## Ideal denoising in an orthonormal basis chosen from a library of bases (1994)

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Venue: | Comptes Rendus Acad. Sci., Ser. I |

Citations: | 99 - 10 self |

### BibTeX

@ARTICLE{Donoho94idealdenoising,

author = {David L. Donoho and Iain M. Johnstone},

title = {Ideal denoising in an orthonormal basis chosen from a library of bases},

journal = {Comptes Rendus Acad. Sci., Ser. I},

year = {1994},

volume = {319},

pages = {1317--1322}

}

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### Abstract

of bases

### Citations

553 |
Ideal spatial adaptation via wavelet shrinkage. Biometrika 81
- Donoho, Johnstone
- 1994
(Show Context)
Citation Context ...or empirically selecting a basis in which to adaptively denoise attains near-ideal performance, in a precise sense. 2 Ideal De-noising in a xed Basis We summarize ideas discussed at greater length in =-=[8]-=-. Suppose we have noisy data as in (1). With the su x [B] onavector denoting the transform of the indicated vector into orthonormal basis B, wehave, by Parseval, that z[B] is a white noise and that ks... |

492 | Entropy-based algorithms for best basis selection - Coifman, Wickerhauser - 1992 |

239 | Wavelet shrinkage: asymptopia
- Donoho, Johnstone, et al.
- 1995
(Show Context)
Citation Context ...rently popular to use a xed orthonormal bases such as wavelet bases for noise removal, by the following thresholding scheme: transform into the basis, apply thresholding, return to the original basis =-=[5, 11, 10]-=-. It has been shown that the success of such a de-noising scheme is directly tied to the extent to which the orthonormal basis compresses the signal to be recovered [6]. Since a given signal may be co... |

142 | Unconditional bases are optimal bases for data compression and for statistical estimation
- Donoho
- 1993
(Show Context)
Citation Context ...n to the original basis [5, 11, 10]. It has been shown that the success of such a de-noising scheme is directly tied to the extent to which the orthonormal basis compresses the signal to be recovered =-=[6]-=-. Since a given signal may be compressed well in one basis and not in others, one naturally expects that working in a single, xed basis (Fourier, Wavelet, etc.) will impose limitations on the kind of ... |

121 | Remarques sur l’analyse de Fourier a fenêtre - Coifman, Meyer - 1991 |

103 | Nonlinear wavelet methods of recovery for signals, densities, and spectra from indirect and noisy data
- Donoho
- 1993
(Show Context)
Citation Context ...t is natural to explore the possibility of using the above adaptive basis ideas for the purpose of noise removal� Coifman and Wickerhauser have informally pursued such ideas� for formal proposals see =-=[7, 13]-=-. In this paper we showthat a certain method for empirically selecting a basis in which to adaptively denoise attains near-ideal performance, in a precise sense. 2 Ideal De-noising in a xed Basis We s... |

72 | Wavelet analysis and signal processing
- Coifman, Meyer, et al.
- 1992
(Show Context)
Citation Context ... xed basis (Fourier, Wavelet, etc.) will impose limitations on the kind of signals which canbe adequately de-noised. In a series of pathbreaking papers, Coifman, Meyer, Wickerhauser and collaborators =-=[2, 3]-=- have developed the powerful ideas of libraries of rapidly constructible orthonormal bases, libraries which arerapidly searchable for \best bases" for representing a signal. Here \best" means most com... |

68 | Simultaneous noise suppression and signal compression using a library of orthonormal bases and the minimum description length criterion
- Saito
- 1994
(Show Context)
Citation Context ...t is natural to explore the possibility of using the above adaptive basis ideas for the purpose of noise removal� Coifman and Wickerhauser have informally pursued such ideas� for formal proposals see =-=[7, 13]-=-. In this paper we showthat a certain method for empirically selecting a basis in which to adaptively denoise attains near-ideal performance, in a precise sense. 2 Ideal De-noising in a xed Basis We s... |

67 | Rates of convergence for minimum contrast estimators - BIRGÉ, MASSART - 1993 |

42 | Fast wavelet techniques for near optimal image processing
- DeVore, Lucier
- 1992
(Show Context)
Citation Context ...rently popular to use a xed orthonormal bases such as wavelet bases for noise removal, by the following thresholding scheme: transform into the basis, apply thresholding, return to the original basis =-=[5, 11, 10]-=-. It has been shown that the success of such a de-noising scheme is directly tied to the extent to which the orthonormal basis compresses the signal to be recovered [6]. Since a given signal may be co... |

21 |
Estimation d’une densité de probabilité par méthode d’ondelettes
- Johnstone, Kerkyacharian, et al.
- 1992
(Show Context)
Citation Context ...rently popular to use a xed orthonormal bases such as wavelet bases for noise removal, by the following thresholding scheme: transform into the basis, apply thresholding, return to the original basis =-=[5, 11, 10]-=-. It has been shown that the success of such a de-noising scheme is directly tied to the extent to which the orthonormal basis compresses the signal to be recovered [6]. Since a given signal may be co... |

17 | Tsybakov, The rate of convergence of nonparametric estimates of maximum likelihood type, Problemy Peredachi Informatsii 21 - Nemirovskiĭ, Polyak, et al. - 1985 |

13 |
Ideal Time-Frequency Denoising
- Donoho, Johnstone
- 1994
(Show Context)
Citation Context ...tein's Unbiased estimate of risk for soft thresholding mentioned in [7] or the Minimum Description Length entropy proposed in [13]. Details for other entropies and thresholding schemes will appear in =-=[9]-=- 2. Improved results. Aversion of our main result dealing with expectations is also possible. Also, working harder, one can get results for <8. 3. Statistical Literature. There is currently a great de... |

3 |
Remarques sur l’analyse de Fourierá fenêtre
- Coifman, Meyer
- 1991
(Show Context)
Citation Context ... xed basis (Fourier, Wavelet, etc.) will impose limitations on the kind of signals which canbe adequately de-noised. In a series of pathbreaking papers, Coifman, Meyer, Wickerhauser and collaborators =-=[2, 3]-=- have developed the powerful ideas of libraries of rapidly constructible orthonormal bases, libraries which arerapidly searchable for \best bases" for representing a signal. Here \best" means most com... |

2 | de Geer, A new approach to least-squares estimation, with applications - Van - 1988 |