Image Processing Reference
In-Depth Information
After some algebraic manipulation we obtain
a k = a k cos( k ωα
) + b k sin( k ωα
) and
b k = b k cos( k ωα
) - a k sin( k ωα
)
(7.18)
The amplitude |
c k is given by
|
|
c
| =
(
a
cos(
k
ω α
) +
b
sin(
ω α
k
))
2
+ (
b
ω α
cos(
k
) -
ω α
a
sin(
k
))
2
k
k
k
k
k
(7.19)
That is,
2
2
|
c
| =
a
+
b
(7.20)
k
k
k
Thus, the amplitude is independent of the shift
. Although shift invariance could be
incorrectly related to translation invariance, actually, as we shall see, this property is
related to rotation invariance in shape description.
7.2.3.4 Discrete computation
Before defining Fourier descriptors, we must consider the numerical procedure necessary
to obtain the Fourier coefficients of a curve. The problem is that Equations 7.11 and 7.13
are defined for a continuous curve. However, given the discrete nature of the image, the
curve c ( t ) will be described by a collection of points . This discretisation has two important
effects. First, it limits the number of frequencies in the expansion. Secondly, it forces
numerical approximation to the integral defining the coefficients.
Figure 7.8 shows an example of a discrete approximation of a curve. Figure 7.8 (a)
shows a continuous curve in a period, or interval, T . Figure 7.8 (b) shows the approximation
of the curve by a set of discrete points. If we try to obtain the curve from the sampled
points, we will find that the sampling process reduces the amount of detail. According to
the Nyquist theorem, the maximum frequency f c in a function is related to the sample
period
by
1
2 f c
=
(7.21)
c(t)
c(t)
Fourier approximation
Sampling points
0
T
0
T
(a) Continuous curve
(a) Discrete approximation
Figure 7.8
Example of a discrete approximation
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