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

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

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.

Keywords

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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References

  1. 1.
    Siegel, C.L.: Symplectic Geometry. Academic Press, New York (1964)zbMATHGoogle Scholar
  2. 2.
    Rao, C.R.: Information and Accuracy attainable in the estimation of statistical parameters. Bull. Calcutta Math. Soc. 37, 81–91 (1945)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Yoshizawa, S., Tanabe, K.: Dual Differential Geometry associated with the Kullback-Leibler Information on the Gaussian Distributions and its 2-parameter Deformations. SUT Journal of Mathematics 35(1), 113–137 (1999)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Ando, T., et al.: Geometric Means. Linear Algebra Appl. 385, 305–334 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Erich, K.: Mathematical Works, Berlin, Walter de Gruyter, ix (2003)Google Scholar
  6. 6.
    Bakas, I.: The Algebraic Structure of Geometric Flows in Two Dimensions., Inst. of Physics, SISSA (October 2005)Google Scholar
  7. 7.
    Barbaresco, F.: Information Intrinsic Geometric Flows. In: MaxEnt 2006 Conference, Paris, vol. 872, pp. 211–218 (June 2006) (published in American Institute of Physics, AIP)Google Scholar
  8. 8.
    Barbaresco, F.: Innovative Tools for Radar Signal Processing Based on Cartan’s Geometry of SPD Matrices and Information Geometry. In: IEEE International Radar Conference, Rome (May 2008)Google Scholar
  9. 9.
    Bruhat, F., Tits, J.: Groupes réductifs sur un corps local. IHES 41, 5–251 (1972)CrossRefzbMATHGoogle Scholar
  10. 10.
    Gromov, M.: Hyperbolic Groups. Essays in Group Theory, Math. Sci. Res. Inst. Publ. 8, New york, pp. 75–263 (1987)Google Scholar
  11. 11.
    Maass, H.: Siegel Modular Forms and Derichlet Series. Lecture Notes in Math., vol. 216. Springer, Berlin (1971)CrossRefGoogle Scholar
  12. 12.
    Cartan, H.: Ouverts fondamentaux pour le groupe modulaire. Séminaire Henri Cartan, tome 10(1), exp. n°3, p. 1–12 (1957)Google Scholar
  13. 13.
    Chentsov, N.N.: Statistical Decision Rules and Optimal Inferences. Trans. of Math. Monog, vol. 53. Amer. Math. Society, Providence (1982)Google Scholar
  14. 14.
    Cartan, E.: Sur les domaines bornés homogènes de l’espace de n variables complexes. Abh. Math. Semin. hamb. Univ. 11, 116–162 (1935)CrossRefzbMATHGoogle Scholar
  15. 15.
    Koecher, M.: Jordan Algebras and their Applications, Univ. of Minnesota, Minneapolis. Lect. Notes (1962)Google Scholar
  16. 16.
    Satake, I.: Algebraic Structures of Symmetric Domains. Kano memorial Lectures, vol. 4. Princeton University Press, Princeton (1980)zbMATHGoogle Scholar
  17. 17.
    Faraut, J., Koranyi, A.: Analysis on Symmetric Cones. Oxford University Press, Oxford (1994)zbMATHGoogle Scholar
  18. 18.
    Bougerol, P.: Kalman Filtering with Random Coefficients and Contractions. SIAM J. Control and Optimization 31(4), 942–959 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Koufany, K.: Analyse et Géométrie des domaines bornés symétriques. HDR, Institut de Mathematiques Elie Cartan, Nancy (November 2006)Google Scholar
  20. 20.
    Björk, A., Hammarling, S.: A Schur method for the square root of a matrix. Linear Algebra and Appl. 52/53, 127–140 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Karcher, H.: Riemannian center of mass and mollifier smoothing. Comm. Pure Applied Math. 30, 509–541 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Fefferman, C.: Monge-Ampère Equations, the Bergman Kernel, and geometry of pseudoconvexs domains. Ann. Of Math. 103, 395–416 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Niculescu, C.P.: An Extension of the Mazur-Ulam Theorem. American Institute of Physics Proc. 729, 248–256 (2004)MathSciNetzbMATHGoogle Scholar
  24. 24.
    Barbaresco, F.: Calculus of Variations and Regularized Spectral Estimation. In: MaxEnt 2000 Conf., American Institute of Physics, AIP, vol. 568, pp. 361–374Google Scholar
  25. 25.
    Berger, M.: « Panoramic View of Riemannian Geometry ». Springer, Heidelberg (2004)zbMATHGoogle Scholar
  26. 26.
    Phong, D.H., Sturm, J.: The Monge-Ampère Equation and geodesics in the space of Kähler Potentials, arXiv:math/0504157v2 [math DG] (May 1, 2005)Google Scholar
  27. 27.
    Arnaudon, M., Li, X.: « Barycentres of measures transported by stochastic flows. Ann. Probab. 33(4), 1509–1543 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Charon, N.: Une nouvelle approche pour la detection de cibles dans les images radar. Technical Report Thales/ENS-Cachan, Master MVA report (September 2008)Google Scholar
  29. 29.
    Lapuyade-Lahorgue, J.: « Evolutions des radars de surveillance côtière », revue REE, vol. 8 (September 2008)Google Scholar
  30. 30.
    Lapuyade-Lahorgue, J.: « Détection de cibles furtives par segmentation statistique et TFAC Doppler », PhD report (French MoD, DGA), ch. 6 (Décember 2008)Google Scholar
  31. 31.
    Bonnabel, S., Sepulchre, R.: Geometric distance and mean for positive semi-definite matrices of fixed rank. arXiv:0807.4462v1, 28 (July 2008)Google Scholar
  32. 32.
    Dhillon, L.S., Tropp, J.A.: Matrix nearness problems with bregman divergences. SIAM J. matrix anal. Appl. 29, 1120–1146 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Ledoux, M.: Géométrie des espaces métriques mesurés: les travaux de Lott, Villani, Sturm. séminaire Bourbaki, 60ème année, vol. 990, pp. 1–21 (March 2008)Google Scholar
  34. 34.
    Varadhan, S.: « Large Deviations ». The Annals of Probability 36(2), 397–419 (2008)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© 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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