Removing Shape-Preserving Transformations in Square-Root Elastic (SRE) Framework for Shape Analysis of Curves

  • Shantanu H. Joshi
  • Eric Klassen
  • Anuj Srivastava
  • Ian Jermyn
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4679)

Abstract

This paper illustrates and extends an efficient framework, called the square-root-elastic (SRE) framework, for studying shapes of closed curves, that was first introduced in [2]. This framework combines the strengths of two important ideas - elastic shape metric and path-straightening methods - for finding geodesics in shape spaces of curves. The elastic metric allows for optimal matching of features between curves while path-straightening ensures that the algorithm results in geodesic paths. This paper extends this framework by removing two important shape preserving transformations: rotations and re-parameterizations, by forming quotient spaces and constructing geodesics on these quotient spaces. These ideas are demonstrated using experiments involving 2D and 3D curves.

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References

  1. 1.
    Grenander, U.: General Pattern Theory. Oxford University Press, Oxford (1993)Google Scholar
  2. 2.
    Joshi, S., Klassen, E., Srivastava, A., Jermyn, I.H.: An efficient representation for computing geodesics between n-dimensional elastic shapes. In: Proc. IEEE Computer Vision and Pattern Recognition (CVPR), Minneapolis, USA (June 2007)Google Scholar
  3. 3.
    Klassen, E., Srivastava, A.: Geodesics between 3D closed curves using path straightening. In: Leonardis, A., Bischof, H., Pinz, A. (eds.) ECCV 2006. LNCS, vol. 3951, pp. 95–106. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  4. 4.
    Klassen, E., Srivastava, A., Mio, W., Joshi, S.H.: Analysis of planar shapes using geodesic paths on shape spaces. IEEE Trans. Pattern Analysis and Machine Intelligence 26(3), 372–383 (2004)CrossRefGoogle Scholar
  5. 5.
    Michor, P.W., Mumford, D.: Riemannian geometries on spaces of plane curves. J. Eur. Math. Soc. 8, 1–48 (2006)MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Mio, W., Srivastava, A.: Elastic-string models for representation and analysis of planar shapes. In: Proc. IEEE Conf. Comp. Vision and Pattern Recognition, pp. 10–15 (2004)Google Scholar
  7. 7.
    Mio, W., Srivastava, A., Joshi, S.H.: On shape of plane elastic curves. International Journal of Computer Vision 73(3), 307–324 (2007)CrossRefGoogle Scholar
  8. 8.
    Palais, R.S.: Morse theory on Hilbert manifolds. Topology 2, 299–349 (1963)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Samir, C., Srivastava, A., Daoudi, M.: Three-dimensional face recognition using shapes of facial curves. IEEE Trans. Pattern Anal. Mach. Intell. 28(11), 1858–1863 (2006)CrossRefGoogle Scholar
  10. 10.
    Sebastian, T.B., Klein, P.N., Kimia, B.B.: On aligning curves. IEEE Transactions on Pattern Analysis and Machine Intelligence 25(1), 116–125 (2003)CrossRefGoogle Scholar
  11. 11.
    Shah, J.: An H2 type riemannian metric on the space of planar curves. In: Larsen, R., Nielsen, M., Sporring, J. (eds.) MICCAI 2006. LNCS, vol. 4190, Springer, Heidelberg (2006)Google Scholar
  12. 12.
    Spivak, M.: A Comprehensive Introduction to Differential Geometry, vol. I & II. Publish or Perish, Inc., Berkeley (1979)Google Scholar
  13. 13.
    Younes, L.: Computable elastic distance between shapes. SIAM Journal of Applied Mathematics 58(2), 565–586 (1998)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Shantanu H. Joshi
    • 1
  • Eric Klassen
    • 2
  • Anuj Srivastava
    • 3
  • Ian Jermyn
    • 4
  1. 1.Dept. of Electrical Engineering, Florida State University, Tallahassee, FL 32310USA
  2. 2.Dept. of Mathematics, Florida State University, Tallahassee, FL 32306USA
  3. 3.Dept. of Statistics, Florida State University, Tallahassee, FL 32306USA
  4. 4.INRIA Sophia Antipolis, B.P. 93, 06902, CedexFrance

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