Stackless Ray-Object Intersections Using Approximate Minimum Weight Triangulations: Results in 2D That Outperform Roped KD-Trees (And Massively Outperform BVHs)

Abstract

Computing ray-object intersections is a key operation of ray tracers. Two wellknown data structures to accelerate this computation are the kd-tree (which partitions space) and the Bounding Volume Hierarchy (BVH, which partitions the primitives). A third type of structure is a Constrained Convex Space Partitioning (CCSP), whichlike the kd-tree -partitions space, but it does this in such a way that the geometric primitives exactly overlap with the boundaries of its cells. As a consequence, it is robust against ill-fitting cells that plague methods with axis-aligned cells (kd-tree, BVH) and it permits an efficient, stackless traversal.Within the computer graphics community, such CCSPs have received some attention in both 2D and 3D, but their construction methods were never directly aimed at minimizing their traversal cost -even having fundamentally opposing goals for Delaunay-type methods. Instead, for an isotropic and translation-invariant ray distribution the traversal cost is minimized when minimizing the weight: the total boundary size of all cells in the structure.We study the two-dimensional case using triangulations as CCSPs and explicitly minimize their total edge length using a simulated annealing process that allows for topological changes and varying vertex count. Standard Delaunay-based triangulation techniques have total edge lengths ranging from 10% higher to twice as high as our optimized triangulations for a variety of scenes, with a similar difference in traversal cost when using the triangulations for ray tracing. Compared to a idealised roped kdtree with stackless traversal, our triangulations require less traversal steps for all scenes that we tested and they are robust against the kd-tree's pathological behaviour when geometry becomes misaligned with the world axes. Moreover, the stackless traversal of our triangulations strongly outperforms a BVH, which always requires a top-down descent in the hierarchy. In fact, we show several scenes where the number of traversal operations for our triangulations decreases as the number of geometric primitives N increases, in contrast to the increasing log N behaviour of a BVH.