Abstract
In order to analyze shapes of continuous curves in ℝ3, we parameterize them by arc-length and represent them as curves on a unit two-sphere. We identify the subset denoting the closed curves, and study its differential geometry. To compute geodesics between any two such curves, we connect them with an arbitrary path, and then iteratively straighten this path using the gradient of an energy associated with this path. The limiting path of this path-straightening approach is a geodesic. Next, we consider the shape space of these curves by removing shape-preserving transformations such as rotation and re-parametrization. To construct a geodesic in this shape space, we construct the shortest geodesic between the all possible transformations of the two end shapes; this is accomplished using an iterative procedure. We provide step-by-step descriptions of all the procedures, and demonstrate them with simple examples.
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.
Download to read the full chapter text
Chapter PDF
Similar content being viewed by others
References
Boothby, W.M.: An Introduction to Differential Manifolds and Riemannian Geometry. Academic Press, Inc., London (1986)
Bronstein, A.M., Bronstein, M.M., Kimmel, R.: Three-dimensional face recognition. International Journal of Computer Vision 64(1), 5–30 (2005)
Dziuk, G., Kuwert, E., Schatzle, R.: Evolution of elastic curves in R n: existence and computation. SIAM J. Math. Anal 33, 1228–1245 (2002)
Eakins, J.P., Shields, K., Boardman, J.: ARTISAN – a shape retrieval system based on boundary family indexing. In: Storage and Retrieval for Still Image and Video Databases IV. Proceedings SPIE, vol. 2670, pp. 17–28 (1996)
Grenander, U., Miller, M.I.: Computational anatomy: An emerging discipline. Quarterly of Applied Mathematics LVI(4), 617–694 (1998)
Kazhdan, M.M.: Shape Representations and Algorithms for 3D Model Retrieval. PhD thesis, Computer Science, Princeton (April 2004)
Klassen, E., Srivastava, A.: A path-straightening method for finding geodesics in shape spaces of closed curves in ℝ3. SIAM Journal of Applied Mathematics, page in review (2005)
Klassen, E., Srivastava, A., Mio, W., Joshi, S.: Analysis of planar shapes using geodesic paths on shape spaces. IEEE Patt. Analysis and Machine Intell. 26(3), 372–383 (2004)
Langer, J., Singer, D.A.: Curve straightening and a minimax argument for closed elastic curves. Topology 24, 75–88 (1985)
Langer, J., Singer, D.A.: Curve straightening in Riemannian manifolds. Ann. Global Anal. Geom. 5, 133–150 (1987)
Michor, P.W., Mumford, D.: Riemannian geometries on spaces of plane curves. Journal of the European Mathematical Society (to appear, 2005)
Mio, W., Srivastava, A.: Elastic string models for representation and analysis of planar shapes. In: Proc. of IEEE Computer Vision and Pattern Recognition (2004)
Mio, W., Srivastava, A., Klassen, E.: Interpolation by elastica in Euclidean spaces. Quarterly of Applied Mathematics LXII(2), 359–378 (2004)
Mokhtarian, F., Abbasi, S., Kittler, J.: Efficient and robust shape retrieval by shape content through curvature scale space. In: Proceedings of First International Conference on Image Database and MultiSearch, pp. 35–42 (1996)
Novotni, M., Klein, R.: A geometric approach to 3D object comparison. In: International Conference on Shape Modeling and Applications, pp. 167–175 (2001)
Osada, R., Funkhouser, T., Chazelle, B., Dobkin, D.: Matching 3D models with shape distributions. In: International Conference on Shape Modeling and Applications, pp. 154–166 (2001)
Palais, R.S.: Morse theory on Hilbert manifolds. Topology 2, 299–340 (1963)
Samir, C., Srivastava, A., Daoudi, M.: Human face recognition using 2D facial curves. In: International Conference on Acoustic, Speech, and Signal Processing, ICASSP (May 2006)
Younes, L.: Optimal matching between shapes via elastic deformations. Journal of Image and Vision Computing 17(5/6), 381–389 (1999)
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
Klassen, E., Srivastava, A. (2006). Geodesics Between 3D Closed Curves Using Path-Straightening. In: Leonardis, A., Bischof, H., Pinz, A. (eds) Computer Vision – ECCV 2006. ECCV 2006. Lecture Notes in Computer Science, vol 3951. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11744023_8
Download citation
DOI: https://doi.org/10.1007/11744023_8
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-33832-1
Online ISBN: 978-3-540-33833-8
eBook Packages: Computer ScienceComputer Science (R0)