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Compressed sensing
Compressed sensing (also known as compressive sensing, compressive sampling, or sparse sampling) is a signal processing technique for efficiently acquiring and reconstructing a signal, by finding solutions to underdetermined linear systems. This is based on the principle that, through optimization, the sparsity of a signal can be exploited to recover it from far fewer samples than required by the Shannon-Nyquist sampling theorem. There are two conditions under which recovery is possible.〔(CS: Compressed Genotyping, DNA Sudoku - Harnessing high throughput sequencing for multiplexed specimen analysis )〕 The first one is sparsity which requires the signal to be sparse in some domain. The second one is incoherence which is applied through the isometric property which is sufficient for sparse signals.〔For most large underdetermined systems of linear equations the minimal 𝓁1-norm solution is also the sparsest solution; See Donoho, David L, Communications on pure and applied mathematics, 59, 797 (2006) 〕〔(M. Davenport, "The Fundamentals of Compressive Sensing", SigView, April 12, 2013. )〕
== Overview ==
A common goal of the engineering field of signal processing is to reconstruct a signal from a series of sampling measurements. In general, this task is impossible because there is no way to reconstruct a signal during the times that the signal is not measured. Nevertheless, with prior knowledge or assumptions about the signal, it turns out to be possible to perfectly reconstruct a signal from a series of measurements. Over time, engineers have improved their understanding of which assumptions are practical and how they can be generalized.
An early breakthrough in signal processing was the Nyquist–Shannon sampling theorem. It states that if the signal's highest frequency is less than half of the sampling rate, then the signal can be reconstructed perfectly. The main idea is that with prior knowledge about constraints on the signal’s frequencies, fewer samples are needed to reconstruct the signal.
Around 2004, Emmanuel Candès, Terence Tao, and David Donoho proved that given knowledge about a signal's sparsity, the signal may be reconstructed with even fewer samples than the sampling theorem requires. This idea is the basis of compressed sensing.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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