**University Lecture Series**

Volume: 22;
2001;
122 pp;
Softcover

MSC: Primary 35; 42; 65; 76;

**Print ISBN: 978-0-8218-2920-2
Product Code: ULECT/22**

List Price: $32.00

AMS Member Price: $25.60

MAA Member Price: $28.80

**Electronic ISBN: 978-1-4704-2169-4
Product Code: ULECT/22.E**

List Price: $30.00

AMS Member Price: $24.00

MAA Member Price: $27.00

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

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*Yves Meyer*

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 Poincaré 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.

#### Readership

Graduate students and researchers working in functional analysis and its applications, in particular to signal and image processing.

#### Table of Contents

# Table of Contents

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

- Cover Cover11 free
- Title v6 free
- Copyright vi7 free
- Contents vii8 free
- Preface ix10 free
- Chapter 1. Still images compression 112 free
- 1.1. Introduction 112
- 1.2. A first glance at compression and denoising. The supernova SN19S7A 112
- 1.3. Atomic decompositions and modeling 314
- 1.4. Wavelets and still image compression: some success stories 819
- 1.5. Sampling, quantization, thresholding and compression 920
- 1.6. A first visit to u + v models for still images 1324
- 1.7. Best-basis algorithms in signal processing 1526
- 1.8. The old JPEG 1728
- 1.9. Karhunen-Loeve expansions 1829
- 1.10. An example where the Karhunen-Loeve approach is ineffective: the ramp function 1930
- 1.11. A second visit to u + v image models 2233
- 1.12. The space BV of functions with bounded variation in the plane 2334
- 1.13. The Osher-Rudin model 2738
- 1.14. The mathematical properties of the Osher-Rudin model 3041
- 1.15. Modeling textures 4253
- 1.16. Wavelet shrinkage 4556
- 1.17. Littlewood-Paley analysis 5061
- 1.18. A survey of wavelet analysis 5768
- 1.19. Littlewood-Paley analysis and wavelet analysis 6576
- 1.20. Quantization issues: Fourier series vs. wavelet series 6677
- 1.21. Fourier series vs. wavelet series: expansions of BV functions 6778

- Chapter 2. The role of oscillations in some nonlinear PDE's 7182
- 2.1. Introduction 7182
- 2.2. Improved Gagliardo-Nirenberg inequalities 7283
- 2.3. Improved Poincare estimates 7990
- 2.4. Wavelet coefficients of integrable functions 8091
- 2.5. A first model case: the nonlinear heat equation 8192
- 2.6. The Navier-Stokes equations 8495
- 2.7. Modeling coherent structures 8899
- 2.8. The nonlinear Schrodinger equation 91102

- Chapter 3. Frequency modulated signals, chirps and the Virgo program 93104
- 3.1. Introduction 93104
- 3.2. Hölder classes with negative exponents 96107
- 3.3. Infinitely oscillating functions 101112
- 3.4. A first definition of n-dimensional chirps 103114
- 3.5. A second definition of chirps 104115
- 3.6. Jaffard's criticism 105116
- 3.7. Chirps and two-microlocal spaces 107118
- 3.8. Wavelets and chirps 109120
- 3.9. A function proposed by Riemann contains infinitely many chirps 111122
- 3.10. A generalized Riemann function 114125

- Conclusion 117128
- Bibliography 119130
- Back Cover Back Cover1134