Image Processing Reference
In-Depth Information
We summarize this result as a theorem.
Theorem 17.1.
When
f
(
r, θ
)
, having its centroid at the origin, is transformed to
f
(
r
,θ
)
through a CT consisting of
θ
=
ϕ
+
θ
, and
r
=
αr
, the complex moments
are invariant up to a multiplication with a constant (complex) scalar,
I
pq
(
f
)=
α
p
+
q
+2
exp(
i
(
p
−
q
)
θ
)
I
pq
(
f
)
(17.23)
This result, albeit with different notation, is due to Reddi [188] and simplifies the
derivation of scale and rotation-invariant forms of moments [42, 89, 112, 190, 234]
discussed below. Most important, however, the theorem shows that the complex mo-
ments are nearly invariant to rotation and scaling because the effects of these are
no more than a multiplicative scalar on the complex moments. Furthermore, one
can always increase the descriptive power of the invariants systematically, by using
high-order complex moments.
Hu's moment invariants
The following quantities
H
1
=
m
20
+
m
02
H
2
=(
m
20
−
m
02
)
2
+4
m
11
m
03
)
2
H
4
=(
m
30
+
m
12
)
2
+(
m
21
+
m
03
)
2
H
5
=(
m
30
−
3
m
12
)
2
+(3
m
21
−
H
3
=(
m
30
−
[(
m
30
+
m
12
)
2
−
3(
m
21
+
m
03
)
2
]+
3
m
12
)(
m
30
+
m
12
)
·
···
[3(
m
30
+
m
12
)
2
−
(
m
21
+
m
03
)
2
]
+(3
m
21
−
m
03
)(
m
21
+
m
03
)
·
[(
m
30
+
m
12
)
2
−
(
m
21
+
m
03
)
2
]+
H
6
=(
m
20
−
m
02
)
·
···
+4
m
11
(
m
30
+
m
12
)(
m
21
+
m
03
)
H
7
=(3
m
21
− m
03
)(
m
30
+
m
12
)
·
[(
m
30
+
m
12
)
2
−
3(
m
21
+
m
03
)
2
]+
···
−
[3(
m
30
+
m
12
)
2
−
(
m
21
+
m
03
)
2
]
(
m
30
−
3
m
12
)(
m
21
+
m
03
)
·
where
m
pq
represent the real (central) moments, were suggested by Hu [112] as
rotation-invariant measures. We will shortly see that this is indeed the case. We write
down the complex moments so as to express
H
1
···
H
7
via the complex moments:
I
11
=
m
20
+
m
02
=
d
I
20
=
m
20
−
m
02
+
i
2
m
11
=
c
+
ic
I
12
=
m
30
+
m
12
−
ib
i
(
m
21
+
m
03
)=
b
−
m
03
)=
a
+
ia
I
30
=
m
30
−
3
m
12
+
i
(3
m
21
−
yielding
