Interactions between Symmetric Cone and Information Geometries: Bruhat-Tits and Siegel Spaces Models for High Resolution Autoregressive Doppler Imagery

  • Frederic Barbaresco
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5416)


Main issue of High Resolution Doppler Imagery is related to robust statistical estimation of Toeplitz Hermitian positive definite covariance matrices of sensor data time series (e.g. in Doppler Echography, in Underwater acoustic, in Electromagnetic Radar, in Pulsed Lidar...). We consider this problem jointly in the framework of Riemannian symmetric spaces and the framework of Information Geometry. Both approaches lead to the same metric, that has been initially considered in other mathematical domains (study of Bruhat-Tits complete metric Space and Upper-half Siegel Space in Symplectic Geometry). Based on Frechet-Karcher barycenter definition and geodesics in Bruhat-Tits space, we address problem of N Covariance matrices Mean estimation. Our main contribution lies in the development of this theory for Complex Autoregressive models (maximum entropy solution of Doppler Spectral Analysis). Specific Blocks structure of the Toeplitz Hermitian covariance matrix is used to define an iterative and parallel algorithm for Siegel metric computation. Based on Affine Information Geometry theory, we introduce for Complex Autoregressive Model, Kähler metric on reflection coefficients based on Kähler potential function given by Doppler signal Entropy. The metric is closely related to Kähler-Einstein manifold and complex Monge-Ampere Equation. Finally, we study geodesics in space of Kähler potentials and action of Calabi and Kähler-Ricci Geometric Flows for this Complex Autoregressive Metric. We conclude with different results obtained on real Doppler Radar Data in HF and X bands : X-band radar monitoring of wake vortex turbulences, detection for Coastal X-band and HF Surface Wave Radars.


Information Geometry Symmetric Cone Geometry Kähler Geometry Bruhat-Tits Space Siegel Space von Mangoldt-Cartan-Hadamard Manifold Mazur-Ulam Theorem Bregman Kernel Kähler-Ricci Flow Calabi Flow Complex Monge-Ampere Equation Complex Autoregressive Model Matrices Mean Doppler Imagery 


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© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Frederic Barbaresco
    • 1
  1. 1.Strategy Technology & Innovation Department,Hameau de RoussignyThales Air SystemsLimoursFrance

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