Image Processing Reference
In-Depth Information
0
0
S
1
S
2
S
3
S
4
T
T
1
2
3
(t)
(t)ds
γ
i
(a) Continuous curve
(b) Riemman sum
Figure 7.13
Integral approximations
m
= 2 +
2
*
a
γ τ
ii
L
i
=1
m
2
k
=
2
*
a
γ τ
cos
s
(7.35)
k
ii
i
L
L
i
=1
m
2
k
= -
2
+
2
*
b
γ τ
sin
s
k
ii
i
k
L
L
i
=1
where
s
i
is the arc length at the
i
th point. Note that
i
s
i
=
(7.36)
r
r
=1
It is important to observe that although the definitions in Equation 7.35 only use the
discrete values of γ (
t
), they obtain a Fourier expansion of γ *(
t
). In the original formulation
(Zahn, 1972), an alternative form of the summations is obtained by rewriting the coefficients
in terms of the increments of the angular function. In this case, the integrals in Equation
7.34 are evaluated for each interval. Thus, the coefficients are represented as a summation
of integrals of constant values as,
s
m
= 2 +
2
i
a
*
ds
i
L
i
=1
s
i
-1
s
m
2
k
=
2
i
a
*
cos
sds
(7.37)
k
L
i
L
i
=1
s
i
-1
s
m
2
k
= -
2
+
2
i
b
*
sin
sds
k
L
i
L
k
i
=1
s
i
-1
By evaluating the integral we obtain
m
= 2 +
2
(-
*
a
ss
)
i
i
i
-1
L
i
=1
