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

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Brand new. We distribute directly for the publisher. 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. Read more Show Less

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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.

Read More Show Less

Product Details

  • ISBN-13: 9780821829202
  • Publisher: American Mathematical Society
  • Publication date: 9/4/2001
  • 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. 1 Still images compression 1
A first glance at compression and denoising. The supernova SN1987A 1
Atomic decompositions and modeling 3
Wavelets and still image compression: some success stories 8
Sampling, quantization, thresholding and compression 9
A first visit to u + v models for still images 13
Best-basis algorithms in signal processing 15
The old JPEG 17
Karhunen-Loeve expansions 18
An example where the Karhunen-Loeve approach is ineffective: the ramp function 19
A second visit to u + v image models 22
The space BV of functions with bounded variation in the plane 23
The Osher-Rudin model 27
The mathematical properties of the Osher-Rudin model 30
Modeling textures 42
Wavelet shrinkage 45
Littlewood-Paley analysis 50
A survey of wavelet analysis 57
Littlewood-Paley analysis and wavelet analysis 65
Quantization issues: Fourier series vs. wavelet series 66
Fourier series vs. wavelet series: expansions of BV functions 67
Ch. 2 The role of oscillations in some nonlinear PDE's 71
Improved Gagliardo-Nirenberg inequalities 72
Improved Poincare estimates 79
Wavelet coefficients of integrable functions 80
A first model case: the nonlinear heat equation 81
The Navier-Stokes equations 84
Modeling coherent structures 88
The nonlinear Schrodinger equation 91
Ch. 3 Frequency modulated signals, chirps and the Virgo program 93
Holder classes with negative exponents 96
Infinitely oscillating functions 101
A first definition of n-dimensional chirps 103
A second definition of chirps 104
Jaffard's criticism 105
Chirps and two-microlocal spaces 107
Wavelets and chirps 109
A function proposed by Riemann contains infinitely many chirps 111
A generalized function 114
Conclusion 117
Bibliography 119
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