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## Compressed sensing (2004)

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### Other Repositories/Bibliography

Citations: | 3625 - 22 self |

### Citations

2728 | Atomic decomposition by basis pursuit,
- Chen, Donoho, et al.
- 1998
(Show Context)
Citation Context ...at at some level of generality, we are discussing here the idea of getting sparse solutions to underdetermined systems of equations using `1 methods, which forms part of a now-extensive body of work: =-=[6, 9, 11, 12, 13, 16, 17, 21, 22, 25, 28, 29]-=-. We expect that many of the authors of the just-cited papers will be contributing to this special issue 1.2 Questions ... Readers may want to ask numerous questions about the result (1.2), starting w... |

2632 | Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,”
- Candes, Romberg, et al.
- 2004
(Show Context)
Citation Context ...ed on: 19 September 2016 Extensions of Compressed Sensing Yaakov Tsaig David L. Donoho October 22, 2004 Abstract We study the notion of Compressed Sensing (CS) as put forward in [14] and related work =-=[20, 3, 4]-=-. The basic idea behind CS is that a signal or image, unknown but supposed to be compressible by a known transform, (eg. wavelet or Fourier), can be subjected to fewer measurements than the nominal nu... |

1671 | Matching pursuit with time-frequency dictionaries. In: - Mallat, Zhang - 1993 |

916 | Greed is good: algorithmic results for sparse approximation,”
- Tropp
- 2004
(Show Context)
Citation Context ...at at some level of generality, we are discussing here the idea of getting sparse solutions to underdetermined systems of equations using `1 methods, which forms part of a now-extensive body of work: =-=[6, 9, 11, 12, 13, 16, 17, 21, 22, 25, 28, 29]-=-. We expect that many of the authors of the just-cited papers will be contributing to this special issue 1.2 Questions ... Readers may want to ask numerous questions about the result (1.2), starting w... |

568 | For most large underdetermined systems of linear equations, the minimal `1 norm solution is also the sparsest solution,”
- Donoho
- 2006
(Show Context)
Citation Context ...at at some level of generality, we are discussing here the idea of getting sparse solutions to underdetermined systems of equations using `1 methods, which forms part of a now-extensive body of work: =-=[6, 9, 11, 12, 13, 16, 17, 21, 22, 25, 28, 29]-=-. We expect that many of the authors of the just-cited papers will be contributing to this special issue 1.2 Questions ... Readers may want to ask numerous questions about the result (1.2), starting w... |

558 |
Sparse approximate solutions to linear systems.
- Natarajan
- 1995
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336 |
Sparse representations in unions of bases,”
- Gribonval, Nielsen
- 2003
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307 | Translation-invariant de-noising,
- Coifman, Donoho
- 1995
(Show Context)
Citation Context ...though the data are not noisy in our examples). To alleviate this phenomenon, we considered the test cases shown earlier, namely Blocks and Bumps, and applied translation-invariant wavelet de-noising =-=[7]-=- to the reconstructed ‘noisy’ signals. Results are shown in panel (c) of Figure 3 and panels (b) and (d) of Figure 6. At least visually, there is a great deal of improvement. 4 Noise-Aware Reconstruct... |

253 | On sparse representations in arbitrary redundant bases.
- Fuchs
- 2004
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189 | n-Widths in Approximation Theory.
- Pinkus
- 1985
(Show Context)
Citation Context ...owed by Garnaev and Gluskin [19], implicitly considered the random signs ensemble in the dual problem of Kolmogorov n-widths. Owing to a duality relationship between Gel’fand and Kolmogorov n-widths (=-=[26]-=-), and a relationship between Gel’fand n-widths and compressed sensing [14, 27] these matrices are suitable for use in the case p = 1. • Donoho [12, 13, 14] considered the uniform Spherical ensemble. ... |

130 | Wavelab and reproducible research,”
- Buckheit, Donoho
- 1995
(Show Context)
Citation Context ...needed. This seems much smaller than the n = ck log(m) one might expect based on (1.2). As a more intuitive representation of this phenomenon, we considered the object Blocks from the Wavelab package =-=[1]-=-. As Figure 2 shows, the object is piecewise constant, and its Haar wavelet transform has relatively few nonzero coefficients. In fact, Blocks has 77 nonzero coefficients for signal length m = 2048. F... |

103 | Just relax : Convex programming methods for subset selection and sparse approximation,”
- Tropp
- 2006
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97 | Highly sparse representations from dictionaries are unique and independent of the sparseness measure,”
- Gribonval, Nielsen
- 2003
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95 |
On the widths of Euclidean balls.
- Garnaev, Gluskin
- 1984
(Show Context)
Citation Context ...d as knowing the N biggest transform coefficients. Examples were sketched for model problems caricaturing imaging and spectroscopy. In related prior work, classical literature in approximation theory =-=[23, 19, 27]-=- (developing the theory of Gel’fand n-widths) deals with closely related problems from an even more abstract 2 viewpoint; see the discussion in [14]. More recently, Gilbert et al. [20] considered n-by... |

95 | Near-optimal sparse fourier representations via sampling. In
- Gilbert, Guha, et al.
- 2002
(Show Context)
Citation Context ...ed on: 19 September 2016 Extensions of Compressed Sensing Yaakov Tsaig David L. Donoho October 22, 2004 Abstract We study the notion of Compressed Sensing (CS) as put forward in [14] and related work =-=[20, 3, 4]-=-. The basic idea behind CS is that a signal or image, unknown but supposed to be compressible by a known transform, (eg. wavelet or Fourier), can be subjected to fewer measurements than the nominal nu... |

54 |
The widths of certain finite-dimensional sets and classes of smooth functions
- Kashin
- 1977
(Show Context)
Citation Context ...d as knowing the N biggest transform coefficients. Examples were sketched for model problems caricaturing imaging and spectroscopy. In related prior work, classical literature in approximation theory =-=[23, 19, 27]-=- (developing the theory of Gel’fand n-widths) deals with closely related problems from an even more abstract 2 viewpoint; see the discussion in [14]. More recently, Gilbert et al. [20] considered n-by... |

26 | Optimally sparse representation from overcomplete dictionaries via ℓ 1 -norm minimization - D, Elad - 2003 |

8 |
For most underdetermined systems of linear equations, the minimal `1norm near-solution approximates the sparsest near-solution
- Donoho
- 2004
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7 |
Xiaoming (2001) Uncertainty Principles and Ideal Atomic Decomposition
- Donoho, Huo
- 2001
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7 | Some results on convex bodies and Banach spaces. In: - Dvoretsky - 1961 |

6 |
2002) A generalized uncertainty principle and sparse representations in pairs of bases
- Elad, Bruckstein
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3 | The dimension of almost-spherical sections of convex bodies - Figiel, Lindenstrauss, et al. - 1977 |

2 |
n-widths and Optimal Recovery in Approximation Theory
- Pinkus
- 1986
(Show Context)
Citation Context ...d as knowing the N biggest transform coefficients. Examples were sketched for model problems caricaturing imaging and spectroscopy. In related prior work, classical literature in approximation theory =-=[23, 19, 27]-=- (developing the theory of Gel’fand n-widths) deals with closely related problems from an even more abstract 2 viewpoint; see the discussion in [14]. More recently, Gilbert et al. [20] considered n-by... |

1 |
Candès and DL Donoho (2004) New tight frames of curvelets and optimal representations of objects with piecewise C2 singularities
- J
(Show Context)
Citation Context ... from 1048 compressed samples (64 coarse-scale samples, 328 compressed samples at each scale). Consider now an example working with a frame rather than an orthobasis, in this case the Curvelets frame =-=[2]-=-. Theory supporting the possible benefits of using this frame for cartoon-like images was developed in [14]. Like the wavelet basis, there is a scale parameter j which specifies the size of the curvel... |

1 |
Estimates for Fourier Minors, with Applications
- Candès, Tao
- 2004
(Show Context)
Citation Context ... transform coefficients consistent with measured data and having the smallest possible `1 norm. We perform a series of numerical experiments which validate in general terms the basic idea proposed in =-=[14, 3, 5]-=-, in the favorable case where the transform coefficients are sparse in the strong sense that the vast majority are zero. We then consider a range of less-favorable cases, in which the object has all c... |