On Shape of Plane Elastic Curves
- 450 Downloads
We study shapes of planar arcs and closed contours modeled on elastic curves obtained by bending, stretching or compressing line segments non-uniformly along their extensions. Shapes are represented as elements of a quotient space of curves obtained by identifying those that differ by shape-preserving transformations. The elastic properties of the curves are encoded in Riemannian metrics on these spaces. Geodesics in shape spaces are used to quantify shape divergence and to develop morphing techniques. The shape spaces and metrics constructed are novel and offer an environment for the study of shape statistics. Elasticity leads to shape correspondences and deformations that are more natural and intuitive than those obtained in several existing models. Applications of shape geodesics to the definition and calculation of mean shapes and to the development of shape clustering techniques are also investigated.
Keywordsplanar shapes shape geodesics mean shape shape analysis clustering shapes
Unable to display preview. Download preview PDF.
- Bookstein, F.L. 1986. Size and shape spaces for landmark data in two dimensions. Statistical Science, 1:181–242.Google Scholar
- Cohen, I., Ayache, N., and Sulger, P. 1992. Tracking points on deformable objects using curvature information. In Lecture Notes in Computer Science, vol. 588.Google Scholar
- do Carmo, Manfredo, 1994. Riemannian Geometry. Birkhauser.Google Scholar
- Dryden, I.L. and Mardia, K.V. 1998. Statistical Shape Analysis. John Wiley & Son.Google Scholar
- Grenander, U. 1993. General Pattern Theory. Oxford University Press.Google Scholar
- Karcher, H. 1977. Riemann center of mass and mollifier smoothing. Comm. Pure Appl. Math., 30:509–541. vGoogle Scholar
- Mennucci, A. and Yezzi, A. 2004. Metrics in the space of curves. Technical report.Google Scholar
- Michor, P. and Mumford, D. Riemannian geometries on spaces of plane curves. J. Eur. Math. Soc. in press.Google Scholar
- Sharon, E. and Mumford, D. 2004. 2D-Shape analysis using conformal mappings. In Proceedings of IEEE Conference on Computer Vision, pp. 350–357.Google Scholar
- Thompson, D.W. 1992. On Growth and Form: The Complete Revised Edition. Dover.Google Scholar
- Younes, L. 1999. Optimal matching between shapes via elastic deformations. Journal of Image and Vision Computing, 17(5/6):381–389.Google Scholar