Graphics Reference
In-Depth Information
Homogeneity Index
As a measure of homogeneity, we compute an homogeneity index. This
index simply calculates the second order entropy because it provides local
information about the behavior of pixel intensity change. The graylevel values
in an image are not independent of each other. One can consider the sequences
of pixels to incorporate the dependency of pixel intensities in estimating the
entropy. In order to compute the entropy of an image, the following theorem
due to Shannon [151, 73] can be stated.
Theorem
Let
p
(
s
i
) be the probability of a sequence
s
i
of graylevels of length
l
, where
a sequence
s
i
of length
l
is defined as a permutation of
l
graylevels. Let us
define
l
i
1
H
(
l
)
=
−
p
(
s
i
) log
2
p
(
s
i
)
,
(4.21)
where the summation is taken over all graylevel sequences of length
l
. Then
H
(
l
)
is a monotonic decreasing function of
l
and
H
(
l
)
lim
l→∞
=
H
, the entropy
of the image. For different values of
l
, we get different orders of entropy.
Case 1:
l
= 1, i.e., sequence of length one. If
l
=1,weget
L−
1
H
(1)
=
−
p
i
log
2
p
i
,
i
=0
where
p
i
is the probability of occurrence of the graylevel
i
.Suchanentropy
is a function of the histogram only and it may be called
global entropy
of the
image. Therefore, different images with identical histograms would have the
same
H
(1)
value, irrespective of their content.
Case 2:
l
= 2, i.e., sequence of length two. Hence,
2
i
1
H
(2)
=
−
p
(
s
i
) log
2
p
(
s
i
)
2
i
(4.22)
1
=
−
p
ij
log
2
p
ij
,
j
where
s
i
is a sequence of length two and
p
ij
is the probability of occurrence of
the graylevels
i
and
j
. Therefore,
H
(2)
can be obtained from the co-occurrence
matrix.
H
(2)
takes into account the spatial distribution of graylevels. There-
fore, two images with identical histograms but different spatial distributions
will result in different entropy,
H
(
i
)
2 may be called local
entropy. Since the second order entropy reflects the local behavior of image,
it is expected that for a homogeneous region/patch, this measure should be
low.
values.
H
(
i
)
,i
≥
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