Joint Registration and Shape Analysis of Curves and Surfaces

Abstract

The shapes of boundaries of objects in images can be conveniently represented as parameterized curves and surfaces. Their analysis, however, is made complicated by the fact that a re-parameterization is shape preserving, and one needs metrics and other analysis tools that are invariant to re-parameterizations. We summarize recent progress in developing Riemannian methods that remove the parameterization variability using equivalence relations and quotient spaces. They exploit the parameterization variability as a tool for registration, that is, matching of points across objects, and provide elastic Riemannian metrics that allow for both simultaneous registration and analysis of shapes. The resulting deformations and shape statistics improve performance over the existing methods in terms of feature preservation and model efficiency. While the original metrics are complicated, we discuss novel mathematical representations, of both curves and surfaces, that reduce shape spaces to standard Hilbert spaces and many of the existing algorithmic tools can be applied. We will illustrate these ideas using examples from shape analysis of curves and surfaces.