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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.Read more
- Establishes the theoretical connection between stochastic processes, splines and wavelets, helping users to select the right kind of basis functions for expanding signals
- Contains a self-contained exposition of generalised functions and generalised stochastic processes, geared towards practical engineering
- Includes many practical examples of signal restoration and biomedical image reconstruction
Reviews & endorsements
"Over the last twenty years, sparse representation of images and signals became a very important topic in many applications, ranging from data compression, to biological vision, to medical imaging. The book An Introduction to Sparse Stochastic Processes by Unser and Tafti is the first work to systematically build a coherent framework for non-Gaussian processes with sparse representations by wavelets. Traditional concepts such as Karhunen-Loève analysis of Gaussian processes are nicely complemented by the wavelet analysis of Levy Processes which is constructed here. The framework presented here has a classical feel while accommodating the innovative impulses driving research in sparsity. The book is extremely systematic and at the same time clear and accessible, and can be recommended both to engineers interested in foundations and to mathematicians interested in applications."
David Donoho, Stanford UniversitySee more reviews
"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
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- Date Published: October 2014
- format: Hardback
- isbn: 9781107058545
- length: 384 pages
- dimensions: 253 x 179 x 20 mm
- weight: 0.95kg
- contains: 47 b/w illus. 12 tables
- availability: In stock
Table of Contents
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
Appendix A. Singular integrals
Appendix B. Positive definiteness
Appendix C. Special functions.
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