Oscillating Patterns in Image Processing and Nonlinear Evolution Equations: The Fifteenth Dean Jacqueline B. Lewis Memorial Lectures
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École Normale Supérieure de Cachan, Cachan, France
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.
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