Section: New Results

Algorithmic Foundations

Computational geometry, Computational topology, Voronoi diagrams, $\alpha$-shapes, Morse theory.

Greedy Geometric Algorithms for Collection of Balls, with Applications to Geometric Approximation and Molecular Coarse-Graining

Participants : Frédéric Cazals, Tom Dreyfus.

In collaboration with S. Sachdeva (Princeton University, USA), and N. Shah (Carnegie Mellon University, USA).

Choosing balls to best approximate a 3D object is a non trivial problem. To answer it, in [18] , we first address the inner approximation problem, which consists of approximating an object ${ℱ}_{𝒪}$ defined by a union of $n$ balls with $k<n$ balls defining a region ${ℱ}_{𝒮}\subset {ℱ}_{𝒪}$. This solution is further used to construct an outer approximation enclosing the initial shape, and an interpolated approximation sandwiched between the inner and outer approximations.

The inner approximation problem is reduced to a geometric generalization of weighted max $k$-cover, solved with the greedy strategy which achieves the classical $1-1/e$ lower bound. The outer approximation is reduced to exploiting the partition of the boundary of ${ℱ}_{𝒪}$ by the Apollonius Voronoi diagram of the balls defining the inner approximation.

Implementation-wise, we present robust software incorporating the calculation of the exact Delaunay triangulation of points with degree two algebraic coordinates, of the exact medial axis of a union of balls, and of a certified estimate of the volume of a union of balls. Application-wise, we exhibit accurate coarse-grain molecular models using a number of balls 20 times smaller than the number of atoms, a key requirement to simulate crowded cellular environments.

Towards Morse Theory for Point Cloud Data

Participants : Frédéric Cazals, Christine Roth.

In collaboration with C. Robert (IBPC / CNRS, Paris, France), and C. Mueller (ETH, Zurich).

Morse theory provides a powerful framework to study the topology of a manifold from a function defined on it, but discrete constructions have remained elusive due to the difficulty of translating smooth concepts to the discrete setting.

Consider the problem of approximating the Morse-Smale (MS) complex of a Morse function from a point cloud and an associated nearest neighbor graph (NNG). While following the constructive proof of the Morse homology theorem, we present novel concepts for critical points of any index, and the associated Morse-Smale diagram [19] .

Our framework has three key advantages. First, it requires elementary data structures and operations, and is thus suitable for high-dimensional data processing. Second, it is gradient free, which makes it suitable to investigate functions whose gradient is unknown or expensive to compute. Third, in case of under-sampling and even if the exact (unknown) MS diagram is not found, the output conveys information in terms of ambiguous flow, and the Morse theoretical version of topological persistence, which consists in canceling critical points by flow reversal, applies.

On the experimental side, we present a comprehensive analysis of a large panel of bi-variate and tri-variate Morse functions whose Morse-Smale diagrams are known perfectly, and show that these diagrams are recovered perfectly.

In a broader perspective, we see our framework as a first step to study complex dynamical systems from mere samplings consisting of point clouds.