Multi-scale Geometric Modeling of Ambiguous Shapes with Toleranced Balls and Compoundly Weighted alpha-shapes

Abstract

Dealing with ambiguous data is a challenge in Science in general and geometry processing in particular. One route of choice to extract information from such data consists of replacing the ambiguous input by a continuum, typically a one-parameter family, so as to mine stable geometric and topological features within this family. This work follows this spirit and introduces a novel framework to handle 3D ambiguous geometric data which are naturally modeled by balls. First, we introduce {\em toleranced balls} to model ambiguous geometric objects. A toleranced ball consists of two concentric balls, and interpolating between their radii provides a way to explore a range of possible geometries. We propose to model an ambiguous shape by a collection of toleranced balls, and show that the aforementioned radius interpolation is tantamount to the growth process associated with an additively-multiplicatively weighted Voronoi diagram (also called compoundly weighted or CW). Second and third, we investigate properties of the CW diagram and the associated CW $\alpha$-complex, which provides a filtration called the $\lambda$-complex. Fourth, we propose a naive algorithm to compute the CW VD. Finally, we use the $\lambda$-complex to assess the quality of models of large protein assemblies, as these models inherently feature ambiguities.

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apport ?
?
de recherche?
ISSN 0249-6399
ISRN INRIA/RR--7306--FR+ENG
Thème BIO
INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE
Multi-scale Geometric Modeling of Ambiguous
Shapes with Toleranced Balls and Compoundly
Weighted α-shapes
Frédéric Cazals — Tom Dreyfus
N° 7306
May 2010
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Unité de recherche INRIA Sophia Antipolis
2004, route des Lucioles, BP 93, 06902 Sophia Antipolis Cedex (France)
Téléphone : +33 4 92 38 77 77 — Télécopie : +33 4 92 38 77 65
Multi-scale Geometric Modeling of Ambiguous Shapes with
Toleranced Balls and Compoundly Weighted α-shapes
Fr´ ed´ eric Cazals∗, Tom Dreyfus†
Th` eme BIO — Syst` emes biologiques
Projet ABS
Rapport de recherche n° 7306 — May 2010 — 29 pages
Abstract:
particular. One route of choice to extract information from such data consists of replacing the ambiguous input
by a continuum, typically a one-parameter family, so as to mine stable geometric and topological features within
this family. This work follows this spirit and introduces a novel framework to handle 3D ambiguous geometric
data which are naturally modeled by balls.
First, we introduce toleranced balls to model ambiguous geometric objects. A toleranced ball consists of two
concentric balls, and interpolating between their radii provides a way to explore a range of possible geometries.
We propose to model an ambiguous shape by a collection of toleranced balls, and show that the aforementioned
radius interpolation is tantamount to the growth process associated with an additively-multiplicatively weighted
Voronoi diagram (also called compoundly weighted or CW). Second and third, we investigate properties of the
CW diagram and the associated CW α-complex, which provides a filtration called the λ-complex. Fourth, we
propose a naive algorithm to compute the CW VD. Finally, we use the λ-complex to assess the quality of models
of large protein assemblies, as these models inherently feature ambiguities.
Dealing with ambiguous data is a challenge in Science in general and geometry processing in
Key-words:
molecular complexes, interfaces, structural biology.
Union of balls, Voronoi diagrams, α-shapes, stability, topological persistence, proteins, macro-
∗INRIA Sophia-Antipolis-M´ editerran´ ee, Algorithms-Biology-Structure; Frederic.Cazals@sophia.inria.fr
†INRIA Sophia-Antipolis-M´ editerran´ ee, Algorithms-Biology-Structure; Sebastien.Loriot@sophia.inria.fr
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Mod´ elisation Multi-echelle de Formes Ambigu¨ es avec des Boules
Tol´ eranc´ ees et les α-shapes ` a Pond´ eration Compos´ ee
R´ esum´ e :
exacerb´ e en g´ eom´ etrie. Une fa¸ con int´ eressante d’extraire de l’information de telles donn´ ees consiste ` a rem-
placer celles-ci par un continuum, typiquement une famille ` a un param` etre, de fa¸ con ` a chercher des structures
g´ eom´ etriques et topologiques stables au sein de cette famille. Ce travail s’inscrit dans cette veine, et propose
un nouveau canevas pour manipuler des donn´ ees 3D ambigu¨ es.
Tout d’abord, nous introduisons les boules toleranc´ ees. Un telle boule est constitu´ ee de deux boules con-
centriques, et interpoler entre leurs rayons permet d’explorer un ensemble de g´ eom´ etries possibles. Nous pro-
posons de mod´ eliser une forme ambigu¨ e par un ensemble de boules toleranc´ ees, et montrons que le proces-
sus d’interpolation ´ evoqu´ e ci-dessus conduit ` a un diagramme de Voronoi additif-multiplicatif (aussi appel´ e ` a
pond´ eration compos´ ee, ou CW). Ensuite, nous nous int´ eressons aux propri´ et´ es du diagramme CW et ` a l’α-shape
associ´ ee, qui repr´ esente une filtration que nous nommons le λ-complexe. Nous poursuivons par un algorithme
na¨ ıf de calcul du diagramme de Voronoi CW. Enfin, nous montrons comment utiliser le λ-complexe pour ´ etudier
la qualit´ e de mod` eles de gros assemblages macro-mol´ eculaires, ces mod` eles pr´ esentant de fa¸ con intrins` eque des
ambigu¨ ıt´ es.
La manipulation de donn´ ees ambigu¨ es est un challenge tout ` a fait g´ en´ eral, qui est particuli` erement
Mots-cl´ es :
complexes macro-mol´ eculaires, interfaces, biologie structurale.
Union de boules, diagrammes de Voronoi, α-shapes, stabilit´ e, persistence topologique, prot´ eines,
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Modeling with toleranced balls3
1Introduction
1.1 Voronoi Diagrams and Applications
The Voronoi diagram of a finite collection of sites equipped with a generalized distance is the cell decomposition
of the ambient space into equivalence classes of points having the same nearest sites for this distance. Voronoi
diagrams are central constructions in science and engineering [OC00], and their versatility actually comes from
two sources. First, the great variability of sites and distances is a first source of diversity. While the most
classical construction is the Euclidean distance diagram for points, the mere class of circles and weighted points
yields as diverse diagrams as power, Apollonius and M¨ obius diagrams–see [BWY06] and Fig. 1. Second, the
information encoded in a Voronoi diagram actually goes beyond the aforementioned cell decomposition. One
can indeed consider the bisectors bounding the cells as the realization of a growth process defined by the
distance, in the sense that the level sets of this distance intersect on the bisectors. This viewpoint motivated
the development of α-complexes and α-shapes [Ede92], a beautiful construction providing a filtration of the
Delaunay triangulation dual of the Voronoi diagram, later complemented by the flow complex [GJ03]. From
a mathematical standpoint, these developments are concerned with the topological changes undergone by the
sub-level sets of the distance, which is the heart of Morse theory [Mil63]. Ideas in this realm also motivated
the development of topological persistence [ELZ02, CSEH05], a subject concerned with the assessment of the
stability of topological features associated with the sub-level sets of a function defined on a topological space.
The success of α-shapes relies on two cornerstones. First, the aforementioned growth process gives access to
a multi-scale analysis of the input sites. For example, the problem of reconstructing a shape from sample points
can be tackled by considering the space-filling diagram consisting of balls grown around the sample points. Alas,
a provably good reconstruction using this strategy requires a uniform sampling, which motivated the definition
of more local growth processes. One may cite conformal α-shapes [CGPZ06], where the growth process depends
on the distance of the samples to the medial axis, and the scale-axis transform [GMPW09] which provides a
hierarchy of skeletal shapes based on a dilation (and retraction) of medial balls. Second, α-shapes inherently
model objects represented by collections of balls—in particular molecules, and encode remarkable geometric
and topological properties [Ede95]. For the particular case of proteins and small macro-molecular complexes,
they have been instrumental to compute molecular surfaces [AE96] and model interfaces [BGNC09, LC10].
Our developments are precisely motivated by the requirement to perform multi-scale analysis in computa-
tional structural biology.
1.2 Contributions
Structural proteomics studies are concerned by the identification and the modeling of molecular machines
operating within the cell [GAG+06], and a major endeavour consists of modeling assemblies involving from tens
[LTSW09] to hundreds [ADV+07] of polypeptide chains. This modeling requires integrating data coming from
several experimental sources, these data being typically noisy and incomplete [AFK+08]. In this context, the
premises just discussed on α-shapes certainly hold: on the one hand, balls are the primitive of choice since we
deal with atoms and molecules; on the other hand, multi-scale analysis are in order since we deal with ambiguous
data and need to accommodate uncertainties on the shapes and positions of proteins within an assembly. In
this context, we make the following contributions.
First, we introduce toleranced balls to model ambiguous geometric shapes. A toleranced ball consists of two
concentric balls—the inner and outer balls, and interpolating between them allows one to replace an arbitrary
ball by one-parameter family of balls. As illustrated on Fig. 2, the inner (outer) ball of a toleranced ball
is meant to accommodate high (low) confidence regions. We note in passing that our approach bears some
similarities with modeling with toleranced parts in engineering, where tolerances are generally accommodated
thanks to Minkowski sums [LWC97]. Second, we show that the growth process associated with this interpolation
is associated with a so-called additively-multiplicatively-weighted Voronoi diagram, also called compoundly
weighted Voronoi diagram—CW VD for short. Third, we investigate properties of this CW diagram and its
dual. Fourth, we present the filtration, called the λ-complex, induced on the dual by the growth process, and
encoding all possible topologies associated with this growth process. For any value of λ, the λ-complex identifies
a list of simplices of the dual complex, together with a label for each of them. This label precisely encodes
the contribution of the simplex to the boundary of the union of the balls grown thanks to the additively-
multiplicatively model. Our analysis generalizes the so-called β-shapes [SCC+06], i.e. the α-shape associated
to an Apollonius diagram [BD05, BWY06, EK06]. Fifth, we present an algorithm to compute the CW VD.
Finally, we present experimental results on toleranced protein models.
RR n° 7306
inria-00497688, version 1 - 7 Jul 2010