Radial Frequency Contours (RFCs, or sometimes termed Fourier Descriptors (FDs)) allows for the parameterization of any closed contour. They were first described here:

• Zahn, C. T., Roskies, R. Z., March 1972. Fourier descriptors for plane closed curves. IEEE Transactions on Computers C-21 (3), 269-281.

RFCs allow you to describe any closed contour shape with a series of values which progressively define the shape, by interpreting the outline of the shape as a waveform which is then broken down into Fourier components.

To do this, one first normalizes the arc length of the shape to 2π. You choose an (arbitrary) starting point on the contour, and plot t vs. Φ, where t is ranging from 0 to 2π as you progress around the shape, and Φ is the curve direction at each point. This creates a function relating cumulative arc length to local contour orientation.

This function is then expanded into a Fourier series. Each of the terms in this expansion represent the amplitude and phase of a particular frequency. The resulting RFCs completely (in the limit) describe the shape - but is invariant over changes in position and size. By looking at only high-order vs. low-order terms, one can break the shape down into low and high frequency details.

An alternative (and often computationally easier) method is to create a function of the radius as a function of angle:

RFCs are useful for generation of parameterized shapes. Given a RFC series, one can reconstruct the original shape:

function theta = cumubend(t, FD)
% THETA = CUMUBEND(T,FD)cumulative angular bend function. 
% FD rows are frequency components 1, 2, 3...; column 1 is amplitude, 2 is phase
% returns the THETA value of the cumulative angular bend function at T
% Feb 2007  dmd, as described by Zahn et al.
 
theta = -t;
 
for freq = 1:length(FD(:,1)),
    amp = FD(freq,1);
    phase = FD(freq,2) /180*pi;
    theta = theta + amp * (cos(freq * t - phase));
end

function points = cumubend2points(FD,steps)
% POINTS = CUMUBEND2POINTS(FD,STEPS)
% FD rows are frequency components 1, 2, 3...; column 1 is amplitude, 2 is phase
% STEPS corresponds to the precision with which to calculate the shape
% returns POINTS, a vector of complex numbers representing the position of each point
 
points = [0];
thispoint = [0];
 
for t = linspace(0,2*pi,steps)
    bendangle = cumubend(t,FD);
    nextpoint = cos(bendangle) + sqrt(-1) * sin(bendangle) + thispoint;
    points = [points nextpoint];
    thispoint = nextpoint;
end

note that i use sqrt(-1) above because there's something wacky with the syntax highlighter on this wiki that breaks when I say 'i'

>> f = [0 0; 0.5 0; 0 0; 0.5 0; 0 0; 1 90];
>> plot(cumubend2points(f,2000));
>> axis off; axis equal;

• various papers using RFCs