Topological modifications and hierarchical representation of cell complexes in arbitrary dimensions

• Published in: Computer vision and image understanding Vol. 121; pp. 2 - 12
• Main Authors: Čomić, Lidija, De Floriani, Leila, Iuricich, Federico, Fugacci, Ulderico
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.04.2014
• Subjects: Algorithms; Cell complexes; Computational topology; Computer vision; Generators; Hierarchical models; Homology; Homology generators; Mathematical models; Operators; Preserves; Representations; Cell complexes; Homology generators; Hierarchical models; Computational topology
• ISSN: 1077-3142; 1090-235X
• DOI: 10.1016/j.cviu.2013.11.011

•A set of basis topologically consistent homology-preserving and homology-modifying operators on cell complexes is introduced.•A multi-resolution model for cell complexes (HCC) is defined.•HCC based on homology-preserving operators is implemented.•Homology generators are computed on the coarsest complex, and propagated to complexes at intermediate resolution using HCC.•Proofs of the correctness are given. 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.