Title
Statistical Analysis of Functions, Curves and Surfaces
Speaker
Sebastian Kurtek, Florida State University
Abstract
In this talk we will consider the problems of functional data analysis and shape analysis of parameterized curves and surfaces. The main underlying issue in these problems is the variability in data due to random parameterizations. Our approach is to form the action of parameterization groups on function spaces of interest and to remove parameterization variability using algebraic quotient operations. This requires metrics and representations that lead to desired invariances. In the case of real-valued functions, we use the Fisher-Rao Riemannian metric and the square-root velocity function (SRVF) representation to perform function registration, and to separate phase and amplitude variability in the given data. For shape analysis of 2D and 3D curves, we have developed an elastic Riemannian metric and a mathematical representation that facilitates elastic shape analysis of curves. For shape analysis of 3D objects using their parameterized surfaces, we introduced a novel mathematical representation that allows their simultaneous registration and comparisons. In all of these applications, an important tool is the computation of geodesic paths on appropriate non-linear manifolds between given objects. The resulting geodesics are used to compute sample statistics and probability models on shape and function manifolds. I will demonstrate these ideas using multiple applications including medical image analysis, analysis of protein backbones, handwriting analysis and others.
