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## Parameter Estimation For Multivariate Generalized Gaussian Distributions (2013)

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7506 |
Matrix Analysis
- Horn, Johnson
- 1985
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Citation Context ...f x. Throughout the paper, we will use several basic results about square matrices, especially regarding the diagonalization of real symmetric and orthogonal matrices. We invite the reader to consult =-=[48]-=- for details about these standard results. Denote as Mp(R) the set of p × p real matrices, SO(p) the set of p × p orthogonal matrices and MT the transpose of M. The identity matrix of Mp(R) will be de... |

3448 | A Theory for Multiresolution Signal Decomposition: The Wavelet Representation
- Mallat
- 1989
(Show Context)
Citation Context ...e statistical properties of various images or features extracted from these images. In particular, the distribution of wavelet or curvelet coefficients has been shown to be modeled accurately by GGDs =-=[12]-=-–[15]. This property has been exploited for many signal and image processing applications including image denoising [16]–[19], contentbased image retrieval [20], [21], image thresholding [22] or textu... |

2336 |
Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields
- Guckenheimer, Holmes
- 1983
(Show Context)
Citation Context ... of fχ are collinear if all the orbits of fχ are bounded in D. We recall here that the orbit of fχ associated with M ∈ D is the trajectory of the dynamical system S defined in (19) starting at M (See =-=[49]-=- for more details about orbits in dynamical systems). Moreover, according to [39], when a function fχ admits an FP, every orbit of fχ is bounded if the following proposition is verified. Proposition I... |

448 |
Continuous Univariate Distributions
- Johnson, Kotz, et al.
- 1995
(Show Context)
Citation Context ...rmly distributed on the unit sphere of R p , and τ is a scalar positive random variable such that τ 2β ( ) p ∼ Γ , 2 2β where Γ(a, b) is the univariate gamma distribution with parameters a and b (see =-=[43]-=- for definition). C. MGGD parameter estimation for known β Let (x1, . . . , xN) be N independent and identically distributed (i.i.d.) random vectors distributed according to an MGGD with parameters M,... |

347 | Adaptive wavelet thresholding for image denoising and compression - Chang, Yu, et al. - 2000 |

295 |
Symmetric Multivariate and Related Distributions
- FANG, KOTZ, et al.
- 1990
(Show Context)
Citation Context ...rs such as [2]–[4]. These properties include various stochastic representations, simulation methods and probabilistic characteristics. GGDs belong to the family of elliptical distributions (EDs) [5], =-=[6]-=-, originally introduced by Kelker in [7] and studied in [8], [9]. Depending on the value of the shape parameter β, multivariate GGDs (MGGDs) and the distributions of spherically invariant random vecto... |

241 | Wavelet-based texture retrieval using generalized Gaussian density and Kullback-Leibler distance
- Do, Vetterli
- 2002
(Show Context)
Citation Context ... shown to be modeled accurately by GGDs [12]–[15]. This property has been exploited for many signal and image processing applications including image denoising [16]–[19], contentbased image retrieval =-=[20]-=-, [21], image thresholding [22] or texture classification in industrial problems [23]. Other F. Pascal is with Supélec/SONDRA, 91192 Gif-sur-Yvette Cedex, France (e-mail:frederic.pascal@supelec.fr) L.... |

214 | Analysis of multiresolution image denoising schemes using generalized-gaussian and complexity priors, in
- Moulin, Liu
- 1999
(Show Context)
Citation Context ...avelet or curvelet coefficients has been shown to be modeled accurately by GGDs [12]–[15]. This property has been exploited for many signal and image processing applications including image denoising =-=[16]-=-–[19], contentbased image retrieval [20], [21], image thresholding [22] or texture classification in industrial problems [23]. Other F. Pascal is with Supélec/SONDRA, 91192 Gif-sur-Yvette Cedex, Franc... |

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

138 |
Robust M-estimators of multivariate location and scatter
- Maronna
- 1976
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Citation Context ...xed point (FP) algorithms have been derived and analyzed in [39], [40] for SIRVs. On the other hand, in the context of robust estimation, the properties of Mestimators have been studied by Maronna in =-=[41]-=-. Unfortunately, Maronna’s conditions are not fully satisfied for MGGDs (see remark II.3). This paper shows that despite the non-applicability of Maronna’s results, the MLE of MGGD parameters exists, ... |

132 | DCT-Domain Watermarking Techniques for Still Images: Detector Performance Analysis and a New Structure”in
- Hernández, Amado, et al.
- 2000
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Citation Context ...n fi defined by fi(y) = y β j Np y + ciy1−βIR +(y) (10) 2 We note here that most values of β encountered in practical applications belong to the interval (0, 1). For instance, β = 0.8 is suggested in =-=[44]-=- as a good choice for most images.PASCAL et al.: PARAMETER ESTIMATION FOR MULTIVARIATE GENERALIZED GAUSSIAN DISTRIBUTIONS 3 hal-00879851, version 1 - 5 Nov 2013 where ci is a positive constant indepe... |

94 |
Distribution theory of spherical distributions and a locationscale parameter generalization,” Sankhyā: The Indian
- Kelker
- 1970
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Citation Context ...lude various stochastic representations, simulation methods and probabilistic characteristics. GGDs belong to the family of elliptical distributions (EDs) [5], [6], originally introduced by Kelker in =-=[7]-=- and studied in [8], [9]. Depending on the value of the shape parameter β, multivariate GGDs (MGGDs) and the distributions of spherically invariant random vectors share common properties (see [10], [1... |

86 |
Generalized multivariate analysis
- FANG, ZHANG
- 1990
(Show Context)
Citation Context ... papers such as [2]–[4]. These properties include various stochastic representations, simulation methods and probabilistic characteristics. GGDs belong to the family of elliptical distributions (EDs) =-=[5]-=-, [6], originally introduced by Kelker in [7] and studied in [8], [9]. Depending on the value of the shape parameter β, multivariate GGDs (MGGDs) and the distributions of spherically invariant random ... |

78 | Covariance structure maximum-likelihood estimates in compound gaussian noise: Existence and algorithm analysis
- Pascal, Chitour, et al.
- 1991
(Show Context)
Citation Context ...sed in [36]–[38]. Several works have analyzed covariance matrix estimators defined under different modeling assumptions. On the one hand, fixed point (FP) algorithms have been derived and analyzed in =-=[39]-=-, [40] for SIRVs. On the other hand, in the context of robust estimation, the properties of Mestimators have been studied by Maronna in [41]. Unfortunately, Maronna’s conditions are not fully satisfie... |

67 |
Covariance matrix estimation for CFAR detection in correlated heavy tailed clutter
- Gini, Greco
- 2002
(Show Context)
Citation Context ...based on (x1, . . . , xN) for a known (4) value of β ∈ (0, 1) 2 . The MGGD is a particular case of elliptical distribution that has received much attention in the literature. Following the results of =-=[45]-=- for real elliptical distributions, by differentiating the log-likelihood of vectors (x1, . . . , xN ) with respect to M, the MLE of the matrix M satisfies the following FP equation M = 2 N N∑ i=1 −gm... |

56 |
Performance analysis of covariance matrix estimates in impulsive noise
- Pascal, Forster, et al.
- 2008
(Show Context)
Citation Context ...tor and (9) is due to the estimation of the scale parameter that equals 1 for the multivariate Gaussian distribution). For β = 0, Eq. (9) reduces to the FP covariance matrix estimator studied in [45]–=-=[47]-=-. Remark II.2 Equation (9) remains unchanged if M is replaced by α M where α is any non-zero real factor. Thus, the solutions of (9) (when there exist) can be determined up to a scale factor α. The no... |

53 |
On the law of frequency of errors
- Subbotin
- 1923
(Show Context)
Citation Context ...I. INTRODUCTION UNIVARIATE and multivariate generalized Gaussian distributions (GGDs) have received much attention in the literature. Historically, this family of distributions has been introduced in =-=[1]-=-. Some properties of these distributions have been reported in several papers such as [2]–[4]. These properties include various stochastic representations, simulation methods and probabilistic charact... |

46 |
Parametric generalized Gaussian density estimation
- Varanasi, Aazhang
- 1989
(Show Context)
Citation Context ...e distributions is clearly an interesting issue. Classical estimation methods that have been investigated for univariate GGDs include the maximum likelihood (ML) method [33] and the method of moments =-=[34]-=-. In the multivariate context, MGGD parameters can be estimated by a least-squares method as in [18] or by minimizing a χ 2 distance between the histogram of the observed data and the theoretical prob... |

31 |
Non-Gaussian random vector identification using spherically invariant random processes
- Rangaswamy, Weiner, et al.
- 1993
(Show Context)
Citation Context ...stic representations, simulation methods and probabilistic characteristics. GGDs belong to the family of elliptical distributions (EDs) [5], [6], originally introduced by Kelker in [7] and studied in =-=[8]-=-, [9]. Depending on the value of the shape parameter β, multivariate GGDs (MGGDs) and the distributions of spherically invariant random vectors share common properties (see [10], [11] for more details... |

28 |
A multivariate generalization of the power exponential family of distributions
- Gómez-Villegas, A, et al.
- 1998
(Show Context)
Citation Context ...e received much attention in the literature. Historically, this family of distributions has been introduced in [1]. Some properties of these distributions have been reported in several papers such as =-=[2]-=-–[4]. These properties include various stochastic representations, simulation methods and probabilistic characteristics. GGDs belong to the family of elliptical distributions (EDs) [5], [6], originall... |

24 |
Multivariate statistical modeling for image denoising using wavelet transforms
- Cho, Bui
- 2005
(Show Context)
Citation Context ...gated for univariate GGDs include the maximum likelihood (ML) method [33] and the method of moments [34]. In the multivariate context, MGGD parameters can be estimated by a least-squares method as in =-=[18]-=- or by minimizing a χ 2 distance between the histogram of the observed data and the theoretical probabilities associated with the MGGD [35]. Estimators based on the method of moments and on the ML met... |

23 |
Complex Elliptically Symmetric Distributions: Survey, New Results and Applications
- Ollila, Tyler, et al.
- 2012
(Show Context)
Citation Context ...7] and studied in [8], [9]. Depending on the value of the shape parameter β, multivariate GGDs (MGGDs) and the distributions of spherically invariant random vectors share common properties (see [10], =-=[11]-=- for more details). MGGDs have been used intensively in the image processing community. Indeed, including Gaussian and Laplacian distributions as special cases, MGGDs are potentially interesting for m... |

20 | Image thresholding based on the EM algorithm and the generalized Gaussian distribution, Pattern Recognition 40 (2
- Bazi, Bruzzone, et al.
- 2007
(Show Context)
Citation Context ... by GGDs [12]–[15]. This property has been exploited for many signal and image processing applications including image denoising [16]–[19], contentbased image retrieval [20], [21], image thresholding =-=[22]-=- or texture classification in industrial problems [23]. Other F. Pascal is with Supélec/SONDRA, 91192 Gif-sur-Yvette Cedex, France (e-mail:frederic.pascal@supelec.fr) L. Bombrun and Y. Berthoumieu are... |

17 |
Maximum likelihood estimation for the exponential power function parameters
- Agrò
- 1995
(Show Context)
Citation Context ...timating the parameters of these distributions is clearly an interesting issue. Classical estimation methods that have been investigated for univariate GGDs include the maximum likelihood (ML) method =-=[33]-=- and the method of moments [34]. In the multivariate context, MGGD parameters can be estimated by a least-squares method as in [18] or by minimizing a χ 2 distance between the histogram of the observe... |

15 |
A globally convergent and consistent method for estimating the shape parameter of a generalized Gaussian distribution
- Song
- 2006
(Show Context)
Citation Context ...ceived much attention in the literature. Historically, this family of distributions has been introduced in [1]. Some properties of these distributions have been reported in several papers such as [2]–=-=[4]-=-. These properties include various stochastic representations, simulation methods and probabilistic characteristics. GGDs belong to the family of elliptical distributions (EDs) [5], [6], originally in... |

15 |
Robust gaussian and non-gaussian matched subspace detection
- Desai, Mangoubi
- 2003
(Show Context)
Citation Context ...sbordeaux.fr; yannick.berthoumieu@ims-bordeaux.fr) J.-Y. Tourneret is with Université de Toulouse, IRIT/INP-ENSEEIHT, (email:jean-yves.tourneret@enseeiht.fr) applications involving GGDs include radar =-=[24]-=-, video coding and denoising [25]–[27] or biomedical signal processing [26], [28], [29]. Finally, it is interesting to note that complex GGDs have been recently studied in [30], [31] and that multivar... |

13 |
Multivariate statistical modelling of images with the curvelet transform
- Boubchir, Fadili
- 2005
(Show Context)
Citation Context ...tistical properties of various images or features extracted from these images. In particular, the distribution of wavelet or curvelet coefficients has been shown to be modeled accurately by GGDs [12]–=-=[15]-=-. This property has been exploited for many signal and image processing applications including image denoising [16]–[19], contentbased image retrieval [20], [21], image thresholding [22] or texture cl... |

12 |
A complex generalized gaussian distribution; characterization, generation, and estimation
- Novey, Adali, et al.
- 2010
(Show Context)
Citation Context ...ving GGDs include radar [24], video coding and denoising [25]–[27] or biomedical signal processing [26], [28], [29]. Finally, it is interesting to note that complex GGDs have been recently studied in =-=[30]-=-, [31] and that multivariate regression models with generalized Gaussian errors have been considered in [32]. Considering the important attention devoted to GGDs, estimating the parameters of these di... |

12 | Exact Maximum Likelihood Estimates for SIRV Covariance Matrix: Existence and Algorithm Analysis - Chitour, Pascal - 2008 |

12 |
An algorithm to compute averages on matrix lie groups,” Signal Processing
- Fiori, Tanaka
- 2009
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Citation Context ... in [47] for assessing the performance of matrix estimators. Note also that other approaches based on computing the mean on the manifold of positive definite matrices could also be investigated [50], =-=[51]-=-. The estimated consistency of Â is verified by computing ||Â − A||. As observed, the estimation performance is the same when a normalization constraint for the scatter matrix is imposed or not. Fig. ... |

11 | Unified framework to regularized covariance estimation in scaled gaussian models - Wiesel - 2012 |

11 |
A new Metric on the manifold of kernel matrices with application to matrix geometric means, NIPS conference
- Sra
- 2012
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Citation Context ...s used in [47] for assessing the performance of matrix estimators. Note also that other approaches based on computing the mean on the manifold of positive definite matrices could also be investigated =-=[50]-=-, [51]. The estimated consistency of Â is verified by computing ||Â − A||. As observed, the estimation performance is the same when a normalization constraint for the scatter matrix is imposed or not.... |

10 |
Multiscale colour texture retrieval using the geodesic distance between multivariate generalized Gaussian models
- Verdoolaege, Backer, et al.
- 2008
(Show Context)
Citation Context ... to be modeled accurately by GGDs [12]–[15]. This property has been exploited for many signal and image processing applications including image denoising [16]–[19], contentbased image retrieval [20], =-=[21]-=-, image thresholding [22] or texture classification in industrial problems [23]. Other F. Pascal is with Supélec/SONDRA, 91192 Gif-sur-Yvette Cedex, France (e-mail:frederic.pascal@supelec.fr) L. Bombr... |

9 | Multivariate Generalized Gaussian Distribution: Convexity and Graphical Models
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(Show Context)
Citation Context ...ween the histogram of the observed data and the theoretical probabilities associated with the MGGD [35]. Estimators based on the method of moments and on the ML method have also been proposed in [36]–=-=[38]-=-. Several works have analyzed covariance matrix estimators defined under different modeling assumptions. On the one hand, fixed point (FP) algorithms have been derived and analyzed in [39], [40] for S... |

8 |
Image denoising based on wavelet shrinkage using neighbor and level dependency. Int
- Cho, Bui, et al.
(Show Context)
Citation Context ...t or curvelet coefficients has been shown to be modeled accurately by GGDs [12]–[15]. This property has been exploited for many signal and image processing applications including image denoising [16]–=-=[19]-=-, contentbased image retrieval [20], [21], image thresholding [22] or texture classification in industrial problems [23]. Other F. Pascal is with Supélec/SONDRA, 91192 Gif-sur-Yvette Cedex, France (e-... |

7 |
Adaptive subband video coding using bivariate generalized Gaussian distribution model
- Coban, Mersereau
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Citation Context ...@ims-bordeaux.fr) J.-Y. Tourneret is with Université de Toulouse, IRIT/INP-ENSEEIHT, (email:jean-yves.tourneret@enseeiht.fr) applications involving GGDs include radar [24], video coding and denoising =-=[25]-=-–[27] or biomedical signal processing [26], [28], [29]. Finally, it is interesting to note that complex GGDs have been recently studied in [30], [31] and that multivariate regression models with gener... |

6 | adaptive wavelet thresholding with context modeling for image denoising - “Spatially - 2000 |

6 |
Wheezing sounds detection using multivariate generalized gaussian distributions
- Cam, Belghith, et al.
- 2009
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Citation Context ...e Toulouse, IRIT/INP-ENSEEIHT, (email:jean-yves.tourneret@enseeiht.fr) applications involving GGDs include radar [24], video coding and denoising [25]–[27] or biomedical signal processing [26], [28], =-=[29]-=-. Finally, it is interesting to note that complex GGDs have been recently studied in [30], [31] and that multivariate regression models with generalized Gaussian errors have been considered in [32]. C... |

5 |
Image and video denoising using adaptative dual-tree discrete wavelet packets
- Yang, Wang, et al.
- 2009
(Show Context)
Citation Context ...bordeaux.fr) J.-Y. Tourneret is with Université de Toulouse, IRIT/INP-ENSEEIHT, (email:jean-yves.tourneret@enseeiht.fr) applications involving GGDs include radar [24], video coding and denoising [25]–=-=[27]-=- or biomedical signal processing [26], [28], [29]. Finally, it is interesting to note that complex GGDs have been recently studied in [30], [31] and that multivariate regression models with generalize... |

3 |
A wavelet-based approach for analyzing industrial stochastic textures with applications
- Scharcanski
- 2007
(Show Context)
Citation Context ...or many signal and image processing applications including image denoising [16]–[19], contentbased image retrieval [20], [21], image thresholding [22] or texture classification in industrial problems =-=[23]-=-. Other F. Pascal is with Supélec/SONDRA, 91192 Gif-sur-Yvette Cedex, France (e-mail:frederic.pascal@supelec.fr) L. Bombrun and Y. Berthoumieu are with Université de Bordeaux, IPB, ENSEIRB-Matmeca, La... |

3 | Alba-Castro, “Generalized Gaussian distributions for sequential data classification
- Bicego, Gonzalo-Jimenez, et al.
- 2008
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Citation Context ...Université de Toulouse, IRIT/INP-ENSEEIHT, (email:jean-yves.tourneret@enseeiht.fr) applications involving GGDs include radar [24], video coding and denoising [25]–[27] or biomedical signal processing =-=[26]-=-, [28], [29]. Finally, it is interesting to note that complex GGDs have been recently studied in [30], [31] and that multivariate regression models with generalized Gaussian errors have been considere... |

3 |
Wavelet-based multivariate approach for multispectral image indexing
- Khelil-Cherif, Benazza-Benyahia
- 2004
(Show Context)
Citation Context ... parameters can be estimated by a least-squares method as in [18] or by minimizing a χ 2 distance between the histogram of the observed data and the theoretical probabilities associated with the MGGD =-=[35]-=-. Estimators based on the method of moments and on the ML method have also been proposed in [36]–[38]. Several works have analyzed covariance matrix estimators defined under different modeling assumpt... |

3 |
On the geometry of multivariate generalized Gaussian models
- Verdoolaege, Scheunders
- 2012
(Show Context)
Citation Context ...e between the histogram of the observed data and the theoretical probabilities associated with the MGGD [35]. Estimators based on the method of moments and on the ML method have also been proposed in =-=[36]-=-–[38]. Several works have analyzed covariance matrix estimators defined under different modeling assumptions. On the one hand, fixed point (FP) algorithms have been derived and analyzed in [39], [40] ... |

3 |
Statistical Distributions in Scientific Work
- Kotz
- 1968
(Show Context)
Citation Context ...ex database are then presented. Conclusions and future works are finally reported in Section VI. A. Definitions II. PROBLEM FORMULATION The probability density function of an MGGD in Rp is defined by =-=[42]-=- p(x|M, m, β) = 1 |M| 1 ( ) T −1 hm,β x M x 2 for any x ∈ Rp , where M is a p × p symmetric real scatter matrix, xT is the transpose of the vector x, and hm,β (·) is a so-called density generator defi... |

2 | matrix variate generalization of the power exponential family of distributions,” Communications in statistics - “A - 2002 |

2 |
generation of correlated non-Gaussian radar clutter
- “Computer
- 1995
(Show Context)
Citation Context ...representations, simulation methods and probabilistic characteristics. GGDs belong to the family of elliptical distributions (EDs) [5], [6], originally introduced by Kelker in [7] and studied in [8], =-=[9]-=-. Depending on the value of the shape parameter β, multivariate GGDs (MGGDs) and the distributions of spherically invariant random vectors share common properties (see [10], [11] for more details). MG... |

2 |
Multivariate exponential power distributions as mixtures of normal distributions with Bayesian applications
- Gómez-Sánchez-Manzano, Gómez-Villegas, et al.
- 2008
(Show Context)
Citation Context ...r in [7] and studied in [8], [9]. Depending on the value of the shape parameter β, multivariate GGDs (MGGDs) and the distributions of spherically invariant random vectors share common properties (see =-=[10]-=-, [11] for more details). MGGDs have been used intensively in the image processing community. Indeed, including Gaussian and Laplacian distributions as special cases, MGGDs are potentially interesting... |

2 |
Bayesian learning of generalized Gaussian mixture models on biomedical images
- Elguebaly, Bouguila
- 2010
(Show Context)
Citation Context ...sité de Toulouse, IRIT/INP-ENSEEIHT, (email:jean-yves.tourneret@enseeiht.fr) applications involving GGDs include radar [24], video coding and denoising [25]–[27] or biomedical signal processing [26], =-=[28]-=-, [29]. Finally, it is interesting to note that complex GGDs have been recently studied in [30], [31] and that multivariate regression models with generalized Gaussian errors have been considered in [... |

2 |
and Gaussianity detection using the complex generalized Gaussian distribution
- “Circularity
- 2009
(Show Context)
Citation Context ...GDs include radar [24], video coding and denoising [25]–[27] or biomedical signal processing [26], [28], [29]. Finally, it is interesting to note that complex GGDs have been recently studied in [30], =-=[31]-=- and that multivariate regression models with generalized Gaussian errors have been considered in [32]. Considering the important attention devoted to GGDs, estimating the parameters of these distribu... |

2 |
Multivariate regression models with power exponential random errors and subset selection using genetic algorithms with information complexity
- Liu, Bozdogan
- 2008
(Show Context)
Citation Context ...], [29]. Finally, it is interesting to note that complex GGDs have been recently studied in [30], [31] and that multivariate regression models with generalized Gaussian errors have been considered in =-=[32]-=-. Considering the important attention devoted to GGDs, estimating the parameters of these distributions is clearly an interesting issue. Classical estimation methods that have been investigated for un... |

2 |
on the manifold of multivariate generalized Gaussian distributions with an application to multicomponent texture discrimination
- “Geodesics
- 2011
(Show Context)
Citation Context ...ose to evaluate the performance of the MLE for the parameters of MGGDs encountered in a real-world application. MGGDs have been used successfully for modeling the wavelet statistics of texture images =-=[37]-=-,8 SUBMITTED TO IEEE TRANS. ON SIGNAL PROCESSING unbiasedness and consistency properties of the scatter matrix estimator and the MLE efficiency of β that have also been observed for synthetic data)... |

2 |
Testing a multivariate model for wavelet coefficients
- Kwitt, Meerwald, et al.
- 2011
(Show Context)
Citation Context ...2013 (b) Fig. 7. Estimation performance for parameter β. (a) Variance of ˆ β versus number of samples N for β = 0.2, ρ = 0.8 and p = 3, (b) Variance of ˆ β versus β for N = 10 000, ρ = 0.8 and p = 3. =-=[52]-=-. In order to analyze the potential of MGGDs for texture modeling, we have considered two images from the VisTex database [53], namely the “Bark.0000” and “Leaves.0008” images displayed in Fig. 8. The... |