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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 well-chosen, 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 n-widths, and information-based complexity. Our basic results concern properties of ` p balls in high-dimensional Euclidean space in the case 0 < p ≤ 1. We estimate the Gel’fand n-widths of such balls, give a criterion for near-optimal subspaces for Gel’fand n-widths, show that ‘most’ subspaces are near-optimal, and show that convex optimization can be used for processing information derived from these near-optimal subspaces. The techniques for deriving near-optimal subspaces include the use of almost-spherical sections in Banach space theory. Key Words and Phrases. Integrated Sensing and Processing. Optimal Recovery. Information- Based Complexity. Gel’fand n-widths. 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