Recently, a new adaptive scheme [1], [2] has been introduced for covariance structure matrix estimation in the context of adaptive radar detection under non Gaussian noise. This latter has been modelled by compound-Gaussian noise, which is the product c of the square root of a positive unknown variable τ (deterministic or random) and an independent Gaussian vector x, c = √ τ x. Because of the implicit algebraic structure of the equation to solve, we called the corresponding solution, the Fixed Point (FP) estimate. When τ is assumed deterministic and unknown, the FP is the exact Maximum Likelihood (ML) estimate of the noise covariance structure, while when τ 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.

This paper deals with covariance matrix estimates in impulsive noise environments. Physical models based on compound noise modeling (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) which 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 M SCM and a normalized version M N , the Fixed Point (FP) estimate M F P , and a theoretical benchmark M T F P . 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.

This paper presents an application of the recent advances in the field of Spherically Invariant Random Vectors (SIRV) modelling for coherency matrix estimation in heterogeneous clutter. The complete description of the POLSAR data set is achieved by estimating the span and the normalized coherency independently. The normalized coherency describes the polarimetric diversity, while the span indicates the total received power. The main advantages of the proposed Fixed Point estimator are that it does not require any "a priori" information about the probability density function of the texture (or span) and it can be directly applied on adaptive neighborhoods. Interesting results are obtained when coupling this Fixed Point estimator with an adaptive spatial support based on the scalar span information. Based on the SIRV model, a new maximum likelihood distance measure is introduced for unsupervised POLSAR classification. The proposed method is tested with both simulated POLSAR data and airborne POLSAR images provided by the RAMSES system. Results of entropy/alpha/anisotropy decomposition, followed by unsupervised classification, allow to discuss the use of the normalized coherency and the span as two separate descriptors of POLSAR data sets.

Abstract-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.