## ON THE MAP ESTIMATION IN THE CONTEXT OF ELLIPTICAL DISTRIBUTIONS

### BibTeX

@MISC{Zozor_onthe,

author = {S. Zozor and C. Vignat},

title = {ON THE MAP ESTIMATION IN THE CONTEXT OF ELLIPTICAL DISTRIBUTIONS},

year = {}

}

### OpenURL

### Abstract

The purpose of this paper is to study the estimation problem of a multivariate elliptically symmetric random variable corrupted by a multivariate elliptically symmetric noise. In this study, the maximum a posteriori (MAP) approach is presented, extending recent works by Alecu et al. [1] and Selesnick [2, 3]: (i) the estimation is performed in a multivariate context, (ii) the corrupting noise is not limited to be Gaussian. This paper also extends our previous work that dealt with the minimum mean square error (MMSE) approach [4]. The MMSE is briefly recalled and the MAP is derived. Then the practical use of the MAP in a general setting is discussed and compared to that of the MMSE and of the Wiener estimator. Several examples illustrate the behaviors of these estimators and exhibit their performances. 1.

### Citations

1455 |
An Introduction to Probability Theory
- Feller
- 1971
(Show Context)
Citation Context ...ting pX over x shows that, provided that integration signs can be exchanged, function fa sums to 1. Moreover, under the condition that dX is a completely monotone function, function fa is nonnegative =-=[6, 13, 14]-=- and hence is a pdf. In terms of random vector, X has then the stochastic representation of a Gaussian scale mixture (GSM) 2 X d =AN where A ∼ fa and N is a Gaussian vector independent of A, of covari... |

803 |
Fundamentals of Statistical Signal Processing: Estimation Theory
- Kay
- 1993
(Show Context)
Citation Context ...ation The well-known Minimum Mean Square Error (MMSE) estimator of X based on the observation Y, i.e. the vector ̂X that minimizes the quadratic error E[‖̂X − X‖ 2 ], is given by the conditional mean =-=[15]-=-, ̂Xmmse = E[X|Y]. We have previously shown that, when both X and Z are wide sense GSM, with mixing functions fa and fb respectively, the MMSE es2 ∼ means “distributed according to” timator writes ∫ R... |

351 | Image denoising using scale mixtures of gaussians in the wavelet domain
- Portilla, Strela, et al.
- 2003
(Show Context)
Citation Context ...ric distributions. A more recent contribution can be found for example in [7, Ch. 2]. The study of such random vectors gave rise to various applications such as radar processing [8], image processing =-=[9, 2]-=-, blind source separation [10] or parameter estimation [11] to cite recent contributions. This paper aims at extending recent results on estimation in the elliptical context. The first study is that o... |

254 |
The Laplace transform
- Widder
- 1941
(Show Context)
Citation Context ...ting pX over x shows that, provided that integration signs can be exchanged, function fa sums to 1. Moreover, under the condition that dX is a completely monotone function, function fa is nonnegative =-=[6, 13, 14]-=- and hence is a pdf. In terms of random vector, X has then the stochastic representation of a Gaussian scale mixture (GSM) 2 X d =AN where A ∼ fa and N is a Gaussian vector independent of A, of covari... |

158 | Symmetric Multivariate and Related Distributions - Fang, Kotz, et al. - 1990 |

64 |
Numerical calculation of stable densities and distribution functions
- Nolan
- 1997
(Show Context)
Citation Context ...ponds to the MAP, while the dashed line represents the MMSE and the dotted line depicts the Wiener estimator. The MMSE is numerically computed via (3), and the code available at the following address =-=[19]-=- is used to numerically compute fa when p ̸= 1 [4]. Furthermore, in the case p = 1, one can easily show that the MAP has an explicit form [1, 2, 3]: X map = ( 1 − √ d+1 σ ‖y‖ ) [0 ; 1] ( 1 − √ d+1 σ ‖... |

37 | Functions of a Complex Variable: Theory and Technique, Hod Books - Carrier, Krook, et al. - 1983 |

23 |
A representation theorem and its applications to spherically-invariant random processes
- Yao
- 1973
(Show Context)
Citation Context ... class of elliptically symmetric distributions. Among the first works on elliptically symmetric distributions in signal processing, one must cite the two almost simultaneous papers by Chu [5] and Yao =-=[6]-=-; the first one deals with the subclass of Gaussian scale mixture (GSM) distributions in the frameworks of optimal estimation, filtering and stochastic control. The second paper provides several stati... |

17 |
Non-Gaussian random vector identification using spherically invariant random processes
- Rangaswamy, Weiner, et al.
- 1993
(Show Context)
Citation Context ...of elliptically symmetric distributions. A more recent contribution can be found for example in [7, Ch. 2]. The study of such random vectors gave rise to various applications such as radar processing =-=[8]-=-, image processing [9, 2], blind source separation [10] or parameter estimation [11] to cite recent contributions. This paper aims at extending recent results on estimation in the elliptical context. ... |

12 | Estimation and decision for linear systems with elliptical random processes - Chu - 1973 |

7 |
The use of the Hankel transform in statistics: I. General theory and examples
- Lord
- 1954
(Show Context)
Citation Context ...( r2 ) dr = Γ(d/2 + 1)/π d 2 , where Γ is the Euler R+ rd+1 dX 1 More rigorously such vector are defined via the characteristic function [7]. The elliptical property is preserved by Fourier transform =-=[12, 6, 4]-=-, and in this paper we restrict to vectors that admit a pdf © EURASIP, 2009 2460Gamma function, removes this indeterminacy, implying that RX is the covariance matrix of X. Finally, note that for any ... |

5 |
The gaussian transform of distributions : definition, computation and application
- Alecu, Voloshynovskiy, et al.
- 2006
(Show Context)
Citation Context ...tically symmetric random variable corrupted by a multivariate elliptically symmetric noise. In this study, the maximum a posteriori (MAP) approach is presented, extending recent works by Alecu et al. =-=[1]-=- and Selesnick [2, 3]: (i) the estimation is performed in a multivariate context, (ii) the corrupting noise is not limited to be Gaussian. This paper also extends our previous work that dealt with the... |

5 |
Exact maximum likelihood estimates for SIRV covariance matrix: existence and algorithm analysis
- Chitour, Pascal
- 2008
(Show Context)
Citation Context ...r example in [7, Ch. 2]. The study of such random vectors gave rise to various applications such as radar processing [8], image processing [9, 2], blind source separation [10] or parameter estimation =-=[11]-=- to cite recent contributions. This paper aims at extending recent results on estimation in the elliptical context. The first study is that of Alecu et al. which dealt with what they call the Gaussian... |

3 | The estimation of laplace random vectors in additive white gaussian noise
- Selesnick
- 2008
(Show Context)
Citation Context ...om vector corrupted by an elliptically distributed random noise. This study extends: (i) the approach of [1], that deals only with scalar GSM (and with a Gaussian corrupting noise); (ii) the study of =-=[2, 3]-=- where the dimension is not restricted to 1, but where the noise is still considered Gaussian and the vectors to estimate restricted to Laplacian or radially exponential distributed. Here, the vector ... |

2 |
An elliptically contoured exponential mixture model for wavelet based image denoising
- Shi, Selesnick
- 2007
(Show Context)
Citation Context ...assume that both covariance matrices RX and RZ and pdfs pX and pZ are known. This problem generalizes [1] in the sense that dimension d is arbitrary and that Z can be nonGaussian. It also generalizes =-=[2, 3]-=- which also restricted Z to be Gaussian and X to be radially exponential or Laplacian respectively. Multiplying the observation by an adequate matrix, as done in [4], the problem can be restricted to ... |

2 |
Revisiting the denoising problem in the context of elliptical distributions
- Zozor, Vignat
- 2008
(Show Context)
Citation Context ...s performed in a multivariate context, (ii) the corrupting noise is not limited to be Gaussian. This paper also extends our previous work that dealt with the minimum mean square error (MMSE) approach =-=[4]-=-. The MMSE is briefly recalled and the MAP is derived. Then the practical use of the MAP in a general setting is discussed and compared to that of the MMSE and of the Wiener estimator. Several example... |

1 |
Blind separation of independent sources using Gaussian mixture model
- Todros, Tabrikian
- 2007
(Show Context)
Citation Context ... contribution can be found for example in [7, Ch. 2]. The study of such random vectors gave rise to various applications such as radar processing [8], image processing [9, 2], blind source separation =-=[10]-=- or parameter estimation [11] to cite recent contributions. This paper aims at extending recent results on estimation in the elliptical context. The first study is that of Alecu et al. which dealt wit... |

1 |
Theory of Multivariate Statistics. New-York
- Bilodeau, Brenner
- 1999
(Show Context)
Citation Context ...ap, ̂Xmmse and ̂Xw will be shown versus the signal-to-noise ratio SNR = E[‖X‖2 ] E[‖Z‖2 Trace(∆) = ] d = σ 2 . The first illustration deals with a vector X following an exponential power distribution =-=[17, 18]-=- X ∼ pX(x) ∝ e−( γxt x) p ( ) d+2 Γ 2 p where γ = ( d Γ dp ) , corrupted by a Gaussian random vector. A d-variate exponential power random vector is a GSM, however the mixing function is expressed via... |

1 |
On the vertical density of the multivariate exponential power distribution
- Kozubowski
- 2002
(Show Context)
Citation Context ...ap, ̂Xmmse and ̂Xw will be shown versus the signal-to-noise ratio SNR = E[‖X‖2 ] E[‖Z‖2 Trace(∆) = ] d = σ 2 . The first illustration deals with a vector X following an exponential power distribution =-=[17, 18]-=- X ∼ pX(x) ∝ e−( γxt x) p ( ) d+2 Γ 2 p where γ = ( d Γ dp ) , corrupted by a Gaussian random vector. A d-variate exponential power random vector is a GSM, however the mixing function is expressed via... |