Modeling Surfaces and Shapes

• Activity Number: 626
• Type: Invited
• Date/Time: Thursday, August 13, 2015 : 8:30 AM to 10:20 AM
• Sponsor: Statistical and Applied Mathematical Sciences Institute
• Author(s): Katharine Turner and Sayan Mukherjee* and Doug Boyer
• Companies: The University of Chicago and Duke University and Duke University
• Keywords: topologogy ; Euler integration ; shape statistics ; sufficient statistics

Abstract

We look at models for using geometry and topology for modeling surfaces and mixtures of subspaces of different dimension. We introduce a statistic, the persistent homology transform (PHT), to model surfaces in 3 dimensions and shapes in two dimensions. This statistic is a collection of persistence diagrams -- multiscale topological summaries used extensively in topological data analysis. We use the PHT to represent shapes and execute operations such as computing distances between shapes or classifying shapes. We prove the map from the space of simplicial complexes in three dimension into the space spanned by this statistic is injective. This implies that the statistic is a sufficient statistic for distributions on the space of "smooth" shapes. We also show that a variant of this statistic, the Euler Characteristic Transform (ECT), admits a simple exponential family formulation which is of use in providing likelihood based inference for shapes and surfaces. We illustrate the utility of this statistic on simulated and real data.