An Introduction to Sparse Stochastic Processes

ISBN-13: 9781107058545 / Angielski / Twarda / 2014 / 384 str.

Michael Unser

Opis

Providing a novel approach to sparsity, this comprehensive book presents the theory of stochastic processes that are ruled by linear stochastic differential equations, and that admit a parsimonious representation in a matched wavelet-like basis. Two key themes are the statistical property of infinite divisibility, which leads to two distinct types of behaviour - Gaussian and sparse - and the structural link between linear stochastic processes and spline functions, which is exploited to simplify the mathematical analysis. The core of the book is devoted to investigating sparse processes, including a complete description of their transform-domain statistics. The final part develops practical signal-processing algorithms that are based on these models, with special emphasis on biomedical image reconstruction. This is an ideal reference for graduate students and researchers with an interest in signal/image processing, compressed sensing, approximation theory, machine learning, or statistics.

Recenzje

'This is a fascinating book that connects the classical theory of generalised functions (distributions) to the modern sparsity-based view on signal processing, as well as stochastic processes. Some of the early motivations given by I. Gelfand on the importance of generalised functions came from physics and, indeed, signal processing and sampling. However, this is probably the first book that successfully links the more abstract theory with modern signal processing. A great strength of the monograph is that it considers both the continuous and the discrete model. It will be of interest to mathematicians and engineers having appreciations of mathematical and stochastic views of signal processing.' Anders Hansen, University of Cambridge

Spis treści

1. Introduction; 2. Roadmap to the book; 3. Mathematical context and background; 4. Continuous-domain innovation models; 5. Operators and their inverses; 6. Splines and wavelets; 7. Sparse stochastic processes; 8. Sparse representations; 9. Infinite divisibility and transform-domain statistics; 10. Recovery of sparse signals; 11. Wavelet-domain methods; 12. Conclusion; Appendix A. Singular integrals; Appendix B. Positive definiteness; Appendix C. Special functions.