Elastic Riemannian frameworks and statistical tools for shape analysis

Interest in shapes of 3D objects naturally leads to shape analysis of curves and surfaces. The theme of this talk is Riemannian frameworks that offer certain distinct advantages. In addition to providing measures for shape comparisons, clustering and retrieval, a Riemannian framework also provides optimal deformations between shapes, statistical averaging of observed shapes, and probabilistic modeling of shapes in different shape classes. The use of such a framework involves the following issues. Firstly, one needs to be invariant to reparameterizations, in addition to the standard shape-preserving transformations of rigid motions and global scalings. The solution is to use mathematical representations (of curves and surfaces) and elastic Riemannian metrics such that re-parameterization groups act by isometries. Secondly, one needs tools for computing geodesic paths between given objects in shape spaces. We have developed a numerical method, called pathstraightening, for this purpose. Next, one needs tools for computing sample statistics. We have adapted definitions and techniques from statistical analysis on Riemannian manifolds for computing means and covariances of shapes. Further, we have used these moments in defining Gaussian-type distributions on shape spaces. These models are useful in statistical shape classification, hypothesis testing and Bayesian shape extractions from images. From the perspective of shape retrieval, these tools contribute in hierarchical organizations of shape databases and shape metrics that relate to probabilistic models for shape classes. This framework is general enough to incorporate other information, such as landmarks, colors, or other annotations along the shapes in the analysis. I will demonstrate these ideas using examples from vision, biometrics, bioinformatics, and medical image analysis. This work has been done in collaboration with several researchers.