Image Processing Reference
In-Depth Information
b
(
m
)exp
N−
1
B
(
n
)=
1
N
imn
2
π
N
−
m
=0
αb
(
m
)exp
N−
1
1
N
−imn
2
π
N
=
m
=0
=
αB
(
n
)
(17.45)
B
(1)
Provided that
|
|
>
0, one can compute
B
(
n
)
|
αB
(
n
)
α
=
(17.46)
B
(1)
|
|
B
(1)
|
B
(1)
which is scale-invariant. If
0 then another FD can be used in the
normalization, yielding more stable scale-invariant features.
|
|≈
Assume that
b
is a rotated version of the boundary, so that
b
= exp(
iθ
)
b
with
θ
being an arbitrary angle. Then
•
b
(
m
)exp
N−
1
B
(
n
)=
1
N
imn
2
π
N
−
m
=0
exp(
iθ
)
b
(
m
)exp
N−
1
1
N
imn
2
π
N
=
−
m
=0
=exp(
iθ
)
B
(
n
)
(17.47)
B
(1)
B
(1)
From this it follows that
|
|
=
|
B
(1)
|
, so that if
|
|
>
0, one can, by
division of both sides of Eq. (17.47), obtain for
n
=1:
B
(1)
=exp(
iθ
)
B
(1)
|
(17.48)
|
B
(1)
|
B
(1)
|
Accordingly, we can use this ratio to normalize
B
(
n
) for
n>
1:
B
(
n
)
B
(1)
|B
(1)
|
iθ
)
B
∗
(1)
|
exp(
iθ
)
B
(
n
)=
B
∗
(1)
|
=exp(
−
B
(
n
)
(17.49)
B
(1)
|
B
(1)
|
B
(1)
to obtain rotation-invariant FD features. As before, if
0 then another
FD can be used in the normalization, yielding more stable rotation-invariant fea-
tures. One can also use
|
|≈
B
(
n
)
|
|
to achieve rotation-invariance, but these features
N
2
annihilate
freedoms, far too much description power of FDs than necessary
for many applications.
≈
•
Because,
B
(1) contains both scale and rotation parameters, to achieve both
scale
and rotation-invariance
in FDs one can normalize the other FDs by this complex
scalar
B
(
n
)
B
(1)
,
with
n
=2
,
3
,
···
N
−
1
,
(17.50)
provided that it is nonzero (else one can use another FD that has a large magni-
tude).
