chapter-8 modified fuzzy c-means (fcm) algorithm for image...
TRANSCRIPT
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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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