Exploring statistical shape analysis with manifolds and fiber bundles

In this talk, we will explore how geometric objects such as manifolds and fiber bundles can aid in studying the statistics of shape space. We will begin by discussing the concept of shape space as a manifold and review the use of Diffusion Maps (DM) in analyzing the underlying structure of the space. We will also present a new parameter tuning method for DM that takes advantage of the semigroup properties of the diffusion kernel. Next, we will consider the shape space as approximately a fiber bundle and review Horizontal Diffusion Maps (HDM). We will introduce a probabilistic model for learning the generative process of data on fiber bundles, and define Gaussian processes on fiber bundles. Throughout the talk, we will use examples from geometric morphometrics to illustrate the use of these methods.