Nonlinear Smoothing and Multiresolution Analysis

Thismonographisintendedasasimpleintroductiontotheso-calledLULU-theory and the practical use of LULU-smoothers leading up to a full Multiresolution Analysis of any ?nite sequence. The attempt has been to present the subject in a way that is retrospectively ordered to some extent, but preserves some of the twisted paths that intuition initially suggested. It is based on the few publications in the subject, some already submitted and lectures given at conferences and on invitation at public and private institutions. The ideas originated more than ?fteen years ago when the author was - volved in practical problems at the Institute for Maritime Technology in Sim- stown and advanced while the author was at the BMI in Stellenbosch. The author hopes that publication of this book will stimulate the interest of other researches and takes responsibility for any remaining typographical or technical errors. A growing love for this particular subject was one of the prime reasons for a return to academic life, and a perceived opportunity to consolidate various p- spectives and results. It is however only the encouragement, guidance and help of severalrespected people that have resulted in the e?ort presented here.

This monograph presents a new theory for analysis, comparison and design of nonlinear smoothers, linking to established practices. Although a part of mathematical morphology, the special properties yield many simple, powerful and illuminating results leading to a novel nonlinear multiresolution analysis with pulses that may be as natural to vision as wavelet analysis is to acoustics. Similar to median transforms, they have the advantages of a supporting theory, computational simplicity, remarkable consistency, full trend preservation, and a Parceval-type identity.

Although the perspective is new and unfamiliar to most, the reader can verify all the ideas and results with simple simulations on a computer at each stage. The framework developed turns out to be a part of mathematical morphology, but the additional specific structures and properties yield a heuristic understanding that is easy to absorb for practitioners in the fields like signal- and image processing.
The book targets mathematicians, scientists and engineers with interest in concepts like trend, pulse, smoothness and resolution in sequences.