Object description - Feature Extraction and Image Processing - page 259

s

( ) = s ( ) r dr - (0) (7.28)

0

Here, the parameter s takes values from zero to L (i.e. the length of the curve). Thus, the
initial and final values of the function are (0) = 0 and ( L ) = -2 , respectively. It is

important to notice that in order to obtain the final value of -2 , the curve must be traced

in a clockwise direction. Figure 7.10 illustrates the relation between the angular function
and the cumulative angular function. In the figure, z (0) defines the initial point in the curve.
The value of ( s ) is given by the angle formed by the inclination of the tangent to z (0) and

that of the tangent to the point z ( s ). If we move the point z ( s ) along the curve, this angle
will change until it reaches the value of -2 . In Equation 7.28, the cumulative angle is

obtained by adding the small angular increments for each point.

y

z(s)

(s)

z(0)

ϕ (0)

(s)

x

Figure 7.10 Angular direction

The cumulative angular function avoids the discontinuities of the angular function.
However, it still has two problems. First, it has a discontinuity at the end. Secondly, its
value depends on the length of curve analysed. These problems can be solved by defining
the normalised function γ *( t ) where

L

*( ) = t + t t (7.29)

2

Here t takes values from 0 to 2π . The factor L /2π normalises the angular function such that
it does not change when the curve is scaled. That is, when t = 2π , the function evaluates the
final point of the function γ ( s ). The term t is included to avoid discontinuities at the end of
the function (remember that the function is periodic). That is, it enforces γ *(0) = γ *(2π ) =
0. Additionally, it causes the cumulative angle for a circle to be zero. This is consistent as
a circle is generally considered the simplest curve and, intuitively, simple curves will have
simple representations.
Figure 7.11 illustrates the definitions of the cumulative angular function with two examples.
Figures 7.11 (b) to (d) define the angular functions for a circle in Figure 7.11 (a). Figures
7.11 (f) to (h) define the angular functions for the rose in Figure 7.11 (e). Figures 7.11 (b)