A Framework for Joint Registration and Statistical Shape Analysis of 2D and 3D Objects

Shape analysis and modeling of 2D and 3D objects has important applications in many branches of science and engineering. The general goals in shape analysis include: derivation of efficient shape metrics, computation of shape templates, representation of dominant shape variability in a shape class, and development of probability models that characterize shape variation within and across classes. While past work on shape analysis is dominated by point representations -- finite sets of ordered or triangulated points on objects' boundaries -- the emphasis has lately shifted to continuous formulations.

The shape analysis of parametrized curves and surfaces introduces an additional shape invariance, the re-parametrization group, in additional to the standard invariants of rigid motions and global scales. Treating re-parametrization as a tool for registration of points across objects, we incorporate this group in shape analysis, in the same way orientation is handled in Procrustes analysis. For shape analysis of parametrized curves, I will describe an elastic Riemannian metric and a mathematical representation, called square-root-velocity-function (SRVF), that allows optimal registration and analysis using simple tools.

This framework provides proper metrics, geodesics, and sample statistics of shapes. These sample statistics are further useful in statistical modeling of shapes in different shape classes. Then, I will describe some preliminary extensions of these ideas to shape analysis of parametrized surfaces, I will demonstrate these ideas using applications from medical image analysis, protein structure analysis, 3D face recognition, and human activity recognition in videos.