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.
Michael Unser is Professor and Director of EPFL's Biomedical Imaging Group, Switzerland. He is a member of the Swiss Academy of Engineering Sciences, a Fellow of EURASIP, and a Fellow of the IEEE. Pouya D. Tafti is a data scientist currently residing in Germany, and a former member of the Biomedical Imaging Group at EPFL, where he conducted research on the theory and applications of probabilistic models for data.
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.
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- ID: 9781107058545
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