## Section: New Results

### Algorithmic foundations

#### Computing the Volume of a Union of Balls: a Certified Algorithm

Participants : Frédéric Cazals, Sébastien Loriot.

In collaboration with H. Kanhere, master student at the Indian Institute of Technology, Bombay.

Balls and spheres are amongst the simplest 3D modeling primitives, and computing the volume of a union of balls is an elementary problem. Although a number of strategies addressing this problem have been investigated in several communities, we are not aware of any robust algorithm, and present the first such algorithm [20] .

Our calculation relies on the decomposition of the volume of the union into convex regions, namely the restrictions of the balls to their regions in the power diagram. Theoretically, we establish a formula for the volume of a restriction, based on Gauss' divergence theorem. The proof being constructive, we develop the associated algorithm. On the implementation side, we carefully analyse the predicates and constructions involved in the volume calculation, and present a certified implementation relying on interval arithmetic. The result is certified in the sense that the exact volume belongs to the interval computed using the interval arithmetic.

Experimental results are presented on hand-crafted models presenting various difficulties, as well as on the 58,898 models found in the 2009-07-10 release of the Protein Data Bank.

#### Reconstructing 3D compact sets

Participant : Frédéric Cazals.

In collaboration with D. Cohen-Steiner, from Geometrica, INRIA Sophia-Antipolis.

Reconstructing a 3D shape from sample points is a central problem faced in medical applications, reverse engineering, natural sciences, cultural heritage projects, etc. While these applications motivated intense research on 3D surface reconstruction, the problem of reconstructing more general shapes hardly received any attention. This paper [19] develops a reconstruction algorithm changing the 3D reconstruction paradigm as follows.

First, the algorithm handles general shapes i.e. compact sets as
opposed to surfaces. Under mild assumptions on the sampling of the
compact set, the reconstruction is proved to be correct in terms of
homotopy type.
Second, the algorithm does not output a single reconstruction but a
nested sequence of *plausible* reconstructions.
Third, the algorithm accommodates topological persistence so as to select the most
stable features only.
Finally, in case of reconstruction failure, it enables the
identification of under-sampled areas, so as to possibly fix the
sampling.

These key features are illustrated by experimental results on challenging datasets, and should prove instrumental in enhancing the processing of such datasets in the aforementioned applications.

#### Robust and Efficient Delaunay triangulations of points on or close to a sphere

Participant : Sébastien Loriot.

In collaboration with Manuel Caroli, Pedro M.M. de Castro, Monique Teillaud, and Camille Wormser. M. Caroli, P. M.M. de Castro and M. Teillaud are with Geometrica, INRIA Sophia-Antipolis. C. Wormser is with the CS Dpt, ETH Zurich.

We propose two approaches for computing the Delaunay triangulation of points on a sphere, or of rounded points close to a sphere [18] . Both approaches are based on the classic incremental algorithm initially designed for the plane. The space of circles gives the mathematical background for this work. We implemented the two approaches in a fully robust way, building upon existing generic algorithms provided by the cgal library. The effciency and scalability of the method is shown by benchmarks