In pattern recognition fourier descriptors (FD) are used to extract meaningful characteristics of a closed curve. Imagine, for instance, you have the fourier descriptors of the contour of a maple and a chestnut leaf and you want to classify another leaf in one of those two classes (you are living in a world with only two kinds of trees). This scenario can be (easily?) solved with FD, because they are tranlation, rotation and scaling invariant.1
You take the vertices of some arbitrary polygon and transform them to complex numbers . The fourier descriptors are the normalized coefficients of DFT of the .
By applying the inverse DFT the original contour can be retrieved. It turns out that only a few low frequency components are enough to get a good approximation.
For every frame in this .gif an additional fourier descriptor is used.