## Section: New Results

### Geometric analysis

Participants : Samuel Hornus, Aurelien Martinet, Cyril Soler, Nicolas Holzschuch, Francois sillion, Elmar Eisemann, Xavier Décoret.

#### Semantic analysis of non coherent geometry

Aurélien Martinet started his PhD in 2003, under the
supervision of Cyril Soler and Nicolas Holzschuch, working on the
automatic extraction of semantic information from non coherent
geometry. This work aims at answering a recurrent need in computer
graphics: most researchers work with 3D scene data into which they
need high level information such as which groups of polygons form
connex shapes, human regognisable objects, have symetries, or even
which groups of polygons look like each other (also known as *instancing information* ). Unfortunately such high level
information is most of the time not present in 3D geometry files,
either because it was lost during format conversions, or because it
was not defined the same way by the designer of the model.

The question to be solved is thus how to automatically retrieve some high
level (also named *semantic* ) information from a *polygon soup* , *i.e* a list of polygons without any information about how these polygons are
related to each other. During the passed year, Aurelien has focused on
developing a new technique for automaticaly extracting instantiation
information in a scene, on the basis of the work he previously performed for
extracting symmetries of objects [8] .
Figure 19 shows an example of an instancing graph
automatically obtained using this method: this structure is a Directed Acylic
Graph where each node is associated to a "generic object" which is
instantiated in the scene, and each edge represents the geometric
transformation of each instance.

#### On Exact Error Bounds for View-Dependent Simplification

In view-dependent simplification, an object is simplified so that
the difference between original and simplified versions *as
seen from a given viewcelll* is bounded by a given error. The error
is the maximum reprojection error, that is the distance between the
projection of a point in image, and the projection of its
counterpart in the simplified version.

To guarantee an error bound, one must know how much a point can be
moved from its original position to satisfy the reprojection error
bound. This defines the *validity region* of the point.
Surprisingly, finding this region is a very difficult geometric
problem. Elmar Eisemann worked on it during his master and found
very important results. For example, the error bound cannot be
checked only at the vertices of the mesh. Also, the maximum
reprojection error is not necessarily observed at one of the corner
of the viewcell. Finally, he showed how to compute exactly the
validity region for the 2D case and opened the way to an extension
to the 3D case. The proof is elegant and very inovative. It
provides the first exact bound on view-dependent simplification
error. In contrast, previously published bounds were often only
approximate (though sufficient for the considered application).
The results have been accepted as a journal paper [6] .