Modified quaternary tree bucketing for Voronoi diagrams with multi-scale generators

When a Voronoi diagram is constructed incrementally, the quaternary-tree bucketing method of Ohya et. al. provides a useful way of spreading out the insertion order of the generator sites and obtaining initial guesses for the place to insertion them. This reduces the average time complexity of constructing Voronoi diagrams for n generator sites from O(n²) to O(n). We extend Ohya´s method to support the construction of multi-scale Voronoi diagrams, using generator sites that have a discrete scale-value, such as those produced by scale-space image analyses. In order to produce a Voronoi diagram at each scale-level, the original bucketing method needed to be repeatedly re-run and produced less efficient insertion orders and initial guesses for the generators. The new method efficiently builds one Voronoi diagram, starting with the set of coarsest scale generators, then adds details in decreasing order of significance. After an introduction and brief summary of Ohya´s method we describe the new algorithm and some example results. The work may have application in both computer vision and computer graphics or visualisation