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Unformatted text preview: Compressed Sensing David L. Donoho Department of Statistics Stanford University September 14, 2004 Abstract Suppose x is an unknown vector in R m (depending on context, a digital image or signal); we plan to acquire data and then reconstruct. Nominally this ‘should’ require m samples. But suppose we know a priori that x is compressible by transform coding with a known transform, and we are allowed to acquire data about x by measuring n general linear functionals – rather than the usual pixels. If the collection of linear functionals is wellchosen, and we allow for a degree of reconstruction error, the size of n can be dramatically smaller than the size m usually considered necessary. Thus, certain natural classes of images with m pixels need only n = O ( m 1 / 4 log 5 / 2 ( m )) nonadaptive nonpixel samples for faithful recovery, as opposed to the usual m pixel samples. Our approach is abstract and general. We suppose that the object has a sparse rep resentation in some orthonormal basis (eg. wavelet, Fourier) or tight frame (eg curvelet, Gabor), meaning that the coefficients belong to an ` p ball for 0 < p ≤ 1. This implies that the N most important coefficients in the expansion allow a reconstruction with ` 2 error O ( N 1 / 2 1 /p ). It is possible to design n = O ( N log( m )) nonadaptive measurements which contain the information necessary to reconstruct any such object with accuracy comparable to that which would be possible if the N most important coefficients of that object were directly observable. Moreover, a good approximation to those N important coefficients may be extracted from the n measurements by solving a convenient linear program, called by the name Basis Pursuit in the signal processing literature. The nonadaptive measurements have the character of ‘random’ linear combinations of basis/frame elements. These results are developed in a theoretical framework based on the theory of optimal re covery, the theory of nwidths, and informationbased complexity. Our basic results concern properties of ` p balls in highdimensional Euclidean space in the case 0 < p ≤ 1. We estimate the Gel’fand nwidths of such balls, give a criterion for nearoptimal subspaces for Gel’fand nwidths, show that ‘most’ subspaces are nearoptimal, and show that convex optimization can be used for processing information derived from these nearoptimal subspaces. The techniques for deriving nearoptimal subspaces include the use of almostspherical sections in Banach space theory. Key Words and Phrases. Integrated Sensing and Processing. Optimal Recovery. Information Based Complexity. Gel’fand nwidths. Adaptive Sampling. Sparse Solution of Linear equations....
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This note was uploaded on 05/28/2010 for the course EE EE564 taught by Professor Runyiyu during the Spring '10 term at Eastern Mediterranean University.
 Spring '10
 RunyiYu

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