Persistent geometry

This thesis is dedicated to geometric inference, and more specifically to the estimation of curvatures of objects in Euclidean space from an approximating set that is close in the Hausdorff distance. In order to extend the filtering property of persistent homology to the realm of geometry, we introduce the framework of persistent geometry. It consists in combining connections between the topology and the curvatures of a subset of R^d provided by integral geometry, such as the principal kinematic formula, with persistence theory thanks to the so-called image persistence modules. We develop a new method to estimate the intrinsic volumes of a set, which are global quantities built from the curvatures of the set; particular intrinsic volumes include boundary area, Euler characteristic, and mean curvature. Our method allows for the recovery of the intrinsic volumes of a set from any approximating set up to an error that is linear with respect to the Hausdorff distance between them. We show that this approximation is valid as long as the estimated set has bounded total curvature and a positive mu-reach for some mu in (0,1). The mu-reach is a relaxation of the reach of Federer defined to extend geometric inference results to possibly non-smooth, non-convex sets. The class of compact sets of R^d having bounded total curvatures and a positive mu-reach for an arbitrary mu in (0,1) is broad, containing compact C^1 submanifolds, compact convex sets, polyhedra, and more generally many compact stratified subsets of R^d.To deal with these possibly singular sets, we use tools from different fields of mathematics, such as non-smooth analysis, geometric measure theory, and Morse theory. In particular, a crucial stepin our reasoning consists in the development of Morse theory for offsets of a set at regular values of its distance function. We show that the topology of sublevel sets of smooth maps restricted to such objects — which are not C^2 manifolds — typically evolves by the gluing of cells around each critical point, just as in the classical Morse theory on C^2 manifolds.