## Section: New Results

Keywords : Geometric inference, computational topology.

### Computation and stability of geometric features

#### Vines and vineyards by updating persistence in linear time

Participant : David Cohen-Steiner.

In collaboration with Herbert Edelsbrunner and Dmitriy Morozov from Duke University.

Persistent homology is the mathematical core of recent work on shape, including reconstruction, recognition, and matching. Its pertinent information is encapsulated by a pairing of the critical values of a function, visualized by points forming a diagram in the plane. The original persistence algorithm computes the pairs from an ordering of the simplices in a triangulation and takes worst-case time cubic in the number of simplices. The main result of this paper [40] is an algorithm that maintains the pairing in worst-case linear time per transposition in the ordering. A side-effect of the algorithm's analysis is an elementary proof of the stability of persistence diagrams in the special case of piecewise-linear functions. We use the algorithm to compute 1-parameter families of diagrams which we apply to folding trajectories of proteins.

#### A sampling theory for compacts sets in Euclidean space

Participants : Frédéric Chazal, David Cohen-Steiner.

In collaboration with A. Lieutier from Dassault Systèmes and LMC/IMAG.

We introduce [39] a parameterized notion of feature size that interpolates between the minimum of the local feature size, and the recently introduced weak feature size. Based on this notion of feature size, we propose sampling conditions that apply to noisy samplings of general compact sets in Euclidean space. These conditions are sufficient to ensure the topological correctness of a reconstruction given by an offset of the sampling. Our approach also yields new stability results for medial axes, critical points and critical values of distance functions.

#### Second fundamental measure of geometric sets and local approximation of curvatures

Participant : David Cohen-Steiner.

In collaboration with J.M. Morvan from IGD (Lyon).

Using the theory of normal cycles, we associate with each geometric subset of a Riemannian manifold a —tensor-valued— curvature measure, which we call its second fundamental measure [25] . This measure provides a finer description of the geometry of singular sets than the standard curvature measures. Moreover, we deal with approximation of curvature measures. We get a local quantitative estimate of the difference between curvature measures of two geometric subsets, when one of them is a smooth hypersurface.