Image Processing Reference
In-Depth Information
i
2
bb
I
20
I
12
=(
c
+
ic
)
I
12
=
c
(
b
2
−
I
12
=(
b
2
−
b
2
)
−
b
2
)+2
c
bb
+
i
[
c
(
b
2
−
b
2
)
i
2
cbb
]
−
3
b
2
b
)
I
30
I
12
=(
a
+
ia
)
I
12
=
ab
(
b
2
−
I
12
=(
b
3
−
3
bb
2
)+
i
(
b
3
−
3
b
)
a
b
(
b
2
−
3
b
2
)+
−
+
i
[
a
b
(
b
2
−
3
b
)
ab
(3
b
2
−
b
2
)]
−
We identify then
H
1
···
H
7
H
1
=
d
=
I
11
H
2
=
c
2
+
c
2
=
2
|
I
20
|
H
3
=
a
2
+
a
2
=
2
|
I
30
|
H
4
=
b
2
+
b
2
=
2
|
I
12
|
H
5
=
ab
(
b
2
3
b
)
a
b
(
b
2
3
b
2
)=
(
I
30
I
12
)
−
−
−
H
6
=
c
(
b
2
− b
2
)+2
c
bb
=
(
I
20
I
12
)
H
7
=
a
b
(
b
2
−
3
b
)
− ab
(3
b
2
− b
2
)=
(
I
30
I
12
)
so that
H
1
=
I
11
(17.24)
I
20
|
2
H
2
=
|
(17.25)
I
30
|
2
H
3
=
|
(17.26)
I
12
|
2
H
4
=
|
(17.27)
I
12
)
H
5
=
(
I
30
·
(17.28)
I
12
)
H
6
=
(
I
20
·
(17.29)
I
12
)
H
7
=
(
I
30
·
(17.30)
By inspection and using the above theorem, it then follows that these scalars are
rotation-invariant. Using the theorem again, the normalized quantities
H
2
H
1
H
2
=
(17.31)
H
3
H
2
.
5
1
H
3
=
(17.32)
H
4
H
2
.
5
1
H
4
=
(17.33)
H
5
H
1
H
5
=
(17.34)
H
6
H
3
.
5
1
H
6
=
(17.35)
H
7
H
1
H
7
=
(17.36)
will make these invariants also scale-invariant.
