Centroidal power diagrams with capacity constraints

Abstract

This article presents a new method to optimally partition a geometric domain with capacity constraints on the partitioned regions. It is an important problem in many fields, ranging from engineering to economics. It is known that a capacity-constrained partition can be obtained as a power diagram with the squared L2 metric. We present a method with super-linear convergence for computing optimal partition with capacity constraints that outperforms the state-of-the-art in an order of magnitude. We demonstrate the efficiency of our method in the context of three different applications in computer graphics and geometric processing: displacement interpolation of function distribution, blue-noise point sampling, and optimal convex decomposition of 2D domains. Furthermore, the proposed method is extended to capacity-constrained optimal partition with respect to general cost functions beyond the squared Euclidean distance.

Weighted linde-buzo-gray stippling

“…In Figure 8 we compare our results to those of Xin et al [2016]. From left to right we show the input image, their stippled result, and our result, both with 10k points 2 .…”

“…The algorithm of de Goes et al [2012] (BNOT) and its optimization [Xin et al 2016] with a speedup of factor 10 represent stateof-the-art methods for generating blue noise point distributions.…”

“…It is, however, not dynamic and needs approximately two minutes for distributing 40k points. More recently, Xin et al [2016] improve the speed of this method by an order of magnitude by restricting kernel functions to squared Euclidean distances. Chen et al [2012] propose a variational framework for producing Blue Noise samples in 2D and on geometry with a variant of CCVT.…”

“…When parameters are changed during the iterations, we reset the hysteresis to its initial value because otherwise the algorithm might stick locally to undesired densities that do not optimally represent the greyscale level. Xin et al [2016]. From left to right: input image, their result, and our result (10k points each).…”

Packing Irregular Objects in 3D Space via Hybrid Optimization

“…Because objects have different volumes, it would be better to allocate an appropriate space for each object initially. We use the centroidal power diagram method [XLC*16] to compute a partition of C with specified volume constraints that each power cell of the partition has a volume proportional to the volume of its associated object. Then the centroids of the power cells are taken as the initial positions for the selected objects.…”