Discrete wavelet transform

Summary

The discrete wavelet transform splits a sequence into a coarse part and a detail part. It uses two filters followed by downsampling by $2$, and then repeats the split on the coarse part. For filters from an orthonormal multiresolution analysis, each step is an orthogonal change of coordinates. So the transform preserves energy and is easy to invert. The outputs are the coefficients of a function in a wavelet basis. The fast wavelet transform computes them in a number of steps proportional to the length of the signal. It is the basis of wavelet compression, for example in the JPEG 2000 image standard.