Morphological Processing of Univariate Gaussian Distribution-Valued Images Based on Poincaré Upper-Half Plane Representation

  • Jesús AnguloEmail author
  • Santiago Velasco-Forero
Part of the Signals and Communication Technology book series (SCT)


Mathematical morphology is a nonlinear image processing methodology based on the application of complete lattice theory to spatial structures. Let us consider an image model where at each pixel is given a univariate Gaussian distribution. This model is interesting to represent for each pixel the measured mean intensity as well as the variance (or uncertainty) for such measurement. The aim of this work is to formulate morphological operators for these images by embedding Gaussian distribution pixel values on the Poincaré upper-half plane. More precisely, it is explored how to endow this classical hyperbolic space with various families of partial orderings which lead to a complete lattice structure. Properties of order invariance are explored and application to morphological processing of univariate Gaussian distribution-valued images is illustrated.


Ordered Poincaré half-plane Hyperbolic partial ordering Hyperbolic complete lattice Mathematical morphology Gaussian-distribution valued image Information geometry image filtering 


  1. 1.
    Angulo, J., Velasco-Forero, S.: Complete lattice structure of Poincaré upper-half plane and mathematical morphology for hyperbolic-valued images. In: Nielsen, F., Barbaresco, F. (eds.) Proceedings of First International Conference Geometric Science of Information (GSI’2013), vol. 8085, pp. 535–542. Springer LNCS (2013)Google Scholar
  2. 2.
    Arnaudon, M., Nielsen, F.: On approximating the riemannian 1-center. Comput. Geom. 46(1), 93–104 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Amari, S.-I., Barndorff-Nielsen, O.E., Kass, R.E., Lauritzen, S.L., Rao, C.R.: Differential geometry in statistical inference. Lecture Notes-Monograph Series, vol. 10, pp. 19–94, Institute of Mathematical Statistics, Hayward (1987)Google Scholar
  4. 4.
    Amari, S.-I., Nagaoka, H.: Methods of information geometry, translations of mathematical monographs. Am. Math. Soc. 191, (2000)Google Scholar
  5. 5.
    Barbaresco, F.: 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 (ETCV’08), Springer LNCS, Heidelberg vol. 5416, pp. 124–163, (2009)Google Scholar
  6. 6.
    Barbaresco, F.: Geometric radar processing based on Fréchet distance: information geometry versus optimal transport theory. In: Proceedings of IEEE International Radar Symposium (IRS’2011), pp. 663–668 (2011)Google Scholar
  7. 7.
    Barbaresco, F.: Information geometry of covariance matrix: cartan-siegel homogeneous bounded domains, Mostow/Berger fibration and fréchet median. In: Nielsen, F., Bhatia, R. (eds.) Matrix Information Geometry, pp. 199–255, Springer, Heidelberg (2013)Google Scholar
  8. 8.
    Bădoiu, M., Clarkson, K.L.: Smaller core-sets for balls. In: Proceedings of the Fourteenth annual ACM-SIAM Symposium on Discrete Algorithms (SIAM), pp. 801–802, ACM, New York(2003)Google Scholar
  9. 9.
    Burbea, J., Rao, C.R.: Entropy differential metric, distance and divergence measures in probability spaces: a unified approach. J. Multivar. Anal. 12(4), 575–96 (1982)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Cǎliman, A., Ivanovici, M., Richard, N.: Probabilistic pseudo-morphology for grayscale and color images. Pattern Recogn. 47, 721–35 (2004)CrossRefGoogle Scholar
  11. 11.
    Cammarota, V., Orsingher, E.: Travelling randomly on the poincaré half-plane with a pythagorean compass. J. Stat. Phys. 130(3), 455–82 (2008)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Cannon, J.W., Floyd, W.J., Kenyon, R., Parry, W.R.: Hyperbolic geometry. Flavors of Geometry, vol. 31, MSRI Publications, Cambridge (1997)Google Scholar
  13. 13.
    Chossat, P., Faugeras, O.: Hyperbolic planforms in relation to visual edges and textures perception. PLoS Comput. Biol. 5(12), p1 (2009)MathSciNetGoogle Scholar
  14. 14.
    Costa, S.I.R., Santos, S.A., Strapasson, J.E.: Fisher information matrix and hyperbolic geometry. In: Proc. of IEEE ISOC ITW2005 on Coding and Complexity, pp. 34–36, (2005)Google Scholar
  15. 15.
    Costa, S.I.R., Santos, S.A., Strapasson, J.E.: Fisher information distance: a geometrical reading, arXiv:1210:2354v1, p. 15 (2012)Google Scholar
  16. 16.
    Dodson, C.T.J., Matsuzoe, H.: An affine embedding of the gamma manifold. Appl. Sci. 5(1), 7–12 (2003)zbMATHMathSciNetGoogle Scholar
  17. 17.
    Frontera-Pons, J., Angulo, J.: Morphological operators for images valued on the sphere. In: Proceedings of IEEE ICIP’12 ( IEEE International Conference on Image Processing), pp. 113–116, Orlando (Florida), USA, October (2012)Google Scholar
  18. 18.
    Fuchs, L.: Partially Ordered Algebraic Systems. Pergamon, Oxford (1963)Google Scholar
  19. 19.
    Guts, A.K.: Mappings of families of oricycles in lobachevsky space. Math. USSR-Sb. 19, 131–8 (1973)CrossRefGoogle Scholar
  20. 20.
    Guts, A.K.: Mappings of an ordered lobachevsky space. Siberian Math. J. 27(3), 347–61 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Heijmans, H.J.A.M.: Morphological Image Operators. Academic Press, Boston (1994)zbMATHGoogle Scholar
  22. 22.
    Heijmans, H.J.A.M., Keshet, R.: Inf-semilattice approach to self-dual morphology. J. Math. Imaging Vis. 17(1), 55–80 (2002)CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Keshet, R.: Mathematical morphology on complete semilattices and its applications to image processing. Fundamenta Informaticæ 41, 33–56 (2000)zbMATHMathSciNetGoogle Scholar
  24. 24.
    Meyer, F.: Vectorial Levelings and Flattenings. In: Mathematical Morphology and its Applications to Image and Signal Processing (Proc. of ISMM’02), pp. 51–60, Kluwer Academic Publishers, Dordrecht (2000)Google Scholar
  25. 25.
    Nielsen, F., Nock, R.: On the smallest enclosing information disk. Inform. Process. Lett. 105, 93–7 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Nielsen, F., Nock. R.: Hyperbolic voronoi diagrams made easy. In: Proceedings of the 2010 IEEE International Conference on Computational Science and Its Applications, pp. 74–80, IEEE Computer Society, Washington (2010)Google Scholar
  27. 27.
    Sachs, Z.: Classification of the isometries of the upper half-plane, p. 14. University of Chicago, VIGRE REU (2011)Google Scholar
  28. 28.
    Sbaiz, L., Yang, F., Charbon, E., Süsstrunk, S., Vetterli, M.: The gigavision camera. In: Proceedings of IEEE ICASSP’09, pp. 1093–1096 (2009)Google Scholar
  29. 29.
    Serra, J.: Image Analysis and Mathematical Morphology. Vol II: theoretical advances, Academic Press, London (1988)Google Scholar
  30. 30.
    Shaked, M., Shanthikumar, G.: Stochastic Orders and Their Applications. Associated Press, New York (1994)zbMATHGoogle Scholar
  31. 31.
    Soille, P.: Morphological Image Analysis. Springer-Verlag, Berlin (1999)CrossRefzbMATHGoogle Scholar
  32. 32.
    Treibergs, A.: The hyperbolic plane and its immersions into \(\mathbb{R}^3\), Lecture Notes in Department of Mathematics, p. 13. University of Utah (2003)Google Scholar

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© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.CMM-Centre de Morphologie MathématiqueMathématiques et Systèmes, MINES ParisTechParisFrance
  2. 2.Department of MathematicsNational University of SingaporeBuona VistaSingapore

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