Approximating the Pathway Axis and the Persistence Diagrams for a Collection of Balls in 3-Space

Eitan Yaffe, Dan Halperin

Research output: Contribution to journalArticlepeer-review

Abstract

Given a collection ℬ of balls in a three-dimensional space, we wish to explore the cavities, voids, and tunnels in the complement space of ∪ℬ. We introduce the pathway axis of ℬ as a useful subset of the medial axis of the complement of ∪ℬ and prove that it satisfies several desirable geometric properties. We present an algorithm that constructs the pathway graph of ∪ℬ, a piecewise-linear approximation of the pathway axis. At the heart of our approach is an approximation scheme that constructs a collection K of same-size balls that approximate ℬ so that the Hausdorff distance between ∪ℬ and ∪ K is bounded by a prescribed parameter. We prove a bound on the ratio between the number of balls in K and the number of balls in ℬ. We employ this bound and the approximation scheme to show how to approximate the persistence diagrams for ∪ℬ, which can be used to extract major topological features such as the large voids and tunnels in the complement of ∪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 ℬ and approximating the persistence diagrams for ∪ℬ. In a companion paper we introduce 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.

Original languageEnglish
Pages (from-to)660-685
Number of pages26
JournalDiscrete and Computational Geometry
Volume44
Issue number3
DOIs
StatePublished - 2010

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