Exactitud de los modelos para el acelerómetro GT3X

Aila et al. [2013]analyzed the correlation between measured performance and the SAM cost of BVHs constructed with several BVH construction algorithms. Their motivation

Figure 2.27: Result of chopped binning of a triangle primitive with a node, which is partitioned into eight bins. The procedure computed the tight bounds (green) of the primitive clipped against the bounds of each bin the primitive overlaps. Each bin computes the tight total bounds of all clipped bounds that fall into it.

was the often made observation that more sophisticated construction algorithms that con- structed BVHs with lower SAM cost improved measurements less than expected or even decreased performance (e.g.,Popov et al. [2009],Walter et al. [2008], andBittner et al. [2015]). At the same time BVHs with similar SAM cost but constructed with different algorithms can give significantly different trace performance. Aila et al. identified end- point-overlap (EPO) as the missing piece of information and proposed the EPO metric to better predict the performance of BVHs in combination with the SAM. The key observation behind EPO is if the start point and/or end point of a ray lies on a primitive we have to at least process all nodes that overlap with these points. If these points overlap with nodes which are not ancestors of the leaves containing the primitives these nodes cause an extra traversal cost simply because of overlap. The aim of EPO is to directly measure this extra traversal cost caused by overlapping nodes. Regarding the derivation of EPOAila et al. [2013]“tried many variations of the general idea (e.g. [Stich et al. 2009], [Popov et al. 2009], or using node areas to approximate triangle areas), but EPO was significantly more descriptive than the alternatives”. EPO assumes secondary diffuse rays. That is, rays start at surface points. Ray origins and ray intersection points are assumed to be uniformly distributed over all primitive surfaces. The probability density of a surface point x ∈SPis

1

A_P, wherePis the set of scene primitives and APis the surface area of the surfaceSPof the union of all primitives inP. The probability of such a random point to be in the bounding volume Bnof a node n is the surface area of the primitive surfaces which overlap Bntimes the probability density:

p(x ∈SP∩ Bn) =

Area

A_P . (2.45)

The EPO metric is interested in surface points which are contained in B_nbut do not belong to any primitive referenced in the subtree of the node n. These points cause extra traversal costs since n needs to be processed simply because they are contained in the overlap with n. This is depicted in Figure2.28. Similar to Aila et al. we denote the set of primitives referenced in the subtree of a node n withQ(n). This allows to define the set of these points asS_P\Q(n)∩ Bn. Thus, the probability of such a random point to be in a node n is

p(x ∈S_P\Q(n)∩ Bn) =

Area

2.5. The Surface Area Metric and Surface Area Heuristic

Figure 2.28: Simple primitive overlap example with a scene consisting of two primitives in two leaves. The primitive in the blue leaf overlaps (orange) with the bounds of the green leaf. Rays starting at or intersecting the overlapping surface inevitably also have to process the green leaf. EPO assumes that the probability of this to occur during traversal is the surface area of the overlapping surface divided by the combined surface area of all scene primitives. 1 2 3 {1} {2} {3}

Figure 2.29: Depiction of EPO for a BVH with three primitives in three leaves. In this example EPO is non-zero because of primitive 3. It overlaps with the bounds of leaf 2 (orange) and the blue inner node (orange and red). As the node bounds overlap region of leaf 1 and 2 (green) does not contain any primitive it has no contribution to EPO. Now, the EPO cost metric for a BVH is defined as the expected extra node processing cost caused by random ray origins or hit points on scene surfaces overlapping with non- ancestral nodes. For a BVH with leaf nodesLand inner nodesIthis cost is

EPO := X n∈I∪L Cn

Area

c_t and c_i are the implementation dependent constants from the SAM. |n| is the number of primitives referenced in a leaf node n. Figure2.29depicts EPO computation in a BVH. Aila et al. [2013]proposed a traversal cost predictor p which uses the SAM and EPO cost of a BVH to give a better approximation of traversal cost than SAM alone. The predictor is simply a convex combination of SAM and EPO:

Here, α ∈ [0, 1] is a scene dependent constant. Given the actual average measured traver- sal cost m for a BVH it is assumed that

m_{∼ p} (2.50)

holds (MacDonald and Booth [1989,1990]). Via the derivation of EPO this predictor is only designed for secondary diffuse rays and scalar traversal. α values as high as 0.98 have been computed byAila et al. [2013]and also by ourselves (see Table5.1) meaning that performance can be almost completely governed by EPO in practice. With such high possible values it becomes clear that minimizing SAM is not enough as EPO must be opti- mized as well.Aila et al. [2013]discovered that top-down greedy SAH-based construction algorithms implicitly reduce node overlap in a way that also minimizes EPO, which gives them an innate superiority over bottom-up or hybrid algorithms. This result is especially important for Chapter5, where we describe an algorithm which exploits this implicit prop- erty. Chapter5also explains computation of α, as one of the chapters contributions is the computation of α values, which result in more accurate performance predictions.