Image Processing Reference
In-Depth Information
as to be expected for such a smooth shape. In this way, we can derive a set numbers that
can be used to recognise the boundary of a shape: a similar ellipse should give a similar set
of numbers whereas a completely different shape will result in a completely different set
of numbers.
We do, however, need to check the result. One way is to take the descriptors of a circle,
since the first harmonic should be the circle's radius. A better way though is to reconstruct
the shape from its descriptors, if the reconstruction matches the original shape then the
description would appear correct. Naturally, we can reconstruct a shape from this Fourier
description since the descriptors are regenerative . The zero-order component gives the
position (or origin) of a shape. The ellipse can be reconstructed by adding in all spatial
components, to extend and compact the shape along the x and y axes, respectively. By this
inversion, we return to the original ellipse. When we include the zero and first descriptor,
then we reconstruct a circle, as expected, shown in Figure 7.7 (b). When we include all
Fourier descriptors the reconstruction, Figure 7.7 (c), is very close to the original, Figure
7.7 (a), with slight difference due to discretisation effects.
(a) Original ellipse
(b) Reconstruction by zero-
and first-order components
(c) Reconstruction by all
Fourier components
Figure 7.7
Reconstructing an ellipse from a Fourier description
But this is only an outline of the basis to Fourier descriptors, since we have yet to
consider descriptors which give the same description whatever an object's position, scale
and rotation. Here we have just considered an object's description that is achieved in a
manner that allows for reconstruction. In order to develop practically useful descriptors,
we shall need to consider more basic properties. As such, we first turn to the use of Fourier
theory for shape description.
7.2.3.2 Fourier expansion
In order to define a Fourier expansion, we can start by considering that a continuous curve
c ( t ) can be expressed as a summation of the form
ct
() =
c f t
()
(7.1)
kk
k
where c k define the coefficients of the expansion and the collection of functions and f k ( t )
define the basis functions. The expansion problem centres on finding the coefficients given
a set of basis functions. This equation is very general and different basis functions can also
be used. For example, f k ( t ) can be chosen such that the expansion defines a polynomial.
Other bases define splines, Lagrange and Newton interpolant functions. A Fourier expansion
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