Cost-Optimal Trees 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.[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 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.

Interactive Isosurface Ray Tracing of Large Octree Volumes

“…Octrees are, in fact, theoretically optimal in terms of fewest traversal steps, assuming objects are contained uniformly within cells of the acceleration structure, with no overlap [2]. The combination of regular, hierarchical nature of the structure affords many different styles of traversal algorithm.…”

Octrees with near optimal cost for ray-shooting

“…The following lemma was proven in [8] for the case d = 2. Its statement and proof extend straightforwardly to higher dimensions.…”

“…We follow the same terminology as [8], and generalize it to encompass any dimension. For the d-cube [0, 1] d and the cells of the decomposition, we borrow the usual terminology of polytopes (vertex, facet, h-face, etc.).…”

“…The following lemma was given in [8] for quadtrees, and rephrased here to encompass any dimension as well.…”

“…As introduced in [8], the dynamic programming algorithm finds the tree that minimizes the cost over all trees with depth at most k, which we denote by T , and simply performs a bottom-up traversal of all the nodes, while maintaining the optimum cost of a tree rooted at that node. The decision whether to keep the subtree of a cell or prune it is based on the cost of the cell vs. the sum of the optimum costs of the subtrees rooted at its 2 d children.…”