Parallel Homology Computation of Meshes

Abstract

In this paper, we propose a method to compute, in parallel, the homology groups of closed meshes (i.e., orientable 2D manifolds without boundary) represented by combinatorial maps. Our experiments illustrate the interest of our approach which is really fast on big meshes and which obtains good speed-up when increasing the number of threads.

Approximating lower-star persistence via 2D combinatorial map simplification

“…The process consists of merging faces if they share a common edge, guaranteeing that the structure of the combinatorial map and the homology of the mesh is preserved throughout the process. This paper extends the work in Damiand and Gonzalez-Diaz (2016) and Damiand and Gonzalez-Diaz (2019) by giving an algorithm to approximate the lower-star persistent homology of meshes within a specified tolerance . First, faces are grouped into clusters according to the parameter .…”

“…In Damiand and Gonzalez-Diaz (2016), the authors proposed an efficient algorithm for computing the homology of meshes represented by 2D combinatorial maps, thereby avoiding the time-consuming step of constructing and modifying boundaries and coboundaries of cells. The process consists of merging faces if they share a common edge, guaranteeing that the structure of the combinatorial map and the homology of the mesh is preserved throughout the process.…”

“…(b) The corresponding 2-map has 20 darts. Images taken from Damiand and Gonzalez-Diaz (2016). strate the utility of our approach, and we conclude with a brief discussion of possible directions for future work in Section 6.…”

Local computation of homology variations over a construction process

“…On choisit ici de présenter les cartes généralisées, telles que définies dans [25]. 4 (α i ) 0≤i≤n telle que :…”

“…. Il est utilisé en modélisation géométrique [4] et dans d'autres domaines comme la médecine [5], la physique [6], l'analyse de données [7,8,9] ou encore en apprentissage automatique [10]. Intuitivement, l'homologie d'un objet représente ses "trous".…”