### ACTIVITIES

Multiresolution meshes: Triangle meshes are used to represent surfaces
in several applications such as Geographic Information Systems (GISs),
Virtual Reality (VR), Computer-Aided Design (CAD),
Finite Element Methods (FEM). Tetrahedral meshes are used for
scientific data visualization and analysis.
All applications mentioned above
can benefit from the use of multiresolution meshes.
The research activity on multiresolution triangle meshes dates back to
1992.
We have proposed models, construction algorithms, query algorithms
in 2-D, in 3-D, and in a dimension-independent setting.
We have also presented algorithms for performing specific operations
of application fields.

Mesh compression: Mesh-based geometric models are often huge in size.
This yields large files and delay in loading and transmission of such
models. Thus, efficient techniques for compressing the description of
a geometric model are required. We have proposed direct and
progressive compression techniques, and studied compact data
structures for multiresolution meshes in both 2-D and 3-D.

Shape reconstruction: In applications, models of spatial objects
need to be built from different data sources and thus need different
techniques. We have worked on parallel algorithms for Delaunay
triangulations (2-D) and Delaunay tetrahedralizations (3-D).
We have also worked on a multiresolution model to represent the
shape of the reconstructed object.

Modeling of non-manifold objects: Non-regular, non-manifold
meshes are used to describe spatial objects consisting of parts of
mixed dimensions, and with a complex topology. In a
multiresolution approach, such meshes can
represent the shape of an object at a coarse level of abstraction,
in which some parts of the object are represented
with a lower dimensionality (e.g., the legs of a chair).
We have studied multiresolution models, data structures, and algorithms
for this case, a multiresolution iconic model
to describe an object through hypergraphs of parts of different
dimensions,
and the decomposition of a non-manifold n-dimensional
object into an assembly of simpler components.