Compressed Sensing with Coherent and Redundant Dictionaries

This article presents novel results concerning the recovery of signals from undersampled data in the common situation where such signals are not sparse in an orthonormal basis or incoherent dictionary, but in a truly redundant dictionary. This work thus bridges a gap in the literature and shows not only that compressed sensing is viable in this context, but also that accurate recovery is possible via an ‘1-analysis optimization problem. We introduce a condition on the measurement/sensing matrix, which is a natural generalization of the now well-known restricted isometry property, and which guarantees accurate recovery of signals that are nearly sparse in (possibly) highly overcomplete and coherent dictionaries. This condition imposes no incoherence restriction on the dictionary and our results may be the rst of this kind. We discuss practical examples and the implications of our results on those applications, and complement our study by demonstrating the potential of ‘1-analysis for such problems.

[1]  P. Dutilleux.  An Implementation of the “algorithme à trous” to Compute the Wavelet Transform , 1989 .

[2]  Zuowei Shen.  Affine systems in L 2 ( IR d ) : the analysis of the analysis operator , 1995 .

[3]  E. Candès,et al.  Near-ideal model selection by ℓ1 minimization , 2008, 0801.0345.

[4]  B. M. Fulk.  MATH , 1992 .

[5]  Bernard Chazelle,et al.  The Fast Johnson--Lindenstrauss Transform and Approximate Nearest Neighbors , 2009, SIAM J. Comput..

[6]  Stéphane Mallat,et al.  Matching pursuits with time-frequency dictionaries , 1993, IEEE Trans. Signal Process..

[7]  Abdelhak M. Zoubir,et al.  Bootstrap Methods in Signal Processing [From the Guest Editors] , 2007 .

[8]  Mike E. Davies,et al.  Iterative Hard Thresholding for Compressed Sensing , 2008, ArXiv.

[9]  A. Ron,et al.  Affine Systems inL2(Rd): The Analysis of the Analysis Operator , 1997 .

[10]  Jian-Feng Cai,et al.  Split Bregman Methods and Frame Based Image Restoration , 2009, Multiscale Model. Simul..

[11]  Michael Elad,et al.  Analysis versus synthesis in signal priors , 2006, 2006 14th European Signal Processing Conference.

[12]  S. Foucart.  A note on guaranteed sparse recovery via ℓ1-minimization , 2010 .

[13]  A. Robert Calderbank,et al.  Why Gabor Frames? Two Fundamental Measures of Coherence and their Geometric Significance , 2009, ArXiv.

[14]  Weiyu Xu,et al.  Improved sparse recovery thresholds with two-step reweighted ℓ1 minimization , 2010, 2010 IEEE International Symposium on Information Theory.

[15]  Laurent Demanet,et al.  Fast Discrete Curvelet Transforms , 2006, Multiscale Model. Simul..

[16]  J. Tropp,et al.  CoSaMP: Iterative signal recovery from incomplete and inaccurate samples , 2008, Commun. ACM.

[17]  Deanna Needell,et al.  Noisy signal recovery via iterative reweighted L1-minimization , 2009, 2009 Conference Record of the Forty-Third Asilomar Conference on Signals, Systems and Computers.

[18]  Mohamed-Jalal Fadili,et al.  The Undecimated Wavelet Decomposition and its Reconstruction , 2007, IEEE Transactions on Image Processing.

[19]  E. Candès.  The restricted isometry property and its implications for compressed sensing , 2008 .

[20]  Emmanuel J. Candès,et al.  Near-Optimal Signal Recovery From Random Projections: Universal Encoding Strategies? , 2004, IEEE Transactions on Information Theory.

[21]  Nir Ailon,et al.  An almost optimal unrestricted fast Johnson-Lindenstrauss transform , 2010, SODA '11.

[22]  S. Mendelson,et al.  Uniform Uncertainty Principle for Bernoulli and Subgaussian Ensembles , 2006, math/0608665.

[23]  D. Donoho,et al.  Redundant Multiscale Transforms and Their Application for Morphological Component Separation , 2004 .

[24]  Emmanuel J. Cand.  The Restricted Isometry Property and Its Implications for Compressed Sensing , 2008 .

[25]  R. DeVore,et al.  A Simple Proof of the Restricted Isometry Property for Random Matrices , 2008 .

[26]  Paige A. Randall.  Sparse recovery via convex optimization , 2009 .

[27]  Rachel Ward,et al.  New and Improved Johnson-Lindenstrauss Embeddings via the Restricted Isometry Property , 2010, SIAM J. Math. Anal..

[28]  S. Muthukrishnan,et al.  Approximation of functions over redundant dictionaries using coherence , 2003, SODA '03.

[29]  Emmanuel J. Candès,et al.  Decoding by linear programming , 2005, IEEE Transactions on Information Theory.

[30]  E. Candès,et al.  Stable signal recovery from incomplete and inaccurate measurements , 2005, math/0503066.

[31]  E. Candès,et al.  New tight frames of curvelets and optimal representations of objects with piecewise C2 singularities , 2004 .

[32]  Ph. Tchamitchian,et al.  Wavelets: Time-Frequency Methods and Phase Space , 1992 .

[33]  Martin Vetterli,et al.  Compressive Sampling [From the Guest Editors] , 2008, IEEE Signal Processing Magazine.

[34]  Abdelhak M. Zoubir,et al.  Bootstrap Methods in Signal Processing , 2007 .

[35]  Pierre Vandergheynst,et al.  Compressed Sensing and Redundant Dictionaries , 2007, IEEE Transactions on Information Theory.

[36]  Joel A. Tropp,et al.  Signal Recovery From Random Measurements Via Orthogonal Matching Pursuit , 2007, IEEE Transactions on Information Theory.

[37]  S. Foucart,et al.  Sparsest solutions of underdetermined linear systems via ℓq-minimization for 0 , 2009 .

[38]  Emmanuel J. Candès,et al.  Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information , 2004, IEEE Transactions on Information Theory.

[39]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[40]  T. Strohmer,et al.  Gabor Analysis and Algorithms , 2012 .

[41]  P. Laguna,et al.  Signal Processing , 2002, Yearbook of Medical Informatics.

[42]  M. Rudelson,et al.  On sparse reconstruction from Fourier and Gaussian measurements , 2008 .

[43]  S. Mallat.  A wavelet tour of signal processing , 1998 .

[44]  Joel A. Tropp,et al.  Greed is good: algorithmic results for sparse approximation , 2004, IEEE Transactions on Information Theory.

[45]  Stephen P. Boyd,et al.  Enhancing Sparsity by Reweighted ℓ1 Minimization , 2007, 0711.1612.

[46]  Aicke Hinrichs,et al.  Johnson‐Lindenstrauss lemma for circulant matrices* * , 2010, Random Struct. Algorithms.

[47]  M. Fornasier,et al.  Iterative thresholding algorithms , 2008 .

[48]  Gitta Kutyniok,et al.  Microlocal Analysis of the Geometric Separation Problem , 2010, ArXiv.