Object description - Feature Extraction and Image Processing - page 258

c i = c ( i τ ) (7.25)

Thus, the height of the rectangle for each pair of coefficients is given by c i cos( k ω i τ ) and

c i sin ( k ω i τ ). Each interval has a length τ = T / m . Thus,

T

m

T
m

ct ( ) cos( k tdt ) c cos( ω τ k i )

i

i =1

0

T

m

T
m

and ct ( ) sin( k tdt ) c sin( ω τ k i ) (7.26)

i

i =1

0

Accordingly, the Fourier coefficients are given by

m m

= 2 = 2 sin( a c cos( k ω τ i ) and b c ω τ k i ) (7.27)

i i

k

m

k

m

i =1 i =1

Here, the error due to the discrete computation will be reduced with increase in the
number of points used to approximate the curve. These equations actually correspond to a
linear approximation to the integral. This approximation is shown in Figure 7.9 (c). In this
case, the integral is given by the summation of the trapezoidal areas. The sum of these
areas leads to Equation 7.26. Notice that b 0 is zero and a 0 is twice the average of the c i
values. Thus, the first term in Equation 7.24 is the average (or centre of gravity) of the
curve.

7.2.3.5 Cumulative angular function
Fourier descriptors can be obtained by using many boundary representations. In a
straightforward approach we could consider, for example, that t and c ( t ) define the angle
and modulus of a polar parameterisation of the boundary. However, this representation is
not very general. For some curves, the polar form does not define a single valued curve,
and thus we cannot apply Fourier expansions. A more general description of curves can be
obtained by using the angular function parameterisation. This function was already defined
in Chapter 4 in the discussion about curvature.
The angular function ϕ ( s ) measures the angular direction of the tangent line as a function
of arc length. Figure 7.10 illustrates the angular direction at a point in a curve. In Cosgriff
(1960) this angular function was used to obtain a set of Fourier descriptors. However, this
first approach to Fourier characterisation has some undesirable properties. The main problem
is that the angular function has discontinuities even for smooth curves. This is because the
angular direction is bounded from zero to 2π . Thus, the function has discontinuities when
the angular direction increases to a value of more than 2π or decreases to be less than zero
(since it will change abruptly to remain within bounds). In Zahn and Roskies' approach
(Zahn, 1972), this problem is eliminated by considering a normalised form of the cumulative
angular function.
The cumulative angular function at a point in the curve is defined as the amount of
angular change from the starting point. It is called cumulative , since it represents the
summation of the angular change to each point. Angular change is given by the derivative
of the angular function ϕ ( s ). We discussed in Chapter 4 that this derivative corresponds to

the curvature κ ( s ). Thus, the cumulative angular function at the point given by s can be

defined as