## OPTIMAL DENOISING IN REDUNDANT BASES

Citations: | 6 - 2 self |

### BibTeX

@MISC{Raphan_optimaldenoising,

author = {Martin Raphan},

title = {OPTIMAL DENOISING IN REDUNDANT BASES},

year = {}

}

### OpenURL

### Abstract

Image denoising methods are often based on estimators chosen to minimize mean squared error (MSE) within the subbands of a multi-scale decomposition. But this does not guarantee optimal MSE performance in the image domain, unless the decomposition is orthonormal. We prove that despite this suboptimality, the expected image-domain MSE resulting from a representation that is made redundant through spatial replication of basis functions (e.g., cycle-spinning) is less than or equal to that resulting from the original nonredundant representation. We also develop an extension of Stein’s unbiased risk estimator (SURE) that allows minimization of the image-domain MSE for estimators that operate on subbands of a redundant decomposition. We implement an example, jointly optimizing the parameters of scalar estimators applied to each subband of an overcomplete representation, and demonstrate substantial MSE improvement over the suboptimal application of SURE within individual subbands.

### Citations

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(Show Context)
Citation Context ...formation. If the pointwise nonlinearity is chosen from a parametric family, Stein’s unbiased risk estimator (SURE) [1] may be used to select the estimator that minimizes the mean squared error (MSE) =-=[2]-=-. The most popular transforms are multi-scale decompositions, and within this family, empirical evidence indicates that redundant representations are more effective than orthonormal representations [3... |

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(Show Context)
Citation Context ...roup: SUREbumps, with undecimated wavelets, optimized for image-domain MSE. We used Eq. (4) to optimize the selection of soft-thresholds for orthonormal wavelet subbands, a method known as SUREshrink =-=[7]-=-. We used the same equation to optimize estimators constructed from the bumps basis, a method which we will refer to as SUREbumps (a similar method, using a different basis, was used with orthonormal ... |

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(Show Context)
Citation Context ...2]. The most popular transforms are multi-scale decompositions, and within this family, empirical evidence indicates that redundant representations are more effective than orthonormal representations =-=[3, 4, 5]-=-. This fact is somewhat mysterious since the estimators are usually optimized for MSE in the transform domain, which, for an overcomplete basis, is not the same as the MSE in the image domain. Eero P.... |

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(Show Context)
Citation Context ...ients with pointwise nonlinear functions, and then applying the inverse linear transformation. If the pointwise nonlinearity is chosen from a parametric family, Stein’s unbiased risk estimator (SURE) =-=[1]-=- may be used to select the estimator that minimizes the mean squared error (MSE) [2]. The most popular transforms are multi-scale decompositions, and within this family, empirical evidence indicates t... |

281 | Curvelets—a surprisingly effective nonadaptive representation for objects with edges
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(Show Context)
Citation Context ...s both to the noise level and to amount of data in each band. It is also likely that improvement could come from use of an oriented basis (e.g., steerable pyramid [3], complex wavelets [9], curvelets =-=[10]-=-). Finally, the image-domain SURE methodology that we have developed is relevant for any estimator that is applied to a transformed version of the data. We are currently pursuing the optimization of m... |

257 |
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(Show Context)
Citation Context ...ity of the basis both to the noise level and to amount of data in each band. It is also likely that improvement could come from use of an oriented basis (e.g., steerable pyramid [3], complex wavelets =-=[9]-=-, curvelets [10]). Finally, the image-domain SURE methodology that we have developed is relevant for any estimator that is applied to a transformed version of the data. We are currently pursuing the o... |

136 | Bivariate shrinkage functions for wavelet-based denoising exploiting interscale dependency - Sendur, Selesnick - 2002 |

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(Show Context)
Citation Context ...2]. The most popular transforms are multi-scale decompositions, and within this family, empirical evidence indicates that redundant representations are more effective than orthonormal representations =-=[3, 4, 5]-=-. This fact is somewhat mysterious since the estimators are usually optimized for MSE in the transform domain, which, for an overcomplete basis, is not the same as the MSE in the image domain. Eero P.... |

72 |
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(Show Context)
Citation Context ...2]. The most popular transforms are multi-scale decompositions, and within this family, empirical evidence indicates that redundant representations are more effective than orthonormal representations =-=[3, 4, 5]-=-. This fact is somewhat mysterious since the estimators are usually optimized for MSE in the transform domain, which, for an overcomplete basis, is not the same as the MSE in the image domain. Eero P.... |

19 | Learning to be Bayesian without supervision
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Citation Context ...races, which is (up to an additive constant) Stein’s unbiased risk estimator (SURE) [1]. This result can be generalized to nonGaussian noise, as well as a variety of non-additive corruption processes =-=[6]-=-. It is common to apply estimators to a linearly transformed version of the image, in which the statistical properties are simplified. Stein’s Lemma is readily extended to this situation. Suppose we h... |

5 | SURE-based wavelet thresholding integrating inter-scale dependencies
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(Show Context)
Citation Context ...e same equation to optimize estimators constructed from the bumps basis, a method which we will refer to as SUREbumps (a similar method, using a different basis, was used with orthonormal wavelets in =-=[8]-=-). As can be seen in Table 1, SUREbumps gives some improvement over SUREshrink in an orthonormal basis. Next, we used Eq. (4) to optimize parameters for the soft-threshold (as in [4]) and the bumps in... |