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 $(x,y)$ of some arbitrary polygon and transform them to complex numbers $(x+iy)$. The fourier descriptors are the normalized coefficients of DFT of the $(x+iy)$.

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.