Skip to main content
Fachgebiete
Automobil + Motoren Bauwesen + Immobilien Business IT + Informatik Elektrotechnik + Elektronik Energie + Nachhaltigkeit Finance + Banking Management + Führung Marketing + Vertrieb Maschinenbau + Werkstoffe Versicherung + Risiko
DE
EN
Bücher
Zeitschriften
Themenseiten
Organisationspsychologie Projektmanagement Marketing Smart Manufacturing
Weitere Formate
Podcasts Webinare Technik Webinare Wirtschaft Kongresse Veranstaltungskalender Awards
Jetzt Einzelzugang starten
Zugang für Unternehmen
Referenzkunden
SLX-Digitalkonferenz: Zukunftswerkstatt 2024

Mehr Umsatz mit Top-Kontakten

Am 7. Mai erfahren Sie, wie Vertriebsteams Leadgenerierung, Leadnurturing und Leadstrategien effizient steuern können.
Erweiterte Suche
Anmelden
nach oben
Erschienen in:
Buchtitelbild

2013 | OriginalPaper | Buchkapitel

1. Supremum/Infimum and Nonlinear Averaging of Positive Definite Symmetric Matrices

verfasst von : Jesús Angulo

Erschienen in: Matrix Information Geometry

Verlag: Springer Berlin Heidelberg

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config
loading …

Abstract

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.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 102.000 Bücher
  • über 537 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 390 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




 

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 67.000 Bücher
  • über 340 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Jetzt Wissensvorsprung sichern!

Jetzt informieren
Nächstes Kapitel The Riemannian Mean of Positive Matrices
Literatur
1.
Afsari, B.: Riemannian Lp center of mass: existence, uniqueness, and convexity. Proc. Am. Math. Soc. 139, 655–674 (2011)
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)
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)
4.
Arnaudon, M., Nielsen, F.: On Approximating the Riemannian 1-Center, arXiv, Hal-00560187, 2011
5.
Arnaudon, M., Dombry, C., Phan, A., Yang, L.: Stochastic algorithms for computing means of probability measures, Preprint arXiv (2011)
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)
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)
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)
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)
10.
Basser, P.J., Mattiello, J., LeBihan, D.: MR diffusion tensor spectroscopy and imaging. Biophys. J. 66, 259–267 (1994)
11.
Bhatia, R.: Positive Definite Matrices. Princeton University Press, Princeton (2007)
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)
13.
Bonnabel, S., Sepulchre, R.: Rank-preserving geometric means of positive semi-definite matrices. arXiv:1007.5494v1, (2010)
14.
Bullen, P.S.: Handbook of Means and Their Inequalities. 2nd edn, Springer, New York (1987)
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)
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)
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)
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)
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)
20.
Fréchet, M.: Les élements aléatoires de nature quelconque dans un espace distancié. Ann. Inst. H. Poincaré 10, 215–310 (1948)
21.
Gonzalez, R.C., Woods, R.E.: Digital Image Processing. Wesley, Boston, USA (1992)
22.
Groß, J.: Löwner partial ordering and space preordering of Hermitian non-negative definite matrices. Linear Algebra Appl. 326, 215–223 (2001)MathSciNetMATHCrossRef
23.
Heijmans, H.J.A.M.: Morphological Image Operators. Academic Press, Boston (1994)MATH
24.
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)
26.
Maragos, P.: Slope transforms: theory and application to nonlinear signal processing. IEEE Trans. Sig. Process. 43(4), 864–877 (1995)
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)
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)MathSciNetMATHCrossRef
29.
Nielsen, F., Nock, R.: Approximating smallest enclosing balls with applications to machine learning. Int. J. Comput. Geometry Appl. 19(5), 389–414 (2009)MathSciNetMATHCrossRef
30.
Osher, S., Rudin, L.I.: Feature-oriented image enhancement using shock filters. SIAM J. Numer. Anal. 27, 919–940 (1990)MATHCrossRef
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)
32.
33.
Serra, J.: Image Analysis and Mathematical Morphology. Academic Press, London (1982)MATH
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)
35.
Serra, J.: The “false colour” problem. In: Proceedings of ISMM’09, pp. 13–23, Springer, New York (2009)
36.
Soille, P.: Morphological Image Analysis. Springer, Berlin (1999)MATH
37.
Stepniak, C.: Ordering of nonnegative definite matrices with applications to comparison of linear models. Linear Algebra Appl. 70, 67–71 (1985)MathSciNetMATHCrossRef
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)
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)
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)
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)CrossRef
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)
43.
Weickert, J., Hagen, H. (eds.): Visualization and Processing of Tensor Fields. Springer, Berlin (2006)
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)
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)
46.
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)
Metadaten
Titel
Supremum/Infimum and Nonlinear Averaging of Positive Definite Symmetric Matrices
verfasst von
Jesús Angulo
Copyright-Jahr
2013
Verlag
Springer Berlin Heidelberg
Buch
Matrix Information Geometry

Print ISBN: 978-3-642-30231-2

Electronic ISBN: 978-3-642-30232-9

Copyright-Jahr: 2013

DOI
https://doi.org/10.1007/978-3-642-30232-9_1

Neuer Inhalt

    Bildnachweise