D79: Detecting and Removing Islands in Graphics-Rendering-Based Computations of Lower Envelopes of Plane Slabs


Geometric algorithms which make use of graphics rendering often require manipulation and adaption of the pixel map of the lower envelope of plane slabs. The complex interaction of the slab geometries may give rise to isolated portions in the pixel map ("islands") which need to be discarded and patched appropriately. Such problems may occur, for instance, when attempting to compute multiplicatively-weighted Voronoi diagrams or straight skeletons in 2D by means of graphics rendering of the lower envelope of plane slabs in 3D. This paper presents general algorithms for detection, labeling, and removal of islands in an input pixel map. Removal, here, means recovery of the correct portions of the pixel map in lieu of the islands. The presented island detection algorithm requires only a constant number of passes over the input pixel map without any dependence on the number of input sites being processed by the geometric algorithm. This paper introduces the concept of black lists for the removal of islands and explains how the presented approach can cope with stacked islands, with no need for looping over the stack of islands for recovery of the correct pixel map underneath it. The discussion is concluded by experimental results obtained with the implementation of the presented algorithms.