Approximating the pathway axis and the persistence diagram of a collection of balls in 3-space.

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Approximating the Pathway Axis and the
Persistence Diagram of a Collection of Balls in 3-Space∗
Eitan Yaffe
School of Computer Science
Tel-Aviv University, Israel
{eitanyaf,danha}@post.tau.ac.il
Dan Halperin
Abstract
Given a collection B of balls in three-dimensional space, each having a radius of at
least 1, we present an approximation scheme that constructs a collection Kε of unit balls
that approximate B, such that the Hausdorff distance between ∪B and ∪Kεis at most ε. We
define the pathway axis as the subset of the medial axis of the complement of ∪B for which
the set of closest balls in B do not have a common intersection. It is the medial axis of the
complement of ∪B without ‘dead-ends’ and therefore it is a good starting point for finding
pathways that lie outside ∪B. The recently introduced persistence diagram of the distance
function from ∪B encodes topological characteristics of the function, giving a measure on the
importance of topological features such as voids or tunnels during a uniform growth process
of B. In this paper we introduce the pathway diagram as a useful subset of the Voronoi
diagram of the centers of the unit balls in Kε, which can be easily and efficiently computed.
We show that the pathway diagram contains an approximation of the pathway axis of B.
We prove a bound on the ratio |Kε|/|B|, namely the ratio between the number of unit balls
in Kε and the number of balls in B. We employ this bound to show how we efficiently
approximate the persistence diagram of ∪B. Finally, we show that our approach is superior
to the standard point-sample approaches for the two problems that we address in this paper:
Approximating the medial axis of the complement of ∪B, and approximating the persistence
diagram of ∪B. In a companion paper we introduce MolAxis, a tool for the identification of
channels in macromolecules, that demonstrates how the pathway diagram and the persistence
diagram are efficiently computed and used to accurately identify pathways in the complement
of molecules modeled as the union of balls.
∗This work has been supported in part by the IST Programme of the EU as Shared-cost RTD (FET Open)
Project under Contract No IST-006413 (ACS - Algorithms for Complex Shapes), by the Israel Science Foundation
(Grant no. 236/06), and by the Hermann Minkowski–Minerva Center for Geometry at Tel-Aviv University.

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1Introduction
Let B be a finite collection of balls in R3and let ∪B denote their union. We assume, without
loss of generality, that every ball in B is not smaller than a unit ball. We wish to capture the
shape of the complement of ∪B. The medial axis of the complement of ∪B is the set of points in
this complement that have more than one closest ball in B. We define the pathway axis of B to
be the set of points in the complement of ∪B for which the collection of closest balls in B do not
have a common intersection. It is the medial axis of the complement of ∪B without ‘dead-ends’
and therefore it is a good starting point for finding pathways that lie outside ∪B. See Figure 1
for a two-dimensional illustration.
The exact medial axis of the complement of the union of balls is a subset of the Voronoi
diagram [6, 8] of the balls and can be computed in an exact manner as shown by Boissonnat
and Delage [7]. We opt for an approximation approach for two main reasons: Simplicity of
implementation and speed of computation. In this paper we offer an approximation scheme that
replaces B by a collection of unit balls Kεsuch that the Hausdorff distance between ∪B and ∪Kε
is not larger than a prescribed ε. We introduce an algorithm that constructs a geometric entity
we call the pathway diagram of the centers of Kε, and prove that the pathway diagram contains
an approximation of the pathway axis of B.
Figure 1:
lection of discs (red, bold line). The dis-
carded portions of the medial axis are col-
ored gray.
The pathway axis of a col-
There are algorithms that approximate the medial axis
of an object from a set of unorganized points sampled
on the surface of the object [2, 14]. Oudot and Boisson-
nat [25] introduce an algorithm for computing the medial
axis that has certified results for smooth shapes. Giesen et
al. [21] approximate a useful subset of the medial axis of a
shape with smooth boundary that captures the topology
of the shape. However, the complement of the union of
a collection of balls is not bounded by a smooth surface,
making it difficult to directly apply the techniques (and
hence have the topological guarantees) obtained in these
papers. In contrast to the aforementioned approaches we
sample a volume with balls instead of sampling a surface
with points.
The λ-medial axis [10], introduced by Chazal and Lieu-
tier, is a subset of the medial axis, that for some “regular”
values of λ remains stable under Hausdorff distance per-
turbation. This leads to an algorithm [10] that constructs
an approximation of the λ-medial axis of an object from a set of noisy unorganized points sam-
pled on or close to the (not necessarily smooth) boundary surface of the object. We apply
theoretical ideas introduced there to prove geometric convergence of our approximation.
A recent result by Lieutier [23] states that under certain conditions the medial axis of an
object and the object itself have the same ‘shape’ (they are homotopy equivalent). Yet due
to the approximate nature of our approach we cannot apply this result.
topological properties of our approximation scheme we utilize the emerging concept of topological
persistence. Edelsbrunner et al. [18] introduce the notion of topological persistence during a
growth process of the union of balls. In that work an efficient algorithm is described that
classifies topological changes during the growth process as topological features or topological
noise depending on their lifetime during the process. The theoretical notion of persistence was
extended independently by Carlsson et al. [9], by Chazal et al. [11], and by Cohen-Steiner et
In order to state
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al. [12]. Cohen-Steiner et al. [12] deal with real-valued functions on a topological space. They
define the persistence diagram of the distance function from ∪B, which encodes topological
characteristics of the function, giving a measure on the importance of topological features.
They show that the persistence diagram of an object is stable under Hausdorff perturbation of
the object.
Figure 2:
bold line) of a collection same-size discs
(light blue) that approximate the discs de-
picted in Figure 1. The discarded portions
of the Voronoi diagram of their centers is
depicted using dotted lines.
The pathway diagram (red,
In this paper we define the pathway diagram as a sub-
set of the Voronoi diagram of the centers of Kε, which is
closely related to the celebrated α-complex introduced by
Edelsbrunner et al. [19]; technically it is the collection of
Voronoi facets dual to simplices that are not included in
the (α = 1)-complex of the centers of Kε. See Figure 2 for
a two-dimensional illustration of a pathway diagram. We
present an algorithm that computes it and prove several
of its properties. As mentioned above, we prove that the
pathway diagram contains an approximation of the path-
way axis. We provide a bound on the number of balls in
Kεas a function of ε and the ratio between the largest and
the smallest ball in B. We employ this bound to show that
the persistence diagram of ∪B can be approximated using
O(|B|/ε4/3) unit balls. Finally, we compare our approach
with a point-sample approach, proving a multiplicative-
factor gain of Ω(1/ε2/3) in the worst case in the number of
sample entities over the point samples for the approxima-
tion of both the pathway axis and the persistence diagram.
In the last section we report on experimental results and
show how the persistence diagram is used to identify the center of the largest chamber within a
molecule, presented as a collection of balls.
In a companion paper [28] we present MolAxis, a new tool designed for the efficient iden-
tification of molecular channels. MolAxis was found to be a very efficient and accurate tool in
identifying transmembrane (TM) channels within proteins and pathways leading from buried
cavities within enzymes to their convex hull. Being very fast, MolAxis has been successfully
used to analyze channel dimensions and lining residues in hundreds of snapshots of Molecular
Dynamic simulation of the human CYP3A4.
More details on the results described in this extended abstract, including proofs omitted for
lack of space, can be found in the thesis [27]:
http://www.cs.tau.ac.il/~eitanyaf/thesis.pdf. The MolAxis paper [28] is available at
http://www.cs.tau.ac.il/~eitanyaf/MolAxis.pdf.
2Preliminaries
Our work builds on a large body of results concerning Voronoi diagrams and the medial axis.
We assume some familiarity with alpha complexes [19]. We borrow notation mainly from the
work of Attali et al. [4] and the work of Chazal and Lieutier [10]. For any set X we denote
by¯ X, Xo, ∂X, Xcand |X| the closure, the interior, the boundary, the complement and the
cardinality of X respectively. B(x,r), Bo(x,r) and S(x,r) denote the closed ball, open ball and
sphere of center x and radius r in R3respectively. We denote the Euclidean distance between
two points x,y ∈ R3by d(x,y). The distance between two subsets A,B of R3is defined to be
d(A,B) = infa∈A,b∈Bd(a,b).
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The one-sided Hausdorff distance between two compact subsets A and B of R3is:
dH(A|B) = sup
x∈Ad(x,B) .
The (symmetric) Hausdorff distance between two compact subsets A and B of R3is the
maximum of the two one-sided distances, namely dH(A,B) = max(dH(A|B),dH(B|A)). We say
that A is a Hausdorff approximation of B with an approximation resolution of ε if the Hausdorff
distance between A and B is not larger than ε. In such a case, we will say for short that A is
an ε-approximation of B.
Let O be a bounded open subset of R3. For any point x ∈ O, we denote by ΓO(x) the set of
closest points to x in the complement Oc, namely ΓO(x) = {y ∈ Oc: d(x,y) = d(x,Oc)}. The
medial axis M[O] of the open set O is the set of points x ∈ O that have at least two closest
boundary points:
M[O] = {x ∈ O : |ΓO(x)| ≥ 2} .
Let E be a finite point set in R3. We denote by V[E] the collection of Voronoi cells of E
which are of dimension 0,1,2 or 3, and by D[E] it dual structure, the Delaunay triangulation of E
which contains simplices which are of dimension 0,1,2 or 3. Note that the Delaunay triangulation
of E is a simplicial complex (see, e.g., [6, 16]).
The rest of the section is devoted to the formal definition of the persistence diagram. We
repeat verbatim the definitions introduced by Cohen-Steiner et al. [12], which we need in order
to state our results in Section 6. The reader is referred to [24] for an accessible introduction to
Homology.
Given a topological space X and an integer k, we denote the k-th singular homology group
of X by Hk(X), and the k-th Betti number of X by βk(X) = dim Hk(X). We work here with
modulo 2 coefficients, so that homology groups are vector spaces over Z2= Z/2Z.
Definition 2.1 [12] Let X be a topological space and f a real function on X. A homological
critical value of f is a real number A for which there exists an integer k such that for all
sufficiently small ε > 0 the map Hk(f−1(−∞,A−ε]) → Hk(f−1(−∞,A+ε]) induced by inclusion
is not an isomorphism.
Definition 2.2 [12] A function f : X → R is tame if it has a finite number of homological
critical values and the homology groups Hk(f−1(−∞,A]) are finite-dimensional for all k ∈ Z
and A ∈ R.
In other words, the homological critical values are the levels where the homology of the sub-
level sets changes. Assuming a fixed integer k we write Fx= Hk(f−1(−∞,x]), and for x < y
we define fy
x : Fx→ Fyto be the map induced by inclusion of the sub-level set of x in that of y.
We write Fy
y is infinite. Let βy
x denote the persistent Betti number for all −∞ ≤ x ≤ y ≤ +∞.
Let f : X → R be a tame function, (ai)i=1...nits homological critical values, and (bi)i=0...n
an interleaved sequence, namely bi−1< ai< bifor all i. We set b−1= a0= −∞ and bn+1=
an+1= +∞. For two integers 0 ≤ i < j ≤ n+1, we define the multiplicity of the pair (ai,aj) to
be µj
bi
define the persistence diagram.
x = im fy
x for the image of Fxin Fy. By convention we set Fy
x= dim Fy
x = {0} whenever x or
i= βbj
bi−1− βbj
bi+ βbj−1
− βbj−1
bi−1. Denoting by¯R the union R ∪ {−∞,+∞} we are ready to
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Definition 2.3 [12] The persistence diagram D(f) ⊂¯R2of f is the set of points (ai,aj),
counted with multiplicity µj
ifor 0 ≤ i < j ≤ n + 1, union all points on the diagonal, counted
with infinite multiplicity.
For points p = (p1,p2) and q = (q1,q2) in¯R2, let ?p−q?∞be the maximum of |p1−q1| and
|p2− q2|. Similarly for functions f and g, let ?f − g?∞= supx|f(x) − g(x)|. Let X and Y be
two multisets of points.
Definition 2.4 [12] The bottleneck distance between X and Y is
dB(X,Y ) = inf
γsup
x
?x − γ(x)?∞,
where x ∈ X and y ∈ Y range over all points and γ ranges over all bijections from X to Y . If
dB(X,Y ) < ε we say that Y is an ε-approximation of X. Cohen-Steiner et al. prove [12] that
small changes in f imply small changes under the bottleneck metric in the persistence diagram.
We state here a weakened version of their main theorem that is sufficient for our needs.
Theorem 2.5 [12] Let A,A′be two subsets of R3such that dH(A,A′) ≤ ε. Let fA,fA′ denote
the distance from A,A′respectively. The persistence diagrams of fA,fA′ satisfy
dB(D(fA),D(fA′)) ≤ ε .
3Constructing the Pathway Diagram
In this section we define and explain what is the pathway diagram of a collection of points in R3
and give a formal description of our algorithm. The algorithm is fairly simple and it proceeds
in two steps. First, we construct a collection Kε of unit balls such that ∪Kε constitutes an
ε-approximation of ∪B. In a second step we construct the pathway diagram of the centers of Kε,
which we denote by Pε. We defer technical implementation details to Appendix B. Theoretical
properties of Kεand the pathway diagram are presented and proved in the following sections.
3.1Pathway Diagram
Let E be a finite point set in R3and let σ be a Delaunay d-simplex of D[E], spanned by the
d + 1 point set T. Let RT denote the radius of the smallest ball that contains all points of T
on its boundary surface. We say that the simplex σ is α-exposed if α > RT [19]. The collection
of α-exposed simplices is a simplicial complex, which is called the α-complex of E. We call the
collection of the dual Voronoi faces of simplices that are not in the α-complex the α-Voronoi
graph of E.
The (α = 1)-Voronoi graph of E plays an important role in this paper, and we refer to it as
the pathway diagram of E. Denoting by K(E) the collection of unit balls centered at points of
E, we can define the pathway diagram of E in a more intuitive manner. It is the set of Voronoi
faces in V[E] that do not intersect ∪K(E). It is a subset of the medial axis of the complement
of ∪K(E) and it contains only flat facets, i.e., patches of planes bounded by simple polygons.
Actually, the only difference between the pathway diagram of E and the whole medial axis of
the complement of ∪K(E) is that the medial axis also contains planar patches bounded by arcs
whenever the medial axis reaches the boundary surface of ∪K(E). The pathway diagram is thus
defined such that it is completely piecewise linear and easy to compute, avoiding the need to
construct facets that are bounded by arcs.
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