Oscillating Patterns in Image Processing and Nonlinear Evolution Equations: The Fifteenth Dean Jacqueline B. Lewis Memorial Lectures

Oscillating Patterns in Image Processing and Nonlinear Evolution Equations: The Fifteenth Dean Jacqueline B. Lewis Memorial Lectures

by Yves Meyer
     
 

ISBN-10: 0821829203

ISBN-13: 9780821829202

Pub. Date: 09/04/2001

Publisher: American Mathematical Society

Image compression, the Navier-Stokes equations, and detection of gravitational waves are three seemingly unrelated scientific problems that, remarkably, can be studied from one perspective. The notion that unifies the three problems is that of ''oscillating patterns'', which are present in many natural images, help to explain nonlinear equations, and are pivotal in

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Overview

Image compression, the Navier-Stokes equations, and detection of gravitational waves are three seemingly unrelated scientific problems that, remarkably, can be studied from one perspective. The notion that unifies the three problems is that of ''oscillating patterns'', which are present in many natural images, help to explain nonlinear equations, and are pivotal in studying chirps and frequency-modulated signals. The first chapter of this book considers image processing, more precisely algorithms of image compression and denoising. This research is motivated in particular by the new standard for compression of still images known as JPEG-2000. The second chapter has new results on the Navier-Stokes and other nonlinear evolution equations. Frequency-modulated signals and their use in the detection of gravitational waves are covered in the final chapter. In the book, the author describes both what the oscillating patterns are and the mathematics necessary for their analysis. It turns out that this mathematics involves new properties of various Besov-type function spaces and leads to many deep results, including new generalizations of famous Gagliardo-Nirenberg and Poincare inequalities. This book is based on the ''Dean Jacqueline B. Lewis Memorial Lectures'' given by the author at Rutgers University. It can be used either as a textbook in studying applications of wavelets to image processing or as a supplementary resource for studying nonlinear evolution equations or frequency-modulated signals. Most of the material in the book did not appear previously in monograph literature.

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Product Details

ISBN-13:
9780821829202
Publisher:
American Mathematical Society
Publication date:
09/04/2001
Series:
University Lecture Series, #22
Pages:
122
Product dimensions:
6.80(w) x 9.90(h) x 0.30(d)

Table of Contents

Preface
Ch. 1Still images compression1
A first glance at compression and denoising. The supernova SN1987A1
Atomic decompositions and modeling3
Wavelets and still image compression: some success stories8
Sampling, quantization, thresholding and compression9
A first visit to u + v models for still images13
Best-basis algorithms in signal processing15
The old JPEG17
Karhunen-Loeve expansions18
An example where the Karhunen-Loeve approach is ineffective: the ramp function19
A second visit to u + v image models22
The space BV of functions with bounded variation in the plane23
The Osher-Rudin model27
The mathematical properties of the Osher-Rudin model30
Modeling textures42
Wavelet shrinkage45
Littlewood-Paley analysis50
A survey of wavelet analysis57
Littlewood-Paley analysis and wavelet analysis65
Quantization issues: Fourier series vs. wavelet series66
Fourier series vs. wavelet series: expansions of BV functions67
Ch. 2The role of oscillations in some nonlinear PDE's71
Improved Gagliardo-Nirenberg inequalities72
Improved Poincare estimates79
Wavelet coefficients of integrable functions80
A first model case: the nonlinear heat equation81
The Navier-Stokes equations84
Modeling coherent structures88
The nonlinear Schrodinger equation91
Ch. 3Frequency modulated signals, chirps and the Virgo program93
Holder classes with negative exponents96
Infinitely oscillating functions101
A first definition of n-dimensional chirps103
A second definition of chirps104
Jaffard's criticism105
Chirps and two-microlocal spaces107
Wavelets and chirps109
A function proposed by Riemann contains infinitely many chirps111
A generalized function114
Conclusion117
Bibliography119

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