The Geometry of Compressed Sensing

Abstract

This book is aimed at presenting concepts, methods and algorithmsableto cope with undersampled and limited data. One such trend thatrecently gained popularity and to some extent revolutionised signalprocessing is compressed sensing. Compressed sensing builds uponthe observation that many signals in nature are nearly sparse (orcompressible, as they are normally referred to) in some domain, andconsequently they can be reconstructed to within high accuracy fromfar fewer observations than traditionally held to be necessary.Apart from compressed sensing this book contains other related approaches.Each methodology has its own formalities for dealing with such problems.As an example, in the Bayesian approach, sparseness promoting priorssuch as Laplace and Cauchy are normally used for penalising improbablemodel variables, thus promoting low complexity solutions. Compressedsensing techniques and homotopy-type solutions, such as the LASSO,utilise l1-norm penalties for obtaining sparse solutions using fewerobservations than conventionally needed. The book emphasizes on therole of sparsity as a machinery for promoting low complexity representationsand likewise its connections to variable selection and dimensionalityreduction in various engineering problems.This book is intended for researchers, academics and practitionerswith interest in various aspects and applications of sparse signalprocessing.