## Section: New Results

### Dynamic Function Bases

Participants : Bruno Lévy, Vincent Nivoliers, Bruno Vallet, Yang Liu, 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

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. However, the approximated Hessian matrix by L-BFGS only contains enough curvature information and it has only r-linear convergence rate in theory. We investigate the state of the art in large-scale optimization technique such as Truncated Newton, Modified Newton and other LMQN methods on CVT computation and integrate the preconditioned LBFGS method which use the true Hessian and is faster than traditional L-BFGS method. The goal of our work is to develop super-linear optimization methods for fast CVT computation both in convergence and computational time. Our algorithm is described in an article (accepted pending revisions in ACM Transactions on Graphics).

Isotropic and anisotropic CVT under Euclidean metric are already well used in computer graphics and numerical simulation. Under other metrics, the behavior and usage of CVT are not well known and studied. For instance, CVT under metric can be a good way to produce nice quadrilateral tiling. Inspired by the concept of CVT, we are pursuing to produce nice 2D quadrilateral meshing and 3D hexahedral meshing with the aid of CVT. We propose a novel generalized CVT concept based on convex distance function and directional and stretch information. The present work includes approximated Voronoi tessellation computation via graphics hardware, dynamical seed-insertion and removal, quad and hexahedral-cell generation.

#### Spectral methods

We continued our research program started in 2006 about spectral geometry processing methods. As a result, we designed the Manifold Harmonics Transform [20] , a formalism to compute on 3D meshes the equivalent of the Fourier transform (or the Discrete Cosine Transform to be more precise). This makes it possible to completely port the signal processing framework from the setting of 2D images to the more complicated setting of manifolds of arbitrary genus and curvature. The approach is also described in the Ph.D. thesis of Bruno Vallet [12] , that he defended this year. We also made the software implementation of this tool available in our Graphite platform. We now start to investigate non-linear spectral mesh processing, shown in Figure 7 and outlined in the next paragraph.

With the ever increasing strive for working with highly detailed meshes, the cost of generating computer solutions for mesh editing problems often becomes exhaustive. It is therefore, desirable to reduce the size of the original problem for increased efficiency. In this aspect we investigate the potential of modal analysis for interactive deformation using the underlying shell model description. By encoding the relationship between input (force excitation) and output (deformation response) in terms of the frequency response function, the deformation problem reduces to representing the deformation in terms of the most significant eigenmodes of the original surface. In this framework, the mesh vertices enjoy six degrees of freedom and the behavior of the deformation is controlled through material properties assigned to the mesh as well as the combination of bending and membrane effects encoded in the shell model. For the discretization, we take advantage of a generic approach which allows the construction and evaluation of several deformation models without much programming effort. The governing deformation modes are the significant modes associated with the generalized eigenvalue problem for the stiffness and mass matrices. As the mesh editing might require large displacements, it is imperative to extend the modal analysis to the nonlinear case where the stiffness varies throughout the deformation process. Besides from direct editing of surface meshes, we plan to investigate the potential of these techniques also in animation reconstruction from video by analyzing the optical flow field.

#### 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
L^{2} 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.