Frédéric Pascal

Covariance Structure Maximum-Likelihood Estimates in Compound Gaussian Noise: Existence and Algorithm Analysis

Recently, a new adaptive scheme [Conte (1995), Gini (1997)] has been introduced for covariance structure matrix estimation in the context of adaptive radar detection under non-Gaussian noise. This latter has been modeled by compound-Gaussian noise, which is the product c of the square root of a positive unknown variable tau (deterministic or random) and an independent Gaussian vector x, c=radictaux. Because of the implicit algebraic structure of the equation to solve, we called the corresponding solution, the fixed point (FP) estimate. When tau is assumed deterministic and unknown, the FP is the exact maximum-likelihood (ML) estimate of the noise covariance structure, while when tau is a positive random variable, the FP is an approximate maximum likelihood (AML). This estimate has been already used for its excellent statistical properties without proofs of its existence and uniqueness. The major contribution of this paper is to fill these gaps. Our derivation is based on some likelihood functions general properties like homogeneity and can be easily adapted to other recursive contexts. Moreover, the corresponding iterative algorithm used for the FP estimate practical determination is also analyzed and we show the convergence of this recursive scheme, ensured whatever the initialization.

Performance Analysis of Covariance Matrix Estimates in Impulsive Noise

This paper deals with covariance matrix estimates in impulsive noise environments. Physical models based on compound noise modeling [spherically invariant random vectors (SIRV), compound Gaussian processes] allow to correctly describe reality (e.g., range power variations or clutter transitions areas in radar problems). However, these models depend on several unknown parameters (covariance matrix, statistical distribution of the texture, disturbance parameters) that have to be estimated. Based on these noise models, this paper presents a complete analysis of the main covariance matrix estimates used in the literature. Four estimates are studied: the well-known sample covariance matrix MSCM and a normalized version MN, the fixed-point (FP) estimate MFP, and a theoretical benchmark MTFP. Among these estimates, the only one of practical interest in impulsive noise is the FP. The three others, which could be used in a Gaussian context, are, in this paper, only of academic interest, i.e., for comparison with the FP. A statistical study of these estimates is performed through bias analysis, consistency, and asymptotic distribution. This study allows to compare the performance of the estimates and to establish simple relationships between them. Finally, theoretical results are emphasized by several simulations corresponding to real situations.

Convergence of Adaptive Discontinuous Galerkin Approximations of Second-Order Elliptic Problems

A residual-type a posteriori error estimator is introduced and analyzed for a discontinuous Galerkin formulation of a model second-order elliptic problem with Dirichlet-Neumann-type boundary conditions. An adaptive algorithm using this estimator together with specific marking and refinement strategies is constructed and shown to achieve any specified error level in the energy norm in a finite number of cycles. The convergence rate is in effect linear with a guaranteed error reduction at every cycle. Results of numerical experiments are presented.

Persymmetric Adaptive Radar Detectors

In the general framework of radar detection, estimation of the Gaussian or non-Gaussian clutter covariance matrix is an important point. This matrix commonly exhibits a particular structure: for instance, this is the case for active systems using a symmetrically spaced linear array with constant pulse repetition interval. We propose using the particular persymmetric structure of the covariance matrix to improve the detection performance. In this context, this work provides two new adaptive detectors for Gaussian additive noise and non-Gaussian additive noise which is modeled by the spherically invariant random vector (SIRV). Their statistical properties are then derived and compared with simulations. The vast improvement in their detection performance is demonstrated by way of simulations or experimental ground clutter data. This allows for the analysis of the proposed detectors on both real Gaussian and non-Gaussian data.

Generalized Robust Shrinkage Estimator and Its Application to STAP Detection Problem

Recently, in the context of covariance matrix estimation, in order to improve as well as to regularize the performance of the Tylers estimator [1] also called the Fixed-Point Estimator (FPE) [2], a “shrinkage” fixed-point estimator has been originally introduced in [3]. First, this work extends the results of [4], [5] by giving the general solution of the “shrinkage” fixed-point algorithm. Secondly, by analyzing this solution, called the generalized robust shrinkage estimator, we prove that this solution converges to a unique solution when the shrinkage parameter (losing factor) tends to 0. This solution is exactly the FPE with the trace of its inverse equal to the dimension of the problem. This general result allows one to give another interpretation of the FPE and more generally, on the Maximum Likelihood approach for covariance matrix estimation when constraints are added. Then, some simulations illustrate our theoretical results as well as the way to choose an optimal shrinkage factor. Finally, this work is applied to a Space-Time Adaptive Processing (STAP) detection problem on real STAP data.

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 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 β ∈ (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.

Statistical Classification for Heterogeneous Polarimetric SAR Images

This paper presents a general approach for high- resolution polarimetric SAR data classification in heterogeneous clutter, based on a statistical test of equality of covariance matrices. The Spherically Invariant Random Vector (SIRV) model is used to describe the clutter. Several distance measures, including classical ones used in standard classification methods, can be derived from the general test. The new approach provide a threshold over which pixels are rejected from the image, meaning they are not sufficiently “close” from any existing class. A distance measure using this general approach is derived and tested on a high-resolution polarimetric data set acquired by the ONERA RAMSES system. It is compared to the results of the classical H-α decomposition and Wishart classifier under Gaussian and SIRV assumption. Results show that the new approach rejects all pixels from heterogeneous parts of the scene and classifies its Gaussian parts.

Exact Maximum Likelihood Estimates for SIRV Covariance Matrix: Existence and Algorithm Analysis

In this paper, we investigate the existence and the algorithm analysis of an adaptive scheme that has been introduced for covariance structure matrix estimation in the context of adaptive radar detection under non-Gaussian noise. This latter has been modeled by spherically invariant random vector (SIRV), which is the product c of the square root of a positive unknown random variable tau and an independent Gaussian vector x,c=radic(tau) x. A similar line of work was undertaken in the context of compound Gaussian noise, and this paper extends the previous results in the case of SIRV modeled noise. More precisely, the fixed-point estimate to be studied verifies a nonlinear algebraic equation (E)x=f(x). The aim of this paper is twofold. First, we prove that (E) admits a unique solution x; secondly, we show that the corresponding iterative algorithm xn+1=f(xn) converges to x for every admissible initial condition.

Asymptotic Properties of Robust Complex Covariance Matrix Estimates

In many statistical signal processing applications, the estimation of nuisance parameters and parameters of interest is strongly linked to the resulting performance. Generally, these applications deal with complex data. This paper focuses on covariance matrix estimation problems in non-Gaussian environments, and particularly the M -estimators in the context of elliptical distributions. First, this paper extends to the complex case the results of Tyler in [D. Tyler, “Robustness and Efficiency Properties of Scatter Matrices,” Biometrika, vol. 70, no. 2, p. 411, 1983]. More precisely, the asymptotic distribution of these estimators as well as the asymptotic distribution of any homogeneous function of degree 0 of the M -estimates are derived. On the other hand, we show the improvement of such results on two applications: directions of arrival (DOA) estimation using the MUltiple SIgnal Classification (MUSIC) algorithm and adaptive radar detection based on the Adaptive Normalized Matched Filter (ANMF) test.

Robust Estimates of Covariance Matrices in the Large Dimensional Regime

This paper studies the limiting behavior of a class of robust population covariance matrix estimators, originally due to Maronna in 1976, in the regime where both the number of available samples and the population size grow large. Using tools from random matrix theory, we prove that, for sample vectors made of independent entries having some moment conditions, the difference between the sample covariance matrix and (a scaled version of) such robust estimator tends to zero in spectral norm, almost surely. This result can be applied to various statistical methods arising from random matrix theory that can be made robust without altering their first order behavior.