Multi-scale Geometric Modeling of Ambiguous Shapes with Toleranced Balls and Compoundly Weighted a-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 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 a-complex, which provides a filtration called the lambda-complex. Fourth, we sketch a naive algorithm to compute the CW VD. Finally, we use the lambdal-complex to assess the quality of models of large protein assemblies, as these models inherently feature ambiguities.