## Section: Research Program

### Geometry Processing for Engineering

**Keywords:**
Mesh processing, parameterization, splines

Geometry processing emerged in the mid-1990's as
a promising strategy to solve the geometric modeling
problems encountered when manipulating meshes composed of hundreds of
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.*, [26]). 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 hundreds of millions of
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 [15].
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 [24]. 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, within the context 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 [19]. This method has become the de-facto standard in automatic unwrapping, and was adopted by several 3D modeling packages (including Maya and Blender). We explored various applications of the method, including normal mapping, mesh completion and light simulation [16].

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 [22]. We also showed the applicability of this method, by proposing the first algorithm that converts a scanned mesh into a Spline surface automatically [18].

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, in 2012 we studied several ways of discretizing partial differential equations on meshes, including Finite Element Modeling and Discrete Exterior Calculus. In 2013, we also explored Spectral Geometry Processing and Sampling Theory (more on this below).