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produce only an object/background (two level) partitioning of the image.
Here in a segmentation problem, such a bi-level thresholding is not adequate.
But, one can consider an algorithm for hierarchical extraction of homogeneous
patches using the conditional entropy thresholding method. The conditional
entropy we can define in terms of the second order co-occurence matrix.
a. Co-occurrence Matrix
Let F =[ f ( x, y )] be an image of size M
×
N , where f ( x, y )isthegray
value at ( x, y ), f ( x, y )
G L =
{
0 , 1 , 2 ,
···
,L
1
}
, the set of graylevels.
The co-occurrence matrix of the image F is an L
L dimensional matrix that
gives us an idea of the transition of intensity between adjacent pixels. In other
words, the ( i, j ) th entry of the matrix gives the number of times the graylevel
“j” follows the graylevel “i” in a specific way.
Let “a” denote the ( i, j )th pixel in F and let “b” be one of eight neigh-
boring pixels of “a”, i.e.,
×
b
a 8 =
{
( i, j
1) , ( i, j +1) , ( i +1 ,j ) , ( i
1 ,j ) , ( i
1 ,j
1) ,
( i
1 ,j +1) , ( i +1 ,j
1) , ( i +1 ,j +1)
}
.
Define t ik =
δ ,
a∈F, b∈a 8
where δ = 1 if the graylevel of “a” is “i” and that of 'b' is 'k', δ =0
otherwise.
Obviously, t ik gives the number of times the gray level 'k' follows graylevel
'i' in any one of the eight directions. The matrix T =[ t ik ] L×L is, therefore, the
co-occurrence matrix of the image F . One may get different definitions of the
co-occurrence matrix by considering different subsets of a 8 , i.e., considering
b
a 8 , where a 8
a 8 .
The co-occurrence matrix may again be either asymmetric or symmetric.
One of the asymmetrical forms can be defined considering
M
N
t ik =
δ
i =1
j =1
with δ =1if f ( i, j )= i and f ( i, j +1) = k ,
f ( i, j )= i and f ( i +1 ,j )= k ,
δ = 0 otherwise.
Here only the horizontally right and vertically lower transitions are consid-
ered. The following definition of t ik gives a symmetrical co-occurence matrix.
M
N
t ik =
δ ,
i =1
j =1
where δ =1 f f ( i, j )= i and f ( i, j +1) = k ,
or f ( i, j )= i and f ( i, j
1) = k ,
or f ( i, j )= i and f ( i +1 ,j )= k ,
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