[2013](see Section2.5.5). They perform local BVH optimizations on a fast to construct low quality BVH in parallel. The resulting BVHs have almost the same traversal perfor-mance as a full SAH-based top-down construction but can be constructed much faster.
Mapping this algorithm to huge datasets should pose a difficult challenge when the BVH does not fit into GPU memory. It would also be interesting to see if the local tree opti-mizations are still effective for much larger trees.
Chapter 9
Final Summary and Discussion
Contents
9.1 Summary . . . 139 9.2 Discussion . . . 141
In this dissertation we presented several contributions on bounding volume hierarchies (BVH) and kd-trees in the context of ray tracing. We now proceed with a summary and discussion of the relevant chapters, and conclude with discussions of overall open prob-lems and future work in the final chapter.
9.1 Summary
The overarching problem of chapters four to six was the construction of BVHs with higher quality. Higher quality was meant in the sense of BVHs with a measurable lower average number of node and primitive intersection tests. A good indicator for hierarchy quality is the surface area metric (SAM). Construction strategies based on the surface heuristic (SAH) are able to build BVHs with lower SAM cost than other strategies.
SAM and SAH both use a geometric conditional probability for intersecting the bounds of a node. In Chapter4we attempted to improve this probability function in order to im-prove SAM and SAH, as well. Our function definition is based on the observation that the conditional probability of intersecting a convex bodyAwith a random ray given that we intersected another convex body B which contains Adepends on the ray direction.
This probability can expose strong directional variance. In contrast to the conventional probability functionpcused for SAM and SAH our alternative functionpaincluded this di-rectional dependence in its derivation. Except for certain types of bounding volumes such as spheres or cubes, wherepadirectly simplifies topc, our probability cannot be evaluated analytically. We identified cases where padegenerates to the conventional probability. In general, we found that pa and pc did not have a large absolute difference in our exper-iments. Ultimately, this also explained why SAH-based construction with pa resulted in essentially the same BVHs than construction with pc.
Additionally, we noted that for random rays the outer body is more likely to be inter-sected from some directions than from others, which effectively changes the probability distribution of random ray directions. We were able to show thatpasimplifies topcwhen this effective non-uniform ray direction distribution is factored into the derivation ofpa. This revealed an alternative derivation for the conventional probabilitypcwhich allowed to interpretpcas accounting for directional variation. This property ofpchas been unrec-ognized so far and might partially explain the lasting success of the surface area heuristic and metric.
Aila et al.[2013]found that SAM is less accurate in predicting the traversal perfor-mance of BVHs then predicting the traversal perforperfor-mance of kd-trees, as it does not suf-ficiently capture the extra BVH traversal costs caused by node overlap. BVH construction has to minimize SAM and EPO cost of a BVH to be sure that measured traversal cost is reduced as well. In Chapter5 we attempted to improve traversal performance by structing BVHs with lower SAM and EPO cost. To achieve this, we examined three con-struction algorithms that use the SAM cost of temporarily constructed SAH-built BVHs to guide BVH construction. This was motivated by research fromAila et al.[2013]which has shown that greedy top-down SAH builders construct BVHs with superior traversal perfor-mance despite the fact that the resulting SAM costs are higher than those created by more sophisticated builders. This comes from the innate property of the top-down builder to implicitly minimize the EPO metric. This property allowed us to implicitly guide construc-tion to choose candidates, which produce lower EPO, as the temporary BVHs have also been constructed with the greedy top-down construction strategy. Our constructed BVHs gave a significant increase in trace performance. We made several observations where temporary BVH construction with spatial splits clearly built BVHs with the lowest SAM and EPO cost, but other construction algorithms gave higher measured traversal perfor-mance. These observations suggest, that the SAM-EPO predictor is not sufficient. While our approach is not suitable for real-time BVH construction, we have shown that the pro-posed algorithm has subquadratic computational complexity in the number of primitives, which still renders it usable in practical applications.
Additionally, we have shown that the uncentered Pearson correlation is more suitable for computation ofαvalues for the SAM-EPO ray tracing performance predictor than the centered Pearson correlation as used byAila et al. [2013]. The centered Pearson corre-lation violates the proportionality assumption between predicted and measured traversal performance. Though the SAM-EPO predictor was initially designed for diffuse rays our improvement in the forecasting abilities also increased its relevance for primary rays.
In Chapter6we attempted to increase the quality of BVHs constructed with bottom-up approaches. The highest quality bottom-up approach is agglomerative clustering, a greedy bottom-up algorithm fromWalter et al.[2008]. This algorithm is guided by the intuition that always clustering nodes which give the cluster bounds with the lowest surface area should be beneficial w.r.t. the SAM cost of the resulting tree. We presented a fully SAM-driven approach for agglomerative clustering. For this, we developed a SAM cost function for the cost of tracing a forest of BVHs. From this function we derived two clustering strate-gies. The first strategy clusters nodes where the cluster has lowest SAM cost. The second strategy clusters nodes which give the largest reduction in the forest cost. Both strategies allow to naturally create leaves as a clustering decision, which was not possible with the original approach in a meaningful way. While our first strategy proved to be detrimental for tree quality, our second strategy on average produced better BVHs than the original