Cost-optimal trees for ray shooting

Predicting and optimizing the performance of ray shooting is a very important problem in computer graphics due to the severe computational demands of ray tracing and other applications, e.g., radio propagation simulation. Aronov and Fortune were the first to guarantee an overall performance within a constant factor of optimal in the following model of computation: build a triangulation compatible with the scene, and shoot rays by locating origin and traversing until hit is found. Triangulations are not a very popular model in computer graphics, but space decompositions like kd-trees and octrees are used routinely. Aronov et al. [1] developed a cost measure for such decompositions, and proved it to reliably predict the average cost of ray shooting. In this paper, we address the corresponding optimization problem, and more generally d-dimensional trees with the cost measure of [1] as the optimizing criterion. We give a construction of quadtrees and octrees which yields cost O(M), where M is the infimum of the cost measure on all trees, for points or for (d - 1)-simplices. Sometimes, a balance condition is important. (Informally, balanced trees ensures that adjacent leaves have similar size.) We also show that rebalancing does not affect the cost by more than a constant multiplicative factor, for both points and (d - 1)-simplices. To our knowledge, these are the only results that provide performance guarantees within approximation factor of optimality for 3-dimensional ray shooting with the octree model of computation.