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5.4 Image Description by Central Moments
Next we study central moments for the shape description.
An image can be interpreted as a two-dimensional density function. So we can
compute the geometric moments
∫∫
pq
p M
=
x ygxy
( ,
)
pq
,
=
0,1, 2,
(11)
gxy .
In the case of digital images, we can replace the integrals by sums:
(, )
with continuous image function
∑∑
p
q
m
=
x y
f
( ,
x y
)
p q
,
=
0,1, 2,
(12)
pq
xy
fxy is the discrete function of the gray levels.
The seven-moment invariants from Hu [19] have the property of being invariant to
translation, rotation and scale. Since our implementation of an algorithm [10] is not
invariant with respect to rotation and scale, the invariant moments are unsuitable for
our image description.
We can consider the central moments, which are the only ones translation
invariant:
(, )
where
∑∑
p
q
m
=
(
x
x
) (
y
y
)
f
( ,
x y
)
p q
,
=
0,1, 2,
(13)
pq
c
c
xy
where
∑∑
∑∑
xf x y
(, )
yf x y
(, )
xy
xy
x
=
and
y
=
.
(14)
∑∑
∑∑
c
c
fxy
(, )
fxy
(, )
xy
xy
For our study we use central moments p m with p and q between 0 and 3 and as
input image the binarized gradient image obtained with the thresholding algorithm of
Otsu. To determine the similarity between two images we use the normalized city-
block distance in equation 6 its the weights
[
]
Fig. 38 shows the dendrogram of this test. If we virtually cut the dendrogram by
the cophenetic similarity measure of 0.0129, we obtain the following groups
G1={Monroe, parrot}, G2={neu3, neu4}, G3={gan128}, G4={neu4_r180},
G5={neu2}, G6={neu1}, and G7={cell}. Compared to the statistical and texture fea-
tures for image description we obtain one more group, but neu4_r180 gets separated
from neu4. The resulting groups seem to represent better the relationship between the
image characteristic and the image segmentation parameters.
ω
=
1/
k
1,
,
k
.
∀∈
i
i
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