Abstract:We propose a new strategy for the parallelization of mesh processing algorithms. Our main contribution is the definition of distributed combinatorial maps (called n-dmaps), which allow us to represent the topology of big meshes by splitting them into independent parts. Our mathematical definition ensures the global consistency of the meshes at their interfaces. Thus, an n-dmap can be used to represent a mesh, to traverse it, or to modify it by using different mesh processing algorithms. Moreover, an nD mesh with a huge number of elements can be considered, which is not possible with a sequential approach and a regular data structure. We illustrate the interest of our solution by presenting a parallel adaptive subdivision method of a 3D hexahedral mesh, implemented in a distributed version. We report space and time performance results that show the interest of our approach for parallel processing of huge meshes.

Abstract. Betti numbers are topological invariants that count the number of holes of each dimension in a space. Cubical complexes are a class of CW complex whose cells are cubes of different dimensions such as points, segments, squares, cubes, etc. They are particularly useful for modeling structured data such as binary volumes. We introduce a fast and simple method for computing the Betti numbers of a three-dimensional cubical complex that takes advantage on its regular structure, which is not possible with other types of CW complexes such as simplicial or polyhedral complexes. This algorithm is also restricted to three-dimensional spaces since it exploits the Euler-Poincaré formula and the Alexander duality in order to avoid any matrix manipulation. The method runs in linear time on a single core CPU. Moreover, the regular cubical structure allows us to obtain an efficient implementation for a multi-core architecture.

International audienceDiscrete gradient vector fields are combinatorial structures that can be used for accelerating the homology computation of CW complexes, such as simplicial or cubical complexes, by reducing their number of cells. Consequently, they provide a bound for the Betti numbers (the most basic homological information). A discrete gradient vector field can eventually reduce the complex to its minimal form, having as many cells of each dimension as its corresponding Betti number, but this is not guaranteed. Moreover, finding an optimal discrete gradient vector field is an NP-hard problem. We describe here a generalization, which we call Homological Discrete Vector Field (HDVF), which can overcome these limitations by allowing cycles under a certain algebraic condition. In this work we define the HDVF and its associated reduction, we study how to efficiently compute a HDVF, we establish the relation between the HDVF and other concepts in computational homology and we estimate the average complexity of its computation. We also introduce five basic operations for modifying a HDVF, which can also be applied to discrete gradient vector fields