Maximal Disjoint Ball Decompositions for shape modeling and analysis

Abstract

A large number of geometric representations have been proposed to address the needs of specific engineering applications. This, in turn, exacerbates the inherent challenges associated with system interoperability for downstream engineering applications.In this paper, we define the Maximal Disjoint Ball Decomposition (MDBD) as the location of the largest d-dimensional closed balls recursively placed in the interior of a d-dimensional domain and show that the proposed decomposition can be used to provide an underlying common analysis framework for geometric models using different representation schemes. Importantly, our decomposition only relies on the ability of an existing geometric representation to compute distances, which must be supported by any valid geometric representation scheme, and does not require an explicit representation conversion. Moreover, MDBD is unique for a given domain up to rigid body transformation, reflection, as well as uniform scaling, and its formulation suggests appealing stability and robustness properties against small boundary modifications.Furthermore, we show that MDBD can be used as a universal shape descriptor to perform shape similarity of models coming from various geometric representation schemes. A salient attribute of this decomposition is that it provides adequate support for key downstream applications for models coming from disparate geometric representations. For example, MDBD can be naturally used to carry out meshless solutions to boundary value problems; efficient collision detection; and 3D mesh generation of models that use any valid geometric representation scheme. Finally, our hierarchical formulation of the proposed Maximal Disjoint Ball Decomposition allows for a choice of model complexity at run-time to match the available computational resources.