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Unformatted text preview: Compressive Sensing Massimo Fornasier and Holger Rauhut Austrian Academy of Sciences Johann Radon Institute for Computational and Applied Mathematics (RICAM) Altenbergerstrasse 69 A-4040, Linz, Austria email@example.com Hausdorff Center for Mathematics, Institute for Numerical Simulation University of Bonn Endenicher Allee 60 D-53115 Bonn, Germany firstname.lastname@example.org April 18, 2010 Abstract Compressive sensing is a new type of sampling theory, which pre- dicts that sparse signals and images can be reconstructed from what was previously believed to be incomplete information. As a main fea- ture, efficient algorithms such as ℓ 1-minimization can be used for recov- ery. The theory has many potential applications in signal processing and imaging. This chapter gives an introduction and overview on both theoretical and numerical aspects of compressive sensing. 1 Introduction The traditional approach of reconstructing signals or images from measured data follows the well-known Shannon sampling theorem , which states that the sampling rate must be twice the highest frequency. Similarly, the fundamental theorem of linear algebra suggests that the number of collected samples (measurements) of a discrete finite-dimensional signal should be at least as large as its length (its dimension) in order to ensure reconstruction. This principle underlies most devices of current technology, such as analog to digital conversion, medical imaging or audio and video electronics. The novel theory of compressive sensing (CS) — also known under the terminology of compressed sensing, compressive sampling or sparse recovery — provides a fundamentally new approach to data acquisition which overcomes this 1 common wisdom. It predicts that certain signals or images can be recovered from what was previously believed to be highly incomplete measurements (information). This chapter gives an introduction to this new field. Both fundamental theoretical and algorithmic aspects are presented, with the awareness that it is impossible to retrace in a few pages all the current developments of this field, which was growing very rapidly in the past few years and undergoes significant advances on an almost daily basis. CS relies on the empirical observation that many types of signals or im- ages can be well-approximated by a sparse expansion in terms of a suitable basis, that is, by only a small number of non-zero coefficients. This is the key to the efficiency of many lossy compression techniques such as JPEG, MP3 etc. A compression is obtained by simply storing only the largest basis coefficients. When reconstructing the signal the non-stored coefficients are simply set to zero. This is certainly a reasonable strategy when full infor- mation of the signal is available. However, when the signal first has to be acquired by a somewhat costly, lengthy or otherwise difficult measurement (sensing) procedure, this seems to be a waste of resources: First, large efforts are spent in order to obtain full information on the signal, and afterwards...
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