## Section: New Results

Keywords : Computational topology, surface meshing, implicit surfaces, point set surfaces, surface learning, geometric probing.

### Surface approximation

#### Reconstruction with Voronoi centered radial basis functions

Participants : Marc Alexa, Pierre Alliez, Marie Samozino, Mariette Yvinec.

We consider [45] , [80] the problem of reconstructing a surface from scattered points sampled on a physical shape. The sampled shape is approximated as the zero level set of a function. This function is defined as a linear combination of compactly supported radial basis functions. We depart from previous work by using as centers of basis functions a set of points located on an estimate of the medial axis, instead of the input data points. Those centers are selected among the vertices of the Voronoi diagram of the sample data points. Being a Voronoi vertex, each center is associated with a maximal empty ball. We use the radius of this ball to adapt the support of each radial basis function. Our method can fit a user-defined budget of centers: the selected subset of Voronoi vertices is filtered using the notion of lambda medial axis, then clustered to fit the allocated budget.

#### Designing quadrangulations with discrete harmonic forms

Participants : Pierre Alliez, David Cohen-Steiner.

In collaboration with Mathieu Desbrun and Yiying Tong from Caltech.

We introduce a framework for quadrangle meshing of discrete manifolds. Based on discrete differential forms, our method hinges on extending the discrete Laplacian operator (used extensively in modeling and animation) to allow for line singularities and singularities with fractional indices. When assembled into a singularity graph, these line singularities are shown to considerably increase the design flexibility of quad meshing. In particular, control over edge alignments and mesh sizing are unique features of our approach. Another appeal of our method is its robustness and scalability from a numerical viewpoint: we solve a sparse linear system to generate a pair of piecewise-smooth scalar fields whose isocontours form a pure quadrangle tiling, with no T-junctions [47] .

#### Periodic global parameterization

Participant : Pierre Alliez.

In collaboration with Nicolas Ray, Bruno Lévy and Wan Chiu Li from Loria, and Alla Sheffer from University of British Columbia.

We present a new globally smooth parameterization method for surfaces of arbitrary topology [31] . Our method does not require any prior partition into charts nor any cutting. The chart layout (i.e., the topology of the base complex) and the parameterization emerge simultaneously from a global numerical optimization process. Given two orthogonal piecewise linear vector fields, our method computes two piecewise linear periodic functions, aligned with the input vector fields, by minimizing an objective function. The bivariate function they define is a smooth parameterization almost everywhere, except in the vicinity of the singular points of the vector field, where both the vector field and the derivatives of the parameterization vanish. Our method can construct quasi-isometric parameterizations at the expense of introducing additional singular points in non-developable regions where the curl of the input vector field is non-zero. We also propose a curvature-adapted parameterization method, that minimizes the curl and removes those additional singular points by adaptively scaling the parameterization. In addition, the same formalism is used to allow smoothing of the control vector fields. We demonstrate the versatility of our method by using it for quad-dominant remeshing and T-spline surface fitting. For both applications, the input vector fields are derived by estimating the principal directions of curvatures.