Sufficient statistics for shapes and surfaces

In this paper we introduce a statistic, the persistent homology transform (PHT), to model objects and surfaces in $\R^3$ and shapes in $\R^2$. 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 a simplicial complexes in $\R^3$ 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 illustrate the utility of this statistic on simulated and real data.