Efficient Homology-Preserving Simplification of High-Dimensional Simplicial Shapes


Simplicial complexes are widely used to discretize shapes. In low dimensions, a 3D shape is represented by discretizing its boundary surface, encoded as a triangle mesh, or by discretizing the enclosed volume, encoded as a tetrahedral mesh. High-dimensional simplicial complexes have recently found their application in topological data analysis. Topological data analysis aims at studying a point cloud P, possibly embedded in a high dimensional metric space, by studying the topological characteristics of the simplicial complexes computed on P. Analyzing such complexes is not feasible due to their size and dimensions and, to this aim, the idea of simplifying a complex while preserving its topological features has been proposed in the literature. Here, we consider the problem of efficiently simplifying simplicial complexes in arbitrary dimensions. We provide a new definition for the edge contraction operator based on a top-based data structure With the objective of preserving structural aspects of a simplicial shape (i.e., its homology), we provide a new algorithm for verifying the link condition on a top-based representation. We implement the simplification algorithm obtained by coupling the new edge contraction and the link condition, on a specific top-based data structure, the Stellar tree, that we use to demonstrate the scalability of our approach.

Computer Graphics Forum