A2SI 3D hole closing Research P-simple points

Discrete surfaces

Involved people  : Gilles Bertrand, Michel Couprie, Rémy Malgouyres, Laurent Perroton.


We used the concept of strong homotopy to propose new approaches of simple surfaces in a cubic mesh. Let X be a subset of Z 3 , such that the complementary of X is composed of two connected components A and B. One says that X is a strong surface if X U A and X U B are respectively strongly homotopic with A and B. This class of discrete surfaces generalizes the concepts of discrete surfaces proposed in the literature   [ BM97 , MB97 , MB99 ].

We studied the structural properties of the various classes of discrete surfaces   [ BC97 ]. In the continuous space, a ``simple'' surface is characterized by the fact that the neighborhood of each point is homeomorphic with a Euclidean disc. A similar characterization for the discrete case consists in requiring that the neighborhood of a point forms a simple closed curve (cycle property), however the usual concepts of neighborhood in Z 3 do not allow such a characterization, even for the discrete planes. We introduced a relation of adjacency between points of Z 3 based on the concept of homotopy, which we named simplicity relation : two points x and y belonging to an object X are bound by the relation of simplicity, if these two points are nonsimple for X and if by removing one of X, one renders the other simple. This relation enabled us to show that strong surfaces (and thus other classes of surfaces) have the structural cycle property evoked above.

Thanks to the simplicity relation and the cycle property, we introduced a new definition of discrete surface, baptized simplicity surface, which itself generalizes the concept of strong surface   [ CB98 ]. We showed that these surfaces are combinatorial manifolds (fig.   7).

We also worked on surface tracking algorithms and on their parallelization   [ Per95 ].

Figure 7: A simplicity surface X and the graph of the simplicity relation for the points of X, this graph makes it possible to extract the combinatorial structure of the surface.