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

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.

Exploration of Empty Space among Spherical Obstacles via Additively Weighted Voronoi Diagram

“…We should mention here, that a similar concept called topological persistence and simplification was proposed by Edelsbrunner et al [ELZ02], but the lifetime of a cavity is measured differently. Perhaps the lifetime proposed by Yaffe et al [YH10] is more close to the approach proposed in this paper.…”

“…Yaffe and Halperin came up with an idea to approximate larger balls by a number of balls having the same size as the smallest ball [YH10]. Thanks to this approximation, they can still use the ordinary Voronoi diagram.…”

Tunnels and Voids in Molecules via Voronoi Diagram

“…In 2007, Kozlikova et al also reported an idea of tunnel extraction using the ordinary Voronoi diagram of atom center points and the Delaunay triangulation [15]. In 2008, Yaffe et al reported a method using the alphashape (which is a derivative structure of the ordinary Voronoi diagram of points) and developed the software MolAxis [16], [17], [18] which attempted to reflect the size differences among different atom types by an approximation with a number of identically sized balls (thus this approach inevitably produced an approximated solution). In 2008, Ho and Gruswitz used a grid-based approach to tunnel extraction and developed a software HOLLOW [19].…”