Shape of Elastic Strings in Euclidean Space
- 172 Downloads
We construct a 1-parameter family of geodesic shape metrics on a space of closed parametric curves in Euclidean space of any dimension. The curves are modeled on homogeneous elastic strings whose elasticity properties are described in terms of their tension and rigidity coefficients. As we change the elasticity properties, we obtain the various elastic models. The metrics are invariant under reparametrizations of the curves and induce metrics on shape space. Analysis of the geometry of the space of elastic strings and path spaces of elastic curves enables us to develop a computational model and algorithms for the estimation of geodesics and geodesic distances based on energy minimization. We also investigate a curve registration procedure that is employed in the estimation of shape distances and can be used as a general method for matching the geometric features of a family of curves. Several examples of geodesics are given and experiments are carried out to demonstrate the discriminative quality of the elastic metrics.
KeywordsShape analysis Shape space Shape geodesics Elastic shapes Shape manifold
Unable to display preview. Download preview PDF.
- Belongie, S., Malik, J., & Puzicha, J. (2002). Shape matching and object recognition using shape context. PAMI, 24, 509–522. Google Scholar
- Cohen, I., Ayache, N., & Sulger, P. (1992). Tracking points on deformable objects using curvature information. In Lecture notes in computer science (vol. 588). Berlin: Springer. Google Scholar
- do Carmo, M. P. (1994). Riemannian geometry. Basel: Birkhauser. Google Scholar
- Grenander, U. (1993). General pattern theory. Oxford: Oxford University Press. Google Scholar
- Joshi, S., Klassen, E., Srivastava, A., & Jermyn, I. (2007). An efficient representation for computing geodesics between n-dimensional elastic shapes. In IEEE conference on computer vision and pattern recognition. Google Scholar
- Klassen, E., & Srivastava, A. (2006). Geodesics between 3D closed curves using path straightening. In European conference on computer vision (ECCV). Google Scholar
- Ling, H., & Jacobs, D. (2007). Shape classification using the inner distance. PAMI, 29(2), 286–299. Google Scholar
- Michor, P., Mumford, D., Shah, J., & Younes, L. (2007). A metric on shape space with explicit geodesics. arXiv:0706.4299v1.
- Mio, W., Bowers, J. C., Hurdal, M. K., & Liu, X. (2007a). Modeling brain anatomy with 3D arrangements of curves. In IEEE 11th international conference on computer vision (pp. 1–8). Google Scholar
- Mumford, D. (2002). Pattern theory: The mathematics of perception. In Proc. of the international congress of mathematicians. Beijing, China. Google Scholar
- Sebastian, T. B., Klein, P. N., & Kimia, B. B. (2004). Recognition of shapes by editing their shock graphs. PAMI, 26(5), 550–571. Google Scholar
- Shah, J. (2006). An H 2 type Riemannian on the space of planar curves. In Workshop on the mathematical foundations of computational anatomy (MICCAI). Google Scholar
- Söderkvist, O. (2001). Computer vision classification of leaves from Swedish trees. Master’s thesis, Linköping University. Google Scholar
- Tu, Z., & Yuille, A. (2004). Shape matching and recognition using generative models and informative features. In European conference on computer vision (ECCV) (pp. 195–209). Google Scholar
- Zheng, X., Chen, Y., Groisser, D., & Wilson, D. (2005). Some new results on non-rigid correspondence and classification of curves. In EMMCVPR (pp. 473–489). Google Scholar