Section: New Results

Dynamic Function Bases

Participants : Bruno Lévy, Vincent Nivoliers, Yang Liu, Feng Sun, Rhaleb Zayer.

In the frame of our GoodShape project, we study geometry processing problems with the specific point of view of computing an optimal function basis. To reach this goal, we explore different strategies, and revisit them with the formalism of numerical optimization. As a mean of computing an efficient function basis, we study Centroidal Voronoi Tessellations and spectral methods, as described in the following two paragraphs. The so-computed function basis will be used as the fundamental tool for new light simulation methods that we try to develop (see below).

Centroidal Voronoi Tessellations

Figure 6. Two examples of a Centroidal Voronoi Tessellation adapted to a background density function computed by our algorithm (in 2D and 3D). The used background density function is defined by = e^{-20(x2 + y2)} + 0.05sin²(x)sin²(y) (with an additional term in z for the 3D example)

Optimization technique for Faster Centroidal Voronoi Tessellation (CVT): CVT is an essential tool in many scientific fields, that can be used to compute the optimal sampling of a given signal. In Figure 6 , we show a CVT adapted to a background density function, computed by our algorithm mentioned below. For large-scale problems, the popular Lloyd relaxation is not fast enough to achieve local minimum due to its linear convergence rate. Our previous work shows that Limited memory Quasi-Newton method (for instance, L-BFGS) is a better method which preserves sparsity and simplicity for our CVT program. We published our efficient CVT algorithm in ACM Transactions on Graphics [12] .

An important application of CVT is isotropic remeshing of 3D models. We developped an efficient algorithm to compute the restricted Voronoi diagram in 3D, i.e. the intersection between a 3D Voronoi diagram and a polygonal mesh embedded in 3D. Our algorithm uses two graph traversals in parallel [16] . A symbolic encoding of vertices configurations allows for numerical opimization with the Newton framework. As a result, meshes of high quality (near equilateral triangles) can be obtained. We also developed a generalization of the algorithm to sample a 3D volume with line segments and graphs. The main application is fitting a skeleton to a 3D model [13] .

We are currently working on extending this framework in two different ways : (1) we introduce a new objective function (L_p -CVT), that approximates the metric. Minimizing this new objective function allows to generate quad-dominant and hex-dominant meshes. (2) we study the problem of anisotropic remeshing from the point of view of embedding the Riemannian manifold defined by the domain to be meshed and its anisotropy into a higher-dimensional space, using Nash's embedding theorem, then meshing this higher-dimensional object isotropically, and finally re-projecting into 3D space.

Spectral methods

Figure 7. Examples of non-linear spectral mesh editing: our new function basis allows to take into account complex deformation modes, that cannot be captured by classical (linear) methods.

We continued our research program started in 2006 about spectral geometry processing methods. We developed a shell based approach for mesh deformation and editing (Figure 7 ). The approach also can take advantage of modal analysis of the surface models and a partitioning approach for efficiently solving the arising eigenvalue problem. cooperation with Alexander Belyaev (Heriot-Watt University, Edinburgh, Scotland, UK), Jens Kerber and Art Tevs (MPI Informatik, Saarbrücken, Germany) . We developed an intuitive artistic tool which allows compressing the depth range of a given scene without compromising the visual quality of surface features. The presented algorithm allows for real time computation, thanks to our implementation on graphics hardware. Hence, besides the interactive design of still results, our method offers the possibility for generating animated Bas-Reliefs.

We presented a course on Spectral Mesh Processing at SIGGRAPH Asia [17] .

Applications to light simulation

Light simulation is a very active topic in the computer graphics community. In the frame of his Ph.D. (started in Oct. 2008), Vincent Nivoliers studies a dynamic basis formulation of the problem. Among the methods used to obtain satisfactory results, radiosity aims at finding an approximate solution to the general light equation problem. The formulation of this problem fits well into the dynamic function basis framework, which could be used to quickly find both a good sampling of the scene, and the best approximation on this sampling. This method would avoid the use of discontinuity meshing, and provide a light solution without requiring hierarchical sampling. The problem of the illumination of a scene can be translated into an integral equation. The general solution of this equation cannot be computed in closed form, therefore, the usual method is to restrict the problem on a specific function space which both approximates the general L² function space of the solution, and has a simple basis on which to project. Most approaches of the problem use hierarchical function basis, to refine the solution were needed, and to compute large-scale interactions with fewer coefficients. In the dynamic function basis formalism, the function basis changes during the optimisation step to fit the solution and enhance the accuracy of the approximation. We experimented the Dynamic Function Basis framework in two different settings, for image approximation, and for sampling direct lighting in the presence of shadows. In this latter configuration, the results are encouraging. The samples are aligned along the direction of the gradient of illumination, and shadow boundaries are well captured.