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

Frédéric Cazals 1, * Tom Dreyfus 1
* Corresponding author
1 ABS - Algorithms, Biology, Structure
CRISAM - Inria Sophia Antipolis - Méditerranée
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 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 -complex, which provides a ltration called the -complex. Fourth, we sketch a naive algorithm to compute the CW VD. Finally, we use the -complex to assess the quality of models of large protein assemblies, as these models inherently feature ambiguities.

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Frédéric Cazals, Tom Dreyfus. Multi-scale Geometric Modeling of Ambiguous Shapes with Toleranced Balls and Compoundly Weighted alpha-shapes. Symposium on Geometry Processing - 2010, Eurographics symposium in cooperation with ACM Siggraph, Jul 2010, Lyon, France. pp.1713-1722. ⟨hal-00796078⟩

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