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.

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.

Biomedical imaging is a vast and diverse field. There are a plethora of imaging devices using light, X-rays, sound waves, magnetic fields, electrons, or protons, to measure structures ranging from nano to macroscale. In many cases, computer software is needed to turn the signals collected by the hardware into a meaningful image. These computer algorithms are similarly diverse and numerous. This survey presents a wide swath of biomedical image reconstruction algorithms under a single framework. It is a coherent, yet brief survey of some six decades of research. The underpinning theory of the techniques are described and practical considerations for designing reconstruction algorithms for use in biomedical systems form the central theme of each chapter. The unifying framework deployed throughout the monograph models imaging modalities as combinations of a small set of building blocks, which identify connections between modalities Thus, the user can quickly port ideas and computer code from one to the next. Furthermore, reconstruction algorithms can treat the imaging model as a black. box, meaning that one algorithm can work for many modalities. This provides a pragmatic approach to designing effective reconstruction algorithms. This monograph is written in a tutorial style that concisely introduces students, researchers and practitioners to the development and design of effective biomedical image reconstruction algorithms.

This book is written in a tutorial style that concisely introduces students, researchers and practitioners to the development and design of effective biomedical image reconstruction algorithms.