Compressed Sensing and Redundant Dictionaries

Rauhut, Holger; Schnass, Karin; Vandergheynst, Pierre

doi:10.1109/tit.2008.920190

Cited by 480 publications

(364 citation statements)

References 23 publications

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“…where C is a given constant, then δ k ≤ ǫ k with high probability [7,9,18]. A similar result for the same family of random matrices holds for the D-RIP [14].…”

Section: Algorithm Guaranteessupporting

confidence: 54%

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Greedy signal space methods for incoherence and beyond

Giryes

Needell

2015

Applied and Computational Harmonic Analysis

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ABSTRACT. Compressive sampling (CoSa) has provided many methods for signal recovery of signals compressible with respect to an orthonormal basis. However, modern applications have sparked the emergence of approaches for signals not sparse in an orthonormal basis but in some arbitrary, perhaps highly overcomplete, dictionary. Recently, several "signal-space" greedy methods have been proposed to address signal recovery in this setting. However, such methods inherently rely on the existence of fast and accurate projections which allow one to identify the most relevant atoms in a dictionary for any given signal, up to a very strict accuracy. When the dictionary is highly overcomplete, no such projections are currently known; the requirements on such projections do not even hold for incoherent or well-behaved dictionaries. In this work, we provide an alternate analysis for signal space greedy methods which enforce assumptions on these projections which hold in several settings including those when the dictionary is incoherent or structurally coherent. These results align more closely with traditional results in the standard CoSa literature and improve upon previous work in the signal space setting.

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“…where C is a given constant, then δ k ≤ ǫ k with high probability [7,9,18]. A similar result for the same family of random matrices holds for the D-RIP [14].…”

Section: Algorithm Guaranteessupporting

confidence: 54%

“…This result also implies perfect recovery in the absence of noise. It was extended also for incoherent redundant dictionaries [9]. An alternative aproach to approximating (2) is to use a greedy strategy.…”

Section: Algorithm 1 Signal Space Cosamp (Sscosamp)mentioning

confidence: 99%

Greedy signal space methods for incoherence and beyond

Giryes

Needell

2015

Applied and Computational Harmonic Analysis

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“…Compressed sensing (CS) aims to find a sparse solution to the problem in (1), where most of the elements of the solution vector x are zero [3,4]. Compressed sensing algorithms can be divided into two broad categories: (i) Single Measurement Vector (SMV) [5], [11] where the solution is a vector; and (ii) Multiple Measurement Vectors (MMV) [9], [10] where the solution is a two-dimensional array, or matrix.…”

Section: Compressed Sensingmentioning

confidence: 99%

“…This problem can be described mathematically as follows: given an original signal y in an n-dimensional space and a set of basis vectors, find, within a preset tolerance, a compact representation of y using the subspace spanned by the basis vectors. The development of pursuit algorithms such as orthogonal matching pursuit [5] from compressed sensing [3,4], with the capability to find a sparse representation, has offered new approaches for tackling the aforementioned problem. Using an over-complete dictionary D consisting of k basis vectors or atoms, the original signal can be decomposed by solving the following system of linear equations:…”

Section: Introductionmentioning

confidence: 99%

A New Approach to Sparse Image Representation Using MMV and K-SVD

Yang

Bouzerdoum

Phung

2009

Advanced Concepts for Intelligent Vision Systems

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A new approach to sparse image representation using MMV and K-SVD AbstractThis paper addresses the problem of image representation based on a sparse decomposition over a learned dictionary. We propose an improved matching pursuit algorithm for Multiple Measurement Vectors (MMV) and an adaptive algorithm for dictionary learning based on multi-Singular Value Decomposition (SVD), and combine them for image representation. Compared with the traditional K-SVD and orthogonal matching pursuit MMV (OMPMMV) methods, the proposed method runs faster and achieves a higher overall reconstruction accuracy. Abstract. This paper addresses the problem of image representation based on a sparse decomposition over a learned dictionary. We propose an improved matching pursuit algorithm for Multiple Measurement Vectors (MMV) and an adaptive algorithm for dictionary learning based on multi-Singular Value Decomposition (SVD), and combine them for image representation. Compared with the traditional K-SVD and orthogonal matching pursuit MMV (OMPMMV) methods, the proposed method runs faster and achieves a higher overall reconstruction accuracy.

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“…For the incomplete data situation, the mathematical technology called compressive sensing, which turned out to be quite successful in sparse signal recovery, was established several years ago by D. Donoho, see [18]. A major breakthrough was achieved when it was proven that it is possible to reconstruct a signal from very few measurements under certain conditions on the signal and the measurement model, see [8,9,10,19,20,18,24,42]. In [12] it was shown that if the sensing operator satisfies the restricted isometry property the solution can be reconstructed exactly by minimization of an 1 constrained problem, provided that the solution is sparse enough.…”

Section: Introductionmentioning

confidence: 99%

Sparsity and Compressed Sensing in Inverse Problems

Herrholz

Lorenz

Teschke

et al. 2014

Extraction of Quantifiable Information From Complex Systems

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This chapter is concerned with two important topics in the context of sparse recovery in inverse and ill-posed problems. In first part we elaborate conditions for exact recovery. In particular, we describe how both 1 -minimization and matching pursuit methods can be used to regularize ill-posed problems and moreover, state conditions which guarantee exact recovery of the support in the sparse case. The focus of the second part is on the incomplete data scenario. We discuss extensions of compressed sensing for specific infinite dimensional ill-posed measurement regimes. We are able to establish recovery error estimates when adequately relating the isometry constant of the sensing operator, the ill-posedness of the underlying model operator and the regularization parameter. Finally, we very briefly sketch how projected steepest descent iterations can be applied to retrieve the sparse solution.

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Compressed Sensing and Redundant Dictionaries

Cited by 480 publications

References 23 publications

Greedy signal space methods for incoherence and beyond

Greedy signal space methods for incoherence and beyond

A New Approach to Sparse Image Representation Using MMV and K-SVD

Sparsity and Compressed Sensing in Inverse Problems

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