Improvements on the kd-tree

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László Szécsi and Balázs Benedek / <strong>Improvements</strong> on the kd-treehits for every primitive is unacceptably slow, in contrastto the results achieved by space subdivision. In the lattercase, we only need to traverse cells along the ray and onlycompute intersections for promising candidates. Best resultsamong the spatial subdivision schemes are delivered by theBSP and kd-trees. The kd-tree we use in this article is a binary,non-balanced, spatial subdivision data structure, withaxis-aligned cutting planes associated to its non-leaf nodes,and subsets of scene objects stored in the leaf nodes.The power of the structure lies in its flexibility. Cuttingplanes can be positioned depending on the location of thescene objects, so at the cost of some calculation the solutionresulting in an optimal traversal time can be chosen. Thecutting planes being axis aligned is a minor limitation, asarbitrarily positioned planes may produce a better tree, butfinding the optimum would be less effective. Furthermore,storing the data describing the cutting planes requires lessmemory space, and it is far easier to compute the ray-planeintersection.2.2. Traversal along a rayDuring the image synthesis a large number of ray-scene intersectionshave to be computed. Compared to the one-timeconstruction of the tree this means such a difference of scale,that it is worth taking every cost just to speed up traversal inmost of the cases.The sequential ray traversal algorithm is based on the spatialproximity search using the kd-tree. First we take the originof the ray, and locate the cell containing it by walkingdown the tree from its root. Within the cell found, we carryout all intersection tests with the objects belonging to thecell. If no intersection within the cell was found, we proceedto the next cell. In order to find it, we use the same methodas before. We calculate the point where the ray leaves thecell, which is exactly where it enters the next. We translateit a tiny bit further along the ray to resolve ambiguity, andrepeat the whole process using the spatial proximity searchwith this next point. We have to remark that the algorithmmay skip cells of extremely little or zero width. Althoughthese may seem useless at the first sight, they can actuallyrightfully appear in kd-trees for scenes where there are numerousaxis-aligned polygons. This may be the case with geometricalscenes, typically boxes and rooms. Another drawbackof this algorithm is that it starts from the root of thetree for every new cell though it is very probable that twocells following each other are near each other in the structure.Therefore one node could be visited many times.The recursive ray traversal algorithm eliminates the maindrawbacks of the sequential ray traversal algorithm and visitsevery node and leaf just only once 2 . We check if the rayintersects the volumes corresponding to the left and rightsub-trees. The sub-trees are traversed in the very same way,if necessary, starting with the one nearer to the origin. Toterminate the recursion the leaves of the tree are handled inthe same manner as above. The implementation of the algorithmneeds a traversal stack to store data about the sub-treesneeded to be processed later.Whichever algorithm we use, we will walk through theleaf cells along the ray, and test possible intersections forthe segment inside the cell. If intersections were found, theclosest is taken, else the ray has to be followed on. Consequently,the objective is to have minimal number of objectsin a cell, and if a ray intersects a cell, it should, with highprobability, also intersect an object within. This, pushed toits extremes, it accomplished when all objects are delimitedby six fitting cutting planes. However, if the bounding boxesof the objects overlap, like in most scenes, then such cutsmay intersects several objects, adding them to both child volumes,resulting in superfluously large list in the leaves, andworse-than-optimal traversal time.2.3. Constructing a kd-tree and possibledecision-making heuristicsThe tree can be built in a recursive way. Processing a volumeinvolves the choice and storage of the cutting plane, andthe processing of the two new sub-volumes. The decision tomake is where to place the cutting plane, and if it is worthsubdividing the volume at all. This may be based on someheuristic scheme, or an estimation of the resulting traversalcost.The first, most obvious method is to cut the volume intotwo equal halves, using the spatial median, similarly to theoctree approach where we care little about the position ofthe objects when subdividing a volume. The resulting treewill of course not be balanced, and it is easy to construct ascene where this method comes near to useless. Similarly tothe octree, spatial median subdivision performs well in caseof evenly distributed objects.Another simple and more promising approach is to makeboth sub-volumes contain the same number of objects. Theposition with this property is called the object median. Tofind it, we have to do a ’select and partition’ median search.This can be considered a modified version of the ’quick sort’algorithm that only sorts the partition containing the halvingelement of the array. This simpler procedure will also separatethe array into elements smaller and greater than themedian, and outperforms ’quick sort’. As the resulting treewould be balanced, its representation could be simple andcompact. Furthermore, a balanced kd-tree can be consideredto be optimal for several tasks, such as proximity search.However, in ray casting, we do not only need to find an object,but to follow a ray through several cells intersected.Therefore, the probability of a sub-volume being hit by aray plays an important role in the expected time cost of therendering algorithm. The object median method disregardsthat aspect. The unfortunate consequence for the optimal tree
László Szécsi and Balázs Benedek / <strong>Improvements</strong> on the kd-treeis, we have to discard the concept of balancedness, and willhave to find the means to store a non-balanced tree in a compactway.Although simple cut heuristics produce inferior traversaltimes, fast construction and compact data structure are advantages.Therefore, they may have some relevance if thestructure is to be built real-time, despite the fact that in globallyilluminated animation the traversal cost tends to be thebottleneck. As the tree construction time rapidly increaseswith the number of objects, but the traversal time for sceneslarge enough is constant, it is not to exclude, that the situationmay change, especially in the case of very high polygonnumber, vertex-based animations. The compact memoryrepresentation used for the balanced tree is definitely to beused somehow in the more sophisticated methods.3. Improvement of the cost function3.1. Previous workA way to find the optimal cut is to consider all reasonablecuts, including cutting off empty space and termination ofthe build, and choose the one that produces the shortest expectedtraversal time. To achieve this we need estimate thattime. Havran proposed the following function, linear withrespect to the number of objects in the sub-volumes:C 1V SA´V SA´le ftChild´V µµŃ L · N SP µ·µSA´rightChild´V µµŃ SP · N R µ℄ (1)Where C V is the cost corresponding to volume V, SA´V µ isthe surface area of volume V, and N L , N R , N SP are the numberof objects completely in the left and right sub-volumes,and the number of objects intersected by the splitting plane,respectively.This means that the expected time for the traversal of avolume is the time needed to carry out the naïve intersectiontest for all objects, multiplied by the probability of a rayhitting the volume. This probability, considering that the volumesare convex, equals the ratio of the surface areas. Obviously,the estimate given by this function does not equal theactual time cost, as the created volumes will be subdividedfurther, and not handled with the naïve algorithm. Havranalso identified this problem and proposed some ideas for thesolution. He stated that the optimal cost function depends onthe distribution of the objects in the actual scene to a greatextent, and thus for a better estimate the cost must be measuredin some way. Although it is possible to build the treeand compute the cost precisely, doing this every time thefunction should be evaluated would lead to computationalexplosion of the construction algorithm. Therefore, in orderto obtain a more effective function, the scene should be characterisedby values that are easily determined, and influencethe cost function.3.2. Non-linear cost estimateIn a recent article we have shown that for scenes with largenumber of random objects, kd-tree traversal is done in constanttime. How can this be brought into consonance with thelinear estimation? How can Havran’s method provide outstandingresults despite this contradiction? If we are low inthe tree, near the leaves, and it is true that the sub-volumeswill go through little to no further subdivision, than the linearestimation is of course perfect. On the other hand, if weare near the root of the tree, meaning that the constant timetraversal statement hold for the sub-trees, then the expectedtraversal time is independent of the cut. Therefore, if the linearestimate would fail, then where we cut is not so importantat all. However, it is possible to construct a more accuratecost estimate, if we are able to account for the gain fromseparating the elements and cutting off empty space. To calculatethat exactly would be hopelessly expensive, but bysimply changing the linear function to a bit more fitting one,we may eliminate some of the inaccuracy on higher levels ofthe tree. It is of course imperative to keep the linearity in thelower regions where it works perfectly. Let us suppose thata cut improves the time by a factor of q 1 on average, andthat a cell containing n 0 elements is not worth dividing anymore. Actually, that means that a cell may contain n 0 objectson average. Using that the cost of traversal, without the adjustmentfor the probability of the volume being hit is givenin the following equation. This function is to be applied tothe number of objects in the sub-volumes in 1:f ńµ n¡q log 2ń n 0µ(2)The value of n 0 is relatively easy to find, and will be determinedby the primitive geometry. The value q, however, isquite an abstraction. It includes both cuts between objectsand cutting off empty space. Actually, it corresponds moreto the subdivision potential of the volume than to the obscureconcept of cost reduction achieved by a single cut. Still, it isnot harmful to overestimate both n 0 and q, as that will get usnearer to the original linear estimate. Therefore, the formulafor the expected number of intersection tests introduced inour previous article 3 can be applied to determine a probableupper bound for the traversal cost of the tree that is beingbuilt, providing a value for q. Naturally, significantly betterresults are only expected for large scenes with high primitivecount, as the linear function is less accurate, and the guessfor q is better in those cases. The previous equation can furtherbe written as:f ńµ n¡ń n 0 µ log 2 q (3)f ńµ ń n 0 µ 1·log 2 q · n 0 ¡ ń n 0 µ log2q (4)As ń n 0 µ log2q 1, the cost may be over-estimated as:f ńµ n¡ń n 0 µ log 2 q ń n 0 µ 1·log 2 q · n 0 (5)
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