Advertisement

Lie Bodies: A Manifold Representation of 3D Human Shape

  • Oren Freifeld
  • Michael J. Black
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7572)

Abstract

Three-dimensional object shape is commonly represented in terms of deformations of a triangular mesh from an exemplar shape. Existing models, however, are based on a Euclidean representation of shape deformations. In contrast, we argue that shape has a manifold structure: For example, summing the shape deformations for two people does not necessarily yield a deformation corresponding to a valid human shape, nor does the Euclidean difference of these two deformations provide a meaningful measure of shape dissimilarity. Consequently, we define a novel manifold for shape representation, with emphasis on body shapes, using a new Lie group of deformations. This has several advantages. First we define triangle deformations exactly, removing non-physical deformations and redundant degrees of freedom common to previous methods. Second, the Riemannian structure of Lie Bodies enables a more meaningful definition of body shape similarity by measuring distance between bodies on the manifold of body shape deformations. Third, the group structure allows the valid composition of deformations. This is important for models that factor body shape deformations into multiple causes or represent shape as a linear combination of basis shapes. Finally, body shape variation is modeled using statistics on manifolds. Instead of modeling Euclidean shape variation with Principal Component Analysis we capture shape variation on the manifold using Principal Geodesic Analysis. Our experiments show consistent visual and quantitative advantages of Lie Bodies over traditional Euclidean models of shape deformation and our representation can be easily incorporated into existing methods.

Keywords

Shape deformation Lie group Statistics on manifolds 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Balan, A., Sigal, L., Black, M., Davis, J., Haussecker, H.: Detailed human shape and pose from images. In: CVPR, pp. 1–8 (2007)Google Scholar
  2. 2.
    Hasler, N., Ackermann, H., Rosenhahn, B., Thormahlen, T., Seidel, H.: Multilinear pose and body shape estimation of dressed subjects from image sets. In: CVPR, pp. 1823–1830 (2010)Google Scholar
  3. 3.
    Salzmann, M., Fua, P.: Linear local models for monocular reconstruction of deformable surfaces. PAMI 33, 931–944 (2011)CrossRefGoogle Scholar
  4. 4.
    Weiss, A., Hirshberg, D., Black, M.: Home 3D body scans from noisy image and range data. In: ICCV, pp. 1951–1958 (2011)Google Scholar
  5. 5.
    Anguelov, D., Srinivasan, P., Koller, D., Thrun, S., Rodgers, J., Davis, J.: SCAPE: Shape completion and animation of people. ACM ToG 24, 408–416 (2005)Google Scholar
  6. 6.
    Sumner, R., Popović, J.: Deformation transfer for triangle meshes. ACM ToG 23, 399–405 (2004)Google Scholar
  7. 7.
    Hasler, N., Stoll, C., Sunkel, M., Rosenhahn, B., Seidel, H.: A statistical model of human pose and body shape. Computer Graphics Forum 28, 337–346 (2009)CrossRefGoogle Scholar
  8. 8.
    Chao, I., Pinkall, U., Sanan, P., Schröder, P.: A simple geometric model for elastic deformations. ACM ToG 29, 38:1–38:6 (2010)CrossRefGoogle Scholar
  9. 9.
    Grenander, U.: General pattern theory: A mathematical study of regular structures. Clarendon Press (1993)Google Scholar
  10. 10.
    Grenander, U., Miller, M.: Pattern theory: From representation to inference. Oxford University Press, USA (2007)zbMATHGoogle Scholar
  11. 11.
    Fletcher, P., Lu, C., Joshi, S.: Statistics of shape via principal geodesic analysis on Lie groups. In: CVPR, vol. 1, pp. 95–101 (2003)Google Scholar
  12. 12.
    Fletcher, P.T., Joshi, S.: Principal Geodesic Analysis on Symmetric Spaces: Statistics of Diffusion Tensors. In: Sonka, M., Kakadiaris, I.A., Kybic, J. (eds.) CVAMIA/MMBIA 2004. LNCS, vol. 3117, pp. 87–98. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  13. 13.
    Balan, A.: Detailed Human Shape and Pose from Images. PhD thesis, Brown Univ., Providence, RI (2010)Google Scholar
  14. 14.
    Robinette, K., Blackwell, S., Daanen, H., Boehmer, M., Fleming, S., Brill, T., Hoeferlin, D., Burnsides, D.: Civilian American and European Surface Anthropometry Resource (CAESAR) final report. AFRL-HE-WP-TR-2002-0169, US AFRL (2002)Google Scholar
  15. 15.
    Grenander, U., Miller, M.: Representations of knowledge in complex systems. J. Royal Stat. Soc. B (Methodological), 549–603 (1994)Google Scholar
  16. 16.
    Freifeld, O., Weiss, A., Zuffi, S., Black, M.: Contour people: A parameterized model of 2D articulated human shape. In: CVPR, pp. 639–646 (2010)Google Scholar
  17. 17.
    Miller, M., Christensen, G., Amit, Y., Grenander, U.: Mathematical textbook of deformable neuroanatomies. PNAS 90, 11944–11948 (1993)zbMATHCrossRefGoogle Scholar
  18. 18.
    Grenander, U., Miller, M.: Computational anatomy: An emerging discipline. Quarterly of Applied Math. 56, 617–694 (1998)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Fletcher, P.T., Joshi, S., Lu, C., Pizer, S.M.: Gaussian Distributions on Lie Groups and Their Application to Statistical Shape Analysis. In: Taylor, C.J., Noble, J.A. (eds.) IPMI 2003. LNCS, vol. 2732, pp. 450–462. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  20. 20.
    Alexa, M.: Linear combination of transformations. ACM ToG 21(3), 380–387 (2002)MathSciNetGoogle Scholar
  21. 21.
    Kilian, M., Mitra, N., Pottmann, H.: Geometric modeling in shape space. ACM ToG 26(3), 64:1–64:8 (2007)Google Scholar
  22. 22.
    Kendall, D.: Shape manifolds, Procrustean metrics, and complex projective spaces. Bull. London Math. Soc. 16, 81–121 (1984)MathSciNetzbMATHCrossRefGoogle Scholar
  23. 23.
    Lee, J.: Introduction to smooth manifolds, vol. 218. Springer (2003)Google Scholar
  24. 24.
    Freifeld, O., Black, M.: Lie bodies: Supplemental material. MPI-IS-TR-005, Max Planck Institute for Intelligent Systems (2012)Google Scholar
  25. 25.
    Murray, R., Li, Z., Sastry, S.: A mathematical introduction to robotic manipulation. CRC Press (1994)Google Scholar
  26. 26.
    Higham, N.: The scaling and squaring method for the matrix exponential revisited. SIAM J. Matrix Analysis and Applications 26(4), 1179–1193 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Karcher, H.: Riemannian center of mass and mollifier smoothing. Comm. Pure and Applied Math. 30, 509–541 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    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

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Oren Freifeld
    • 1
  • Michael J. Black
    • 2
  1. 1.Division of Applied MathematicsBrown UniversityProvidenceUSA
  2. 2.Max Planck Institute for Intelligent SystemsTübingenGermany

Personalised recommendations