Thresholding of Medical Images - Medical Image Processing: Advanced Fuzzy Set Theoretic Techniques - page 116

6.3.3 Fuzzy Clustering Method

Jawahar and Ray [13] thresholded the image using the fuzzy c means
clustering algorithm. Clustering is grouping similar objects into a group and
dissimilar objects into another group. Thresholding may be looked upon
as clustering where the image may be clustered into two or three or more
depending on the grey level of objects. Fuzzy clustering may be looked as
partitioning a set of ' n ' sample points X = { x 1 , x 2 , x 3 , …, x n } into ' c ' classes. If
h i is the histogram and p i is the probability of the distribution of the grey
values, i ∈ {0, 1, 2, 3, …, L − 1}, the cluster means for each class is

∑
∑

L − 1

hi () i

τ

μ

i j

(6.3)

v =

i = 0

j

L − 1

τ

hi μ ()

ij

i 0

=

j = 1, 2 for the background and object region. The objective function
J

∑

2

= ∑

L − 1

() (, ) 2

τ

= h idiv

ij

μ

- τ ≥ 1 controls the fuzziness in partition - can
be iteratively minimized by computing the means from Equation 6.3 and
updating the membership as

j

j 1 i = 0

1

μ () i =

O

+ ⎣ (, )(, ) /(

2 τ − 1 )

⎦

1 divdiv /

O B

and

μ () i =− 1 μ () i

B O

where d represents any distance function between the grey level i and class
mean.
The mean and membership values are updated iteratively until there is no
appreciable change in μ O and μ B .
As said earlier that as medical images contain uncertainties, fuzzy meth-
ods are very useful. So, examples on medical image thresholding using
fuzzy divergence, fuzzy geometry and fuzzy clustering are shown.

Example 6.2

An example of thresholding medical images are shown in Figure 6.2
using fuzzy divergence, fuzzy geometry and fuzzy clustering on
the medical images. Figure 6.2b is the result using fuzzy divergence,
Figure 6.2c is the result using fuzzy geometry, and Figure 6.2d is the
result using fuzzy clustering.