Image Processing Reference
In-Depth Information
then we obtain the standard trigonometric form given by
a
0
ct
() =
+
(
a
cos (
k
t
) +
b
sin (
k
t
))
(7.11)
k
k
2
k
=1
The coefficients of this expansion, a k and b k , are known as the Fourier descriptors . These
control the amount of each frequency that contributes to make up the curve. Accordingly,
these descriptors can be said to describe the curve since they do not have the same values
for different curves. Notice that according to Equations 7.7 and 7.10 the coefficients of the
trigonometric and exponential form are related to by
ajb
-
2
ajb
+
2
k
k
k
k
- (7.12)
The coefficients in Equation 7.11 can be obtained by considering the orthogonal property
in Equation 7.3. Thus, one way to compute values for the descriptors is
c
=
and
c
=
k
k
T
T
= 2
= 2
a
ct
( ) cos(
k t dt
)
and
b
ct
( ) sin(
k t dt
)
(7.13)
k
k
T
T
0
0
In order to obtain the Fourier descriptors, a curve can be represented by the complex
exponential form of Equation 7.2 or by the sin/cos relationship of Equation 7.11. The
descriptors obtained by using either of the two definitions are equivalent, and they can be
related by the definitions of Equation 7.12. Generally, Equation 7.13 is used to compute the
coefficients since it has a more intuitive form. However, some works have considered the
complex form (e.g. Granlund (1972)). The complex form provides an elegant development
of rotation analysis.
7.2.3.3 Shift invariance
Chain codes required special attention to give start point invariance. Let us see if that is
required here. The main question is whether the descriptors will change when the curve is
shifted. In addition to Equations 7.2 and 7.11, a Fourier expansion can be written in
another sinusoidal form. If we consider that
|
c
| =
a
2
+
b
2
and
= tan
-1
(
b
/
a
)
(7.14)
k
k
k
k
k
k
then the Fourier expansion can be written as
a
0
(7.15)
ct
() =
+
|
c
| cos(
k
t
+
)
k
k
2
k
=0
Here | c k | is the amplitude and
ϕ k is the phase of the Fourier coefficient. An important
property of the Fourier expansion is that | c k | does not change when the function c ( t ) is
shifted (i.e. translated), as in Section 2.6.1. This can be observed by considering the
definition of Equation 7.13 for a shifted curve c ( t +
). Here,
represents the shift value.
Thus,
T
T
= 2
= 2
a
ct
(
+
) cos(
k t dt
)
and
b
ct
(
+
) sin(
k t dt
)
k
k
T
T
0
0
(7.16)
By defining a change of variable by t = t
+
, we have
T
T
= 2
= 2
a
ct
( ) cos(
k t
-
ωα
k
)
dt
and
b
ct
( ) sin(
k t
ωα
-
k
)
dt
k
k
T
T
0
0
(7.17)
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