An Introduction to Sparse Stochastic Processes
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.
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Roadmap to the book
Mathematical context and background
Continuousdomain innovation models
Operators and their inverses
Splines and wavelets
Sparse stochastic processes
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adjoint algorithm analysis B-spline basis functions Brownian motion characteristic function characterization classical compound-Poisson condition continuous continuous-domain converges convolution corresponding D(Rd decay deﬁned deﬁnition denoising denoted derivative Dirac impulses discrete domain equation equivalent Figure filter finite ﬁrst first-order formula Fourier transform Fourier-domain fractional Gaussian given Green’s function Haar wavelet id distributions impulse response infinite-dimensional infinitely divisible innovation model integral inverse iterative k∈Zd kernel Laplace Lévy density Lévy exponent Lévy noise Lévy processes linear Lp Rd MAP estimator matrix measure MMSE MMSE estimator non-Gaussian nuclear spaces null space optimal parameter Poisson polynomial positive definite Proposition random variables reconstruction representation S(Rd sampling scale scale-invariant Section self-similar sequence shift-invariant signal solution sparsity specified splines stationary stationary processes statistical stochastic processes symmetric test functions Theorem theory tion underlying vector wavelet basis wavelet coefficients wavelet-domain white noise whitening operator zero