## Section: New Results

### Geometry Processing

Figure 4. A: triangulated surface (scanned mesh); B: periodic global parameterization; C: automatically constructed Spline surface. #### Optimization with dynamic function bases

We have presented in  a new framework for numerical optimization with dynamic function bases, and some preliminary results in Geometry Processing (we have also started to study this formulation in the context of light simulation). We consider the problem of the numerical approximation of the solutions of Partial Derivative Equations (or integro-differential equations in the case of global illumination). In its most general form, this class of problem can be expressed by the following equation: Lf= g where Lis a linear operator, fis the unknown function, and the function gis the right hand side. The classic Finite Element formulation (Galerkin) projects this equation onto a linear function basis . In our setting, the approximation of the solution is also represented in a function basis , but all those functions depend on an additional unknown vector of parameters p. For instance, we suppose that the function fis a bivariate piecewise linear function, defined on the faces of a Delaunay triangulation. This setting corresponds to exactly one basis function per vertex kof the triangulation, and the vector of parameters pthen corresponds to all the coordinates ( xk, yk) at all the vertices of the triangulation. The function fis then given by we will first explore the problem of minimizing the residual : The main difficulty comes from the non-linear dependencies introduced by the additional vector of parameters p. The other difficulty is that in the general case, the expression of the energy functional Fdepends on the value of the parameters p( Fis piecewise defined). To compute the fixed points of F, we have designed a general framework, based on Newton's algorithm: In practice, we will instanciate this general framework into different algorithms. We can now imagine a research program to instanciate our general framework in increasingly complex settings, that we have started studying in cooperation with our partners in AIMShape and ARC Georep :

• 1D: Laplace equation, Poisson equation ;

• 1D + t: Heat equation;

• 2D, L= Id : this trivial case corresponds to a function approximation problem. We have started to study the problem and an application for converting bitmap images into vector representations;

• 2D + t: fluid simulation with Navier-stokes;

• 3D, L= Id : Mesh-to-Spline conversion, this result is presented below.

• 3D: light simulation; optimizing for the parameters pcorresponds to a numerical approach to computing the visibility complex;

• 3D + t: various physics simulations.

We have presented in  a new method to automatically convert a mesh surface of arbitrary genus into a Spline (corresponding to the 3 D, L= Id case of our framework). The method is outlined in Figure 4 -A,B,C. Figure A shows the initial mesh model; Figure B shows the coordinate system computed by our method; Figure C shows the fitted Spline surface and its control mesh. A quadrilateral chart layout (i.e., the structure of the B-Spline basis functions) and the parameterization emerge simultaneously from a global numerical optimization process. Given the principal directions of curvature on the surface, our method computes two piecewise linear periodic functions aligned with these directions, by minimizing an objective function. The possible applications of our method comprise quad-dominant remeshing, texture mapping and T-spline surface fitting.

#### Interactive mesh editing

We have proposed in  a new topological data structure for representing a set of polygonal curves embedded in a meshed surface. In our representation, the vertices of the curve do not necessarily correspond to the vertices of the surface. The partition of the surface yielded by the intersecting curves is efficiently represented as a "cut-graph". The cut-graph stores combinatorial information of the network of curves. In our approach, the combinatorial form of information is systematically preferred to geometrical information since it improves both robustness and efficiency. Thanks to the topological data structure and algorithms, the cut-graph can be sketched through iterations of designing and erasing curves on the mesh surface in a "nondestructive" way, i.e. without modifying the mesh until the cutting operation is committed. We also demonstrate several prototype curve design tools inspired by 2D vector and bitmap graphics paradigms. We show how to sketch the cut-graph and how these tools can be combined.

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