## Section: Scientific Foundations

### Geometry Processing

Participants : Bruno Lévy, Yang Liu, Vincent Nivoliers, Nicolas Ray, Rhaleb Zayer.

Geometry processing recently emerged (in the middle of
the 90's) as a promising strategy to solve the geometric modeling
problems encountered when manipulating meshes composed of hundred
millions of elements. Since a mesh
may be considered to be a *sampling* of a surface - in other
words a *signal* - the *digital signal processing*
formalism was a natural theoretic background for this
subdomain (see e.g., [21] ). Researchers
of this domain then studied different aspects of this formalism applied to
geometric modeling.

Although many advances have been made in the
geometry processing area, important problems still
remain open. Even if shape acquisition and filtering is much easier than 30 years ago,
a scanned mesh composed of hundred million triangles cannot
be used directly in real-time visualization or complex
numerical simulation. For this reason, automatic methods to
convert those large meshes into higher level representations
are necessary. However, these automatic methods do not exist
yet. For instance, the pioneer Henri Gouraud often mentions in his talks that the *data
acquisition* problem is still open. Malcolm Sabin, another pioneer
of the “Computer Aided Geometric Design” and “Subdivision”
approaches, mentioned during several conferences of the domain
that constructing the optimum control-mesh of a subdivision surface
so as to approximate a given surface is still an open problem. More
generally, converting a mesh model into a higher level
representation, consisting of a set of equations, is a difficult
problem for which no satisfying solutions have been proposed. This is
one of the long-term goals of international initiatives, such as the
AIMShape European network of excellence.

Motivated by gridding application for finite elements modeling for oil and gas exploration, in the frame of the Gocad project, we started studying geometry processing in the late 90's and contributed to this area at the early stages of its development. We developed the LSCM method (Least Squares Conformal Maps) in cooperation with Alias Wavefront [6] . This method has become the de-facto standard in automatic unwrapping, and was adopted by several 3D modeling packages (including Maya and Blender). We experimented various applications of the method, including normal mapping, mesh completion and light simulation [2] .

However, classical mesh parameterization requires to partition the considered object into a set of topological disks. For this reason, we designed a new method (Periodic Global Parameterization) that generates a continuous set of coordinates over the object [8] . We also showed the applicability of this method, by proposing the first algorithm that converts a scanned mesh into a Spline surface automatically [7] . Both algorithms are demonstrated in Figure 2 .

We are still not fully satisfied with these results, since the method remains quite complicated. We think that a deeper understanding of the underlying theory is likely to lead to both efficient and simple methods. For this reason, we studied last year several ways of discretizing partial differential equations on meshes, including Finite Element Modeling and Discrete Exterior Calculus. This year, we also explored Spectral Geometry Processing and Sampling Theory (more on this below).