Topological modifications and hierarchical representation of cell complexes in arbitrary dimensions

We propose a set of atomic modeling operators for simplifying and refining cell complexes in arbitrarydimensions. Such operators either preserve the homology of the cell complex, or they modify it in a con-trolled 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 newoperators, we define a hierarchical model for cell complexes, that we call aHierarchical Cell Complex(HCC), and we discuss its properties. AnHCCimplicitly encodes a virtually continuous set of complexesobtained from the original complex through the application of our operators. Then, we describe theimplementation of a version of theHCCbased on the subset of the proposed modeling operators whichpreserve homology. We apply the homology-preservingHCCto enhance the efficiency in extractinghomology generators at different resolutions. To this aim, we propose an algorithm which computeshomology generators on the coarsest representation of the original complex, and uses the hierarchicalmodel 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 proposedapproach