Approximate Message Passing Algorithms For Compressed Sensing
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Approximate Message Passing Algorithms for Compressed Sensing
Compressed sensing refers to a growing body of techniques that `undersample' high-dimensional signals and yet recover them accurately. Such techniques make fewer measurements than traditional sampling theory demands: rather than sampling proportional to frequency bandwidth, they make only as many measurements as the underlying `information content' of those signals. However, as compared with traditional sampling theory, which can recover signals by applying simple linear reconstruction formulas, the task of signal recovery from reduced measurements requires nonlinear, and so far, relatively expensive reconstruction schemes. One popular class of reconstruction schemes uses linear programming (LP) methods; there is an elegant theory for such schemes promising large improvements over ordinary sampling rules in recovering sparse signals. However, solving the required LPs is substantially more expensive in applications than the linear reconstruction schemes that are now standard. In certain imaging problems, the signal to be acquired may be an image with $10^6$ pixels and the required LP would involve tens of thousands of constraints and millions of variables. Despite advances in the speed of LP, such methods are still dramatically more expensive to solve than we would like. In this thesis we focus on a class of low computational complexity algorithms known as iterative thresholding. We study them both theoretically and empirically. We will also introduce a new class of algorithms called approximate message passing or AMP. These schemes have several advantages over the classical thresholding approaches. First, they take advantage of the statistical properties of the problem to improve the convergence rate and predictability of the algorithm. Second, the nice properties of these algorithms enable us to make very accurate theoretical predictions on the asymptotic performance of LPs as well. It will be shown that more traditional techniques such as coherence and restricted isometry property are not able to make such precise predictions.
Turbo Message Passing Algorithms for Structured Signal Recovery
This book takes a comprehensive study on turbo message passing algorithms for structured signal recovery, where the considered structured signals include 1) a sparse vector/matrix (which corresponds to the compressed sensing (CS) problem), 2) a low-rank matrix (which corresponds to the affine rank minimization (ARM) problem), 3) a mixture of a sparse matrix and a low-rank matrix (which corresponds to the robust principal component analysis (RPCA) problem). The book is divided into three parts. First, the authors introduce a turbo message passing algorithm termed denoising-based Turbo-CS (D-Turbo-CS). Second, the authors introduce a turbo message passing (TMP) algorithm for solving the ARM problem. Third, the authors introduce a TMP algorithm for solving the RPCA problem which aims to recover a low-rank matrix and a sparse matrix from their compressed mixture. With this book, we wish to spur new researches on applying message passing to various inference problems. Provides an in depth look into turbo message passing algorithms for structured signal recovery Includes efficient iterative algorithmic solutions for inference, optimization, and satisfaction problems through message passing Shows applications in areas such as wireless communications and computer vision
Compressed Sensing in Radar Signal Processing
Author: Antonio De Maio
language: en
Publisher: Cambridge University Press
Release Date: 2019-10-17
Learn about the most recent theoretical and practical advances in radar signal processing using tools and techniques from compressive sensing. Providing a broad perspective that fully demonstrates the impact of these tools, the accessible and tutorial-like chapters cover topics such as clutter rejection, CFAR detection, adaptive beamforming, random arrays for radar, space-time adaptive processing, and MIMO radar. Each chapter includes coverage of theoretical principles, a detailed review of current knowledge, and discussion of key applications, and also highlights the potential benefits of using compressed sensing algorithms. A unified notation and numerous cross-references between chapters make it easy to explore different topics side by side. Written by leading experts from both academia and industry, this is the ideal text for researchers, graduate students and industry professionals working in signal processing and radar.