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

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Emerging Trends in Visual Computing (ETVC 2008)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 5416))

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Abstract

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.

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Author information

Authors and Affiliations

  1. Strategy Technology & Innovation Department,Hameau de Roussigny, Thales Air Systems, F-91470, Limours, France

    Frederic Barbaresco

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  1. Frederic Barbaresco
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Editor information

Editors and Affiliations

  1. LIX, Ecole Polytechnique, Route de Saclay, 91128, Palaiseau Cedex, France

    Frank Nielsen

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Barbaresco, F. (2009). Interactions between Symmetric Cone and Information Geometries: Bruhat-Tits and Siegel Spaces Models for High Resolution Autoregressive Doppler Imagery. In: Nielsen, F. (eds) Emerging Trends in Visual Computing. ETVC 2008. Lecture Notes in Computer Science, vol 5416. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-00826-9_6

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  • DOI: https://doi.org/10.1007/978-3-642-00826-9_6

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-00825-2

  • Online ISBN: 978-3-642-00826-9

  • eBook Packages: Computer ScienceComputer Science (R0)

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