted from CT

xxviii Bibliography

images and its application to biopsy path planning in virtual bronchoscopy. Lecture


[117] Morrin MM, Farrell RJ, Raptopoulos V, McGee JB, Bleday R, Kruskal JB: Role of

virtual computed tomographic colonography in patients with colorectal cancers and

obstructing colorectal lesions. Diseases of the Colon & Rectum 43:303–311 (2000)

[118] Mumford D, Shah J: Optimal approximations of piecewise smooth functions and

associated variational problems. Communications on Pure and Applied Mathematics

42:577–685 (1989)

[119] Murata R, Endo Y, Haruyama H, Miyamoto S: On fuzzy c-means on data with

tolerance. Lecture Notes in Computer Science 3885:361–371 (2006)

[120] Nagy L, Benyo B: Filtering and contrast enhancement on subtracted direct digital

angiograms. 26th Annual International Conference of IEEE Engineering in Medicine

and Biology Society, San Francisco 1533–1536 (2004)

[121] Neri E, Bonanomi G, Vignali C, Cioni R, Ferrari M, Petruzzi P, Bartolozzi C: Spiral

CT virtual endoscopy of abdominal arteries: clinical applications. Abdominal Imaging

25:59–61 (2000)

[122] Neri E, Caramella D, Panconi D, Berrettini S, Sellari Franceschini S, Forli F, Bar-

tolozzi C: Virtual endoscopy of the middle ear. European Radiology 11:41–49 (2001)

[123] Nielson GM, Hamann B: The asymptotic decider: resolving the ambiguity in march-

ing cubes. IEEE Conference on Visualisation San Diego 83–91 (1991)

[124] Nielson GM: On marching cubes. IEEE Transactions on Visualisation and Computer

Graphics 9:283–297 (2003)

[125] Noordam JC, van den Brook WHAM, Buydens LMC: Multivariate image segmenta-

tion with cluster size insensitive fuzzy c-means. Chemometrics and Intelligent Labo-

ratory Systems 64:65–78 (2002)

Bibliography xxix

[126] Osher S, Sethian JA: Fronts propagating with curvature dependent speed: algorithms

based on Hamilton-Jacobi formulations. Journal of Computational Physics 79:12-49

(1988)

[127] Osher S, Paragios N (Eds.): Geometric level set methods in imaging, vision, and

graphics. Springer-Verlag:New York (2003)

[128] Pal NR, Bezdek JC, Tsao ECK: Generalized clustering networks and Kohonen’s self-

organizing scheme. IEEE Transactions on Neural Networks 4:549–557 (1993)

[129] Pal NR, Bezdek JC: On cluster validity for the fuzzy c-means model. IEEE Trans-

actions on Fuzzy Systems 3:370–379 (1995)

[130] Pal NR, Bezdek JC, Hathaway R: Sequential competitive learning and the fuzzy

c-means clustering algorithms. Neural Networks 9:787–796 (1996)

[131] Pal NR, Pal K, Bezdek JC: A mixed c-means clustering model. 6th IEEE Int’l Con-

ference on Fuzzy Systems - FUZZ-IEEE Barcelona 11–21 (1997)

[132] Pal NR, Pal K, Keller JM, Bezdek JC: A possibilistic fuzzy c-means clustering algo-

rithms. IEEE Transactions on Fuzzy Systems 13:517–530 (2005)

[133] Pedrycz W, Loia V, Senatore S: P-FCM: a proximity–based fuzzy clustering. Fuzzy

Sets and Systems 148:21–41 (2004)

[134] Pedrycz W, Vukovich G: Fuzzy clustering with supervision. Pattern Recognition

37:1339–1349 (2004)

[135] Pham DL, Prince JL: Partial volume estimation and the fuzzy c-means algorithm.

Int’l Conference on Image Processing 819–822 (1998)

[136] Pham DL, Prince JL: An adaptive fuzzy C-means algorithm for image segmentation

in the presence of intensity inhomogeneity. Pattern Recognition Letters 20:57–68

(1999)

xxx Bibliography

[137] Pham DL, Prince JL: Adaptive fuzzy segmentation of magnetic resonance images.

IEEE Transactions on Medical Imaging 18:737–752 (1999)

[138] Pham DL: Spatial models for fuzzy clustering. Computer Vision and Image Under-

standing 84:285–297 (2001)

[139] Pham DL: Unsupervised tissue classification in medical images using edge-adaptive

clustering. 25th Int’l Conference of IEEE Engineering in Medicine and Biology So-

ciety Cancun 634–637 (2003)

[140] Pham DL, Bazin PL: Simultaneous boundary and partial volume estimation in med-

ical images. Lecture Notes in Computer Science 3216:119–126 (2004)

[141] Pitiot A, Toga AW, Thompson PM: Adaptive elastic segmentation of brain MRI

via shape-model-guided evolutionary programming. IEEE Transactions on Medical

Imaging 21:910–923 (2002)

[142] Rodel R, Rodenwaldt J, Hommerich CP: Inner surface imaging of laryngeal and tra-

cheal stenosis by spiral-CT: role of a new diagnostic procedure. Laryngorhinootologie

79:584–590 (2000)

[143] Rogalla P, Ruckert JC, Schmidt B, Witt C, Meiri N, Hamm B: Virtual bronchoscopy.

Radiologe 41:261–268 (2001)

[144] Rohde V, Krombach GA, Struffert T, Gilsbach JM: Virtual MRI endoscopy: detec-

tion of anomalies of the ventricular anatomy and its possible role as a presurgical

planning tool for endoscopic third ventriculostomy. Acta Neurochirurgica 143:1085–

1091 (2001)

[145] Roozbahani RG, Ghassemian MH, Sharafat AR: Estimating gain fields in multispec-

tral MRI. IEEE Transactions on Biomedical Engineering 47:1610–1615 (2000)

[146] Rose K: Deterministic annealing, clustering and optimization. PhD thesis, California

Institute of Technology (1991)

Bibliography xxxi

[147] Ruan S, Jaggi C, Xue J, Fadili J, Bloyet D: Brain tissue classification of magnetic reso-

nance images using partial volume modeling. IEEE Transactions on Medical Imaging

16:22–32 (2000)

[148] Ruan S, Moretti B, Fadili J, Bloyet D: Fuzzy Markovian segmentation in application

of magnetic resonance images. Computer Vision and Image Understanding 85:54–69

(2002)

[149] Ruspini EH: A new approach to clustering. Information and Control 16:22–32 (1969)

[150] Sabry Hassouna M, Farag AA, Falk R: Virtual fly-over: a new visualization technique

for virtual colonoscopy. Lecture Notes in Computer Science 4190:381–388 (2006)

[151] Sahin G, Dogan BE, Demirtas M: Virtual MR arthroscopy of the wrist joint: a new

intraarticular perspective. Skeletal Radiology 33:9–14 (2004)

[152] Sanchez E: Medical diagnosis and composite fuzzy relations. In: Gupta MM, Ragade

RK, Yager RR (eds): Advances in Fuzzy Set Theory and Aplications. North-Holland:

Amsterdam 437–444 (1979)

[153] Santago P, Gage HD: Quantification of MR brain images by mixture density and

partial volume modeling. IEEE Transactions on Medical Imaging 12:566–574 (1993)

[154] Sata N, Kurihara K, Koizumi M, Tsukahara M, Yoshizawa K, Nagai H: CT vir-

tual pancreatoscopy: a new method for diagnosing intraductal papillary mucinous

neoplasm (IPMN) of the pancreas. Abdominal Imaging 31:326–331 (2006)

[155] Sata N, Endo K, Shimura K, Koizumi M, Nagai H: A new 3D-diagnosis strategy for

duodenal malignant lesions using multi-detector row CT, CT virtual duodenoscopy,

duodenography and 3D multi-cholangiography. Abdominal Imaging in press (2008),

doi 10.1007/s00261-006-9121-0

[156] Sethian JA: Level set methods and fast marching methods. University Press: Cam-

bridge (1999)

xxxii Bibliography

[157] Seymour NE: Virtual reality in general surgical training. European Surgery 37/5:298–

303 (2005)

[158] Shattuck DW, Sandor-Leahy SR, Schaper KA, Rottenberg DA, Leahy RM: Mag-

netic resonance image tissue classification using a partial volume model. NeuroImage

13:856–876 (2001)

[159] Shin BS, Lim HS: An efficent navigation method for virtual endoscopy using volume

ray casting. Lecture Notes in Computer Science 2659:60–69 (2003)

[160] Siddiqui K, Lauriere YB, Tannenbaum A, Zucker SW: Area and length minimizing

flows for shape segmentation. IEEE Transactions on Image Processing 7:433–443

(1998)

[161] Sled JG, Zijdenbos AP, Evans AC: A nonparamtertic method for automatic correction

of intensity nonuniformities. IEEE Transactions on Medical Imaging 17:87–97 (1998)

[162] Siyal MY, Yu L: An intelligent modified fuzzy c-means based algorithm for bias field

estimation and segmentation of brain MRI. Pattern Recognition Letters 26:2052–

2062 (2005)

[163] Sonka M, Hlavac V, Boyle R: Image processing, analysis, and machine vision, Thop-

son Publishing (1998)

[164] Steinhaus H: Sur la division des corp materiels en parties. Bulletin de l’Academie

Polonaise des Science C1 III., IV:801–804 (1956)

[165] Stenzl A, Kolle D, Eder R, Stoger A, Frank R, Bartsch G: Virtual reality of the lower

urinary tract in women. Int’l Urogynecology Journal 10:248–253 (1999)

[166] Studholme C, Cardenas V, Song E, Ezekiel F, Maudsley A, Weiner M: Accurate

template-based correction of brain MRI intensity distortion with application to de-

mentia and aging. IEEE Transactions on Medical Imaging 23:99–110 (2004)

Bibliography xxxiii

[167] Sun ZH: Multislice CT angiography in post-aortic stent grafting: optimization of

scanning protocols for virtual intravascular endoscopy. Int’l Journal of Computer

Assisted Radiology and Surgery 3:19–26 (2008)

[168] Suri JS, Liu K, Singh S, Laxminarayan SN, Zeng X, Reden L: Shape recovery algo-

rithms using level sets in 2-D/3-D medical imagery: a state-of-the-art review. IEEE

Transactions on Information Technology in Biomedicine 6:8–28 (2002)

[169] Suri JS, Liu K, Reden L, Laxminarayan S: A review of MR vascular image processing

algorithms: acquisition and prefiltering. IEEE Transactions on Information Technol-

ogy in Biomedicine 6:324–337 (2008)

[170] Suri JS, Liu K, Reden L, Laxminarayan S: A review of MR vascular image processing

algorithms: skeleton versus nonskeleton approaches. IEEE Transactions on Informa-

tion Technology in Biomedicine 6:338–350 (2008)

[171] Szekely G, Kelemen A, Brechbuhler C, Gerig G: Segmentation of 2-D and 3-D objects

from MRI volume data using constrained elastic deformations of flexible Fourier

contour and surface models. Medical Imaging Analysis 1:19–34 (1996)

[172] Takagi T, Sugeno M: Fuzzy identification of systems and its applications to modeling

and control. IEEE Transactions on System, Man and Cybernetics 15:116-132 (1985)

[173] Tanaka H, Guo P, Turksen IB: Portfolio selection based on fuzzy probabilities and

possibility distributions. Fuzzy Sets and Systems 111:389–397 (2000)

[174] Tao X, Prince JL, Davatzikos C: Using a statistical shape model to extract sulcal

curves on the outer cortex of the human brain. IEEE Transactions on Medical Imag-

ing 21:513–524 (2002)

[175] Timm H, Borgelt C, Doring C, Kruse R: An extension to possibilistic fuzzy cluster

analysis. Fuzzy Sets and Systems 147:3–16 (2004)

xxxiv Bibliography

[176] Theodoridis S, Koutroumbas K: Pattern Recognition. 2nd edition. Elsevier Academic

Press, San Diego, CA (2003)

[177] Tiede U, von Sternberg-Gospos N, Steiner P, Hohne KH: Virtual endoscopy using

cubic QuickTime-VR panorama views. Lecture Notes in Computer Science 2489:186–

192 (2002)

[178] Tolias YA, Panas SM, Tsoukalas LH: Generalized fuzzy indices for similarity match-

ing. Fuzzy Sets and Systems 120:255–270 (2001)

[179] Truyen R, Deschamps T, Cohen LD: Clinical evaluation of an automatic path tracker

for virtual colonoscopy. Lecture Notes in Computer Science 2208:169–176 (2001)

[180] Tsao ECK, Bezdek JC, Pal NR: Fuzzy Kohonen clustering networks. Pattern Recog-

nition 27:757–764 (1994)

[181] D’Urso P, Giordani P: A weighted fuzzy c-means clustering model for fuzzy data.

Computational Statistics & Data Analysis 50:1496–1523 (2006)

[182] Vapnik V: Statistical learning theory. Wiley: New York (1998)

[183] Vemuri P, Kholmovski EG, Parker DL, Chapman BE: Coil sensitivity estimation for

optimal SNR reconstruction and intensity inhomogeneity correction in phased array

MR imaging. Lecture Notes in Computer Science 3565:603–613 (2005)

[184] Vovk U, Pernus F, Likar B: Intensity inhomogeneity correction of multispectral MR

images. NeuroImage 32:54–61 (2006)

[185] Vovk U, Pernus F, Likar B: A review of methods for correction of intensity inhomo-

geneity in MRI. IEEE Transactions on Medical Imaging 26:405–421 (2007)

[186] Wells WM, Grimson WEL, Kikinis R, Jolesz FA: Adaptive segmentation of MRI

data. IEEE Transactions on Medical Imaging 15:429–442 (1996)

Bibliography xxxv

[187] Windham MP: Cluster validity for fuzzy clustering algorithms. Fuzzy Sets and Sys-

tems 5:177–185 (1981)

[188] Wood BJ, Razavi P: Virtual endoscopy: a promising new technology. American

Family Physician 66:107–112 (2002)

[189] Wu KL, Yang MS: Alternative c-means clustering algorithms. Pattern Recognition

35:2267–2278 (2002)

[190] Xu C, Prince JL: Snakes, shapes, and gradient vector flow. IEEE Transactions on

Image Processing 7:359–369 (1998)

[191] Xie Z, Wang S, Chung FL: An enhanced possibilistic c-means clustering algorithm.

Soft Computing 12:593–611 (2008)

[192] Yair E, Zeger K, Gersho A: Competitive learning and soft competition for vector

quantization design. IEEE Transactions on Signal Processing 40:294–309 (1992)

[193] Yang MS, Wu KL: A similarity-based robust clustering method. IEEE Transactions

on Pattern Recognition and Machine Intelligence 26:434–448 (2004)

[194] Yezzi A, Kichenassami S, Kumar A, Olver P, Tannenbaum A: A geometric snake

model for segmentation of medical imagery. IEEE Transactions on Medical Imaging

16:199–209 (1997)

[195] Yu J: General c-means clustering model. IEEE Transactions on Pattern Recognition

and Machine Intelligence 27:1197–1211 (2005)

[196] Zadeh LA: Fuzzy sets. Information and Control 8:338–353 (1965)

[197] Zhan Y, Shen DG, Zeng JC, Sun L, Fichtinger G, Moul JW, Davatzikos C: Targeted

prostate biopsy using statistical image analysis. IEEE Transactions on Medical Imag-

ing 26:779–788 (2007)

xxxvi Bibliography

[198] Zhang Y, Brady M, Smith S: Segmentation of brain MR images through a hid-

den Markov random field model and the expectation-maximization algorithm. IEEE


[199] Zhang B, Zhang C, Yi X: Competitive EM algorithm for finite mixture models.

Pattern Recognition 37:131–144 (2004)

[200] Zhang DQ, Chen SC: A novel kernelized fuzzy c-means algorithm with application

in medical image segmentation. Artificial Intelligence in Medicine 32:37–50 (2004)

[201] Zhao Q, Song J, Wu Y: Improved fuzzy c-means segmentation algorithm for images

with intensity inhomogeneity. Advances in Soft Computing 41:150–159 (2007)

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