Image Processing Reference
In-Depth Information
B
(
n
)exp
imn
2
π
N
,
N
−
1
2
N
−
1
b
(
m
)=
with
K
∈
1
,
2
,
·≤
,
(17.41)
2
N
−
1
2
n
=
−
where we assumed that
N
is odd, for simplicity. The synthesis can be truncated at
K
N/
2, in which case we obtain a smoother approximation of the boundary curve
B
(
n
)exp
imn
2
π
N
,
K
N
−
1
2
b
(
m
)=
with
K
∈
1
,
2
,
·
,
(17.42)
n
=
−K
because
b
lacks high-frequency terms. Such a truncation is often utilized in image
analysis, e.g., to reduce the number of FDs, which are in turn utilized in shape-based
recognition or discrimination. Smoothing via truncations is illustrated by Fig. 17.7
where we show the boundary approximation of the continents by FDs. The total
available FDs is equal to the number of boundary samples. Note that intricate curves
need more samples, for a faithful reconstruction.
The FDs have a number of desirable properties that we list below.
•
The centroid of the boundary is given by
B
(0)
b
(
m
)exp
=
N−
1
N−
1
1
N
im
0
2
π
N
1
N
B
(0) =
−
b
(
m
)
(17.43)
m
=0
m
=0
•
All FDs except
B
(0) are translation-invariant. To see this, we apply a translation
Δb
=
Δx
+
iΔy
to the boundary and obtain the new coordinates as
b
(
m
)=
b
(
m
)+
Δb
having the FDs:
N−
1
B
(
n
)=
1
N
imn
2
π
b
(
m
) exp(
−
N
)
m
=0
N−
1
1
N
imn
2
π
=
(
b
(
m
)+
Δb
) exp(
−
N
)
m
=0
N−
1
N−
1
1
N
imn
2
π
1
N
imn
2
π
=
b
(
m
) exp(
−
N
)+
Δb
·
exp(
−
N
)
m
=0
m
=0
N−
1
1
N
imn
2
π
=
b
(
m
) exp(
−
N
)+
Δbδ
(
n
)
m
=0
=
B
(
n
)+
Δb
·
δ
(
n
)
(17.44)
Accordingly, to obtain shift-invariant FDs one can ignore
B
(0), the centroid.
•
Assume that
b
is a scaled version of the boundary, so that
b
=
αb
with
α
being
a positive real scalar. Then
