Image Processing Reference
In-Depth Information
function and the Fourier descriptors. The curves in Figures
7.14
(a) and (e) represent the
same object (the contour of an F-14 fighter), but the curve in Figure
7.14
(e) was scaled and
rotated. We can see that the angular function changes significantly, whilst the normalised
function is very similar but with a shift due to the rotation. The Fourier descriptors shown
in Figures
7.14
(d) and (h) are quite similar since they characterise the same object. We can
see a clear difference between the normalised angular function for the object presented in
Figure
7.14
(i) (the contour of a different plane, a B1 bomber). These examples show that
Fourier coefficients are indeed invariant to scale and rotation, and that they can be used to
characterise different objects.
7.2.3.6 Elliptic Fourier descriptors
The cumulative angular function transforms the two-dimensional description of a curve
into a one-dimensional periodic function suitable for Fourier analysis. In contrast
, elliptic
Fourier descriptors
maintain the description of the curve in a two-dimensional space
(Granlund, 1972). This is achieved by considering that the image space defines the complex
plane. That is, each pixel is represented by a complex number. The first co-ordinate represents
the real part whilst the second co-ordinate represents the imaginary part. Thus, a curve is
defined as
c
(
t
) =
x
(
t
) +
jy
(
t
) (7.44)
Here we will consider that the parameter
t
is given by the arc-length parameterisation.
Figure
7.15
shows an example of the complex representation of a curve. This example
y(t)
Imaginary
2T
Real
0
T
x(t)
Figure 7.15
Example of complex curve representation
