  In this paper we introduce a statistic, the persistent homology transform
(PHT), to model surfaces in $\mathbb{R}^3$ and shapes in $\mathbb{R}^2$. 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 simplicial
complexes in $\mathbb{R}^3$ into the space spanned by this statistic is
injective. This implies that the statistic is a sufficient statistic for
probability densities on the space of piecewise linear 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.
