CompressiveSampling

# CompressiveSampling - Compressive sampling Emamnuel J Cands Abstract Conventional wisdom and common practice in acquisition and reconstruction of

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Compressive sampling Emamnuel J. Candès Abstract. Conventional wisdom and common practice in acquisition and reconstruction of images from frequency data follow the basic principle of the Nyquist density sampling theory. This principle states that to reconstruct an image, the number of Fourier samples we need to acquire must match the desired resolution of the image, i.e. the number of pixels in the image. This paper surveys an emerging theory which goes by the name of “compressive sampling” or “compressed sensing,” and which says that this conventional wisdom is inaccurate. Perhaps surprisingly, it is possible to reconstruct images or signals of scienti±c interest accurately and sometimes even exactly from a number of samples which is far smaller than the desired resolution of the image/signal, e.g. the number of pixels in the image. It is believed that compressive sampling has far reaching implications. For example, it suggests the possibility of new data acquisition protocols that translate analog information into digital form with fewer sensors than what was considered necessary. This new sampling theory may come to underlie procedures for sampling and compressing data simultaneously. In this short survey, we provide some of the key mathematical insights underlying this new theory, and explain some of the interactions between compressive sampling and other ±elds such as statistics, information theory, coding theory, and theoretical computer science. Mathematics Subject Classifcation (2000). Primary 00A69, 41-02, 68P30; Secondary 62C65. Keywords. Compressive sampling, sparsity, uniform uncertainty principle, underdertermined systems of linear equations, ` 1 -minimization, linear programming, signal recovery, error cor- rection. 1. Introduction One of the central tenets of signal processing is the Nyquist/Shannon sampling theory: the number of samples needed to reconstruct a signal without error is dictated by its bandwidth – the length of the shortest interval which contains the support of the spectrum of the signal under study. In the last two years or so, an alternative theory of “compressive sampling” has emerged which shows that super-resolved signals and images can be reconstructed from far fewer data/measurements than what is usually considered necessary. The purpose of this paper is to survey and provide some of the key mathematical insights underlying this new theory. An enchanting aspect of compressive sampling it that it has signi±cant interactions and bearings on some ±elds in the applied sciences and engineering such as statistics, information theory, coding The author is partially supported by an NSF grant CCF–515362. Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006 © 2006 European Mathematical Society

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2 Emmanuel J. Candès theory, theoretical computer science, and others as well. We will try to explain these connections via a few selected examples.
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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.

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CompressiveSampling - Compressive sampling Emamnuel J Cands Abstract Conventional wisdom and common practice in acquisition and reconstruction of

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