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{{pp-semi|small=yes}}

{{pp-semi|small=yes}}

'''Compressed sensing''' (also known as '''compressive sensing''', '''compressive sampling''', or '''sparse sampling''') is a [[signal processing]] technique for efficiently acquiring and reconstructing a [[Signal (electronics)|signal]], by finding solutions to [[Underdetermined system|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 [[Nyquist–Shannon sampling theorem]]. There are two conditions under which recovery is possible.<ref>[http://nuit-blanche.blogspot.com/2009/09/cs.html CS: Compressed Genotyping, DNA Sudoku – Harnessing high throughput sequencing for multiplexed specimen analysis].</ref> The first one is [[sparsity]], which requires the signal to be sparse in some domain. The second one is [[Coherence (signal processing)|incoherence]], which is applied through the isometric property, which is sufficient for sparse signals.<ref>{{cite journal | last1 = Donoho | first1 = David L. | year = 2006 | title =  For most large underdetermined systems of linear equations the minimal 1-norm solution is also the sparsest solution| url = https://semanticscholar.org/paper/3424286d6d39de51080ddd683646565545d015e2| journal = Communications on Pure and Applied Mathematics | volume = 59 | issue = 6| pages = 797–829 | doi = 10.1002/cpa.20132 }}</ref><ref>M. Davenport, [http://www.brainshark.com/brainshark/brainshark.net/portal/title.aspx?pid=zCdz10BfTRz0z0 "The Fundamentals of Compressive Sensing"], SigView, April 12, 2013.</ref>

'''Compressed sensing''' (also known as '''compressive sensing''', '''compressive sampling''', or '''sparse sampling''') is a [[signal processing]] technique for efficiently acquiring and reconstructing a [[Signal (electronics)|signal]], by finding solutions to [[Underdetermined system|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 [[Nyquist–Shannon sampling theorem]]. There are two conditions under which recovery is possible.<ref name=":0">[http://nuit-blanche.blogspot.com/2009/09/cs.html CS: Compressed Genotyping, DNA Sudoku – Harnessing high throughput sequencing for multiplexed specimen analysis].</ref> The first one is [[sparsity]], which requires the signal to be sparse in some domain. The second one is [[Coherence (signal processing)|incoherence]], which is applied through the isometric property, which is sufficient for sparse signals.<ref name=":1">{{cite journal | last1 = Donoho | first1 = David L. | year = 2006 | title =  For most large underdetermined systems of linear equations the minimal 1-norm solution is also the sparsest solution| url = https://semanticscholar.org/paper/3424286d6d39de51080ddd683646565545d015e2| journal = Communications on Pure and Applied Mathematics | volume = 59 | issue = 6| pages = 797–829 | doi = 10.1002/cpa.20132 }}</ref><ref name=":2">M. Davenport, [http://www.brainshark.com/brainshark/brainshark.net/portal/title.aspx?pid=zCdz10BfTRz0z0 "The Fundamentals of Compressive Sensing"], SigView, April 12, 2013.</ref>

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 Nyquist–Shannon sampling theorem. There are two conditions under which recovery is possible. 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.

压缩感知(也称为压缩传感、压缩采样或稀疏采样)是一种'''<font color="#ff8000">信号处理signal processing</font>'''技术，可通寻找'''<font color="#ff8000">欠定线性系统underdetermined linear systeme</font>'''的解决方案，来有效地获取和重构信号。这是基于以下原理：通过优化，可以利用信号的稀疏性从远少于'''<font color="#ff8000"> 奈奎斯特—香农nyquist-Shannon</font>'''采样定理所要求的样本中恢复信号。在两种情况下可以进行恢复。<ref name=":0" />第一种是'''<font color="#ff8000">稀疏性sparsity</font>'''，它要求信号在某个域上是稀疏的。第二种是'''<font color="#ff8000">不相干性incoherence</font>'''，它是通过等距特性来实现的，对于稀疏信号来说足够了。<ref name=":1" /><ref name=":2" />

 

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== Overview ==<br>

总览

总览

[[File:Underdetermined equation system.svg|200px|alt=Underdetermined linear equation system|Underdetermined linear equation system]]

[[File:Underdetermined equation system.svg|200px|Underdetermined linear equation system|链接=Special:FilePath/Underdetermined_equation_system.svg]]

Underdetermined linear equation system

Underdetermined linear equation system

解决方案/重构方法

解决方案/重构方法

[[File:Orthogonal Matching Pursuit.gif|500px|thumb|right|Example of the retrieval of an unknown signal (gray line) from few measurements (black dots) using the knowledge that the signal is sparse in the Hermite polynomials basis (purple dots show the retrieved coefficients).]]

[[File:Orthogonal Matching Pursuit.gif|500px|thumb|right|Example of the retrieval of an unknown signal (gray line) from few measurements (black dots) using the knowledge that the signal is sparse in the Hermite polynomials basis (purple dots show the retrieved coefficients).|链接=Special:FilePath/Orthogonal_Matching_Pursuit.gif]]

Example of the retrieval of an unknown signal (gray line) from few measurements (black dots) using the knowledge that the signal is sparse in the Hermite polynomials basis (purple dots show the retrieved coefficients).

Example of the retrieval of an unknown signal (gray line) from few measurements (black dots) using the knowledge that the signal is sparse in the Hermite polynomials basis (purple dots show the retrieved coefficients).

反复重新加权<math>l_{1}</math>最小化

反复重新加权<math>l_{1}</math>最小化

[[File:IRLS.png|thumb|iteratively reweighted l1 minimization method for CS]]

[[File:IRLS.png|thumb|iteratively reweighted l1 minimization method for CS|链接=Special:FilePath/IRLS.png]]

iteratively reweighted l1 minimization method for CS

iteratively reweighted l1 minimization method for CS

=====Edge-preserving total variation (TV) based compressed sensing<ref name ="EPTV">{{cite journal | last1 = Tian | first1 = Z. | last2 = Jia | first2 = X. | last3 = Yuan | first3 = K. | last4 = Pan | first4 = T. | last5 = Jiang | first5 = S. B. | year = 2011 | title = Low-dose CT reconstruction via edge preserving total variation regularization | url = | journal = Phys Med Biol | volume = 56 | issue = 18| pages = 5949–5967 | doi=10.1088/0031-9155/56/18/011| pmid = 21860076 | pmc = 4026331 | arxiv = 1009.2288 | bibcode = 2011PMB....56.5949T }}</ref>=====

=====Edge-preserving total variation (TV) based compressed sensing<ref name ="EPTV">{{cite journal | last1 = Tian | first1 = Z. | last2 = Jia | first2 = X. | last3 = Yuan | first3 = K. | last4 = Pan | first4 = T. | last5 = Jiang | first5 = S. B. | year = 2011 | title = Low-dose CT reconstruction via edge preserving total variation regularization | url = | journal = Phys Med Biol | volume = 56 | issue = 18| pages = 5949–5967 | doi=10.1088/0031-9155/56/18/011| pmid = 21860076 | pmc = 4026331 | arxiv = 1009.2288 | bibcode = 2011PMB....56.5949T }}</ref>=====

基于边缘保留的总变分（TV）的压缩感知

基于边缘保留的总变分（TV）的压缩感知

[[File:Edge preserving TV.png|thumb|Flow diagram figure for edge preserving total variation method for compressed sensing]]

[[File:Edge preserving TV.png|thumb|Flow diagram figure for edge preserving total variation method for compressed sensing|链接=Special:FilePath/Edge_preserving_TV.png]]

压缩感知的边缘保持总变分方法流程图

压缩感知的边缘保持总变分方法流程图

[[File:Augmented Lagrangian.png|thumb|right|Augmented Lagrangian method for orientation field and iterative directional field refinement models]]

[[File:Augmented Lagrangian.png|thumb|right|Augmented Lagrangian method for orientation field and iterative directional field refinement models|链接=Special:FilePath/Augmented_Lagrangian.png]]

Augmented Lagrangian method for orientation field and iterative directional field refinement models

Augmented Lagrangian method for orientation field and iterative directional field refinement models