Image Processing Reference
In-Depth Information
values merely require division by the shape's area). In general, the
centre of mass
(,
xy
)
can be calculated from the ratio of the first-order to the zero-order components as
m
m
m
m
10
00
01
00
x
=
=
y
(7.78)
The first ten
x
-axis moments of a smooth shape are shown in Figure
7.21
. The moments
rise exponentially so are plotted in logarithmic form. Evidently, the moments provide a set
of descriptions of the shape: measures that can be collected together to differentiate between
different shapes.
30
20
log
(ellipse_moment
p,0
)
10
0
5
10
p
Figure 7.21
Horizontal axis ellipse moments
Should there be an intensity transformation that
scales
brightness by a particular factor,
say α , such that a new image
I
′ (
x
,
y
) is a transformed version of the original one
I
(
x
,
y
)
given by
I
′ (
x
,
y
) = α
I
(
x
,
y
)
(7.79)
Then the transformed moment values
m
pq
are related to those of the original shape
m
pq
by
mm
=
(7.80)
pq
pq
Should it be required to distinguish
mirror symmetry
(reflection of a shape about a chosen
axis), then the rotation of a shape about the, say,
x
axis gives a new shape
I
(
x
,
y
) which is
the reflection of the shape
I
(
x
,
y
) given by
I
(
x
,
y
) =
I
(-
x
,
y
) (7.81)
The transformed moment values can be given in terms of the original shape's moments as
= (- 1) (7.82)
However, we are usually concerned with more basic invariants than mirror images, namely
invariance to
position
,
size
and
rotation
. Given that we now have an estimate of a shape's
centre (in fact, a reference point for that shape), the
centralised moments
,
m
p
m
pq
pq
µ
pq
, which are
invariant to
translation
, can be defined as
Σ Σ
p
q
µ
=
( - ) ( - )
xx yyIxyA
( , )
(7.83)
pq
xy
