*Comparing 3D objects using shape analysis techniques*

The ever-growing availability of data acquired by different devices has been inducing an exponential growth of the volume of the available data and has been deeply modifying the research approach in shape analysis and reasoning. These data are often stored using different data formats and have to be analysed, interpreted and stored with significant computational efforts and experts’ commitment. Therefore it is everyday more urgent the definition of strategies for the description and representation of data in an efficient way, as well as the definition of tools that are able to perform computations and similarity quantification over this kind of information even in presence of partial data or affected by noise.

In this talk, I will present a brief overview on the existing computational techniques for the topological analysis of 3D data based on Morse theory and persistent homology. On the basis of these achievements it is possible to effectively simplify the data and to efficiently store their structure in a compact and salient description. Finally, I will present recent results on the description and comparison of large, multi-variate 3D data and I will discuss the applicative challenges I foresee we will face in the next years.

*Discrete Tomography and Applications.*

Image reconstruction by means of X-rays bases on the early work of J. Radon (1917), who made use of continuous geometry and analytical properties to provide the theoretical tools to attack this inverse problem.

However, in real applications, discretization in several steps is always required, so that the reconstruction problem must be reconsidered from a discrete point of view. This leads to Discrete Tomography, where the term "discrete" can be variously interpreted.

In particular, Discrete Tomography deals with the reconstruction of images when only a few projection directions can be considered. Differently from the continuous model, the usual approaches mainly base on linear algebra and discrete geometric models. A related problem is that several different solutions could satisfy the same set of projections.

After presenting some of the main issues connected to discrete image reconstruction, the talk will focus on some uniqueness results. Examples and applications will be discussed and commented.

**Bernd Gärtner** (ETH, Zürich)

*The Many Facets of Smallest Enclosing Balls*

Every useful new technique in computational geometry must be applicable to the problem of computing the convex hull of a set of points in the plane. This folklore statement (to be taken with a grain of salt) has a counterpart in the subfield of geometric optimization, in the form that every new technique must be applicable to the problem of computing the smallest enclosing ball of a set of points in d-dimensional space. Indeed, the latter problem is an excellent showcase for classic and new techniques in geometric optimization, such as random sampling, multiplicative weight update, core sets, sparse approximation, and pruning / screening of data.

In this talk, I will survey some of these techniques and try to explain them in the simplest possible way, using smallest enclosing balls.

*Mean curvature of unstructured point clouds: a varifold approach*

Varifolds are tools from geometric measure theory which were introduced by Almgren in the context of Plateau’s problem. They carry both spatial and tangential informations, and they have nice properties in a variational context. It appears that many discrete surfaces, e.g. point clouds, meshes, pixel or voxel approximations, can be associated with a notion of "discrete varifolds". The talk will be devoted to approximation properties of such discrete varifolds, and to a new definition of approximated mean curvature which arises in this context. Numerical evaluations on various 2D and 3D point clouds will be provided. This is a joint work with Blanche Buet (Paris 11) and Gian Paolo Leonardi (Modena).