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

DOI:HAL: http://hal.archives-ouvertes.fr/inria-00497688/en/
Source: OAI

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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### Keywords

3D ambiguous geometric data

additively-multiplicatively weighted Voronoi diagram

ambiguous data

ambiguous input

associated CW $\alpha$-complex

balls

concentric balls

CW diagram

CW VD

geometry processing

large protein assemblies

model ambiguous geometric objects

naive algorithm

novel framework

possible geometries

toleranced ball

toleranced balls

topological features

{\em toleranced balls}