Image Processing Reference
In-Depth Information
illustrates two periods of each component of the curve. Generally,
T
= 2π , thus the fundamental
frequency is ω = 1. It is important to notice that this representation can be used to describe
open curves. In this case, the curve is traced twice in opposite directions. In fact, this
representation is very general and can be extended to obtain the elliptic Fourier description
of irregular curves (i.e. those without derivative information) (Montiel, 1996), (Montiel,
1997).
In order to obtain the elliptic Fourier descriptors of a curve, we need to obtain the
Fourier expansion of the curve in Equation 7.44. The Fourier expansion can be performed
by using the complex or trigonometric form. In the original work (Granlund, 1972), the
expansion is expressed in the complex form. However, other works have used the trigonometric
representation (Kuhl, 1982). Here, we will pass from the complex form to the trigonometric
representation. The trigonometric representation is more intuitive and easier to implement.
According to Equation 7.5 we have that the elliptic coefficients are defined by
c
k
=
c
xk
+
jc
yk
(7.45)
where
T
T
=
1
=
1
c
xte
( )
-
jk
t
dt
and
c
yte
()
-
jk
t
dt
(7.46)
xk
T
yk
T
0
0
By following Equation 7.12, we notice that each term in this expression can be defined by
a pair of coefficients. That is,
ajb
-
2
ajb
-
2
xk
xk
yk
yk
c
=
c
=
xk
yk
(7.47)
ajb
+
2
=
+
ajb
xk
xk
yk
yk
c
=
c
xk
-
yk
-
2
Based on Equation 7.13 the trigonometric coefficients are defined as
T
T
=
2
=
2
a
xt
( ) cos(
k t dt
)
and
b
xt
( ) sin(
k t dt
)
xk
xk
T
T
0
0
(7.48)
T
T
=
2
=
2
a
yt
( ) cos(
k t dt
)
and
b
yt
( ) sin(
k t dt
)
yk
yk
T
T
0
0
That according to Equation 7.27 can be computed by the discrete approximation given by
m
m
=
2
=
2
sin(
a
x
cos(
k
ω τ
i
) and
b
x
ω τ
k
i
)
xk
i
xk
i
m
m
i
=1
i
=1
(7.49)
m
m
=
2
=
2
sin(
a
y
cos(
k
ω τ
i
) and
b
y
ω τ
k
i
)
yk
i
yk
i
m
m
i
=1
i
=1
where
x
i
and
y
i
define the value of the functions
x
(
t
) and
y
(
t
) at the sampling point
i
. By
considering Equations 7.45 and 7.47 we can express
c
k
as the sum of a pair of complex
numbers. That is,
c
k
=
A
k
-
jB
k
and
c
-
k
=
A
k
+
jB
k
(7.50)
