Advertisement

Anisotropic Distributions on Manifolds: Template Estimation and Most Probable Paths

  • Stefan SommerEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9123)

Abstract

We use anisotropic diffusion processes to generalize normal distributions to manifolds and to construct a framework for likelihood estimation of template and covariance structure from manifold valued data. The procedure avoids the linearization that arise when first estimating a mean or template before performing PCA in the tangent space of the mean. We derive flow equations for the most probable paths reaching sampled data points, and we use the paths that are generally not geodesics for estimating the likelihood of the model. In contrast to existing template estimation approaches, accounting for anisotropy thus results in an algorithm that is not based on geodesic distances. To illustrate the effect of anisotropy and to point to further applications, we present experiments with anisotropic distributions on both the sphere and finite dimensional LDDMM manifolds arising in the landmark matching problem.

Keywords

Template estimation Manifold Diffusion Geodesics Frame bundle Most probable paths Anisotropy 

References

  1. 1.
    Andersson, L., Driver, B.K.: Finite dimensional approximations to wiener measure and path integral formulas on manifolds. J. Funct. Anal. 165(2), 430–498 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Fletcher, P., Lu, C., Pizer, S., Joshi, S.: Principal geodesic analysis for the study of nonlinear statistics of shape. IEEE Trans. Med. Imaging 23(8), 995–1005 (2004)CrossRefGoogle Scholar
  3. 3.
    Fujita, T., Kotani, S.I.: The Onsager-Machlup function for diffusion processes. J. Math. Kyoto Univ. 22(1), 115–130 (1982)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Hsu, E.P.: Stochastic Analysis on Manifolds. American Mathematical Society, Providence (2002)zbMATHCrossRefGoogle Scholar
  5. 5.
    Huckemann, S., Hotz, T., Munk, A.: Intrinsic shape analysis: geodesic PCA for Riemannian manifolds modulo isometric Lie group actions. Statistica Sin. 20, 1–100 (2010)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Joshi, S., Davis, B., Jomier, B.M., B, G.G.: Unbiased diffeomorphic atlas construction for computational anatomy. NeuroImage 23, 151–160 (2004)CrossRefGoogle Scholar
  7. 7.
    Joshi, S., Miller, M.: Landmark matching via large deformation diffeomorphisms. IEEE Trans. Image Process. 9(8), 1357–1370 (2000)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Kendall, D.G.: Shape manifolds, procrustean metrics, and complex projective spaces. Bull. Lond. Math. Soc. 16(2), 81–121 (1984)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Micheli, M.: The differential geometry of landmark shape manifolds: metrics, geodesics, and curvature. Ph.D. thesis, Brown University, Providence, USA (2008)Google Scholar
  10. 10.
    Michor, P.W.: Topics in Differential Geometry. American Mathematical Society, Providence (2008)zbMATHCrossRefGoogle Scholar
  11. 11.
    Michor, P.W., Mumford, D.: Riemannian geometries on spaces of plane curves. J. Eur. Math. Soc. 8, 1–48 (2004)MathSciNetGoogle Scholar
  12. 12.
    Mok, K.P.: On the differential geometry of frame bundles of Riemannian manifolds. J. Fur Die Reine Angew. Math. 1978(302), 16–31 (1978)MathSciNetGoogle Scholar
  13. 13.
    Nye, T.: Construction of distributions on Tree-Space via diffusion processes. Mathematisches Forschungsinstitut Oberwolfach (2014). http://www.mfo.de/document/1440a/preliminary_OWR_2014_44.pdf
  14. 14.
    Pennec, X.: Intrinsic statistics on Riemannian manifolds: basic tools for geometric measurements. J. Math. imaging Vis. 25(1), 127–154 (2006)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Siddiqi, K., Pizer, S.: Medial Representations: Mathematics, Algorithms and Applications. Computational Imaging and Vision, 1st edn. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  16. 16.
    Sommer, S.: Horizontal dimensionality reduction and iterated frame bundle development. In: Nielsen, F., Barbaresco, F. (eds.) GSI 2013. LNCS, vol. 8085, pp. 76–83. Springer, Heidelberg (2013) CrossRefGoogle Scholar
  17. 17.
    Sommer, S.: Diffusion processes and PCA on manifolds. Mathematisches Forschungsinstitut Oberwolfach (2014). http://www.mfo.de/document/1440a/preliminary_OWR_2014_44.pdf
  18. 18.
    Sommer, S., Lauze, F., Hauberg, S., Nielsen, M.: Manifold valued statistics, exact principal geodesic analysis and the effect of linear approximations. In: Daniilidis, K., Maragos, P., Paragios, N. (eds.) ECCV 2010, Part VI. LNCS, vol. 6316, pp. 43–56. Springer, Heidelberg (2010) CrossRefGoogle Scholar
  19. 19.
    Strichartz, R.S.: Sub-Riemannian geometry. J. Differ. Geom. 24(2), 221–263 (1986). http://projecteuclid.org/euclid.jdg/1214440436 zbMATHMathSciNetGoogle Scholar
  20. 20.
    Tipping, M.E., Bishop, C.M.: Probabilistic principal component analysis. J. Roy. Stat. Soc. B 61(3), 611–622 (1999)zbMATHMathSciNetCrossRefGoogle Scholar
  21. 21.
    Vaillant, M., Miller, M., Younes, L., Trouv, A.: Statistics on diffeomorphisms via tangent space representations. NeuroImage 23(Supplement 1), S161–S169 (2004)CrossRefGoogle Scholar
  22. 22.
    Vialard, F.-X., Risser, L.: Spatially-varying metric learning for diffeomorphic image registration: a variational framework. In: Golland, P., Hata, N., Barillot, C., Hornegger, J., Howe, R. (eds.) MICCAI 2014, Part I. LNCS, vol. 8673, pp. 227–234. Springer, Heidelberg (2014) Google Scholar
  23. 23.
    Younes, L.: Shapes and Diffeomorphisms. Springer, Heidelberg (2010) zbMATHCrossRefGoogle Scholar
  24. 24.
    Zhang, M., Fletcher, P.: Probabilistic principal geodesic analysis. In: NIPS, pp. 1178–1186 (2013)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.DIKUUniversity of CopenhagenCopenhagenDenmark

Personalised recommendations