## Section: Scientific Foundations

### Guiding principles

After having introduced the *geometry processing* and *light simulation*
scientific domains, we now present the principles that we use to
design a common mathematical framework that can be applied to
both domains.
Early approaches to geometry processing and light simulation
were driven by a Signal Processing approach. In other words, the solution of
the problem is obtained after applying a *filtering scheme*
multiple times. This is for instance the case of the mesh
smoothing operator defined by Taubin in his pioneering
work [21] . Recent approaches still inherit
from this background. Even if the general trend moves to
Numerical Analysis, much work in geometry
processing still studies the coefficients of the gradient of the objective
function *one by one* . This intrinsically refers to *descent* methods
(e.g., Gauss-Seidel), which are not the most efficient,
and do not converge in general when applied to meshes larger
than a certain size (in practice, the limit appears to be around 10^{4} facets).

In the approach we develop in the ALICE project-team, geometry processing and light simulation are systematically restated as a (possibly non-linear and/or constrained) functional optimization problem. As a consequence, studying the properties of the minimum is easier: the minimizer of a multivariate function can be more easily characterized than the limit of multiple applications of a smoothing operator. This simple remark makes it possible to derive properties (existence and uniqueness of the minimum, injectivity of a parameterization, and independence to the mesh).

Besides helping to characterize the solution, restating the geometric problem as a numerical optimization problem has another benefit. It makes it possible to design efficient numerical optimization methods, instead of the iterative relaxations used in classic methods.

Richard Feynman (Nobel Prize in physics) mentions in his lectures that
physical models are a “smoothed” version of reality. The global
behavior and interaction of multiple particles is captured by physical
entities of a larger scale. According to Feynman, the striking
similarities between equations governing various physical phenomena
(e.g., Navier-Stokes in fluid dynamics and Maxwell in electromagnetism)
is an illusion that comes from the way the phenomena are modeled and
represented by “smoothed” larger-scale values (i.e., *fluxes* in
the case of fluids and electromagnetism). Note that those larger-scale
values do not necessarily directly correspond to a physical
intuition, they can reside in a more abstract “computational”
space. For instance, representing lighting by the coefficients of a
finite element is a first step in this direction. More generally, our approach consists
in trying to get rid of the limits imposed by the classic view of the existing
solution mechanisms. The traditional approaches are based on an
intuition driven by the laws of physics. Instead of
trying to mimic the physical process, we try to restate the problem
as an abstract numerical computation problem, on which more
sophisticated methods can be applied (a plane flies like a bird, but it
does not flap its wings). We try to consider the problem from a
computational point of view, and focus on the link between
the numerical simulation process and the properties of the solution
of the Rendering Equation. Note also that the numerical computation problems yielded
by our approach lie in a high-dimensional space (millions of
variables). To ensure that our solutions scale-up to scientific and
industrial data from the real world, our strategy is to try to always
use the best formalism and the best tool. The best formalism comprises
Finite Elements theory, differential geometry, topology, and the best
tools comprise recent hardware, such as GPU (Graphic Processing
Units), with the associated highly parallel algorithms. To implement our strategy,
we develop algorithmic, software and hardware architectures, and
distribute these solutions in both open-source software
(Graphite ) and industrial software (Gocad , DVIZ ).