الگوهای نوسانی در پردازش تصویر و معادلات تکاملی غیرخطی: پانزدهمین سخنرانی‌های یادبود دکتر جکولین بی. لوئیس

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. 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. 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 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.--jacket. Still Images Compression -- A First Glance At Compression And Denoising. The Supernova Sn1987a -- Atomic Decompositions And Modeling -- Wavelets And Still Image Compression: Some Success Stories -- Sampling, Quantization, Thresholding And Compression -- A First Visit To U + V Models For Still Images -- Best-basis Algorithms In Signal Processing -- The Old Jpeg -- Karhunen-loeve Expansions -- An Example Where The Karhunen-loeve Approach Is Ineffective: The Ramp Function -- A Second Visit To U + V Image Models -- The Space Bv Of Functions With Bounded Variation In The Plane -- The Osher-rudin Model -- The Mathematical Properties Of The Osher-rudin Model -- Modeling Textures -- Wavelet Shrinkage -- Littlewood-paley Analysis -- A Survey Of Wavelet Analysis -- Littlewood-paley Analysis And Wavelet Analysis -- Quantization Issues: Fourier Series Vs. Wavelet Series -- Fourier Series Vs. Wavelet Series: Expansions Of Bv Functions -- The Role Of Oscillations In Some Nonlinear Pde's -- Improved Gagliardo-nirenberg Inequalities -- Improved Poincare Estimates -- Wavelet Coefficients Of Integrable Functions -- A First Model Case: The Nonlinear Heat Equation -- The Navier-stokes Equations -- Modeling Coherent Structures -- The Nonlinear Schrodinger Equation -- Frequency Modulated Signals, Chirps And The Virgo Program -- Holder Classes With Negative Exponents -- Infinitely Oscillating Functions -- A First Definition Of N-dimensional Chirps -- A Second Definition Of Chirps -- Jaffard's Criticism -- Chirps And Two-microlocal Spaces. Yves Meyer. Includes Bibliographical References. 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