Covariance matrix estimation via geometric barycenters and its application to radar training data selection

Abstract

This study deals with the problem of covariance matrix estimation for radar signal processing applications. The authors propose and analyse a class of estimators that do not require any knowledge about the probability distribution of the sample support and exploit the characteristics of the positive-definite matrix space. Any estimator of the class is associated with a suitable distance in the considered space and is defined as the geometric barycenter of some basic covariance matrix estimates obtained from the available secondary data set. Then, the authors introduce an adaptive detection structure, exploiting the new covariance matrix estimators, based on two stages. The former consists of a data selector screening among the training data, whereas the latter is a conventional adaptive matched filter taking the final decision about the target presence. At the analysis stage, the authors assess the performance of the proposed two-stage scheme in terms of probability of correct outliers excision, constant false alarm rate behaviour and detection probability. The analysis is conducted both on simulated data and on the challenging KASSPER datacube.

Positive Definite Matrix Space Based Detector With Limited Training Samples for Multiple Target Situations

“…In Figure 1, from covariance matrix C to positive-definite matrix S, R(C) = 1 is transformed to R(S) = N , which means expansion of the eigenvalues of C. The clutter power estimation P is arithmetic mean of eigenvalues of S, which acts as compressors of the eigenvalues, and hence they de-emphasise the effect of outliers [34,35]. Additionally, besides the aforementioned mathematical operation, we find that the clutter power estimation method proposed in PDLT is robust under the condition of limited training samples (small N ), which is an significant way to resist multiple target situations.…”

“…Proof of Theorem 1. See [34]. (3) After getting the renewed positive-definite matrix S, the arithmetic mean of its diagonal elements is used to form the clutter power:…”

“…In [32,33], an algorithm based on the Riemannian medians and means of secondary matrices for radar target detection is presented. Besides, a class of geometric barycenter (GB) based signal processing methods has been illustrated to estimate the covariance matrix [34][35][36][37]. In detail, any detector of this category is bound with an appropriate distance in the positive-definite matrix space.…”

A Geometric Approach to Covariance Matrix Estimation and its Applications to Radar Problems

“…In [27] and [29] also an extension of the conventional Ordered Statistic (OS) framework [30] is proposed relying on the Riemannian p-mean computation of Toepliz or Toeplitz-BlockToeplitz space-time covariance matrices. Besides, in [31] and [32], covariance estimates defined through geometric barycenters/medians (associated with specific distances in the space of Hermitian matrices) of structured covariance estimates are exploited both for training data selection and adaptive radar detection highlighting significant gains with respect to the classic sample covariance matrix. Finally, in [33], [34], [35], [36], [37], [38] other interesting geometric-inspired procedures are devised.…”

Median matrices and their application to radar training data selection

“…For a detailed overview of this research activity, see also [11] and references therein. Finally, in [12], covariance estimates defined through geometric barycenters are exploited both for training data selection and for adaptive radar detection.…”