Section: Overall Objectives
Overall Objectives
History
ALICE is one of the three INRIA projects proposed by former ISA members. This section summarizes the scientific evolution of the research groups within ISA that yielded those three project proposals. More specifically, since VEGAS and ALICE both do research in geometry, this section explains the two different visions of geometry developed by these two project proposals.
One of the principal research orientations of the ISA project was computer vision and augmented reality. A clearly identified group headed by MarieOdile Berger developed this approach, so it was natural for them to propose the creation of the MAGRITE project.
The other principal research orientation of the ISA project was physicallybased light simulation. The main challenges of this domain are both geometrical (visibility complex, surface intersections) and computational (numerical resolution of an integral equation). To deal with the geometrical problems, a 'geometry research group' was created within ISA. The missions of this group were the following three ones:

To generate the surfacic geometry of the scene from volumic Constructive Solid Geometry descriptions, design new intersection algorithms . More precisely, given the equation of two surfaces, the goal is to obtain a parameterization of the intersection;

Find ways of attaching photometric properties to the geometric objects in the scene. For the numerical simulation of light (i.e., energy transfers), it is necessary to find parameterizations with a constant Jacobian (i.e., energypreserving parameterizations);

Optimize pointtopoint visibility requests (they are massively issued by light simulation algorithms).
To make the scope of this research as general as possible, the two main classes of surfacic representations were considered, i.e., algebraic surfaces and meshed models. When using algebraic surfaces, to represent complex objects, piecewise defined surfaces are used. The geometric continuity between the charts is the main challenge of the Geometric Design domain of research. In other words, Geometric Design is concerned with G^{k} continuity (Geometric Continuity), defined as follows: a surface of class G^{k} is a surface for which a parameterization of class C^{k} exists. To construct a G^{1} continuous object with a set of parametric polynomial surfaces defined over triangles, it was proved that at least degree 4 is required [22] . We considered the main class of algebraic surfaces used to represent geometry in the Geometric Design community, i.e., rational fractions called NURBS (NonUniform Rational BSplines). However, it quickly appeared to us that the solution of the mathematical problems expressed with NURBS (surface intersection and energypreserving parameterization) do not have a closed form in general. As a consequence, two complementary approaches were developed in parallel by ISA:

The first approach considered an exact solution of a simplified version of the problems, and limited the study to polynomial surfaces of degree 2 (i.e., quadrics). The main advantage is that closed forms for both the problem of surface intersection (point 1 of the research program on the previous page) and energypreserving parameterization (point 2) could be derived. Using Galois's group theory and projective geometry, it was possible to derive a general expression of the intersection of two quadrics, with a provable minimum number of square roots. This elegant theoretical result answered several open questions and was welcomed by the Computational Geometry community. As a matter of fact, the projective geometry background acquired by the group was also successfully applied to start the study of the visibility complex (point 3). The VEGAS project headed by Sylvain Lazard and Sylvain Petitjean continues to develop this approach;

The second approach kept the initial specification of the problem: the geometry is represented by a set of highorder surfaces or by meshed models. The group already knows that no closed form can be derived, for this reason, we developed approximated solution mechanisms, based on applied mathematics and numerical analysis. Being able to process industrialscale models (with millions of primitives) was also one of the major preoccupations of the group. Applied to mesh models, the solutions we developed had different applications in texture mapping (point 2 of the research program). More generally, our solutions were welcomed by the Geometry Processing community, a new discipline of Computer Graphics that recently emerged. In addition, we also started to apply our geometry processing tools to the numerical simulation of light. These two aspects  Geometry Processing and the numerical simulation of light  are the directions of research proposed by the ALICE project.
Overall objectives
ALICE is a new project in Computer Graphics. The fundamental aspects of this domain concern the interaction of light with the geometry of the objects. The lighting problem consists in designing accurate and efficient numerical simulation methods for the light transport equation. The geometrical problem consists in developing new solutions to transform and optimize geometric representations . Our original approach to both issues is to restate the problems in terms of numerical optimization . We try to develop solutions that are provably correct , numerically stable and scalable .

By provably correct, we mean that some properties/invariants of the initial object need to be preserved by our solutions.

By numerically stable, we mean that our solutions need to be resistant to the degeneracies often encountered in industrial data sets.

By scalable, we mean that our solutions need to be applicable to data sets of industrial size.
To reach these goals, our approach consists in transforming the physical or geometric problem into a numerical optimization problem, studying the properties of the objective function and designing efficient minimization algorithms.
The main applications of our results concern Scientific Visualization. We develop cooperations with researchers and people from the industry, who experiment applications of our general solutions to various domains, comprising CAD, industrial design, oil exploration and plasma physics. Our solutions are distributed in both opensource software ( Graphite ) and industrial software ( Gocad ).