Modern imaging relies heavily on digital image manipulation. The algorithms to enhance pictures or to automatically detect items in pictures are all inherently mathematical.
Jean Baptiste Joseph Fourier (French, 1768-1830) lends his name to one of the most important techniques in image processing, Fourier Transforms, which converts images (or signals in general) into frequency representations. This alternate view of images makes it easier to analyze and manipulate the data mathematically.
The general one-dimensional Fourier ransform of a function x(t) is:
For image processing, we require a two-dimensional variation:
Fast Fourier Transforms (FFT) are versions of the general Fourier Transforms where the mathematics is broken into portions (discretized) to allow efficient computation on computers.