Compressed sensing

Abstract

Suppose is x an unknown vector in Rm(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 = 0(m1/4 log5/2 (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 ℓp ball for 0 < p ≤ 1, 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. © 2006 IEEE.