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CHAPTER-8

MODIFIED FUZZY C-MEANS (FCM) ALGORITHM

FOR IMAGE SEGMENTATION

8.1 Pros and Cons of Fuzzy c-means (FCM) clustering algorithm

The FCM algorithm is a very powerful method of clustering. Due to its

flexibility, FCM has proven a powerful tool to analyze real life data, both

categorical and numerical. The closeness of fuzzy membership function to the

qualitative nature of human perception makes it a first choice of any practical

problem solution methodologies. From programming point of view, FCM is

relatively straightforward. The objective function of FCM algorithm is intuitive

and easy to grasp. For data sets consists of hyper-spherically shaped well separated

clusters, FCM discovers these clusters accurately. Also, because of fuzzy based

approach, FCM always converges to a solution with consistent membership values.

However, there are several shortcomings of FCM algorithm. FCM requires

the number clusters to look for which is a priori. The initialization process requires

the parameters to be set. Some of the drawbacks of FCM are:

1. It requires the number of clusters to look for.

2. The initialization of FCM requires some parameters to be set and

inefficient parameters can lead to local minima problem.

3. FCM looks for clusters of same type.

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4. The objective function is not good for clusters of unequal size lying very

close to each other.

5. The iterative nature consumes a lot of computational time particularly in

case of high dimensional data such as image data.

Despite these shortcomings of FCM, several problem domains employ

FCM in the solution process. Due to the wide-spread application of FCM, several

extensions and modifications of FCM exist in literature. A few of them are

discussed in the next section.

8.2 Extensions and modifications of FCM algorithm

Gustafson et al. [184] introduced the idea of fuzzy covariance clustering.

The principal of Gustafson-Kessel (GK) algorithm is that it allows each cluster to

have its own partition matrix-norm and thus finding clusters of different ellipsoidal

shape. It can detect hyper-ellipsoidal clusters of differing shape. The obvious

limitation of GK algorithm is that it only looks for hyper-ellipsoidal shaped

clusters.

Fuzzy c-Elliptotypes (FCE) algorithm was introduced Bezdek et al. [31] to

detect clusters that have shape of lines or planes. The principal of FCE algorithm is

to discount Euclidean distances for points lying along the main eigenvectors of a

clusters while computing Euclidean distance to full extent for other points. A

distance measure is computed from a weighted combination of two distance

measures as follows [31]

����� � �� ��� �� � �� � ���� �

�

�

�

����

�

Here, �� �� is the Euclidean distance and �� �

� is defined as [31]

�� �� ���� � ����� � ���� � ��� �����

�

���

Where��� �� �!, and ��� is the "#$ eigenvector of the covariance matrix, %�

of cluster�&. The FCE algorithm only searches for linear structures, which is not

applicable to all kind of data.

Another extension of FCM algorithm is the Fuzzy c-Shell clustering.

Several variants of shell clustering exist in literature. A complete review of them

can be found in Hoopner et al. [185]. The idea of fuzzy c-shell clustering is based

on the distance measure it uses. The cluster is described by its centre point and

radius, pi and ri respectively. The distance measure is defines as [186]

�������� � ��� ����� � ���� � ����

The distance measure is responsible for detecting the type clusters and there

are different distance measures of detection for different shaped clusters.

For image processing application, the images are preprocessed for edge

detection and the edge pixels are fed to the algorithm for boundary detection. Shell

clustering is computationally expensive as the update procedure requires iteration

of non-linear equations. Also, the result depends on the efficiency of the edge

detection method.

A modification FCM algorithm is the Possibilistic c-means (PCM)

algorithm. The objective function of conventional FCM algorithm is modified by

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adding an extra term, which forces the membership matrix '�� to be as large as

possible. The update procedure for cluster center also changes [187], [188]. The

PCM algorithm is very much dependent on the initialization; it may or may not

converge to a solution. The determination of the possible large value of

membership depends on the choice of a positive number used in the added term.

Some other notable extensions and modification of FCM are: High-

contrast approach [189], Competitive agglomerative fuzzy clustering [190],

Credibilistic fuzzy c-means algorithm [191].

8.3 The Proposed Modification of FCM algorithm for image segmentation

In this section, a modified FCM algorithm suitable image segmentation is

introduced. The limitations of conventional FCM algorithm are discussed in the

previous section. Some improvement attempts, made by various researchers, are

also discussed in the previous section. Most of them focus on a particular type of

clusters. Also, some of them are computationally expensive and/or inefficient in

handling high dimensional data such as digital image. Limitations of FCM

algorithm are already discussed in the previous sections. Here, emphasis is given

on three major drawbacks of FCM algorithm.

Firstly, the number clusters have to be set before the initialization of the

FCM algorithm. There is no generalized method for determining the number of

clusters. The choice is very much application specific and requires sound

knowledge of the problem domain. Conventional FCM algorithm is based on

Euclidean distance, which can only be used to detect spherical structural clusters.

FCM is an iterative algorithm which tries to find the optimal value of the objective

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function iteratively. The iterative nature makes FCM algorithm computationally

expensive for image segmentation particularly in segmentation of colour images.

The high dimension of colour image also aids to the local minima problem. The

local minima problem occurs when small clusters, lying close to very large

clusters, are missed by the FCM algorithm. One by one these limitations are

described and remedies are suggested in the next section. The modification is

supported by the results of application of the modified FCM algorithm in

segmentation Pap smear images and the results are also validated with cluster

validity measures.

8.3.1 Determining the Number of Clusters

Determination of number of clusters is the first parameter for FCM

algorithm initialization. The number of clusters is the most important parameter.

When clustering real data without a priori information about the structures in the

data, one usually has to make assumptions about the number of underlying

clusters. The algorithm then searches for that particular number of clusters,

regardless of whether they are present in the data or not. Several attempts have

been made, though the methodologies pertain to a class of problem. Huntsberger et

al. [55] suggested a fixed number of clusters at each iteration. Celenk et al. [192]

proposed a method of mathematical evaluation criterion for determining the

number of clusters. Bensaid et al. [193] suggested a method of using the validity

measure in synergy with the objective function, rather than using it as an

optimization criterion. Some more suggestions can be found in [194], [195].

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This thesis presents a simple yet effective measure for determining the

number of clusters for FCM algorithm initialization for segmentation of Pap smear

images. First, the histogram of an image is created. The histogram is smoothed

several times, keeping the shape and properties of the original histogram intact.

The repetitive smoothing will also removes the noise and spurious peaks, which is

an added advantage of the method. Then the histogram is divided into a number of

buckets and the sum of pixels in each bucket is computed. Next, successive

differences between each of the bucket pairs are calculated. The signs of the

differences at each peak or valley will change once. The number of qualified

number of cluster centers can be determined from the symbolic change and the

value of difference. The generation of histogram for RGB colour image is

described in the Section 8.3.3.

8.3.2 Mahalanobis Distance measure

The Euclidean distance used in classical FCM algorithm is responsible for

finding out the distance between the points and the cluster center. The norm matrix

in the objective function of FCM is an identity matrix and hence it is not written in

the objective function definition. The identity norm matrix corresponds to

Euclidean distance. FCM algorithm with Euclidean distance measure cannot detect

arbitrary shaped clusters except hyper-spherical shaped ones. A simple solution is

use diagonal norm matrix, which detects hyper-ellipsoidal clusters. But the

orientation of the clusters remains fixed; axes of the hyper-ellipsoids are parallel to

the coordinate exes. Another distance measure Mahalanobis distance measure

[196] can be used. The use of Mahalanobis distance enables the FCM algorithm to

detect hyper-ellipsoidal clusters with arbitrary orientation. The types of clusters

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generated by these three distance measure can be described with following figure

[197].

Figure 8.1: Shape and orientations of clusters for different distance measures

(Picture courtesy: R. Babuska, “Fuzzy Modeling for Control”, Kluwer Academic

Publishers, 1998 [197])

This thesis proposed a replacement of the Euclidean distance measure by

Mahalanobis distance measure in the objective function of the FCM algorithm.

Before detailing the replacement strategy, a formal definition of Mahalanobis

distance is given below.

Mahalanobis distance: Suppose there are two distinct groups which are to

be labeled as G1 and G2. Consider a number (say, p) of relevant characteristics of

individuals in these groups. Let X denote a (random) vector that contains the

measurements made on a given individual or entity under study. A common

assumption is to take the p-dimensional random vector X as having the same

variation about its mean within either group. Then the difference between the

groups can be considered in terms of the difference between the mean vectors of X,

in each group relative to the common within-group variation. A measure of this

type is the Mahalanobis squared distance defined by [196]

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�� �(� � (��)�*��(� � (��

The superfix + denotes matrix transpose. � denotes the covariance matrix

of X in each group.

This definition can be easily incorporated in the objective function of the

FCM clustering algorithm. The objective function of FCM algorithm is as follows

2

1 1

( , )c n

mik k i

i k

J U V u x v= =

= −��

The term ��� � ,��� is the Euclidean distance. This term is replaced by

Mahalanobis distance term. So the modified objective function is:

-�.� /� ��'��0���� � ,��12*���� � ,����3

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4

���

The term 2*� is the inverse of the covariance matrix of �� and ,�. The membership update function changes to:

'�� �5 6����� � ,��12�*���� � ,��������� � ,��12�*���� � ,����7

� 0*�84���

The terms, 2�*� and 2�*� are the inverse of covariance matrix of �� and ,�, and �� and ,� respectively.

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8.3.3 Histogram induced Cluster center computation

The FCM algorithm is an iterative algorithm which is computationally

expensive particularly in image segmentation. The computation of cluster center

takes up much of the computation time at each iteration. In image segmentation

problem, the membership value for each pixel needs to be calculated, and the

number of pixels in an image is very large. Another concern is that the high

dimensionality of the image may create local minima problem for FCM algorithm.

The solution is to reduce the dimension of the data so that the computation time is

minimized and local minima problem is avoided. A very common approach to this

is to limit the data items to a certain number. Another approach is to sample an

image or to take window average. Here, a new method is proposed based on the

histogram of an image, using it in the cluster center computation rather than the

pixel values itself.

The normalized histogram of an image defined is defined in the Section 5.1

of Chapter 5. The Normalize module stretches an image's pixel values to cover the

entire pixel value range (0-255). RGB colour images have three components for

each pixel and thus there is three histograms for an RGB colour image. An RGB

function is used to normalize the range given by the following equation [198]

9 ��9 � 9:;<�=�9:>� � 9:;<�� ? @AA

B ��B � B:;<�=�B:>� � B:;<�� ? @AA

C ��C � C:;<�=�C:>� � C:;<�� ? @AA�

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The function processes each colour channel R, G and B and computes the

minimum and maximum value in each of the three colour channel. When these

values are found, the image is reprocessed by subtracting the minimum value of

each channel from each pixel and dividing by its max-min range (3 times for each

RGB pixel). This stretches the pixel value to the full 0-255 pixel value range. To

our naked eyes, the image appears to have increased in contrast. The normalized

histogram is denoted by������, r is the pixel value. The cluster center computation

is modified by replacing pixel term �� by the histogram������. The cluster center

computation formula changes to

,� 5 '��03��� �����5 '��03���

8.3.4 Optimization for Covariance matrix computation

The modification of the FCM algorithm is based on the three major

drawbacks and their solutions has already been discussed in the previous sections.

The inclusion of Mahalanobis distance and histogram information in the FCM

algorithm makes it more efficient in the sense that it can now detect clusters of

arbitrary shape with varying size and the computational time is minimized by a

great extent. Also the reduced data dimension voids the possibility of local minima

problem. However, some concerns still remain which need to be redressed. The

computational overhead of calculating the covariance matrix and its determinant

affects the overall performance of the modified FCM algorithm. The modified

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FCM algorithm is optimized as follows. The calculation of covariance matrix of

each cluster is done by using the following equation

B� 5 (��0��� � ,����� � ,��13��� 5 (��03���

B; is the fuzzy covariance matrix of�;#$ cluster. To avoid the calculation of

determinant of the covariance at each iteration, a regulating factor of covariance

matrix is added to the modified objective function as proposed by Liu et al. [199].

Here, the term is��D EF� �B��. The membership function changes to

'�� �5 GH��� � ,��1B� ��� � ,�� � EF� �B��H���� � ,�1B� ��� � ,� � EF� �B���I

� 0*�84���

8.3.5 The complete modified FCM algorithm for image segmentation

Three major drawbacks of FCM algorithm are addressed and solutions are

suggested in the previous section. The complete modified FCM algorithm has been

described below:

Input: source image

Output: segmented image

Algorithm steps:

Step1: Initialization: Create the normalized histogram of the source image

using the RGB function. Find the number of clusters using the method described in

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Section 8.3.1. Set the value of : and�J. � K : K �, initialize�.L, such that

'��L �� ;� ��� @� M N� "� ��� @�M <

Step 2: Calculate the cluster center using

,� 5 '��03��� �����5 '��03���

Step 3: Calculate the covariance matrix B� Step 4: Calculate the new membership values using

'�� �5 GH��� � ,��1B� ��� � ,�� � EF� �B��H���� � ,�1B� ��� � ,� � EF� �B���I

� 0*�84���

Step 5: Compare Uk+1 and Uk, if | Uk+1- Uk|�� then stop, otherwise go to step 2;

where � is the pre-specified small number representing the smallest acceptable

change in U.

8.4 Application of modified FCM algorithm in segmentation of Pap smear

images

The modified FCM algorithm is applied in the segmentation of Pap smear

images. Here, normalized histograms of the Pap smear are created and smoothed

according to the modified FCM algorithm. The membership values of the pixels

are constituted the same way as described in Section 7.3.1 of Chapter 7. The same

set of cluster validity measures are used to validate the clustering result as in

Chapter 7.

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An attempt is made to improve the shape analysis method described in the

previous chapter. Below, the improvement in shape analysis method is discussed.

8.5 Improved Shape analysis of cervical cells

The simple coded shape study has some major drawbacks. To eliminate

those shortcomings, Dutta Majumdar’s Shape Theory [200] is used to study the

shape of cervical cells. Next section describes Dutta Majumdar’s Shape Theory in

detail. Dutta Mjumdar’s Shape Theory consists of two parts and they are:

Generalized Transformation Theory: In this approach alignment on the

basis of some invariant landmark points on boundary of the Region of Interest

(ROI) is performed. If by translation and/or scaling and/or rotation, landmark

points of one image can be mapped on the corresponding landmark points of other

image, then two figures are of same shape. In case of inexact matching,

approximation can be done by considering closest match. Geometrically invariant

points are selected by considering points of high curvature on the boundary of the

region of interest (ROI).

A geometrical figure X in RK space consisting of N control points can be

represented by OP�Q�R matrix, now from the concept of shape, two figures X and

O’ have the same shape if they are related by the following rigid body

transformation equation, X’=βXΓ+ΙΝνΤ , where

Γ : K x K is a rotation matrix and |Γ|=1.

ΙΝ : N x 1 of one.

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ν : K x 1 is a translation vector.

β : isotropic scaling factor and >= 0

It is possible to formulate an approximate co-ordinate transformation for

mapping between two sets of landmarks in a least square sense using Taylor series

expansion. For two sets of landmark points (xm, ym) and (xm´, ym´), m = 1,2,…,n,

one set can be expressed in terms of other as follows:

x´ = q0+q1x+ q2y+ q3x2+ q4xy+ q5y2+….

y´ = r0+r1x+ r2y+ r3x2+ r4xy+ r5y2+….

Shape Metric and Distance Measures: Several attempts have been made

to represent an object in term of its morphological structure. Here a set theoretic

approach is described based on the works of Majumdar et al. [199] to define shape

and shape distance.

The shape of an object can be defined as a subset O in 9� if

(i) O is closed and bounded.

(ii) Interior of O is non-empty and connected.

(iii) Closure property holds on interior of�O.

This representation of shape remains invariant with respect to translation,

rotation and scaling. Moreover, another object S in R2 is of same shape to object

O���9� if it preserves translation, rotation and scaling invariance. In term of set

theory, these three transformations can be represented as follows:

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Translation: S� � T��� � �>�� �U� � �V�P��� U��OW

Rotation: S� � TX����� X@���OW where X� & X@ are rotation around x and

y axes respectively.

Scaling: S� � T�"�� "U�P��� U���OW�

Each of translation, dilation and rotation defines an equivalence relation

on�Y. If 9 is an equivalence relation on Y and objects Z�C� %�Y then:

Reflexivity: �Z� Z����9 �

Symmetry: �Z� C����9����C� Z����9�

Transitivity: if �Z� C����9�[��C� %����9�\]�<��Z� %����9 holds under each

of translation, rotation and dilation of shape transformation.

Distance �� between shape O and S in Y is defined as follows:

���O� S� � :��� �O � S����S � O�!��������������������������������

Where :��is Lebesgue measure [201] in 9� and �� satisfies following

rules:

�̂ ���O� S�����_��

^̂� ���O� S� � �_�; �̀><��a<bU�; �̀O� �S�

^̂ �̂ ���O� S� � �����S� O��

iv) ���O� S� �� ����S� c�������O� c����

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The shape is described on the basis of its structural features using certain

chain codes with respect to one reference point. Reference points are obtained from

the intersection points of the major axis with the contour of the region, which is

invariant under translation, rotation, and dilation of the region and the major axis is

unique.

Figure 8.2: Cell image with two principal axes.

The centroid ��d� Ud� of the contour is given by Q number of points as defined by

the following equation:

�� ==== n

j jg

n

j jg yn

yxn

x11

11

and in polar co-ordinate the major axis is defined as:

2

1

))sin(cos(),( �=

−+=n

jjjrccf θθθθ

where �� �_�\a��e_.�

The slope of axis (∝) is found from the best linear fit solution by minimizing f (θ,

c) as:

�

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�

��

�

==

=

−−−

−−=

n

jgj

n

jgj

n

jgjgj

yyxx

yyxx

1

2

1

2

1

)()(

))((22tan α

2

1 1

cos 2 * ( ) ( )n n

j g j gj j

x x y yα= =

− − − +� �

�=

≥−−n

jgjgj yyxxSin

1

0))((*22 α

To extract the feature of the boundary of the region of interest (ROI) it is

helpful to represent the closed contour with a set of direction. The direction code

may be taken among < selected points on the contour, which has same distance

between any two consecutive points. The direction d makes an angle (i-d) 45° with

direction i, where real number� ����\a�e�><��;� � ���@� � �e�. Let �0 �������

3

where :� �Z� C, be the contour starting from each reference point A and B and

are denoted by dA and dB respectively. If d2 is a rotation of d1 then �� ��� � �

for any real number, . For all j we can write �� ��� �� ������&

The distance function� f, in terms of the direction code between the contour

of interest and the model is defined as [199]

))(8),min(()( 211

2121 jj

n

jjj ddddddD �

=

−=

The normalized value of D is D/n and the shape similarity measure between

the two shapes is given by [199]

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�� �� � f=<

Smaller value of D indicates higher degree of similarity [199]

Figure 8.3: Chain code representation

Figure 8.4: A contour with major axis and two intersecting points of the major axis

with the contour: two reference points A and B

8.6 Results

The modified FCM clustering algorithm is used to segment the Pap smear

images into its constituent parts namely: cytoplasm and nucleus. The

segmentation is validated by the cluster validity measures.

The segmented images are analyzed with the Shape theory to trace the

boundary of the cell nuclei. Segmentation results are shown in the Figure 8.5.

The tracing of cell nuclei boundary with Dutta Majumdar’ s Shape theory

[220] is shown in the Figure 8.6.

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The seven classes of Pap smear cell images are segmented with modified

FCM clustering algorithm. Also cell nuclei tracing is carried out on the seven

classes of images. The segmentation and cell nuclei tracing result is shown in

Table 8.1.

(a) (b)

(c) (d) Figure 8.5: Modified FCM segmentation (a) and (c) Original images, (b) and (d)

Segmented images

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(a) (b)

(c) (d)

Figire 8.6: Improved cell nuclei tracing (a) and (c) Original image, (b) and (d)

traced cell nuclei with white dots

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Table 8.1: Seven class of Cervical cell segmentation and cell nuclei tracing

Cell

Class

Original Image

1

2

3

4

5

6

7

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: Seven class of Cervical cell segmentation and cell nuclei tracing

Image Segmented Image Traced cell nucleus

boundary

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: Seven class of Cervical cell segmentation and cell nuclei tracing

Traced cell nucleus

boundary

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The clustering process is validated by three cluster validity measures as

before. The modified FCM segmentation is carried out on both the images with

cluster of cells and the images with single cell. The cluster validity measure values

are given below. The fuzzifier : is kept at a value of 1.2 just like in the

conventional FCM algorithm used in Chapter 6.

Table 8.2: Cluster validity measure values

Image Number of

Clusters

(c)

Partition

coefficient

(PC)

Partition

Entropy

(PE)

Compactness and

Separation index

(SC)

Figure 8.5

(b)

3

0.89

0.48

2.23

Figure 8.5

(d)

3

0.87

0.51

2.11

The cluster validity measure values for the seven classes of cervical cells

are given in the Table 8.3.

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Table 8.3: Cluster validity measure values for seven

Cell Class

Segmented image

1

2

3

4

5

6

7

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: Cluster validity measure values for seven classes of cervical cell image

Segmented image Number of

Clusters

Partition Coefficient

(PC)

Partition Entropy

(PE)

Compactness

Separation index (SC)

3

0.69

0.55

3

0.63

0.61

3

0.71

0.68

3

0.68

0.64

3

0.73

0.58

3

0.79

0.73

3

0.81

0.71

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classes of cervical cell image

Compactness and

Separation index (SC)

0.34

0.56

0.78

0.53

0.83

0.82

0.87

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8.7 Discussion and Conclussion

The FCM clustering algorithm has an edge over hard clustering and

conventional non-fuzzy image segmentation methods, which has already been

illustrated in the previous chapter. However, it suffers from some major drawbacks

and in image segmentation process those drawbacks can be seen. The proposed

modification of the conventional FCM algorithm is devoid of these shortcomings.

The initialization of the modified FCM algorithm is a simple yet effective

one. Use of basic image properties makes it easy to grasp and implement. For high

dimensional data like RGB colour space, the use of the normalized histogram

makes it easier for the programmers to set the other parameters.

The replacement of Euclidean distance by Mahalanobis distance enhances

the cluster detection capacity of FCM by a great extent. Accurate detection of

different types of clusters is very essential in image segmentation problem. The

computation of covariance matrix is a computationally costly procedure for a large

dataset such colour image. The goodness attained by the modified FCM can be

marred by this computational overhead. But, the optimization of the covariance

matrix calculation proposed in this thesis nullifies the bad effect of this

phenomenon.

The histogram induced cluster center calculation is very effective and

highly time saving as well as space saving. The data reduction helps FCM to

overcome the burden of time consuming iterative nature. Also, it helps to avoid the

local minima problem. For an (M x N) image, the data set is reduced to L from (M

x N). So, for an (512 x512) image size and 8- bit gray image, the computing time is

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improved by 1024 times when we neglect the time needed to compute the

histogram and mark the pixels.

The shape analysis is more robust than the one used in Chapter 7. The

representation of shape remains invariant with respect to translation, rotation and

scaling. The use of reflexive, transitive and symmetric relations among the shapes

along with shape distance based Lebesgue measure [200] enables the tracing of

uncertain boundaries more accurately.

The results obtained in this chapter when compared to the result obtained in

the previous chapter shows the robustness of the proposed modification FCM in

image segmentation. The segmentation results shown in the Figure 7.2 in Chapter

7 and the results in Figure 8.5 in this chapter clearly show the superiority of the

modified FCM algorithm. The cluster validity measure values in Table 7.2 and 7.3,

and values in Table 8.2 and 8.3 also differ by good extent.

The seven classes of cervical cells are segmented using both methods and

results are shown in Table 7.1 in Chapter 7 and in Table 8.2 in this chapter. It can

be seen that the modified initialization of the cluster numbers accurately

determines the number clusters to look for, which 3 in case of Pap smear images.

The proposed method is robust in the sense that it avoids the three

fundamental problem faced by conventional FCM algorithm. The affinity of FCM

algorithm to noise and outlier point is a mention-worthy concern. Though some

efforts have been made in this study in the form of repetitive smoothing of

histogram, it may not suit images with different modalities. Also histogram

smoothing has its own disadvantages.

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The data reduction is effective in saving time and avoiding local minima,

though in some cases it may lead to loss of data.

Some of the future works include: testing of the modified FCM in other

problem domain, giving sufficient amount of attention to noise removal, ensuring

minimal loss of data, establishing a mathematical model for determining the

initialization parameters of FCM algorithm, evaluation of the method with some

more cluster validity measures suitable for practical application.

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