Topological modifications and hierarchical representation of cell complexes in arbitrary dimensions

Abstract

We propose a set of atomic modeling operators for simplifying and refining cell complexes in arbitrary dimensions. Such operators either preserve the homology of the cell complex, or they modify it in a controlled way. We show that such operators form a minimally complete basis for updating cell complexes, and we compare them with various operators previously proposed in the literature. Based on the new operators, we define a hierarchical model for cell complexes, that we call a Hierarchical Cell Complex (HCC), and we discuss its properties. An HCC implicitly encodes a virtually continuous set of complexes obtained from the original complex through the application of our operators. Then, we describe the implementation of a version of the HCC based on the subset of the proposed modeling operators which preserve homology. We apply the homology-preserving HCC to enhance the efficiency in extracting homology generators at different resolutions. To this aim, we propose an algorithm which computes homology generators on the coarsest representation of the original complex, and uses the hierarchical model to propagate them to complexes at any intermediate resolution, and we prove its correctness. Finally, we present experimental results showing the efficiency and effectiveness of the proposed approach. © 2014 Elsevier Ltd. All rights reserved.