Image Processing Reference
In-Depth Information
Clearly, the zero-order
centralised
moment is again the shape's area. However, the first-
order centralised moment µ
01
is given by
Σ Σ
µ
=
( - )
yyIxyA
1
( , )
01
xy
Σ Σ
Σ Σ
=
yI x y
(,
)
A
-
yI x y
(,
)
A
xy
xy
Σ Σ
=
01
my
-
( , )
I x y
A
(7.84)
xy
m
m
01
00
=
m
-
m
01
00
= 0
= µ
10
Clearly, neither of the first-order centralised moments has any description capability since
they are both zero. Going to higher order, one of the second-order moments, µ
20
, is
Σ Σ
µ
=
( - )
xxIxyA
2
( , )
20
xy
ΣΣ
=
(
x
2
- 2
xx
+
x
2
) (
I x y
, )
A
xy
2
m
m
mm
m
m
10
10
00
=
- 2
+
m
(7.85)
20
10
00
00
2
m
m
10
00
=
m
-
20
and this has descriptive capability.
The use of moments to describe an ellipse is shown in Figure
7.22
. Here, an original
(a) Original ellipse
(b) Translated ellipse
(c) Rotated ellipse
02
= 2.4947 · 10
6
02
= 2.4947 · 10
6
µ
02
= 6.4217 · 10
5
µ
20
= 2.4947 · 10
6
µ
µ
20
= 6.4217 · 10
5
20
= 6.4217 · 10
5
µ
µ
(d) 2nd order centralised moments
of original ellipse
(e) 2nd order centralised moments
of translated ellipse
(f) 2nd order centralised moments
of rotated ellipse
Figure 7.22
Describing a shape by centralised moments
