Persistent homology transform for modeling shapes and surfaces

We introduce a statistic, the persistent homology transform (PHT), to model surfaces in R³ and shapes in R². 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 provide a constructive proof that the map from the space of simplicial complexes in R³ into the space spanned by this statistic is injective. This implies that we can use it to determine a metric on the space of piecewise linear shapes. Stability results justify that we can approximate this metric using finitely many persistence diagrams. We illustrate the utility of this statistic on simulated and real data.