Octrees with near optimal cost for ray-shooting

Abstract

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. in [B. Aronov, H. Brönnimann, A.Y. Chang, Y.-J. Chiang, Cost prediction for ray shooting, in: Proc. 18th Annu. ACM Sympos. Comput. Geom., ACM, New York, 2002, pp. 293-302; Computational Geometry, submitted for publication] 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 on octrees with the same cost measure as the optimizing criterion. More generally, we solve the generalization for generalized octrees in any d dimensions with scenes made up of (d − 1)-dimensional simplices. We give a construction of trees which yields cost O(M), where M is the infimum of the cost measure on all trees. 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. These are the first and only known results that provide performance guarantees on the approximation factor for 3-dimensional ray shooting with this realistic model of computation. Our results have been validated experimentally by Aronov et al. in [B. Aronov, H. Brönnimann, A.Y. Chang, Y.-J. Chiang, Cost-driven octree construction schemes: an experimental study, in:

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