Proceedings of the International Conference ``Wavelets and Applications'' Toulouse, France - June 1992, Y. Meyer and S. Roques, editors (Editions Frontieres, Gif-sur-Yvette, France, 1993) 593-597

Abstract: Wavelets provide a class of methods for localized signal decomposition. This is the first step in the development of localized system theories to parallel LTI (linear time invariant) systems. Given a mother wavelet, it is needed to develop a class of operators that have the corresponding wavelets as fixed points and are linear over a given resolution. There should be criteria that one may apply for the design of optimal operators such that when applied to a signal, the result is the "closest" signal at the resolution for which the operator was designed. Then filtering or control is feasible. This paper discusses progress in this direction for discrete signals. In particular a class of nonlinear operators (the WMMR, weighted majority with minimum range operators) that have as fixed points the PICO(N) signals (signals that are piecewise constant with the smallest length of a constant segment being N). In addition, for a fixed window width of 2N+1, the WMMR filters are linear over the Haar basis, where the length of the Haar wavelet is 2N+2. For filtering and control these filters may be optimized for edge enhancement and with respect to the ranked residuals, a figure of merit that reflects local changes in signal and noise. These properties allow the design of optimal filters/operators for filtering of PICO(N) signals under certain conditions. Applications to 1-dimensional (biological) and 2-dimensional (X-ray) data will be discussed.

@inproceedings{long:haar,
author={Longbotham, H. Richardson, W. and Gokhman, D.},
title={Design of optimal linear operators for the Haar basis},
booktitle={Progress in Wavelet Analysis and Applications},
note={Proceedings of the International Conference on Wavelets and Applications,
Toulouse, France, June 1992},
editor={Yves Meyer and Sylvie Roques},
publisher={Editions Frontieres},
pages={593--597},
year=1993}