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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