Image Processing Reference

In-Depth Information

entropy, difference variance, and difference entropy, are computed for each window. For

four angular directions, a set of four values is obtained for each of ten measures. The mean

of each of the ten measures, averaged over four values, along with gray value, homogeneity,

and edge value of the pixel, comprise the set of 13 features which is used as feature vector

of the corresponding pixel.

A limitation of the c-means algorithm is that it can only achieve a local optimum solution

that depends on the initial choice of the centroids. Consequently, computing resources may

be wasted in that some initial centroids get stuck in regions of the input space with a scarcity

of data points and may therefore never have the chance to move to new locations where

they are needed. To overcome this limitation of the c-means algorithm, next a method

is described to select initial centroids, which is based on discriminant analysis maximizing

some measures of class separability (Otsu, 1979). It enables the algorithm to converge to

an optimum or near optimum solutions (Maji and Pal, 2008).

Prior to describe the new method for selecting initial centroids, next a quantitative mea-

sure of class separability (Otsu, 1979) is provided that is given by

P
1
(T)P
2
(T)[m
1
(T)−m
2
(T)]
2

P
1
(T)σ
2
1

J(T) =

(2.7)

(T) + P
2
(T)σ
2
2

(T)

where

(T) =
T

z=0

(T) =
L−1

z=T+1

P

h(z);

P

h(z) = 1−P

(T)

1

2

1

T

L−1

1

P
1
(T)

1

P
2
(T)

m
1
(T) =

zh(z);

m
2
(T) =

zh(z)

z=0

z=T+1

T

L−1

1

P
1
(T)

1

P
2
(T)

σ
2
1

[z−m
1
(T)]
2
h(z);

σ
2
2

[z−m
2
(T)]
2
h(z)

(T) =

(T) =

z=0

z=T+1

Here, L is the total number of discrete values ranging between [0, L−1], T is the threshold

value, which maximizes J(T), and h(z) represents the percentage of data having feature

value z over the total number of discrete values of the corresponding feature. To maximize

J(T), the means of the two classes should be as well separated as possible and the variances

in both classes should be as small as possible.

Based on the concept of maximization of class separability, the method for selecting initial

centroids is described next. The main steps of this method proceeds as follows.

1. The data set X ={x
1
,, x
j
,, x
n
}with x
j
∈ℜ
m
are first discretized to

facilitate class separation method. Suppose, the possible value range of a feature

f
m
in the data set is (f
m,min
, f
m,max
), and the real value that the data element x
j

takes at f
m
is f
mj
, then the discretized value of f
mj
is

) = (L−1)×
f

−f

min

f
m,max
−f
m,min

mj

m

,

Discretized(f

(2.8)

mj

where L is the total number of discrete values ranging between [0, L−1].

2. For each feature f
m
, calculate h(z) for 0≤z < L.

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