Image Processing Reference
In-Depth Information
direction, or by detecting the lack of linear symmetry, to describe textures lacking
direction. Measuring the lack of linear symmetries has been frequently used as a
way of detecting corners in image processing.
10.2 Real and Complex Moments in 2D
In image analysis there are a variety of occasions when we need to quantitate func-
tions by comparing them to other functions. Assuming that the integral exists, the
quantity
=
x
p
y
q
κ
(
x, y
)
dxdy
x
p
y
q
,κ
m
pq
(
κ
)=
(10.9)
with
p
and
q
being nonnegative integers, is the
real moment
p, q
of the function
κ
.
If
κ
has a finite extension, then the real moments defined as above are projections
of an integrable function onto the vector space of polynomials. It follows from the
Weierstrass theorem [193, 209], that the vector space of the polynomials is powerful
enough to approximate a finite extension function
κ
to a desired degree of accuracy.
In that, the approximation property of moments is comparable to the FCs, although
the polynomial basis of moments is not orthogonal, whereas the Fourier basis is.
Nonetheless, moments are widely used in applications. If
κ
is a positive function
then it is possible to view it as a probability distribution, after a normalization with
m
00
. Accordingly,
1
m
00
c
=(
x, y
)
T
(
m
10
,m
01
)
T
=
(10.10)
represents the centroid or the mean vector of the function
κ
. The quantity
x
p
y
q
m
10
m
00
m
01
m
00
m
pq
(
κ
)=
−
−
,κ
=
x
p
y
q
m
10
m
00
m
01
m
00
−
−
κ
(
x, y
)
dxdy
(10.11)
related to real moments, is called the central moment
p, q
of the function
κ
. Both real
moments and central moments have been utilized as tools to quantitate the shape of
a finite extension image region. We will discuss this further in Sect. 17.4.
Another type of moment, which we will favor over real moments in what follows,
is
=
(
x
+
iy
)
p
(
x
iy
)
p
(
x
+
iy
)
q
,κ
iy
)
q
κ
(
x, y
)
dxdy
(10.12)
I
pq
(
κ
)=
(
x
−
−
with
p
and
q
being nonnegative integers. This is the
complex moment
p, q
of the func-
tion
κ
. Notice that complex moments are linear combinations of the real moments,
e.g.,
I
20
=
m
20
− m
02
+
i
2
m
11
I
11
=
m
20
+
m
02
