Supremum/Infimum and Nonlinear Averaging of Positive Definite Symmetric Matrices

  • Jesús Angulo


Mathematical morphology is a nonlinear image processing methodology based on the computation of supremum (dilation operator) and infimum (erosion operator) in local neighborhoods called structuring elements. This chapter deals with definition of supremum and infimum operators for positive definite symmetric (PDS) matrices, which are the basic ingredients for the extension mathematical morphology to PDS matrices-valued images. The problem is tackled under three different paradigms. Firstly, total orderings using lexicographic cascades of eigenvalues as well as kernelized distances to matrix references are studied. Secondly, by decoupling the shape and the orientation of the ellipsoid associated to each PDS matrix, the supremum and infimum can be obtained by using a marginal supremum/infimum for the eigenvalues and a geometric matrix mean for the orthogonal basis. Thirdly, an estimate of the supremum and infimum associated to the Löwner ellipsoids are computed as the asymptotic cases of nonlinear averaging using the original notion of counter-harmonic mean for PDS matrices. Properties of the three introduced approaches are explored in detail, including also some numerical examples.


Orthogonal Basis Total Ordering Complete Lattice Mathematical Morphology Morphological Operator 
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  1. 1.
    Afsari, B.: Riemannian Lp center of mass: existence, uniqueness, and convexity. Proc. Am. Math. Soc. 139, 655–674 (2011)Google Scholar
  2. 2.
    Alvarez, L., Guichard, F., Lions, P.-L., Morel, J-M.: Axioms and fundamental equations of image processing. Arch. Ration. Mech. 123(3), 199–257 (1993)Google Scholar
  3. 3.
    Angulo, J.: Pseudo-morphological image diffusion using the counter-harmonic paradigm. In: Proceedings of Acivs’2010 (2010 Advanced Concepts for Intelligent Vision Systems), LNCS vol. 6474, Part I, pp. 426–437. Springer, New York (2010)Google Scholar
  4. 4.
    Arnaudon, M., Nielsen, F.: On Approximating the Riemannian 1-Center, arXiv, Hal-00560187, 2011Google Scholar
  5. 5.
    Arnaudon, M., Dombry, C., Phan, A., Yang, L.: Stochastic algorithms for computing means of probability measures, Preprint arXiv (2011)Google Scholar
  6. 6.
    Arsigny, V., Fillard, P., Pennec, X., Ayache, N.: Geometric means in a novel vector space structure on symmetric positive-definite matrices. SIAM. J. Matrix Anal. Appl. 29, 328–347 (2007)Google Scholar
  7. 7.
    Baksalary, J.K., Pukelsheim, F.: On the Löwner, minus, and start partial orderings of nonnegative definite matrices and their squares. Linear Algebra Appl. 151, 135–141 (1991)Google Scholar
  8. 8.
    Barbaresco, F.: New foundation of radar doppler signal processing based on advanced differential geometry of symmetric spaces: doppler matrix CFAR and radar application. In: Proceedings of International Radar Conference, Bordeaux, France (2009)Google Scholar
  9. 9.
    Barbaresco, F.: Geometric radar processing based on Fréchet distance: information geometry versus optimal transport theory. In: Proceedings of International Radar Conference, Washington, USA (2011)Google Scholar
  10. 10.
    Basser, P.J., Mattiello, J., LeBihan, D.: MR diffusion tensor spectroscopy and imaging. Biophys. J. 66, 259–267 (1994)Google Scholar
  11. 11.
    Bhatia, R.: Positive Definite Matrices. Princeton University Press, Princeton (2007)Google Scholar
  12. 12.
    Bonnabel, S., Sepulchre, R.: Geometric distance and mean for positive semi-definite matrices of fixed rank. SIAM. J. Matrix Anal. Appl. 31, 1055–1070 (2009)Google Scholar
  13. 13.
    Bonnabel, S., Sepulchre, R.: Rank-preserving geometric means of positive semi-definite matrices. arXiv:1007.5494v1, (2010)Google Scholar
  14. 14.
    Bullen, P.S.: Handbook of Means and Their Inequalities. 2nd edn, Springer, New York (1987)Google Scholar
  15. 15.
    Burgeth, B., Bruhn, A., Didas, S., Weickert, J., Welk, M.: Morphology for tensor data: ordering versus PDE-based approach. Image Vis. Comput. 25(4), 496–511 (2007)Google Scholar
  16. 16.
    Burgeth, B., Papenberg, N., Bruhn, A., Welk, M., Weickert, J.: Mathematical morphology for matrix fields induced by the loewner ordering in higher dimensions. Sig. Process. 87(2), 277–290 (2007)Google Scholar
  17. 17.
    Culver, W.J.: On the existence and uniqueness of the real logarithm of a matrix. Proc. American Math. Soc. 7(5), 1146–1151 (1966)Google Scholar
  18. 18.
    Ennis, D.B., Kindlmann, G.: Orthogonal tensor invariants and the analysis of diffusion tensor magnetic resonance images. Magn. Reson. Med. 55(1), 136–46 (2006)Google Scholar
  19. 19.
    Förstner, W., Gülch, E.: A fast operator for detection and precise location of distinct points, corners and centres of circular features. In: Proceedings of ISPRS Intercommission Conference on Fast Processing of Photogrammetric Data, pp. 281–304. Interlaken, Switzerland (1987)Google Scholar
  20. 20.
    Fréchet, M.: Les élements aléatoires de nature quelconque dans un espace distancié. Ann. Inst. H. Poincaré 10, 215–310 (1948)Google Scholar
  21. 21.
    Gonzalez, R.C., Woods, R.E.: Digital Image Processing. Wesley, Boston, USA (1992)Google Scholar
  22. 22.
    Groß, J.: Löwner partial ordering and space preordering of Hermitian non-negative definite matrices. Linear Algebra Appl. 326, 215–223 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Heijmans, H.J.A.M.: Morphological Image Operators. Academic Press, Boston (1994)zbMATHGoogle Scholar
  24. 24.
    Karcher, H.: Riemann center of mass and mollifier smoothing. Comm. Pure Appl. Math. 30, 509–541 (1997)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Kindlmann, G., San José Estépar, R., Niethammer, M., Haker, S., Westin, C.-F.: Geodesic-loxodromes for diffusion tensor interpolation and difference measurement. In: Proceedings of the 10th international conference on Medical image computing and computer-assisted intervention (MICCAI’07), (2007)Google Scholar
  26. 26.
    Maragos, P.: Slope transforms: theory and application to nonlinear signal processing. IEEE Trans. Sig. Process. 43(4), 864–877 (1995)Google Scholar
  27. 27.
    Mitra, S.K., Bhimasankaram, P., Malik, S.B.: Matrix partial orders, shorted operators and applications. Series in Algebra vol. 10, World Scientific, New Jersey (2010)Google Scholar
  28. 28.
    Moakher, M.: A differential geometric approach to the geometric mean of symmetric positive-definite matrices. SIAM. J. Matrix Anal. Appl. 26, 735–747 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Nielsen, F., Nock, R.: Approximating smallest enclosing balls with applications to machine learning. Int. J. Comput. Geometry Appl. 19(5), 389–414 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Osher, S., Rudin, L.I.: Feature-oriented image enhancement using shock filters. SIAM J. Numer. Anal. 27, 919–940 (1990)zbMATHCrossRefGoogle Scholar
  31. 31.
    Peeters, T.H.J.M., Rodrigues, P.R., Vilanova, A., ter Haar Romeny, B.M.: Analysis of distance/similarity measures for diffusion tensor imaging. In: Laidlaw, D.H., Weickert, J. (eds.) Visualization and Processing of Tensor Fields: Advances and Perspectives, pp. 113–136. Springer, Berlin (2009)Google Scholar
  32. 32.
    Peyré, G.: Numerical Tours of Signal Processing., (2012)
  33. 33.
    Serra, J.: Image Analysis and Mathematical Morphology. Academic Press, London (1982)zbMATHGoogle Scholar
  34. 34.
    Serra, J.: Anamorphoses and function lattices (Multivalued Morphology). In: Dougherty (ed.) Mathematical Morphology in Image Processing, pp. 483–523, Marcel-Dekker, New York (1992)Google Scholar
  35. 35.
    Serra, J.: The “false colour” problem. In: Proceedings of ISMM’09, pp. 13–23, Springer, New York (2009)Google Scholar
  36. 36.
    Soille, P.: Morphological Image Analysis. Springer, Berlin (1999)zbMATHGoogle Scholar
  37. 37.
    Stepniak, C.: Ordering of nonnegative definite matrices with applications to comparison of linear models. Linear Algebra Appl. 70, 67–71 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  38. 38.
    van Vliet, L.J.: Robust local max-min filters by normalized power-weighted filtering. In: Proceedings of IEEE 17th International Conference of the Pattern Recognition (ICPR’04), vol 1, pp. 696–699, (2004)Google Scholar
  39. 39.
    Velasco-Forero, S., Angulo, J.: Supervised ordering in \(R^n\): Application to morphological processing of hyperspectral images. IEEE Trans. Image Process. 20(11), 3301–3308 (2011)Google Scholar
  40. 40.
    Velasco-Forero, S., Angulo, J.: Mathematical morphology for vector images using statistical depth. In: Proceedings of ISMM’11 (2011 International Symposium on Mathematical Morphology), LNCS 6671, pp. 355–366, Springer, Berlin (2011)Google Scholar
  41. 41.
    Vemuri, B.C., Liu, M., Amari, S.-I., Nielsen, F.: Total bregman divergence and its applications to DTI analysis. IEEE Trans. Med. Imaging 30(2), 475–483 (2011)CrossRefGoogle Scholar
  42. 42.
    Weickert, J., Brox, T.: Diffusion and regularization of vector- and matrix-valued images. In: Inverse Problems, Image Analysis, and Medical, Imaging, vol. 313, pp. 251–268, AMS, Providence (2002)Google Scholar
  43. 43.
    Weickert, J., Hagen, H. (eds.): Visualization and Processing of Tensor Fields. Springer, Berlin (2006)Google Scholar
  44. 44.
    Welk, M.: Families of generalised morphological scale spaces. In: Proceedings of 4th International Conference of Scale-Space Methods in Computer Vision LNCS, vol. 2695, pp. 770–784, Springer, New York (2003)Google Scholar
  45. 45.
    Westin, C.F., Peled, S., Gudbjartsson, H., Kikinis, R., Jolesz, F.A.: Geometrical diffusion measures for MRI from tensor basis analysis. In: Proceedings of ISMRM ’97, p. 1742, (1997)Google Scholar
  46. 46.
    Yang, L.: Riemannian median and its estimation. LMS J. Comput. Math. 13, 461–479 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Yang, L.: Médianes de mesures de probabilité dans les variétés riemanniennes et applications à la détection de cibles radar. Thèse de Doctorat, Université de Poitiers, France (2011)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.CMM-Centre de Morphologie MathématiqueMathématiques et Systèmes, MINES ParisTechCedexFrance

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