Bounding interval hierarchy

A bounding interval hierarchy (BIH) is a partitioning data structure similar to that of bounding volume hierarchies or kd-trees. Bounding interval hierarchies can be used in high performance (or real-time) ray tracing and may be especially useful for dynamic scenes.

The BIH was first presented under the name of SKD-Trees,[1] presented by Ooi et al., and BoxTrees,[2] independently invented by Zachmann.

Overview

Bounding interval hierarchies (BIH) exhibit many of the properties of both bounding volume hierarchies (BVH) and kd-trees. Whereas the construction and storage of BIH is comparable to that of BVH, the traversal of BIH resemble that of kd-trees. Furthermore, BIH are also binary trees just like kd-trees (and in fact their superset, BSP trees). Finally, BIH are axis-aligned as are its ancestors. Although a more general non-axis-aligned implementation of the BIH should be possible (similar to the BSP-tree, which uses unaligned planes), it would almost certainly be less desirable due to decreased numerical stability and an increase in the complexity of ray traversal.

The key feature of the BIH is the storage of 2 planes per node (as opposed to 1 for the kd tree and 6 for an axis aligned bounding box hierarchy), which allows for overlapping children (just like a BVH), but at the same time featuring an order on the children along one dimension/axis (as it is the case for kd trees).

It is also possible to just use the BIH data structure for the construction phase but traverse the tree in a way a traditional axis aligned bounding box hierarchy does. This enables some simple speed up optimizations for large ray bundles [3] while keeping memory/cache usage low.

Some general attributes of bounding interval hierarchies (and techniques related to BIH) as described by [4] are:

Operations

Construction

To construct any space partitioning structure some form of heuristic is commonly used. For this the surface area heuristic, commonly used with many partitioning schemes, is a possible candidate. Another, more simplistic heuristic is the "global" heuristic described by [4] which only requires an axis-aligned bounding box, rather than the full set of primitives, making it much more suitable for a fast construction.

The general construction scheme for a BIH:

Potential heuristics for the split plane candidate search:

Ray traversal

The traversal phase closely resembles a kd-tree traversal: One has to distinguish 4 simple cases, where the ray

For the third case, depending on the ray direction (negative or positive) of the component (x, y or z) equalling the split axis of the current node, the traversal continues first with the left (positive direction) or the right (negative direction) child and the other one is pushed onto a stack.

Traversal continues until a leaf node is found. After intersecting the objects in the leaf, the next element is popped from the stack. If the stack is empty, the nearest intersection of all pierced leaves is returned.

It is also possible to add a 5th traversal case, but which also requires a slightly complicated construction phase. By swapping the meanings of the left and right plane of a node, it is possible to cut off empty space on both sides of a node. This requires an additional bit that must be stored in the node to detect this special case during traversal. Handling this case during the traversal phase is simple, as the ray

Properties

Numerical stability

All operations during the hierarchy construction/sorting of the triangles are min/max-operations and comparisons. Thus no triangle clipping has to be done as it is the case with kd-trees and which can become a problem for triangles that just slightly intersect a node. Even if the kd implementation is carefully written, numerical errors can result in a non-detected intersection and thus rendering errors (holes in the geometry) due to the missed ray-object intersection.

Extensions

Instead of using two planes per node to separate geometry, it is also possible to use any number of planes to create a n-ary BIH or use multiple planes in a standard binary BIH (one and four planes per node were already proposed in [4] and then properly evaluated in [5]) to achieve better object separation.

See also

References

Papers

  1. ↑ Nam, Beomseok; Sussman, Alan. A comparative study of spatial indexing techniques for multidimensional scientific datasets
  2. ↑ Zachmann, Gabriel. Minimal Hierarchical Collision Detection
  3. ↑ Wald, Ingo; Boulos, Solomon; Shirley, Peter (2007). Ray Tracing Deformable Scenes using Dynamic Bounding Volume Hierarchies
  4. 1 2 3 Wächter, Carsten; Keller, Alexander (2006). Instant Ray Tracing: The Bounding Interval Hierarchy
  5. ↑ Wächter, Carsten (2008). Quasi-Monte Carlo Light Transport Simulation by Efficient Ray Tracing

External links

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