Compressed sensing

Suppose x is an unknown vector in Ropfm (a digital image or signal); we plan to measure n general linear functionals of x and then reconstruct. If x is known to be compressible by transform coding with a known transform, and we reconstruct via the nonlinear procedure defined here, the number of measurements n can be dramatically smaller than the size m. Thus, certain natural classes of images with m pixels need only n=O(m1/4log5/2(m)) nonadaptive nonpixel samples for faithful recovery, as opposed to the usual m pixel samples. More specifically, suppose x has a sparse representation in some orthonormal basis (e.g., wavelet, Fourier) or tight frame (e.g., curvelet, Gabor)-so the coefficients belong to an lscrp ball for 0<ples1. The N most important coefficients in that expansion allow reconstruction with lscr2 error O(N1/2-1p/). It is possible to design n=O(Nlog(m)) nonadaptive measurements allowing reconstruction with accuracy comparable to that attainable with direct knowledge of the N most important coefficients. Moreover, a good approximation to those N important coefficients is extracted from the n measurements by solving a linear program-Basis Pursuit in signal processing. The nonadaptive measurements have the character of "random" linear combinations of basis/frame elements. Our results use the notions of optimal recovery, of n-widths, and information-based complexity. We estimate the Gel'fand n-widths of lscrp balls in high-dimensional Euclidean space in the case 0<ples1, and give a criterion identifying near- optimal subspaces for Gel'fand n-widths. We show that "most" subspaces are near-optimal, and show that convex optimization (Basis Pursuit) is a near-optimal way to extract information derived from these near-optimal subspaces

[1]  J. Peetre,et al.  Interpolation of normed abelian groups , 1972 .

[2]  Charles A. Micchelli,et al.  A Survey of Optimal Recovery , 1977 .

[3]  T. Figiel,et al.  The dimension of almost spherical sections of convex bodies , 1976 .

[4]  Henryk Wozniakowski,et al.  A general theory of optimal algorithms , 1980, ACM monograph series.

[5]  C. Micchelli,et al.  Optimal Sequential and Non-Sequential Procedures for Evaluating a Functional. , 1980 .

[6]  A. Melkman n-WIDTHS OF OCTAHEDRA , 1980 .

[7]  B. Carl Entropy numbers, s-numbers, and eigenvalue problems , 1981 .

[8]  C. Schütt Entropy numbers of diagonal operators between symmetric Banach spaces , 1984 .

[9]  A. Pinkus n-Widths in Approximation Theory , 1985 .

[10]  V. Milman,et al.  Asymptotic Theory Of Finite Dimensional Normed Spaces , 1986 .

[11]  G. Pisier The volume of convex bodies and Banach space geometry , 1989 .

[12]  M. Kon,et al.  The adaption problem for approximating linear operators , 1990 .

[13]  D. Pollard Empirical Processes: Theory and Applications , 1990 .

[14]  S. Szarek Spaces with large distance to l∞n and random matrices , 1990 .

[15]  Stanislaw J. Szarek,et al.  Condition numbers of random matrices , 1991, J. Complex..

[16]  Ingrid Daubechies,et al.  Ten Lectures on Wavelets , 1992 .

[17]  B. M. Fulk MATH , 1992 .

[18]  D. Donoho Unconditional Bases Are Optimal Bases for Data Compression and for Statistical Estimation , 1993 .

[19]  Y. Meyer Wavelets and Operators , 1993 .

[20]  D. Donoho Unconditional Bases and Bit-Level Compression , 1996 .

[21]  Erich Novak,et al.  On the Power of Adaption , 1996, J. Complex..

[22]  Axthonv G. Oettinger,et al.  IEEE Transactions on Information Theory , 1998 .

[23]  Martin Vetterli,et al.  Data Compression and Harmonic Analysis , 1998, IEEE Trans. Inf. Theory.

[24]  Michael A. Saunders,et al.  Atomic Decomposition by Basis Pursuit , 1998, SIAM J. Sci. Comput..

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

[26]  R. DeVore,et al.  Nonlinear Approximation and the Space BV(R2) , 1999 .

[27]  E. Candès,et al.  Curvelets: A Surprisingly Effective Nonadaptive Representation for Objects with Edges , 2000 .

[28]  Thomas Kühn,et al.  A Lower Estimate for Entropy Numbers , 2001, J. Approx. Theory.

[29]  M. Ledoux The concentration of measure phenomenon , 2001 .

[30]  Xiaoming Huo,et al.  Uncertainty principles and ideal atomic decomposition , 2001, IEEE Trans. Inf. Theory.

[31]  D. Donoho Sparse Components of Images and Optimal Atomic Decompositions , 2001 .

[32]  Sudipto Guha,et al.  Near-optimal sparse fourier representations via sampling , 2002, STOC '02.

[33]  Rémi Gribonval,et al.  Sparse representations in unions of bases , 2003, IEEE Trans. Inf. Theory.

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

[35]  Jean-Jacques Fuchs,et al.  On sparse representations in arbitrary redundant bases , 2004, IEEE Transactions on Information Theory.

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

[37]  J. Tropp JUST RELAX: CONVEX PROGRAMMING METHODS FOR SUBSET SELECTION AND SPARSE APPROXIMATION , 2004 .

[38]  Charles A. Micchelli,et al.  Optimal estimation of linear operators from inaccurate data: A second look , 1993, Numerical Algorithms.

[39]  D. Donoho For most large underdetermined systems of linear equations the minimal 𝓁1‐norm solution is also the sparsest solution , 2006 .

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

[41]  Michael Elad,et al.  Stable recovery of sparse overcomplete representations in the presence of noise , 2006, IEEE Transactions on Information Theory.

[42]  Joel A. Tropp,et al.  Just relax: convex programming methods for identifying sparse signals in noise , 2006, IEEE Transactions on Information Theory.

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

[44]  F. Sebert,et al.  Compressed Sensing MRI with Random B1 field , 2007 .

[45]  D. Donoho,et al.  Sparse MRI: The application of compressed sensing for rapid MR imaging , 2007, Magnetic resonance in medicine.

[46]  E.J. Candes,et al.  An Introduction To Compressive Sampling , 2008, IEEE Signal Processing Magazine.