Image Processing Reference
In-Depth Information
A(S) = 2316
P(S) = 498.63
C(S) = 0.11
I(S) = 2.24
IR(S) = 6.67
A(S) = 6104
P(S) = 310.93
C(S) = 0.79
I(S) = 1.85
IR(S) = 1.91
A(S) = 4917
P(S) = 259.27
C(S) = 0.91
I(S) = 1.00
IR(S) = 1.03
(a) Descriptors for the circle
(b) Descriptors for the convoluted
region
(c) Descriptors for the ellipse
Figure 7.20
Basic region descriptors
moments can be considered as a global region description. Moments for image analysis
were again originally introduced in the 1960s (Hu, 1962) (an exciting time for computer
vision researchers too!) and an excellent and fairly up-to-date review is available (Prokop,
1992).
Moments are actually often associated more with
statistical
pattern recognition than
with
model-based
vision since a major assumption is that there is an
unoccluded
view of
the target shape. Target images are often derived by thresholding, usually one of the
optimal forms that can require a single object in the field of view. More complex applications,
including handling occlusion, could presuppose
feature extraction
by some means, with a
model to in-fill for the missing parts. However, moments do provide a global description
with invariance properties and with the advantages of a compact description aimed to avoid
the effects of noise. As such, they have proved popular and successful in many applications.
The
two-dimensional Cartesian moment
is actually associated with an order that starts
from low (where the lowest is zero) up to higher orders. The moment of order
p
and
q
,
m
pq
of a function
I
(
x
,
y
), is defined as
m
=
x
pq
y I x y d x dy
( , )
(7.74)
pq
-
-
For discrete images, Equation 7.74 is usually approximated by
Σ Σ
pq
m
=
x
y I x y
( , )
A
(7.75)
pq
xy
These descriptors have a
uniqueness
property in that if the function satisfies certain conditions,
then moments of all orders exist. Also, and conversely, the set of descriptors uniquely
determines the original function, in a manner similar to reconstruction via the inverse
Fourier transform. However, these moments are descriptors, rather than a specification
which can be used to reconstruct a shape. The
zero-order moment
,
m
00
, is
Σ Σ
m
00
=
( , )
I x y
A
(7.76)
xy
which represents the total mass of a function. Notice that this equation is equal to Equation
7.67 when
I
(
x
,
y
) takes values of zero and one. However, Equation 7.76 is more general
since the function
I
(
x
,
y
) can take a range of values. In the definition of moments, these
values are generally related to density. The two
first-order moments
,
m
01
and
m
10
, are given
by
Σ Σ
Σ Σ
m
=
xIxy
( , )
A m
=
yIxy
( , )
A
(7.77)
10
01
xy
xy
For binary images, these values are proportional to the shape's centre co-ordinates (the
