Multivariate Generalized Gaussian Distribution Research Articles

Multivariate Statistical Modeling for Multitemporal SAR Change Detection Using Wavelet Transforms and Integrating Subband Dependencies

In this paper, we propose a new method for automatic change detection in multi-temporal fully polarimetric synthetic aperture radar (PolSAR) images based on multivariate statistical wavelet subband modeling. The proposed method allows us to take into account the correlation structure between subbands by modeling the wavelet coefficients through multi-variate probability distributions. Three types of correlation are investigated: inter-scale, inter-orientation, and inter-polarization dependences. The multivariate generalized Gaussian distribution (MGGD) is used to model the interdependencies between wavelet coefficients at different orientations, scales, and polarizations. Kullback-Leibler similarity measures are computed and used to generate the change map. Simulated and real multilook PolSAR data are employed to assess the performance of the method and are compared to the multivariate Gaussian distribution (MGD) based method. We show that the information embedded in the correlation between subbands improves the accuracy of the change map, leading to better performance. Moreover, the MGGD represents better the correlations between wavelet coefficients and outperforms the MGD.

Incipient fault detection for geological drilling processes using multivariate generalized Gaussian distributions and Kullback–Leibler divergence

Early detection of downhole faults in geological drilling processes can effectively reduce downtime and prevent catastrophic drilling accidents. Considering that the changes of drilling signals in early faults are difficult to observe while the data distribution may have significant deviations, a systematic incipient fault detection method is proposed for drilling processes based on the Multivariate Generalized Gaussian Distributions (MGGDs) and Kullback-Leibler Divergence (KLD). The contributions are twofold: (1) A new dissimilarity index is proposed for incipient fault detection by estimating the KLD between the MGGD of the historical data under the normal condition and that of the real-time data; (2) an adaptive alarm limit design approach is proposed for updating the alarm limits that adapt to the increasing of the drilling depth. Industrial case studies from a real drilling project are presented to demonstrate the effectiveness and practicability of the proposed method.

Fault detection for geological drilling processes using multivariate generalized Gaussian distribution and Kullback Leibler divergence

The presence of downhole faults compromises the safety and also leads to increased maintenance costs in complex geological drilling processes. In order to achieve timely and accurate detection of downhole faults, a systematic fault detection method is proposed based on the Multivariate Generalized Gaussian Distribution (MGGD) and the Kullback Leibler Divergence (KLD). Uncorrelated components are obtained from the original drilling process signals using the principle component analysis; then, the distribution of components is estimated using the MGGD; afterwards, the KLD is calculated based on a deduced analytic formula; last, the downhole faut is detected by comparing the calculated KLD with the alarm threshold obtained from normal data. The effectiveness and practicality of the proposed method are demonstrated by application to a real drilling process.

Blind Speech Extraction Based on Rank-Constrained Spatial Covariance Matrix Estimation With Multivariate Generalized Gaussian Distribution

In this article, we propose a new blind speech extraction (BSE) method that robustly extracts a directional speech from background diffuse noise by combining independent low-rank matrix analysis (ILRMA) and efficient rank-constrained spatial covariance matrix (SCM) estimation. To achieve more accurate BSE than ILRMA, which assumes each source to be a point source (rank-1 spatial model), the proposed method restores the lost spatial basis for the full-rank SCM of diffuse noise. We adopt the multivariate complex generalized Gaussian distribution (GGD) as the statistical generative model to express various types of observed signal. To estimate the model parameters for an arbitrary shape parameter of the multivariate GGD, we derive a new inequality for rank-constrained SCMs. Also, we propose new acceleration methods to accomplish much faster extraction than conventional blind source separation methods. In BSE experiments using simulated and real recorded data, we confirm that the proposed method achieves more accurate and faster speech extraction than conventional methods.

Kullback–Leibler Divergence Between Multivariate Generalized Gaussian Distributions

The Kullback–Leibler divergence (KLD) between two multivariate generalized Gaussian distributions (MGGDs) is a fundamental tool in many signal and image processing applications. Until now, the KLD of MGGDs has no known explicit form, and it is in practice either estimated using expensive Monte-Carlo stochastic integration or approximated. The main contribution of this letter is to present a closed-form expression of the KLD between two zero-mean MGGDs. Depending on the Lauricella series, a simple way of calculating numerically the KLD is exposed. Finally, we show that the approximation of the KLD by Monte-Carlo sampling converges to its theoretical value when the number of samples goes to the infinity.

Unsupervised weld defect classification in radiographic images using multivariate generalized Gaussian mixture model with exact computation of mean and shape parameters

In industry, the welding inspection is considered as a mandatory stage in the process of quality assurance/quality control. This inspection should satisfy the requirements of the standards and codes governing the manufacturing process in order to prevent unfair harm to the industrial plant in construction. For this purpose, in this paper, a software specially conceived for computer-aided diagnosis in weld radiographic testing is presented, where a succession of operations of preprocessing, image segmentation, feature extraction and finally defects classification is carried out on radiographic images. The last operation which is the main contribution in this paper consists in an unsupervised classifier based on a finite mixture model using the multivariate generalized Gaussian distribution (MGGD). This classifier is newly applied on a dataset of weld defect radiographic images. The parameters of the nonzero-mean MGGD-based mixture model are estimated using the Expectation-Maximization algorithm where, exact computations of mean and shape parameters are originally provided. The weld defect database represent four weld defect types (crack, lack of penetration, porosity and solid inclusion) which are indexed by a shape geometric descriptor composed of geometric measures. An outstanding performance of the proposed mixture model, compared to the one using the multivariate Gaussian distribution, is shown, where the classification rate is improved by 3.2% for the whole database, to reach more than 96%. The efficiency of the proposed classifier is mainly due to the flexible fitting of the input data, thanks to the MGGD shape parameter.

Transformation of the Multivariate Generalized Gaussian Distribution for Image Editing.

Multivariate generalized Gaussian distributions (MGGDs) have aroused a great interest in the image processing community thanks to their ability to describe accurately various image features, such as image gradient fields. However, so far their applicability has been limited by the lack of a transformation between two of these parametric distributions. In this paper, we propose a novel transformation between MGGDs, consisting of an optimal transportation of the second-order statistics and a stochastic-based shape parameter transformation. We employ the proposed transformation between MGGDs for a color transfer and a gradient transfer between images. We also propose a new simultaneous transfer of color and gradient, which we apply for image color correction.

Adaptive independent vector analysis for multi-subject complex-valued fMRI data

BackgroundComplex-valued fMRI data can provide additional insights beyond magnitude-only data. However, independent vector analysis (IVA), which has exhibited great potential for group analysis of magnitude-only fMRI data, has rarely been applied to complex-valued fMRI data. The main challenges in this application include the extremely noisy nature and large variability of the source component vector (SCV) distribution. New methodTo address these challenges, we propose an adaptive fixed-point IVA algorithm for analyzing multiple-subject complex-valued fMRI data. We exploited a multivariate generalized Gaussian distribution (MGGD)- based nonlinear function to match varying SCV distributions in which the MGGD shape parameter was estimated using maximum likelihood estimation. To achieve our de-noising goal, we updated the MGGD-based nonlinearity in the dominant SCV subspace, and employed a post-IVA de-noising strategy based on phase information in the IVA estimates. We also incorporated the pseudo-covariance matrix of fMRI data into the algorithm to emphasize the noncircularity of complex-valued fMRI sources. ResultsResults from simulated and experimental fMRI data demonstrated the efficacy of our method. Comparison with existing method(s)Our approach exhibited significant improvements over typical complex-valued IVA algorithms, especially during higher noise levels and larger spatial and temporal changes. As expected, the proposed complex-valued IVA algorithm detected more contiguous and reasonable activations than the magnitude-only method for task-related (393%) and default mode (301%) spatial maps. ConclusionsThe proposed approach is suitable for decomposing multi-subject complex-valued fMRI data, and has great potential for capturing additional subject variability.

A New Riemannian Averaged Fixed-Point Algorithm for MGGD Parameter Estimation

Multivariate generalized Gaussian distribution (MGGD) has been an attractive solution to many signal processing problems due to its simple yet flexible parametric form, which requires the estimation of only a few parameters, i.e., the scatter matrix and the shape parameter. Existing fixed-point (FP) algorithms provide an easy to implement method for estimating the scatter matrix, but are known to fail, giving highly inaccurate results, when the value of the shape parameter increases. Since many applications require flexible estimation of the shape parameter, we propose a new FP algorithm, Riemannian averaged FP (RA-FP), which can effectively estimate the scatter matrix for any value of the shape parameter. We provide the mathematical justification of the convergence of the RA-FP algorithm based on the Riemannian geometry of the space of symmetric positive definite matrices. We also show using numerical simulations that the RA-FP algorithm is invariant to the initialization of the scatter matrix and provides significantly improved performance over existing FP and method-of-moments (MoM) algorithms for the estimation of the scatter matrix.

A statistical reduced-reference method for color image quality assessment

Although color is a fundamental feature of human visual perception, it has been largely unexplored in the reduced-reference (RR) image quality assessment (IQA) schemes. In this paper, we propose a natural scene statistic (NSS) method, which efficiently uses this information. It is based on the statistical deviation between the steerable pyramid coefficients of the reference color image and the degraded one. We propose and analyze the multivariate generalized Gaussian distribution (MGGD) to model the underlying statistics. In order to quantify the degradation, we develop and evaluate two measures based respectively on the Geodesic distance between two MGGDs and on the closed-form of the Kullback Leibler divergence. We performed an extensive evaluation of both metrics in various color spaces (RGB, HSV, CIELAB and YCrCb) using the TID 2008 benchmark and the FRTV Phase I validation process. Experimental results demonstrate the effectiveness of the proposed framework to achieve a good consistency with human visual perception. Furthermore, the best configuration is obtained with CIELAB color space associated to KLD deviation measure.

Parameter Estimation For Multivariate Generalized Gaussian Distributions

Due to its heavy-tailed and fully parametric form, the multivariate generalized Gaussian distribution (MGGD) has been receiving much attention for modeling extreme events in signal and image processing applications. Considering the estimation issue of the MGGD parameters, the main contribution of this paper is to prove that the maximum likelihood estimator (MLE) of the scatter matrix exists and is unique up to a scalar factor, for a given shape parameter \beta\in(0,1). Moreover, an estimation algorithm based on a Newton-Raphson recursion is proposed for computing the MLE of MGGD parameters. Various experiments conducted on synthetic and real data are presented to illustrate the theoretical derivations in terms of number of iterations and number of samples for different values of the shape parameter. The main conclusion of this work is that the parameters of MGGDs can be estimated using the maximum likelihood principle with good performance.

Multivariate Generalized Gaussian Distribution: Convexity and Graphical Models

We consider covariance estimation in the multivariate generalized Gaussian distribution (MGGD) and elliptically symmetric (ES) distribution. The maximum likelihood optimization associated with this problem is non-convex, yet it has been proved that its global solution can be often computed via simple fixed point iterations. Our first contribution is a new analysis of this likelihood based on geodesic convexity that requires weaker assumptions. Our second contribution is a generalized framework for structured covariance estimation under sparsity constraints. We show that the optimizations can be formulated as convex minimization as long the MGGD shape parameter is larger than half and the sparsity pattern is chordal. These include, for example, maximum likelihood estimation of banded inverse covariances in multivariate Laplace distributions, which are associated with time varying autoregressive processes.

On the Geometry of Multivariate Generalized Gaussian Models

This paper concerns the geometry of the zero-mean multivariate generalized Gaussian distribution (MGGD) and the calculation of geodesic distances on the MGGD manifold. The MGGD is a suitable distri...

Geodesics on the Manifold of Multivariate Generalized Gaussian Distributions with an Application to Multicomponent Texture Discrimination

We consider the Rao geodesic distance (GD) based on the Fisher information as a similarity measure on the manifold of zero-mean multivariate generalized Gaussian distributions (MGGD). The MGGD is shown to be an adequate model for the heavy-tailed wavelet statistics in multicomponent images, such as color or multispectral images. We discuss the estimation of MGGD parameters using various methods. We apply the GD between MGGDs to color texture discrimination in several classification experiments, taking into account the correlation structure between the spectral bands in the wavelet domain. We compare the performance, both in terms of texture discrimination capability and computational load, of the GD and the Kullback-Leibler divergence (KLD). Likewise, both uni- and multivariate generalized Gaussian models are evaluated, characterized by a fixed or a variable shape parameter. The modeling of the interband correlation significantly improves classification efficiency, while the GD is shown to consistently outperform the KLD as a similarity measure.

Multivariate statistical modeling for image denoising using wavelet transforms

Recently a variety of efficient image denoising methods using wavelet transforms have been proposed by many researchers. In this paper, we derive the general estimation rule in the wavelet domain to obtain the denoised coefficients from the noisy image based on the multivariate statistical theory. The multivariate distributions of the original clean image can be estimated empirically from a sample image set. We define a parametric multivariate generalized Gaussian distribution (MGGD) model which closely fits the sample distribution. Multivariate model makes it possible to exploit the dependency between the estimated wavelet coefficients and their neighbours or other coefficients in different subbands. Also it can be shown that some of the existing methods based on statistical modeling are subsets of our multivariate approach. Our method could achieve high quality image denoising. Among the existing image denoising methods using the same type of wavelet (Daubechies 8) filter, our results produce the highest peak signal-to-noise ratio (PSNR).