Image Processing Reference
In-Depth Information
In this section, we present the results of different c-means algorithms on pixel classification
of brain MR images, that is, the results of clustering based on only gray value of pixels.
Above 100 MR images with different sizes and 16 bit gray levels are tested with different
c-means algorithms. All the brain MR images are collected from Advanced Medicare and
Research Institute, Salt Lake, Kolkata, India. The comparative performance of different c-
means is reported with respect to DB, and Dunn index, as well as the β index (Pal, Ghosh,
and Sankar, 2000), which are reported next.
Davies-Bouldin (DB) Index:
The Davies-Bouldin (DB) index (Bezdek and Pal, 1988) is a function of the ratio of sum of
within-cluster distance to between-cluster separation and is given by
max
i=k
S(v
i
) + S(v
k
)
d(v
i
, v
k
)
c
1
c
DB =
i=1
for 1≤i, k≤c. The DB index minimizes the within-cluster distance S(v
i
) and maximizes
the between-cluster separation d(v
i
, v
k
). Therefore, for a given data set and c value, the
higher the similarity values within the clusters and the between-cluster separation, the lower
would be the DB index value. A good clustering procedure should make the value of DB
index as low as possible.
Dunn Index:
Dunn index (Bezdek and Pal, 1988) is also designed to identify sets of clusters that are
compact and well separated. Dunn index maximizes
Dunn = min
i
min
i=k
d(v
i
, v
k
)
max
l
S(v
l
)
for 1≤i, k, l≤c.
A good clustering procedure should make the value of Dunn index as high as possible.
β
Index:
The β-index of Pal et al. (Pal et al., 2000) is defined as the ratio of the total variation and
within-cluster variation, and is given by
c
n
i
c
n
i
c
N
M
;
||x
ij
−v||
2
; M =
||x
ij
−v
i
||
2
;
β =
where
N =
n
i
= n;
i
=1
j
=1
i
=1
j
=1
i
=1
n
i
is the number of objects in the ith cluster (i = 1, 2,, c), n is the total number of
objects, x
ij
is the jth object in cluster i, v
i
is the mean or centroid of ith cluster, and v is
the mean of n objects. For a given image and c value, the higher the homogeneity within
the segmented regions, the higher would be the β value. The value of β increases with c.
Consider the image of Fig. 2.3 as an example, which represents an MR image (I-20497774)
of size 256×180 with 16 bit gray levels. So, the number of objects in the data set of IMAGE-
20497774 is 46080. Table 2.1 depicts the values of DB index, Dunn index, and β index of
FCM and RFCM for different values of c on the data set of I-20497774 considering only
gray value of pixel. The results reported here with respect to DB and Dunn index confirm
that both FCM and RFCM achieve their best results for c = 4 (background, gray matter,
white matter, and cerebro-spinal fluid). Also, the value of β index, as expected, increases
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