Automatic Convexification of Space using BSP-trees

Convex representations of shapes have several nice properties that can be exploited to generate efficient geometric algorithms. At the same time, extending algorithms from convex to non-convex shapes is non-trivial and often leads to more expensive solutions. An alternative and sometimes more efficient solution is to transform the non-convex problem into a collection of convex problems using a convexification approach. In this paper, we address the issue of building convexification of 3D spatial domains. This process is non-trivial, since it might lead to many convex cells and the computations are subject to numerical errors. In particular, we give the first fully automatic convexification technique of non-convex polyhedral meshes that leads to most of the time to a small increase in the number of convex cells. The basic idea of our technique is to use the leaves of a binary space partition tree (BSP-tree) to create the cells that we use for filling up the space between the non-convex polyhedral mesh and its convex hull. We show an application of our ideas to volume rendering of unstructured grids. CR Categories: I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling—Geometric algorithms