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Approximating the pathway axis and the persistence diagrams for a collection of balls in 3-space. (English) Zbl 1211.52004

Summary: Given a collection \(\mathcal B\) of balls in a three-dimensional space, we wish to explore the cavities, voids, and tunnels in the complement space of \(\cup \mathcal B\). We introduce the pathway axis of \(\mathcal B\) as a useful subset of the medial axis of the complement of \(\cup \mathcal B\) and prove that it satisfies several desirable geometric properties. We present an algorithm that constructs the pathway graph of \(\cup \mathcal B\), a piecewise-linear approximation of the pathway axis. At the heart of our approach is an approximation scheme that constructs a collection \({\mathcal{K}}\) of same-size balls that approximate \(\mathcal B\) so that the Hausdorff distance between \(\cup \mathcal B\) and \(\cup{\mathcal{K}}\) is bounded by a prescribed parameter. We prove a bound on the ratio between the number of balls in \({\mathcal{K}}\) and the number of balls in \(\mathcal B\). We employ this bound and the approximation scheme to show how to approximate the persistence diagrams for \(\cup \mathcal B\), which can be used to extract major topological features such as the large voids and tunnels in the complement of \(\cup \mathcal B\). We show that our approach is superior in terms of complexity to the standard point-sample approaches for the two problems that we address in this paper: approximating the pathway axis of \(\mathcal B\) and approximating the persistence diagrams for \(\cup \mathcal B\). In a companion paper [E. Yaffe, D. Fishelovitch, H. J. Wolfson, D. Halperin, and R. Nussinov, Bioinform. 73.1, 72–86 (2008)], we introduced MolAxis, a tool for the identification of channels in macromolecules that demonstrates how the pathway graph and the persistence diagrams are used to identify plausible pathways in the complement of molecules.

MSC:

52A15 Convex sets in \(3\) dimensions (including convex surfaces)
05C90 Applications of graph theory
52C45 Combinatorial complexity of geometric structures
52B55 Computational aspects related to convexity
92E10 Molecular structure (graph-theoretic methods, methods of differential topology, etc.)
68U05 Computer graphics; computational geometry (digital and algorithmic aspects)
65D18 Numerical aspects of computer graphics, image analysis, and computational geometry
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