Oscillating Patterns In Image Processing And Nonlinear Evolution Equations

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, moreprecisely 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 theiruse 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 bythe 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 inmonograph 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. 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.

Abstract Biological vision is a rather fascinating domain of research. Scientists of various origins like biology, medicine, neurophysiology, engineering, math ematics, etc. aim to understand the processes leading to visual perception process and at reproducing such systems. Understanding the environment is most of the time done through visual perception which appears to be one of the most fundamental sensory abilities in humans and therefore a significant amount of research effort has been dedicated towards modelling and repro ducing human visual abilities. Mathematical methods play a central role in this endeavour. Introduction David Marr's theory v^as a pioneering step tov^ards understanding visual percep tion. In his view human vision was based on a complete surface reconstruction of the environment that was then used to address visual subtasks. This approach was proven to be insufficient by neuro-biologists and complementary ideas from statistical pattern recognition and artificial intelligence were introduced to bet ter address the visual perception problem. In this framework visual perception is represented by a set of actions and rules connecting these actions. The emerg ing concept of active vision consists of a selective visual perception paradigm that is basically equivalent to recovering from the environment the minimal piece information required to address a particular task of interest.

This book publishes a collection of original scientific research articles that address the state-of-art in using partial differential equations for image and signal processing. Coverage includes: level set methods for image segmentation and construction, denoising techniques, digital image inpainting, image dejittering, image registration, and fast numerical algorithms for solving these problems.

This book constitutes the refereed proceedings of the First International Conference on Scale Space Methods and Variational Methods in Computer Vision, SSVM 2007, emanated from the joint edition of the 4th International Workshop on Variational, Geometric and Level Set Methods in Computer Vision, VLSM 2007 and the 6th International Conference on Scale Space and PDE Methods in Computer Vision, Scale-Space 2007, held in Ischia Italy, May/June 2007.

Similar to the way in which computer vision and computer graphics act as the dual fields that connect image processing in modern computer science, the field of image processing can be considered a crucial middle road between the vision and graphics fields. Research Developments in Computer Vision and Image Processing: Methodologies and Applications brings together various research methodologies and trends in emerging areas of application of computer vision and image processing. This book is useful for students, researchers, scientists, and engineers interested in the research developments of this rapidly growing field.

The updated 2nd edition of this book presents a variety of image analysis applications, reviews their precise mathematics and shows how to discretize them. For the mathematical community, the book shows the contribution of mathematics to this domain, and highlights unsolved theoretical questions. For the computer vision community, it presents a clear, self-contained and global overview of the mathematics involved in image procesing problems. The second edition offers a review of progress in image processing applications covered by the PDE framework, and updates the existing material. The book also provides programming tools for creating simulations with minimal effort.

This book contains 71 original, scienti?c articles that address state-of-the-art researchrelatedto scale space and variationalmethods for image processing and computer vision. Topics covered in the book range from mathematical analysis of both established and new models, fast numerical methods, image analysis, segmentation, registration, surface and shape construction and processing, to real applications in medical imaging and computer vision. The ideas of scale spaceandvariationalmethodsrelatedtopartialdi?erentialequationsarecentral concepts. The papers re?ect the newest developments in these ?elds and also point to the latest literature. All the papers were submitted to the Second International Conference on Scale Space and Variational Methods in Computer Vision, which took place in Voss, Norway, during June 1–5, 2009. The papers underwent a peer review process similar to that of high-level journals in the ?eld. We thank the authors, the Scienti?c Committee, the Program Committee and the reviewers for their hard work and helpful collaboration. Their contribution has been crucial for the e?cient processing of this book, and for the success of the conference.