## TWO DENOISING SURE-LET METHODS FOR COMPLEX OVERSAMPLED SUBBAND DECOMPOSITIONS

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@MISC{Gauthier_twodenoising,

author = {Jérôme Gauthier and Laurent Duval and Jean-christophe Pesquet},

title = {TWO DENOISING SURE-LET METHODS FOR COMPLEX OVERSAMPLED SUBBAND DECOMPOSITIONS},

year = {}

}

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### Abstract

Redundancy in wavelets and filter banks has the potential to greatly improve signal and image denoising. Having developed a framework for optimized oversampled complex lapped transforms, we propose their association with the statistically efficient Stein’s principle in the context of mean square error estimation. Under Gaussian noise assumptions, expectations involving the (unknown) original data are expressed using the observation only. Two forms of Stein’s Unbiased Risk Estimators, derived in the coefficient and the spatial domain respectively, are proposed, the latter being more computationally expensive. These estimators are then employed for denoising with linear combinations of elementary threshold functions. Their performances are compared to the oracle, and addressed with respect to the redundancy. They are finally tested against other denoising algorithms. They prove competitive, yielding especially good results for texture preservation. 1.

### Citations

747 | Adapting to Unknown Smoothness via Wavelet Shrinkage
- Donoho, Johnstone
- 1995
(Show Context)
Citation Context ...trated its usefulness in signal and image processing in a wavelet transformed domain, leading to Stein’s Unbiased Risk Estimator (SURE) wavelet shrinkage (SureShrink) proposed by Donoho and Johnstone =-=[3]-=-. This approach does not require a priori knowledge of the underlying signal: assumptions only lie on the noise. It additionally benefits from the wavelet concentration property, which tends to better... |

328 |
Estimation of the Mean of a Multivariate Normal Distribution
- Stein
- 1981
(Show Context)
Citation Context ... prove competitive, yielding especially good results for texture preservation. 1. INTRODUCTION Estimation of unknown data from noisy observations is a central problem in statistics. Stein’s principle =-=[1]-=- allowed for instance a remarkable improvement of the standard estimator of a multivariate normal mean by shrinking the standard estimator towards zero [2]. This principle demonstrated its usefulness ... |

151 | Bivariate shrinkage functions for wavelet-based denoising exploiting interscale dependency
- Sendur, Selesnick
- 2002
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Citation Context ... other denoising algorithms, namely Curvelets 2 (redundancy: ∼ 7.3), SureShrink CS [3] with cycle-spinning (redundancy: 3 jmax + 1 = 13 where jmax = 4 is the decomposition level), bivariate shrinkage =-=[12]-=- (BiShrink) 3 and undecimated wavelets SURE-LET: UWT SURE-LET (same redundancy: 13) [7]. We chose N = 4 and k ′ = 3 (thus the redundancy is: 9). Table 3 reports the median PSNR on 20 realizations. FBS... |

33 |
The SURE-LET approach to image denoising
- Blu, Luisier
- 2007
(Show Context)
Citation Context ... yet effective soft or hard thresholding from earlier works have been improved for instance in [5, 6]with a linear parameterization of thresholds using elementary functions. Recently, Blu and Luisier =-=[7]-=- associated redundant wavelet transforms and the SURE principle to such a linear expansion of thresholds, leading to a method minimizing the risk in the image domain instead of the transform domain. F... |

25 |
Building robust wavelet estimators for multicomponent images using Stein’s principle
- Benazza-Benyahia, Pesquet
- 2005
(Show Context)
Citation Context ...efits from the wavelet concentration property, which tends to better represent structured signal out of a noisy environment. Since then, Stein’s principle has been exploited in more involved settings =-=[4]-=-. The relatively simple yet effective soft or hard thresholding from earlier works have been improved for instance in [5, 6]with a linear parameterization of thresholds using elementary functions. Rec... |

19 | Learning to be Bayesian without supervision
- Raphan, Simoncelli
- 2007
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Citation Context ...T METHOD 5.1 Principle We now consider that the thresholding functions F and G are linear combinations of K elementary thresholding functions F (k) and G (k) weighted by a vector of parameters a ∈ RK =-=[5, 10]-=-. This combination has been called a Linear Expansion of Thresholds in [7]. The main interest of this approach is that the design of the estimators minimizing εc and εs can be achieved in a straightfo... |

15 |
A new wavelet estimator for image denoising
- Pesquet, Leporini
- 1997
(Show Context)
Citation Context ...nt. Since then, Stein’s principle has been exploited in more involved settings [4]. The relatively simple yet effective soft or hard thresholding from earlier works have been improved for instance in =-=[5, 6]-=-with a linear parameterization of thresholds using elementary functions. Recently, Blu and Luisier [7] associated redundant wavelet transforms and the SURE principle to such a linear expansion of thre... |

13 | A nonlinear Stein based estimator for multichannel image denoising
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Citation Context ... other words, it can be written: Θ(w) = (ϑ j(w j)) 1≤ j≤L ′, for all w = (w j) 1≤ j≤L ′ ∈ CL′ . For all j ∈ {1,...,L ′}, ϑ j is a scalar function from C to C. More complex operators can be considered =-=[9]-=-, but in the scope of this paper we have considered scalar operators that lead to a simple minimization process as seen in Section 5. We also define the functions θ j : R2 → C, such that: ϑ j(w j) = θ... |

6 | Optimal denoising in redundant bases
- Raphan, Simoncelli
(Show Context)
Citation Context ...nt. Since then, Stein’s principle has been exploited in more involved settings [4]. The relatively simple yet effective soft or hard thresholding from earlier works have been improved for instance in =-=[5, 6]-=-with a linear parameterization of thresholds using elementary functions. Recently, Blu and Luisier [7] associated redundant wavelet transforms and the SURE principle to such a linear expansion of thre... |

3 |
Estimation with quadratic loss,”Proc
- James, Stein
- 1961
(Show Context)
Citation Context ...ral problem in statistics. Stein’s principle [1] allowed for instance a remarkable improvement of the standard estimator of a multivariate normal mean by shrinking the standard estimator towards zero =-=[2]-=-. This principle demonstrated its usefulness in signal and image processing in a wavelet transformed domain, leading to Stein’s Unbiased Risk Estimator (SURE) wavelet shrinkage (SureShrink) proposed b... |

3 | Oversampled inverse complex lapped transform optimization
- Gauthier, Duval, et al.
(Show Context)
Citation Context ...res are especially important, it may be more effective to resort to transformations that better preserve high frequency directional details. An example of such a transform was used in a previous work =-=[8]-=-, in which we proposed a practical way to compute optimized synthesis filter banks (FBs) associated with a complex analysis FB. The contributions of this work are twofold. First, we extend the results... |

1 |
Inversion and optimization of oversampled complex filter banks: application to directional filtering of images
- Gauthier, Duval, et al.
- 2008
(Show Context)
Citation Context ... on oversampled filter banks and SURE-LET in the coefficient (as in Section 5.2) and sample (as in Section 5.3) domain, respectively. The analysis and the associated optimized synthesis FBs are as in =-=[8, 11]-=-. 7.1 FB-SURE-LET-C and FB-SURE-LET-S vs. oracle The analysis FB is parametrized here by: k = 3 (overlapping factor), k ′ = 3 (redundancy) and N = 4 (downsampling factor). Two versions of the Lena ima... |