Issue No. 09 - Sept. (2012 vol. 34)
S. Kurtek , Dept. of Stat., Florida State Univ., Tallahassee, FL, USA
E. Klassen , Dept. of Math., Florida State Univ., Tallahassee, FL, USA
J. C. Gore , Inst. of Imaging Sci., Vanderbilt Univ., Nashville, TN, USA
Zhaohua Ding , Inst. of Imaging Sci., Vanderbilt Univ., Nashville, TN, USA
A. Srivastava , Dept. of Stat., Florida State Univ., Tallahassee, FL, USA
This paper presents a novel Riemannian framework for shape analysis of parameterized surfaces. In particular, it provides efficient algorithms for computing geodesic paths which, in turn, are important for comparing, matching, and deforming surfaces. The novelty of this framework is that geodesics are invariant to the parameterizations of surfaces and other shape-preserving transformations of surfaces. The basic idea is to formulate a space of embedded surfaces (surfaces seen as embeddings of a unit sphere in IR3) and impose a Riemannian metric on it in such a way that the reparameterization group acts on this space by isometries. Under this framework, we solve two optimization problems. One, given any two surfaces at arbitrary rotations and parameterizations, we use a path-straightening approach to find a geodesic path between them under the chosen metric. Second, by modifying a technique presented in , we solve for the optimal rotation and parameterization (registration) between surfaces. Their combined solution provides an efficient mechanism for computing geodesic paths in shape spaces of parameterized surfaces. We illustrate these ideas using examples from shape analysis of anatomical structures and other general surfaces.
Shape, Space vehicles, Three dimensional displays, Vectors, Orbits, Extraterrestrial measurements, geodesics., Shape analysis, Riemannian distance, parameterization invariance, path-straightening
A. Srivastava, Zhaohua Ding, E. Klassen, J. C. Gore and S. Kurtek, "Elastic Geodesic Paths in Shape Space of Parameterized Surfaces," in IEEE Transactions on Pattern Analysis & Machine Intelligence, vol. 34, no. , pp. 1717-1730, 2012.