Abstract
Klassen et al. [9] recently developed a theoretical formulation to model shape dissimilarities by means of geodesics on appropriate spaces. They used the local geometry of an infinite dimensional manifold to measure the distance dist(A,B) between two given shapes A and B. A key limitation of their approach is that the computation of distances developed in the above work is inherently unstable, the computed distances are in general not symmetric, and the computation times are typically very large. In this paper, we revisit the shooting method of Klassen et al. for their angle-oriented representation. We revisit explicit expressions for the underlying space and we propose a gradient descent algorithm to compute geodesics. In contrast to the shooting method, the proposed variational method is numerically stable, it is by definition symmetric, and it is up to 1000 times faster.
This work was supported by the German Research Foundation, grant #CR-250/1-1.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Bergtholdt, M., Schnörr, C.: Shape Priors and Online Appearance Learning for Variational Segmentation and Object Recognition in Static Scenes. In: Kropatsch, W.G., Sablatnig, R., Hanbury, A. (eds.) DAGM 2005. LNCS, vol. 3663, pp. 342–350. Springer, Heidelberg (2005)
Bürgisser, P., Clausen, M., Shokrollahi, M.A.: Algebraic Complexity Theory. In: Grundlehren der Mathematischen Wissenschaften, vol. 315. Springer, Heidelberg (1997)
Chern, S., Chen, W., Lam, K.S.: Lectures on Differential Geometry. World Scientific, Singapore (1999)
Cremers, D., Kohlberger, T., Schnörr, C.: Shape statistics in kernel space for variational image segmentation. Pattern Recognition 36(9), 1929–1943 (2003)
Cremers, D., Soatto, S.: A pseudo-distance for shape priors in level set segmentation. In: Paragios, N. (ed.) IEEE 2nd Int. Workshop on Variational, Geometric and Level Set Methods, Nice, pp. 169–176 (2003)
do Carmo, M.P.: Differential Geometry of Curves and Surfaces, 503 pages. Prentice-Hall, Englewood Cliffs (1976)
Dryden, I.L., Mardia, K.V.: Statistical Shape Analysis. Wiley, Chichester (1998)
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)
Klassen, E., Srivastava, A., Mio, W., Joshi, S.H.: Analysis of planar shapes using geodesic paths on shape spaces. IEEE Trans. Pattern Anal. Mach. Intell. 26(3), 372–383 (2003)
Marques, J.S., Abrantes, A.J.: Shape alignment – optimal initial point and pose estimation. Pattern Recognition Letters 18(1), 49–53 (1997)
Michor, P., Mumford, D.: Riemannian geometries on spaces of plane curves. J. of the European Math. Society (2003)
Mokhtarian, F., Abbasi, S., Kittler, J.: Efficient and robust retrieval by shape content through curvature scale space (1996)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Schmidt, F.R., Clausen, M., Cremers, D. (2006). Shape Matching by Variational Computation of Geodesics on a Manifold. In: Franke, K., Müller, KR., Nickolay, B., Schäfer, R. (eds) Pattern Recognition. DAGM 2006. Lecture Notes in Computer Science, vol 4174. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11861898_15
Download citation
DOI: https://doi.org/10.1007/11861898_15
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-44412-1
Online ISBN: 978-3-540-44414-5
eBook Packages: Computer ScienceComputer Science (R0)