On the Fast Construction of Spatial Hierarchies for Ray Tracing

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On the Fast Construction of Spatial Hierarchies
for Ray Tracing

• Vlastimil Havran1,2
• Robert Herzog1 Hans-Peter Seidel1
• 1 MPI Informatik, Saarbruecken, Germany
• 2 Czech Technical University in Prague

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Talk Outline

• Previous work
• H-trees
• AH-trees
• Performance results

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Previous Work
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Previous Work

• Kd-trees in O(N log N)
• Bounding volume hierarchies BVH in O(N log N)
• Two-level data structures for hierarchical motion
  (Wald et al. 03)
• Parallel tree traversal (Szecsi et al. 03)
• Concurrent papers this year (EGSR06, EG06,
  RT06)

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Approaches to Animated Scenes

• Set of frames, planned motion or not ?
• Some objects are moving (1, 50, or 100?)
• Two Approaches
• Rebuild data structures from scratch for every
  frame
• Update data structures for new object positions

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Most Objects Are Moving .

• Rebuild from scratch? Why possibly better?
• High quality of data structures for ray tracing
• Fast rebuild from scratch is useful in general,
  also for larger static scenes
• Fast rebuild from scratch has not been addressed
  well in CG publications

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Spatial Sorting

• Spatial sorting is a base of ray tracing, it is
  an extension to quicksort.
• Clearly, sorting is O(N log N)
• Why then radix-sort is O(N) ?
• Project goal could we make faster spatial
  sorting in radix-sort like manner?

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Some Other Issues

• Definition object 3D geometric primitive,
  which defines a bounding box, and operation
  ray-object intersection (point normal)
• Binary-Interpolation Search search over the
  monotonically increasing values in array, using
  two phases
• interpolation search O(log log N)
• binary search O(log N)

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Intro to Spatial KD-trees (SKD-trees)

• Using two splitting planes
• Proposed by Ooi et al. in 1987 as spatial index
  for objects as extension to kd-trees

Overlapping Configuration
Disjoint Configuration
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Spatial KD-trees Properties

• Each object is referenced just once
• It is not a spatial subdivision (spatial regions
  can overlap)
• More memory efficient than BVH (only two planes
  in a node, not six planes)
• Interesting candidate for ray tracing

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Cost Model for Ray Tracing
Typical cost model for a spatial hierarchy
C C_T C_L C_R C C_TS N_TS
C_LO N_LO C_Access N_Access

• C_T cost of traversing interior nodes
• C_L cost of incidence operation in leaves
• C_R cost of accessing the data from internal or
  external memory

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Cost Model Based Construction

• It is also referred to as SAH (surface area
  heuristics)
• Local greedy decision, where to put splitting
  planes recursion stopping criteria
• Cost N_L Prob_L N_R Prob_R
• Probability given by surface area of the box of
  the left and on the right child node
• Problem is evaluation of N_L and N_R

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Normalized Cost Function for Many Objects
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SKD-tree Cost Model

• Using buckets to find out minimum of the cost (at
  most 100 buckets)
• Objects are put to buckets according to their
  centroids
• Recording also minima and maxima of object boxes
  in buckets
• Sweep-plane algorithm to compute tight left and
  right bounding boxes

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Performance of Pure SKD-trees

• It is rather low, not competitive with kd-trees
• Low performance means too many traversal steps
  and ray-object intersections
• Need for better bounding of objects
• A hybrid data structures using bounding
  primitives and SKD-tree nodes

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How Many Bounding Volume Primitives in 3D ?

• Assumption axis-aligned bounding planes
• In total 63 possibilities
• 1 plane, 6 cases corresponds to kd-tree
• 2 planes, 15 cases we use only parallel planes
  (3 cases in 3D)
• 3 planes, 20 cases
• 4 planes, 15 cases
• 5 planes, 6 cases
• 6 planes, 1 case corresponds to BVH

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H-trees SKD-trees Bounding
Volumes Primitives
Seven types of internal nodes
3 types (x,y,z axis)
1 type (6 planes)
3 types (x,y,z axis)
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H-trees Construction Algorithm

• Select an axis (X, or Y, or Z)
• Decide if to put an bounding node based on the
  cost (details in the paper)
• Select the number of buckets N (-,10,100)
• Distribute the objects to the buckets - keep
  minima and maxima in 3 axis in the bucket
• By sweep-plane compute cost function for all N
  buckets, select minimum cost
• Distribute the objects into left and right child
• Recurse until having one object in a node (leaf)

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H-trees Results

• Tests on SPD scenes (30 scenes) by Eric Haines
  other 12 scenes of different distributions
• Used by recursive ray tracing and also only ray
  casting (primary rays)
• In paper results for only 9 scenes.

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Reference kd-trees

• Termination criteria 4 objects in a leaf
• No split clipping operation
• O(N log N) complexity
• Highly optimized C code (but no SSE, no
  multithreading)
• Construction about 1,0 seconds for 100,000
  objects for 3.5 leaf references per object on a
  PC
• Ray tracing about 300,000 rays per second for
  completely random rays ( incoherent rays)

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H-trees Results

• Parts of the cost model
• N_TS number of traversal steps
• N_LO number of ray object intersection tests
• N_Access number of memory accesses
• Timings for H-trees construction
• 2.4 to 12 times faster than kd-trees
• Timings for tracing rays
• Comparable with kd-trees

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AH-trees Motivation

• Avoid resorting
• Modification of method Reif and Tate, 1998, for a
  set of points in 3D, achieving time complexity
  O(N log log N)
• Mimic radix-sort and hence possibly decrease
  complexity
• Use cost model based on SAH
• Use H-trees nodes

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AH-trees Avoiding Resorting

• Use buckets in 3D uniform grid in 3D
• Classify objects as small and oversize

Small object allows to limit the object extent
by a box if we know about the presence of an
object in a grid cell
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AH-trees Construction Algorithm

• Select a grid resolution, construct grid
• Classify objects as small and oversize
• Construct H-trees over small objects in
  predefined position given by 3D grid, in leaves
• Recurse by AH-trees construction algorithm
• Recurse by H-trees construction algorithm
• Create references to a leaf
• Decide to which level belong oversize objects
• Create ternary child nodes processing oversize
  objects, creating AH-trees or H-trees

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AH-trees Ternary child example
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AH-trees Results

• Parts of the cost model
• N_TS number of traversal steps
• N_LO number of ray object intersection tests
• N_Access number of memory accesses
• Timings for AH-trees construction
• 4 to 20 times faster than kd-trees
• Timings for tracing rays
• Up to 4 times slower than kd-trees for skewed
  object distribution
• Comparable to kd-trees for moderately uniform
  object distribution

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AH-trees Consideration

• The deeper in the hierarchy in a single grid, the
  resolution of cost function evaluation is smaller
• The spatial sorting is approximate using the
  object centroids
• There is a tradeoff between time complexity for
  construction and ray tracing performance
• For limited object size O(N log log N)

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Ray Traversal Algorithm

• C/C switch for more node types (no problem, CPU
  branch prediction fails anyway)
• Interval-based technique similar to kd-trees (but
  overlapping intervals)
• The source code is straightforward
• For AH-trees the ternary child subtrees are
  traversed first

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Conclusion

• Hybrid spatial hierarchy
• H-trees introduced as SKD-tree nodes bounding
  volume nodes
• AH-trees as way of approximate sorting for
  H-trees, achieving smaller construction time and
  worse performance (tradeoff)
• Performance results in the paper

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Future Work

• Implementation work ray packets for H-trees
• Research work tuning between construction time
  and performance for AH-trees
• Hybrid trees with more types of nodes

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Take Home Message

• Efficient ray tracing
• Efficient spatial hierarchy (tree)
• Efficient performance cost model
• Efficient spatial hierarchy
• top-down construction
• cost-based model decision (with SAH)
• termination criteria
• State of the art ?

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Poppers model of natural sciences
Karl Popper, 1934 (The Logic of Scientific
Discovery)
real inputs
experiments
design
induction
Falsifiable hypothesis
deduction
implementation
analysis
deduction
Asymptotic bounds