The Basic Geometric Structures of Electromagnetic Digital Information: Statistical characterization of the digital measurement of spatio-Doppler and polarimetric fluctuations of the radar electromagnetic wave

The aim is to describe new geometric approaches to define the statistics of spatio-temporal and polarimetric measurements of the states of an electromagnetic wave, using the works of Maurice Fr{e}chet, Jean-Louis Koszul and Jean-Marie Souriau, with in particular the notion of average state of this digital measurement as a Fr{e}chet barycentre in a metric space and a model derived from statistical mechanics to define and calculate a maximum density of entropy (extension of the notion of Gaussian) to describe the fluctuations of the electromagnetic wave. The article will illustrate these new tools with examples of radar application for Doppler, spatio-temporal and polarimetric measurement of the electromagnetic wave by introducing a distance on the covariance matrices of the electromagnetic digital signal, based on Fishers metric from Information Geometry.

Branch-and-Bound Method for Just-in-Time Optimization of Radar Search Patterns

Electronic phased-array radars offer new possibilities for radar search pattern optimization by using bi-dimensional beam-forming and beam-steering. Radar search pattern optimization can be approximated as a set cover problem and solved using integer programming, while accounting for localized clutter and terrain masks in detection constraints. We present a set cover problem approximation for time-budget minimization of radar search patterns, under constraints of range, detection probability and direction-specific scan update rates. Branch&Bound is a classical optimization procedure for solving combinatorial problems. It is known mainly as an exact algorithm, but features interesting characteristics, making it particularly fit for solving optimization problems in real-time applications and producing just-in-time solutions.

Reparameterization invariant metric on the space of curves

This paper focuses on the study of open curves in a manifold M, and proposes a reparameterization invariant metric on the space of such paths. We use the square root velocity function (SRVF) introduced by Srivastava et al. in [11] to define a reparameterization invariant metric on the space of immersions M = Imm([0,1], M) by pullback of a metric on the tangent bundle TM derived from the Sasaki metric. We observe that such a natural choice of Riemannian metric on TM induces a first-order Sobolev metric on M with an extra term involving the origins, and leads to a distance which takes into account the distance between the origins and the distance between the SRV representations of the curves. The geodesic equations for this metric are given, as well as an idea of how to compute the exponential map for observed trajectories in applications. This provides a generalized theoretical SRV framework for curves lying in a general manifold M .

Medians and means in Riemannian geometry: existence, uniqueness and computation

This paper is a short summary of our recent work on the medians and means of probability measures in Riemannian manifolds. Firstly, the existence and uniqueness results of local medians are given. In order to compute medians in practical cases, we propose a subgradient algorithm and prove its convergence. After that, Frechet medians are considered. We prove their statistical consistency and give some quantitative estimations of their robustness with the aid of upper curvature bounds. We also show that, in compact Riemannian manifolds, the Frechet medians of generic data points are always unique. Stochastic and deterministic algorithms are proposed for computing Riemannian p-means. The rate of convergence and error estimates of these algorithms are also obtained. Finally, we apply the medians and the Riemannian geometry of Toeplitz covariance matrices to radar target detection.