An Introduction to Sparse Stochastic Processes

Michael Unser, Pouya D. Tafti

• Publisher: Cambridge University Press
• Year: 2014
• Print ISBN: 9781107058545
• eBook ISBN: 9781316056875
• Topic: Technology & Engineering
• Subtopic: Signals & Signal Processing
• Index: Technology & Engineering

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.

Table of contents

• Cover
• Half-Title
• Endorsements page
• Title page
• Copyright information
• Dedication
• Table of contents
• Preface
• Notation
• 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
• References
• Index