My research works

You can have a look to my habilitation (called habilitation à diriger des recherches, only available in french for the moment sorry) to read a detailled abstract of my main research activites.

My research works are about the study of combinatorial and topological models (for example combinatorial maps, generalized maps, simplicial sets, map chain...). The first part of my works are theoretical and concern the study of the properties of these models, the definition of generic models in any dimension, the optimization of these models to answer specific needs, and the links between algebraic topology for example to compute topological invariants from these models.

A second part of my works are about the utilization of theoretical results in image processing, in geometrical modelling and in animation/simulation. For that, we define algorithms based on the properties of our models, for example to use topological criteria during 3D segmentation algorithms. We have worked to define 2D and 3D segmentation algorithms, to propose modifying operations (for example the merging or the split of regions) and to define algorithms to compute topological invariants (Euler characteristic, homology group...).

Each time, we implement our research results in different computer softwares in order to test and compare our solutions and thus show the practical interest of our researches :

Moka: a 3D topological based geometrical modeller

To validate our model, we have developed a geometrical modeler software based on a generalized map kernel. This software allows us to test easily a new algorithm. It is the core of several other works (building modelling, geological evolution study...). It contains many different operations, including our operations of topological invariant compuations.

2D Topological maps and 3D Topological maps

These two softwares allows to build the 2D (resp. 3D) combinatorial map from a 2D (resp. 3D) image. They propose segmentation algorithms that are based on topological maps. The principle of these algorithms is to use "split and merge" methods in a similar way than methods based on region adjacency graphs (RAG), but by using the specificities of topological maps in order to use topological criteria during the segmentation. For example, it is possible to control the evolution of Betti numbers in order to guide the result of the segmentation. The 2D software proposes also some polygonal reconstruction methods, possibly in multi-thread mode and some methods of deformable partition. The 3D software proposes some operations allowing to modify manually the result of the segmentation.

nD combinatorial maps, nD generalized maps and Linear cell complexes in CGAL

These three packages are integrated in CGAL, an important library of computational geometry algorithms and allow to represent combinatorial and generalized maps in any dimension, and the geometrical overlay that embed these maps in a linear way. Some basic operations (insertion/removal) exist and a demo illustrates some possible geometrical operations such that for example the computation of a 3D Voronoi diagramm.

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