Image Processing Reference
In-Depth Information
and complex contour representations. However, Fourier expansions can be developed for
other curve representations (Van Otterloo, 1991).
In addition to the curve's definition, a factor that influences the development and properties
of the description is the choice of Fourier expansion. If we consider that the trace of a curve
defines a periodic function, then we can opt to use a Fourier series expansion. However, we
could also consider that the description is not periodic. Thus, we could develop a representation
based on the Fourier transform. In this case, we could use alternative Fourier integral
definitions. Here, we will develop the presentation based on expansion in Fourier series.
This is the common way used to describe shapes in pattern recognition.
It is important to notice that although a curve in an image is composed of discrete pixels,
Fourier descriptors are developed for continuous curves. This is convenient since it leads
to a discrete set of Fourier descriptors. Additionally, we should remember that the pixels in
the image are actually the sampled points of a continuous curve in the scene. However, the
formulation leads to the definition of the integral of a continuous curve. In practice, we do
not have a continuous curve, but a sampled version. Thus, the expansion is actually
approximated by means of numerical integration.
7.2.3.1 Basis of Fourier descriptors
In the most basic form, the co-ordinates of boundary pixels are
x
and
y
point co-ordinates.
A Fourier description of these essentially gives the set of spatial frequencies that fit the
boundary points. The
first
element of the Fourier components (the d.c. component) is
simply the average value of the
x
and
y
co-ordinates,
giving the co-ordinates of the centre
point of the boundary, expressed in complex form. The
second
component essentially gives
the radius of the circle that best fits the points. Accordingly, a circle can be described by
its zero- and first-order components (the d.c. component and first harmonic). The
higher
order components increasingly describe detail, as they are associated with higher frequencies.
This is illustrated in Figure
7.6
. Here, the Fourier description of the ellipse in Figure
7.6
(a) is the frequency components in Figure
7.6
(b), depicted in logarithmic form for
purposes of display. The Fourier description has been obtained by using the ellipse boundary
points' co-ordinates. Here we can see that the low order components dominate the description,
log
(|Fcv
n
|)
n
(a) Original ellipse
(b) Fourier components
Figure 7.6
An ellipse and its Fourier description
