Nonlinear Discrete Wavelet Transforms over Finite Sets and an Application to Binary Image Compression

Abstract

The discrete wavelet transform was originally a linear operator that works on signals that are modeled as functions from the integers into the real or complex numbers. However, many signals have discrete function values. This paper builds on two recent developments: the extension of the discrete wavelet transform to finite-valued signals and the research of nonlinear wavelet transforms triggered by the introduction of the lifting scheme by Sweldens. It defines discrete wavelet transforms as bijective, translation-invariant decompositions of signals that are functions from the integers into any finite set. Such transforms are essentially nonlinear, but they can be calculated very time efficiently since only discrete arithmetic is needed. Properties of these generalized discrete wavelet transforms are given along with an elaborate example of such a transform. In addition, the paper presents some ideas to find explicit examples of discrete wavelet transforms over finite sets. These ideas are used to show that, in case the finite set is a ring, there are much more nonlinear transforms than linear transforms. Finally, the paper exploits this increased number of transforms to do lossless compression of binary images.