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## Sparsity from binary hypothesis testing and application to non-parametric estimation (2008)

Venue: | in European Signal Processing Conference, EUSIPCO’08 |

Citations: | 4 - 3 self |

### Citations

3199 | A Wavelet Tour of Signal Processing. - Mallat - 1998 |

1278 | Denosing by soft thresholding
- Donoho
- 1995
(Show Context)
Citation Context ... λ. For the sparse transform, it is customary to use the DWT. Regarding the thresholding function, the soft thresholding function is a good choice for its properties of smoothness and adaptation (see =-=[5]-=-). This function is defined for every real value x by δλ(x) = { x− sgn(x)λ if |x| > λ, 0 elsewhere, (4) where sgn(x) = 1 (resp. -1) if x > 0 (resp. x < 0). Let Yi and Θi, i = 1, 2, . . . , N , be the ... |

1266 | Ideal spatial adaptation by wavelet shrinkage
- Donoho, Johnstone
- 1994
(Show Context)
Citation Context ...ting unknown signals in non-necessarily white or Gaussian noise. 1. INTRODUCTION Sparsity has gained much interest in the signal processing community since Donoho and Johnstone’s seminal work in 1994 =-=[1]-=-. The reason is the existence of transforms that are sparse in the following sense: a sparse transform makes it possible to represent many signals by coefficients that have small or null amplitudes, e... |

512 | Image denoising using scale mixtures of Gaussians in the wavelet domain
- Portilla, Strela, et al.
- 2003
(Show Context)
Citation Context ...dependent detection thresholds are proposed in section 3.3 for the estimation of signals by soft thresholding. The resulting approach has performance measurements close to those obtained with BLS-GSM =-=[4]-=-, a reference amongst the best up-to-date parametric techniques. Section 4 relates the notion of sparsity proposed in this paper to estimators of the noise standard deviation, especially for situation... |

184 |
Translation invariant denoising,"
- Coifman, Donoho
- 1995
(Show Context)
Citation Context ...section by experimental results dedicated to image denoising. We consider the Stationary Wavelet Transform (SWT), particularly suitable for denoising because it is translation invariant and redundant =-=[10, 7]-=-. The SWT is performed with the Symlet wavelet of order 8 (‘sym8’ in the Matlab Wavelet toolbox). The decomposition levels are j = 1, 2, . . . , J with J = 4. We denoise the 512× 512 ‘Lena’ image addi... |

114 |
A Wavelet Tour of Signal Processing, Second edition,
- Mallat
- 1998
(Show Context)
Citation Context ... The expression of the minimax threshold is useless in the sequel. Several authors have suggested that the universal and minimax thresholds are actually too large for many practical applications (see =-=[6, 7]-=- among others). By considering the notion of sparsity of section 2, another threshold is proposed in [2] and this threshold, called the detection threshold, yields better performance measurements than... |

84 | Wavelet shrinkage denoising using the non-negative garrote. - Gao - 1998 |

38 |
Understanding waveshrink variance and bias estimation, Biometrika
- Bruce, Gao
- 1996
(Show Context)
Citation Context ... The expression of the minimax threshold is useless in the sequel. Several authors have suggested that the universal and minimax thresholds are actually too large for many practical applications (see =-=[6, 7]-=- among others). By considering the notion of sparsity of section 2, another threshold is proposed in [2] and this threshold, called the detection threshold, yields better performance measurements than... |

22 | Neoclassical minimax problems, thresholding and adaptive function estimation.
- Donoho, Johnstone
- 1995
(Show Context)
Citation Context ...) and (O) are satisfied, the non-parametric estimation of the signal can be carried out as proposed in this paper. We plan to study such an approach in connection to results - such as those stated in =-=[16, 17]-=-, among others - about sparsity and Besov spaces. To complete this concluding prospects, note also that the soft thresholding function can successfully be replaced by the smooth shrinkage function pro... |

13 | Wavelets and the theory of non-parametric function
- Johnstone
(Show Context)
Citation Context ...nce p∗ is concerned, we consider the leat favourable case p∗ = 1/2. This model is acceptable to describe the statistical behaviour of the wavelet coefficients for smooth or piecewise regular signals (=-=[1, 8]-=-). Since the threshold of the thresholding function is aimed at distinguishing small from large coefficients, section 2 leads to choose this threshold equal to the so-called detection threshold λd(N) ... |

11 |
High-Order Wavelet Packets and Cumulant Field Analysis,”
- Leporini, Pesquet
- 1999
(Show Context)
Citation Context ...esses. More specifically, consider some signal in additive and independent strictly stationary noise. Do not assume that noise is either Gaussian or white. According to results such as those given in =-=[13]-=-, [14] and [15], the sequences of coefficients returned by the wavelet, wavelet packet and M -band wavelet packet transforms of the input noise tend to be white and Gaussian in a distributional sense ... |

9 |
Detection threshold for non-parametric estimation
- Atto, Pastor, et al.
- 2008
(Show Context)
Citation Context ...s an introduction to forthcoming work on the topic. In future work, a theoretical upper-bound on the MSE should be derived for comparison with the upper-bound established for the detection threshold (=-=[2]-=-) and the upperbounds given for the universal and minimax thresholds ([1]), in the context of soft thresholding estimation in the wavelet domain. We also would like to highlight how the contents of th... |

8 | A sharp upper bound for the probability of error of likelihood ratio test for detecting signals in white gaussian noise - Pastor, Gay, et al. - 2002 |

8 | Central Limit Theorems for Wavelet Packet Decompositions of Stationary Random Processes,”
- Atto, Pastor
- 2010
(Show Context)
Citation Context ... More specifically, consider some signal in additive and independent strictly stationary noise. Do not assume that noise is either Gaussian or white. According to results such as those given in [13], =-=[14]-=- and [15], the sequences of coefficients returned by the wavelet, wavelet packet and M -band wavelet packet transforms of the input noise tend to be white and Gaussian in a distributional sense specif... |

5 |
A theoretical result for processing signals that have unknown distributions and priors in white gaussian noise
- Pastor
- 2008
(Show Context)
Citation Context ... alternative to the MAD estimator could be derived from sparsity assumptions such as (A) and (O). Indeed, with sparsity assumptions that embrace (A) and (O), the noise standard deviation is proved in =-=[11]-=- to be the only positive real number satisfying a specific convergence criterion when the sample size and the minimum amplitude of the signals tend to infinity and the observations are independent. Th... |

5 |
Smooth Sigmoid Wavelet Shrinkage For Non-Parametric Estimation.
- Atto, Pastor, et al.
- 2008
(Show Context)
Citation Context ...thers - about sparsity and Besov spaces. To complete this concluding prospects, note also that the soft thresholding function can successfully be replaced by the smooth shrinkage function proposed in =-=[18]-=-. This shrinkage function is not only continuous but also avoids the over-smoothing and important estimation bias incurred by using the soft thresholding function [19]. In a forthcoming paper, we will... |

4 | Algorithms and applications for estimating the standard deviation of awgn when observations are not signal-free
- Pastor, Amehraye
- 2007
(Show Context)
Citation Context ...probability distributions nor the probabilities of occurrence of the alternative hypotheses. Estimators of the noise standard deviation, derived from this theoretical result, are proposed in [11] and =-=[12]-=-. They have been tested in applications different from those considered in the present paper. Therefore, the performance of these estimators should be studied and compared to that of the MAD estimator... |

1 |
distributions for wavelet packet coefficients of band-limited stationary random processes
- “Limit
(Show Context)
Citation Context ...cifically, consider some signal in additive and independent strictly stationary noise. Do not assume that noise is either Gaussian or white. According to results such as those given in [13], [14] and =-=[15]-=-, the sequences of coefficients returned by the wavelet, wavelet packet and M -band wavelet packet transforms of the input noise tend to be white and Gaussian in a distributional sense specified in [1... |