Abstract

Statistical analysis of shapes of objects is an important topic area. Here tools are developed on a foundation of appropriate mathematical representations and shape metrics. While a large body of prior work is based on registered landmarks, more complex data demands more general solutions. In this talk we will look at some recent developments, with a special focus on the registration problem. Registration is the process of identifying points across objects. It not only preserves important structures in the data but also leads to more parsimonious statistical models for capturing shape variability. In case of parametrized curves and surfaces, the registration step is akin to finding optimal parametrizations. Taking three fundamentally different examples: (1) real-valued function data, (2) parametrized curves in Euclidean spaces, and (3) parametrized surfaces in R3, I will describe a Riemannian framework that achieves the following goals. It provides an analysis of shapes of objects that is invariant to standard similarity transformations and, additionally, to parametrizations of these objects. This framework, called elastic shape analysis, incorporates an optimal registration of points across objects while simultaneously providing proper metrics, geodesics, and sample statistics of shapes. These sample statistics are further useful in statistical modeling of shapes in different shape classes. I will demonstrate these ideas using applications from medical image analysis, protein structure analysis, 3D face recognition, and human activity recognition in videos.