Candes-Wakin-IEEE-SPM-2008-March

Candes-Wakin-IEEE-SPM-2008-March - DIGITAL VISION An...

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© DIGITAL VISION Digital Object Identifier 10.1109/MSP.2007.914731 C onventional approaches to sampling signals or images follow Shannon’s cel- ebrated theorem: the sampling rate must be at least twice the maximum fre- quency present in the signal (the so-called Nyquist rate). In fact, this principle underlies nearly all signal acquisition protocols used in consumer audio and visual electronics, medical imaging devices, radio receivers, and so on. (For some signals, such as images that are not naturally bandlimited, the sam- pling rate is dictated not by the Shannon theorem but by the desired temporal or spatial resolution. However, it is common in such systems to use an antialiasing low-pass filter to bandlimit the signal before sampling, and so the Shannon theorem plays an implicit role.) In the field of data conversion, for example, standard analog-to-digital converter (ADC) technology implements the usual quantized Shannon representation: the signal is uniformly sampled at or above the Nyquist rate. [ Emmanuel J. Candès and Michael B. Wakin ] An Introduction To Compressive Sampling [ A sensing/sampling paradigm that goes against the common knowledge in data acquisition ] 1053-5888/08/$25.00©2008IEEE IEEE SIGNAL PROCESSING MAGAZINE [ 21 ] MARCH 2008 Authorized licensed use limited to: ULAKBIM UASL - DOGU AKDENIZ UNIV. Downloaded on April 04,2010 at 14:52:22 EDT from IEEE Xplore. Restrictions apply.
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This article surveys the theory of compressive sampling, also known as compressed sensing or CS, a novel sensing/sampling paradigm that goes against the common wisdom in data acquisi- tion. CS theory asserts that one can recover certain signals and images from far fewer samples or measurements than tradition- al methods use. To make this possible, CS relies on two princi- ples: sparsity , which pertains to the signals of interest, and incoherence , which pertains to the sensing modality. Sparsity expresses the idea that the “information rate” of a continuous time signal may be much smaller than suggested by its bandwidth, or that a discrete-time signal depends on a number of degrees of freedom which is comparably much smaller than its (finite) length. More precisely, CS exploits the fact that many natural signals are sparse or compressible in the sense that they have concise rep- resentations when expressed in the proper basis 9 . Incoherence extends the duality between time and fre- quency and expresses the idea that objects having a sparse representation in 9 must be spread out in the domain in which they are acquired, just as a Dirac or a spike in the time domain is spread out in the frequency domain. Put differently, incoherence says that unlike the signal of interest, the sampling/sensing waveforms have an extremely dense representation in 9 . The crucial observation is that one can design efficient sensing or sampling protocols that capture the useful infor- mation content embedded in a sparse signal and condense it into a small amount of data. These protocols are nonadaptive and simply require correlating the signal with a small num-
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Candes-Wakin-IEEE-SPM-2008-March - DIGITAL VISION An...

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