This document discusses acceleration data structures (ADS) for ray tracing. It introduces the concepts of ray tracing and ADS, describes different types of ADS including kd-trees and bounding volume hierarchies (BVHs), and discusses methods for building ADS including the surface area heuristic (SAH) and techniques that approximate local ray termination to improve the SAH. The objectives are to develop efficient heuristics to approximate local ray termination for use in constructing spatial data structures like BSPs, BIHs and BVHs to accelerate ray tracing queries.

1. Local Ray Termination
Matthias Moulin
Computer Graphics Research Group
Department of Computer Science
KU Leuven

2. Contents
1. Ray Tracing concept
2. ADS concept
3. ADS types
4. ADS build
5. Prior Work
6. Objectives
7. Architecture
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3. 1. Ray Tracing concept
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4. closest-hit query
1.
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5. X
closest-hit query
1.
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6. X
any-hit queries
1.
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7. X
1.
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8. X
closest-hit query
1.
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9. X
X’
closest-hit query
1.
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10. X
X’
any-hit queries
1.
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11. X
X’
1.
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12. X
X’
closest-hit query
1.
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13. X
X’
X”
closest-hit query
1.
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14. X
X’
X”
any-hit queries
1.
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15. X
X’
X”
1.
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16. 2. ADS concept
2D space
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17. 2. ADS concept
2D space
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18. 2. ADS concept
2D space
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19. 2. ADS concept
2D space partitioning
Geometric
primitives
kd-tree acceleration
data structure
ADS build
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20. 2. ADS concept
2D space partitioning
Geometric
primitives
kd-tree acceleration
data structure
ADS build
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21. 2. ADS concept
2D space partitioning
Geometric
primitives
kd-tree acceleration
data structure
ADS build
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22. 2. ADS concept
2D space partitioning kd-tree acceleration
data structure
ADS traversal
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23. 2. ADS concept
2D space partitioning kd-tree acceleration
data structure
X
ADS traversal
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24. 2. ADS concept
2D space partitioning kd-tree acceleration
data structure
X
ADS traversal
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25. 2. ADS concept
2D space partitioning kd-tree acceleration
data structure
X
ADS traversal
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26. 3. ADS types
Regular grid
spatial partitioning scheme
non-hierarchical
(non-)axis aligned voxels
[Fujimoto et al. 1986]
Constrained tetrahedralization
spatial partitioning scheme
non-hierarchical
non-axis aligned tetrahedra
[Lagae et al. 2008]
(Binary) Space Partition ((B)SP)
spatial partitioning scheme
hierarchical
binary/n-ary tree
(non-)axis aligned voxels
[Bentley and Friedman 1979]
[Kaplan 1985] kd-trees
[Glassner 1988] octrees
[Ize et al. 2008] BSPs
Bounding Volume Hierarchy (BVH)
object partitioning scheme
hierarchical
binary/n-ary tree
(non-)axis aligned voxels
voxels tight in 6 planes
[Rubin & Whitted 1980]
Bounding Interval Hierarchy (BIH)
hybrid partitioning scheme
hierarchical
binary/n-ary tree
(non-)axis aligned voxels
voxels tight in 1 plane
[Ooi et al. 1987]
[Havran et al. 2006]
[Woop et al. 2006]
[Wächter and Keller 2006]
Spatial Split BVH (SBVH)
hybrid partitioning scheme
hierarchical
binary/n-ary tree
(non-)axis aligned voxels
voxels tight in 6 planes
[Stich et al. 2009]
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27. 3. ADS types
Regular grid
spatial partitioning scheme
non-hierarchical
(non-)axis aligned voxels
[Fujimoto et al. 1986]
Constrained tetrahedralization
spatial partitioning scheme
non-hierarchical
non-axis aligned tetrahedra
[Lagae et al. 2008]
(Binary) Space Partition ((B)SP)
spatial partitioning scheme
hierarchical
binary/n-ary tree
(non-)axis aligned voxels
[Bentley and Friedman 1979]
[Kaplan 1985] kd-trees
[Glassner 1988] octrees
[Ize et al. 2008] BSPs
Bounding Volume Hierarchy (BVH)
object partitioning scheme
hierarchical
binary/n-ary tree
(non-)axis aligned voxels
voxels tight in 6 planes
[Rubin & Whitted 1980]
Bounding Interval Hierarchy (BIH)
hybrid partitioning scheme
hierarchical
binary/n-ary tree
(non-)axis aligned voxels
voxels tight in 1 plane
[Ooi et al. 1987]
[Havran et al. 2006]
[Woop et al. 2006]
[Wächter and Keller 2006]
Spatial Split BVH (SBVH)
hybrid partitioning scheme
Hierarchical
binary/n-ary tree
(non-)axis aligned voxels
voxels tight in 6 planes
[Stich et al. 2009]
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28. 3. ADS types
BSP candidate partition
Parent voxel 𝑉P
Left child voxel 𝑉L
Right child voxel 𝑉R
Splitting plane(s) 𝑆 (𝑆L, 𝑆R)
𝑆 = 𝑆L = 𝑆R
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29. 3. ADS types
BSP candidate partition BVH candidate partition
Parent voxel 𝑉P
Left child voxel 𝑉L
Right child voxel 𝑉R
Splitting plane(s) 𝑆 (𝑆L, 𝑆R)
𝑆 = 𝑆L = 𝑆R
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30. 3. ADS types
BSP candidate partition BVH candidate partition BIH candidate partition
Parent voxel 𝑉P
Left child voxel 𝑉L
Right child voxel 𝑉R
Splitting plane(s) 𝑆 (𝑆L, 𝑆R)
𝑆L𝑆R𝑆 = 𝑆L = 𝑆R
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31. 3. ADS types
BSP candidate partition BVH candidate partition BIH candidate partition
Parent voxel 𝑉P
Left child voxel 𝑉L
Right child voxel 𝑉R
Splitting plane(s) 𝑆 (𝑆L, 𝑆R)
𝑆L𝑆R𝑆 = 𝑆L = 𝑆R
SBVH candidate partitions
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32. 3. ADS types
BSP candidate partition BVH candidate partition BIH candidate partition
Parent voxel 𝑉P
Left child voxel 𝑉L
Right child voxel 𝑉R
Splitting plane(s) 𝑆 (𝑆L, 𝑆R)
𝑆L𝑆R𝑆 = 𝑆L = 𝑆R
SBVH candidate partitions
(𝑉P, 𝑆) ⟼ (𝑉L, 𝑉R)
Candidate partition of 𝑉P is fully determined by 𝑆21/03/2017

33. 4. ADS build
𝐶SAH 𝑉L, 𝑉R 𝑆, 𝑉P = 𝑐 + 𝑝L. 𝑁L + 𝑝R. 𝑁R
Assumptions about the ray distribution:
• isotropic distribution of rays originating from outside the scene
• rays do not intervene with geometric primitives
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34. 4. ADS build
𝐶SAH 𝑉L, 𝑉R 𝑆, 𝑉P = 𝑐 + 𝑝L. 𝑁L + 𝑝R. 𝑁R
Surface Area Heuristic (SAH)
• BVHs [Goldsmith & Salmon 1987], BSPs/kd-trees [MacDonald & Booth 1990]
• BIHs [Havran et al. 2006] [Woop et al. 2006], SBVHs [Stich et al. 2009]
Improvements
• split clipping & automatic termination [Havran & Bittner 2002]
• favouring empty space cut offs [Hurley et al. 2002] [Wald & Havran 2006]
• mailboxing [Hunter 2008]
Alternatives
• interior ray distribution [Fabianowski et al. 2009]
• fixed origin/direction ray distribution [Havran & Bittner 1999]
• local ray termination [Reinhard et al. 1996] [Havran 2000] [Moulin et al. 2015] [Moulin 2015]
• global ray termination [Choi et al. 2012]
• actual ray distribution [Bittner & Havran 2009] [Vinkler et al. 2012]
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35. 4. ADS build
𝐶SAH 𝑉L, 𝑉R 𝑆, 𝑉P = 𝑐 + 𝑝L. 𝑁L + 𝑝R. 𝑁R
Surface Area Heuristic (SAH)
• BVHs [Goldsmith & Salmon 1987], BSPs/kd-trees [MacDonald & Booth 1990]
• BIHs [Havran et al. 2006] [Woop et al. 2006], SBVHs [Stich et al. 2009]
Improvements
• split clipping & automatic termination [Havran & Bittner 2002]
• favouring empty space cut offs [Hurley et al. 2002] [Wald & Havran 2006]
• mailboxing [Hunter 2008]
Alternatives
• interior ray distribution [Fabianowski et al. 2009]
• fixed origin/direction ray distribution [Havran & Bittner 1999]
• local ray termination [Reinhard et al. 1996] [Havran 2000] [Moulin et al. 2015] [Moulin 2015]
• global ray termination [Choi et al. 2012]
• actual ray distribution [Bittner & Havran 2009] [Vinkler et al. 2012]
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36. 4. ADS build
no ray termination local ray termination global ray termination
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37. 4. ADS build
no ray termination local ray termination global ray termination
𝐶SAH 𝑉L, 𝑉R 𝑆, 𝑉P 𝐶RTSAH 𝑉L, 𝑉R 𝑆, 𝑉P
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38. 4. ADS build
𝐶SAH 𝑉L, 𝑉R 𝑆, 𝑉P = 𝑐 + 𝑝L. 𝑁L + 𝑝R. 𝑁R
Assumptions about the ray distribution:
• isotropic distribution of rays originating from outside the scene
• rays do not intervene with geometric primitives
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39. 4. ADS build
𝐶SAH 𝑉L, 𝑉R 𝑆, 𝑉P = 𝑐 + 𝑝L. 𝑁L + 𝑝R. 𝑁R
Assumptions about the ray distribution:
• isotropic distribution of rays originating from outside the scene
• rays do not intervene with geometric primitives
𝐶RTSAH 𝑉L, 𝑉R 𝑆, 𝑉P = 𝑐
+ 𝑝jL + 𝑝L⟶R + 𝑝R⟶L. 𝒱R⟶L . 𝑁L
+ 𝑝jR + 𝑝R⟶L + 𝑝L⟶R. 𝒱L⟶R . 𝑁R
Assumptions about the ray distribution:
• isotropic distribution of rays originating from outside the parent voxel
• rays do intervene with geometric primitives
local ray termination
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40. 4. ADS build
𝐶SAH 𝑉L, 𝑉R 𝑆, 𝑉P = 𝑐 + 𝑝L. 𝑁L + 𝑝R. 𝑁R
• Piercing probabilities: 𝑝L 𝑝R
𝐶RTSAH 𝑉L, 𝑉R 𝑆, 𝑉P = 𝑐
+ 𝑝jL + 𝑝L⟶R + 𝑝R⟶L. 𝒱R⟶L . 𝑁L
+ 𝑝jR + 𝑝R⟶L + 𝑝L⟶R. 𝒱L⟶R . 𝑁R
• Piercing probabilities: 𝑝jL 𝑝jR 𝑝L⟶R 𝑝R⟶L (𝑝jϵ)
• Visibility probabilities: 𝒱L⟶R 𝒱R⟶L
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41. 4. ADS build
𝐶SAH 𝑉L, 𝑉R 𝑆, 𝑉P = 𝑐 + 𝑝L. 𝑁L + 𝑝R. 𝑁R
• Piercing probabilities: 𝑝L 𝑝R
𝐶RTSAH 𝑉L, 𝑉R 𝑆, 𝑉P = 𝑐
+ 𝑝jL + 𝑝L⟶R + 𝑝R⟶L. 𝒱R⟶L . 𝑁L
+ 𝑝jR + 𝑝R⟶L + 𝑝L⟶R. 𝒱L⟶R . 𝑁R
• Piercing probabilities: 𝑝jL 𝑝jR 𝑝L⟶R 𝑝R⟶L (𝑝jϵ)
• Visibility probabilities: 𝒱L⟶R 𝒱R⟶L
𝐶RTSAH 𝑉L, 𝑉R 𝑆, 𝑉P = 𝑐 + 𝑝vL. 𝑁L + 𝑝vR. 𝑁R
• Combined probabilities: 𝑝vL 𝑝vR
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42. 4. ADS build
𝐶SAH 𝑉L, 𝑉R 𝑆, 𝑉P = 𝑐 + 𝑝L. 𝑁L + 𝑝R. 𝑁R
• Piercing probabilities: 𝑝L 𝑝R
𝐶RTSAH 𝑉L, 𝑉R 𝑆, 𝑉P = 𝑐
+ 𝑝jL + 𝑝L⟶R + 𝑝R⟶L. 𝒱R⟶L . 𝑁L
+ 𝑝jR + 𝑝R⟶L + 𝑝L⟶R. 𝒱L⟶R . 𝑁R
• Piercing probabilities: 𝑝jL 𝑝jR 𝑝L⟶R 𝑝R⟶L (𝑝jϵ)
• Visibility probabilities: 𝒱L⟶R 𝒱R⟶L
𝐶RTSAH 𝑉L, 𝑉R 𝑆, 𝑉P = 𝑐 + 𝑝vL. 𝑁L + 𝑝vR. 𝑁R
BSPs, BVHs,
BIHs, SBVHs
BSPs, BIHs
BSPs, BVHs,
BIHs, SBVHs
• Combined probabilities: 𝑝vL 𝑝vR
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43. 6. Objectives
• Develop heuristics to approximate local ray termination efficiently
i.e. without additional pre-processing or ray (sub)sampling
[Havran 2000] [Choi et al. 2012] [Bittner & Havran 2009] [Vinkler et al. 2012]
[Feltman et al. 2012] [Ogaki & Derouet-Jourdan 2015, 2016]
• Use these heuristics for:
• approximating plane visibility probabilities in RTSAH
to construct BSPs, BIHs and (S)BVHs for accelerating closest-hit queries;
• approximating box hit probabilities in SRDH
to construct BVHs for accelerating any-hit queries.
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44. 7. Architecture: Binned Partitioners
Binned Partitioner
Build
𝑝, 𝒞 𝑝, 𝒞 𝑝
¬
𝒪 𝑁 log 𝑁 with 𝐵 ≪ 𝑁
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45. 7. Architecture: Binned Partitioners
Binned Partitioner
Partition Strategy
Build
Cost Strategy
𝑉L, 𝑉R, 𝐺L, 𝐺R
𝒞 𝑝, 𝒞 𝑝
¬
𝑝, 𝒞 𝑝, 𝒞 𝑝
¬
𝒪 𝑁 log 𝑁 with 𝐵 ≪ 𝑁
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46. 7. Architecture: Binned Partitioners
Binned Partitioner
Partition Strategy
Build
Cost Model
Cost Strategy
𝒞 𝑝, 𝒞 𝑝
¬
𝑝, 𝒞 𝑝, 𝒞 𝑝
¬
SAH, SIROH𝑝L, 𝑝R, (𝑝P)
𝑉L, 𝑉R, 𝐺L, 𝐺R
𝒪 𝑁 log 𝑁 with 𝐵 ≪ 𝑁
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47. 7. Architecture: Binned Partitioners
Binned Partitioner
Partition Strategy
Build
Permeability Strategy RTSAH Cost Model
Permeability Approximation
ℱℱ⊥, ℱℱ// ⟶ 𝑝jL, 𝑝jR, 𝑝jϵ, 𝑝LR
Cost Model
Cost Strategy
OR
𝒱LR, 𝒱RL
𝒞 𝑝, 𝒞 𝑝
¬
𝑝, 𝒞 𝑝, 𝒞 𝑝
¬
SAH, SIROH
RTSAH for
BSP, BIH
𝑝L, 𝑝R, (𝑝P)
𝑉L, 𝑉R, 𝐺L, 𝐺R
𝒪 𝑁 log 𝑁 with 𝐵 ≪ 𝑁
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48. 7. Architecture: Binned Partitioners
Binned Partitioner
Partition Strategy
Build
Permeability Strategy RTSAH Cost Model
Permeability Approximation
RTSAH Cost ModelMC iterations
Permeability Transfer Strategy
Permeability Approximation
ℱℱ⊥, ℱℱ// ⟶ 𝑝jL, 𝑝jR, 𝑝jϵ, 𝑝LR
Cost Model
Cost Strategy
OR
OR
𝒱LR, 𝒱RL
𝑝vL, 𝑝vR
𝒞 𝑝, 𝒞 𝑝
¬
𝑝, 𝒞 𝑝, 𝒞 𝑝
¬
SAH, SIROH𝑝L, 𝑝R, (𝑝P)
𝑉L, 𝑉R, 𝐺L, 𝐺R
RTSAH for
BSP, BIH
RTSAH for
BSP, BIH,
(S)BVH
𝒪 𝑁 log 𝑁 or 𝒪 𝑁 log 𝑁 2
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