Abstract
A general formulation for geodesic distance propagation of surfaces is presented. Starting from a surface lying on a 3-manifold in IR4, we set up a partial differential equation governing the propagation of surfaces at equal geodesic distance (on the 3-manifold) from the given original surface. This propagation scheme generalizes a result of Kimmel et al. [11] and provides a way to compute distance maps on manifolds. Moreover, the propagation equation is generalized to any number of dimensions. Using an eulerian formulation with level-sets, it gives stable numerical algorithms for computing distance maps. This theory is used to present a new method for surface matching which generalizes a curve matching method [5]. Matching paths are obtained as the orbits of the vector field defined as the sum of two distance maps’ gradient values. This surface matching technique applies to the case of large deformation and topological changes.
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
R. Abraham and J.E. Marsden and T.S. Ratiu, Manifolds, tensor analysis and applications, Springer Verlag, 1988.
G. Borgefors, Distance Transformations in Arbitrary Dimensions, Journ. of CV-GIP, vol. 27, p. 321–345, 1984.
V. Caselles, R. Kimmel, R. and G. Sapiro, Geodesic Active Contours, Int. Journal of Computer Vision, vol 22, No 1, February 1997.
I. Cohen and N. Ayache and P. Sulger, Tracking Points on Deformable Objects Using Curvature Information, Proc. of ECCV, p. 458–466, May 1992.
I. Cohen and I. Herlin, Curves Matching Using Geodesic Paths, Proceedings of CVPR’98, page 741–746.
M.P. do Carmo, Differential Geometry of Curves and Surfaces, Ed. Prentice-Hall, Englewood Cliffs, 1976.
C.L. Epstein and M. Gage, The curve shortening flow, Wave motion: Theory, modeling and Computation, Ed. Springer-Verlag, 1987.
O. Faugeras and R. Keriven, Variational Principles, Surface Evolution, PDE’s, level-set methods and the Stereo Problem, INRIA, RR-3021, 1996.
A. Guéziec and N. Ayache, Smoothing and matching of 3D space curves, Int. J. of Comp. Vision, vol. 12, No 1, p. 79–104, February 1994.
G. Hermosillo, O. Faugeras, and J. Gomes, Cortex Unfolding Using Level Set Methods, INRIA, RR-3663, April 1999.
R. Kimmel, A. Amir and A.F. Bruckstein, Finding shortest paths on surfaces using levelset propagation, Journal of PAMI, vol. 17, No 6, p 635–640, 1995.
R. Kimmel and A. Bruckstein, Tracking Level Sets by Level Sets: A Method for Solving the Shape from Shading Problem, J. of CVIU, vol. 62, No 1, p. 47–58, July 1995.
R. Kimmel and J. Sethian, Computing Geodesic Paths on Manifolds, Proceedings of the National Academy of Sciences July 1998.
S. Osher and J. Sethian, Fronts Propagating with Curvature Dependent Speed: Algorithms Based on Hamilton-Jacobi Formulations, J. of Comput. Physics, vol 79, p. 12–49, 1998.
B. Serra and M. Berthod, Optimal subpixel matching of contour chains and segments, Proc. of ICCV, p. 402–407, June 1995.
J. Sethian, Level Set Methods: Evolving Interfaces in Geometry, Fluid Mechanics, Computer Vision and Materials Sciences, Cambridge University Press, 1996.
M. Spivak, A comprehensive introduction to differential geometry, Vol I, Publish or Perish, Berkeley, 1971.
Thompson, P. and Toga, A, A surface-based technique for warping 3-dimensional images of the brain. IEEE Trans. on Med. Imag, p 402–417, 1996
T. Pajdla and L. Van Gool, Matching of 3D curves using semi differential invariants, Proc. of ICCV, p. 390–395, June 1995.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2000 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Huot, E.G., Yahia, H.M., Cohen, I., Herlin, I.L. (2000). Surface Matching with Large Deformations and Arbitrary Topology: A Geodesic Distance Evolution Scheme on a 3-Manifold. In: Computer Vision - ECCV 2000. ECCV 2000. Lecture Notes in Computer Science, vol 1842. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45054-8_50
Download citation
DOI: https://doi.org/10.1007/3-540-45054-8_50
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-67685-0
Online ISBN: 978-3-540-45054-2
eBook Packages: Springer Book Archive