Riemannian Medians and Means With Applications to Radar Signal Processing

Abstract

We develop a new geometric approach for high resolution Doppler processing based on the Riemannian geometry of Toeplitz covariance matrices and the notion of Riemannian <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> -means. This paper summarizes briefly our recent work in this direction. First of all, we introduce radar data and the problem of target detection. Then we show how to transform the original radar data into Toeplitz covariance matrices. After that, we give our results on the Riemannian geometry of Toeplitz covariance matrices. In order to compute <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i> -means in practical cases, we propose deterministic and stochastic algorithms, of which the convergence results are given, as well as the rate of convergence and error estimates. Finally, we propose a new detector based on Riemannian median and show its advantage over the existing processing methods.