An introduction to sparse stochastic processes

Michael Unser and Pouya Tafti

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Mondriaan process mapped onto the surface of an exponential-spline monopole

2D Screen Saver

Pseudo-color display of a realization of a Mondrian process

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Cover design by Annette Unser

ISBN 9781107058545

Abstract

Sparse stochastic processes are continuous-domain processes that admit a parsimonious representation in some matched wavelet-like basis. Such models are relevant for image compression, compressed sensing, and, more generally, for the derivation of statistical algorithms for solving ill-posed inverse problems.

This book introduces an extended family of sparse processes that are specified by a generic (non-Gaussian) innovation model or, equivalently, as solutions of linear stochastic differential equations driven by white Lévy noise. It presents the mathematical tools for their characterization. The two leading threads of the exposition are

  • the statistical property of infinite divisibility, which induces two distinct types of behavior—Gaussian vs. sparse—at the exclusion of any other;
  • 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 the investigation of sparse processes, including the complete description of their transform-domain statistics. The final part develops signal-processing techniques that are based on these models. This leads to a reinterpretation of popular sparsity-promoting processing schemes—such as total-variation denoising, LASSO, and wavelet shrinkage—as MAP estimators for specific types of sparse processes. It also suggests alternative Bayesian recovery procedures that minimize the estimation error. The framework is illustrated with the reconstruction of biomedical images (deconvolution microscopy, MRI, X-ray tomography) from noisy and/or incomplete data.

The book is mostly self-contained. It is targeted to an audience of graduate students and researchers with an interest in signal/image processing, compressed sensing, approximation theory, machine learning, and statistics.

Audio: Sparve vs. Gaussian

All the three signals have the same spectral contents (a-minor chord)

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