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INFOMAGR Advanced Graphics Jacco Bikker - November 2019 - February 2020 Lecture 5 - The Perfect BVH Welcome! , = (, ) , + , , ,

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SLIDE 1

𝑱 π’š, π’šβ€² = 𝒉(π’š, π’šβ€²) 𝝑 π’š, π’šβ€² + ΰΆ±

𝑻

𝝇 π’š, π’šβ€², π’šβ€²β€² 𝑱 π’šβ€², π’šβ€²β€² π’†π’šβ€²β€²

INFOMAGR – Advanced Graphics

Jacco Bikker - November 2019 - February 2020

Lecture 5 - β€œThe Perfect BVH”

Welcome!

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SLIDE 2

Today’s Agenda:

β–ͺ Building Better BVHs β–ͺ Refitting β–ͺ Fast BVH Construction β–ͺ The Top-level BVH

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SLIDE 3

Better BVHs

Advanced Graphics – The Perfect BVH 3

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SLIDE 4

Better BVHs

Advanced Graphics – The Perfect BVH 4

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SLIDE 5

Better BVHs

Advanced Graphics – The Perfect BVH 5

What Are We Trying To Solve?

A BVH is used to reduce the number of ray/primitive intersections. But: it introduces new intersections. The ideal BVH minimizes: β–ͺ # of ray / primitive intersections β–ͺ # of ray / node intersections.

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Better BVHs

Advanced Graphics – The Perfect BVH 6

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Better BVHs

Advanced Graphics – The Perfect BVH 7

BVH versus kD-tree

The BVH better encapsulates geometry. βž” This reduces the chance of a ray hitting a node. βž” This is all about probabilities! What is the probability of a ray hitting a random triangle? What is the probability of a ray hitting a random node? This probability is proportional to surface ar area.

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Better BVHs

Advanced Graphics – The Perfect BVH 8

Route 1: 10% up-time, $1000 fine Route 2: 100% up-time, $100 fine

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Better BVHs

Advanced Graphics – The Perfect BVH 9

Optimal Split Plane Position

The ideal split minimizes the expected cost of a ray intersecting the resulting nodes. This expected cost is based on: β–ͺ Number of primitives that will have to be intersected β–ͺ Probability of this happening The cost of a split is thus: π΅π‘šπ‘“π‘”π‘’ βˆ— π‘‚π‘šπ‘“π‘”π‘’ + π΅π‘ π‘—π‘•β„Žπ‘’ βˆ— π‘‚π‘ π‘—π‘•β„Žπ‘’

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SLIDE 10

Better BVHs

Advanced Graphics – The Perfect BVH 10

Optimal Split Plane Position

The ideal split minimizes the expected cost of a ray intersecting the resulting nodes. This expected cost is based on: β–ͺ Number of primitives that will have to be intersected β–ͺ Probability of this happening The cost of a split is thus: π΅π‘šπ‘“π‘”π‘’ βˆ— π‘‚π‘šπ‘“π‘”π‘’ + π΅π‘ π‘—π‘•β„Žπ‘’ βˆ— π‘‚π‘ π‘—π‘•β„Žπ‘’

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Better BVHs

Advanced Graphics – The Perfect BVH 11

Optimal Split Plane Position

Or, more concisely: π΅π‘šπ‘“π‘”π‘’ βˆ— π΅π‘šπ‘“π‘”π‘’

1

βˆ— π‘‚π‘šπ‘“π‘”π‘’

1

+ π΅π‘ π‘—π‘•β„Žπ‘’

1

βˆ— π‘‚π‘ π‘—π‘•β„Žπ‘’

1

+ π΅π‘ π‘—π‘•β„Žπ‘’ βˆ— π΅π‘šπ‘“π‘”π‘’

2

βˆ— π‘‚π‘šπ‘“π‘”π‘’

2

+ π΅π‘ π‘—π‘•β„Žπ‘’

2

βˆ— π‘‚π‘ π‘—π‘•β„Žπ‘’

2

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Better BVHs

Advanced Graphics – The Perfect BVH 12

Optimal Split Plane Position

Which positions do we consider? Object subdivision may happen over x, y or z axis. The cost function is constant between primitive centroids. βž” For N primitives: 3(𝑂 βˆ’ 1) possible locations βž” For a 2-level tree: (3(𝑂 βˆ’ 1))2 configurations

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Better BVHs

Advanced Graphics – The Perfect BVH 13

SAH and Termination

A split is β€˜not worth it’ if it doesn’t yield a cost lower than the cost of the parent node, i.e.: π΅π‘šπ‘“π‘”π‘’ βˆ— π‘‚π‘šπ‘“π‘”π‘’ + π΅π‘ π‘—π‘•β„Žπ‘’ βˆ— π‘‚π‘ π‘—π‘•β„Žπ‘’ β‰₯ 𝐡 βˆ— 𝑂 This provides us with a natural and

  • ptimal termination criterion.

(and it solves the problem

  • f the Bad Artist)
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SLIDE 14

Better BVHs

Advanced Graphics – The Perfect BVH 14

Optimal Split Plane Position

Evaluating (3(𝑂 βˆ’ 1))2 configurations? Solution: apply the surface area heuristic (SAH) in a greedy manner*.

*: Heuristics for Ray Tracing using Space Subdivision, MacDonald & Booth, 1990.

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Better BVHs

Advanced Graphics – The Perfect BVH 15

Optimal Split Plane Position

Comparing naΓ―ve versus SAH: β–ͺ SAH will cut #intersections in half; β–ͺ expect ~2x better performance. SAH & kD-trees: β–ͺ Same scheme applies.

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Better BVHs

Advanced Graphics – The Perfect BVH 16

Median Split

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Better BVHs

Advanced Graphics – The Perfect BVH 17

Surface Area Heuristic

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Better BVHs

Advanced Graphics – The Perfect BVH 18

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Better BVHs

Advanced Graphics – The Perfect BVH 19

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Better BVHs

Advanced Graphics – The Perfect BVH 20

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SLIDE 21

Today’s Agenda:

β–ͺ Building Better BVHs β–ͺ Refitting β–ͺ Fast BVH Construction β–ͺ The Top-level BVH

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Refitting

Advanced Graphics – The Perfect BVH 22

Summary of BVH Characteristics

A BVH provides significant freedom compared to e.g. a kD-tree: β–ͺ No need for a 1-to-1 relation between bounding boxes and primitives β–ͺ Bounding boxes may overlap β–ͺ Bounding boxes can be altered, as long as they fit in their parent box β–ͺ A BVH can be very bad but still valid Some consequences / opportunities: β–ͺ We can rebuild part of a BVH β–ͺ We can combine two BVHs into one β–ͺ We can refit a BVH

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Refitting

Advanced Graphics – The Perfect BVH 23

Refitting

Q: What happens to the BVH of a tree model, if we make it bend in the wind? A: Likely, only bounds will change; the topology of the BVH will be the same (or at least similar) in each frame. Refitting: Updating the bounding boxes stored in a BVH to match changed primitive coordinates.

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Refitting

Advanced Graphics – The Perfect BVH 24

Refitting

Updating the bounding boxes stored in a BVH to match changed primitive coordinates. Algorithm:

  • 1. For each leaf, calculate the bounds
  • ver the primitives it represents
  • 2. Update parent bounds
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Refitting

Advanced Graphics – The Perfect BVH 25

Refitting - Suitability

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Refitting

Advanced Graphics – The Perfect BVH 26

Refitting – Practical

Order of nodes in the node array: We will never find the parent of node X at a position greater than X. Therefore:

for( int i = N-1; i >= 0; i-- ) nodeArray[i].AdjustBounds();

N-1 BVH node array Root node Level 1

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SLIDE 27

Today’s Agenda:

β–ͺ Building Better BVHs β–ͺ Refitting β–ͺ Fast BVH Construction β–ͺ The Top-level BVH

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Binning

Advanced Graphics – The Perfect BVH 28

Rapid BVH Construction

Refitting allows us to update hundreds of thousands of primitives in real-

  • time. But what if topology changes significantly?

Rebuilding a BVH requires 3π‘‚π‘šπ‘π‘•π‘‚ split plane evaluations. Options:

  • 1. Do not use SAH (significantly lower quality BVH)
  • 2. Do not evaluate all 3 axes (minor degradation of BVH quality)
  • 3. Make split plane selection independent of 𝑂
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Binning

Advanced Graphics – The Perfect BVH 29

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Binning

Advanced Graphics – The Perfect BVH 30

Binned BVH Construction* Binned construction: Evaluate SAH at N discrete intervals.

*: On fast Construction of SAH-based Bounding Volume Hierarchies, Wald, 2007

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Binning

Advanced Graphics – The Perfect BVH 31

Binned BVH Construction

Detailed algorithm: 1. Calculate spatial bounds 2. Calculate object centroid bounds 3. Calculate intervals (efficiently and accurately!) 4. Populate bins 5. Sweep: evaluate cost, keep track of counts 6. Use best position

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Binning

Advanced Graphics – The Perfect BVH 32

Binned BVH Construction

Performance evaluation: 472ms 7.88M triangles (12 cores @ 2Ghz)*.

*: Parallel BVH Construction using Progressive Hierarchical Refinement, Henrich et al., 2016.

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Binning

Advanced Graphics – The Perfect BVH 33

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Today’s Agenda:

β–ͺ Building Better BVHs β–ͺ Refitting β–ͺ Fast BVH Construction β–ͺ The Top-level BVH

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Top-level BVH

Advanced Graphics – Real-time Ray Tracing 35

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Top-level BVH

Advanced Graphics – Real-time Ray Tracing 36

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Top-level BVH

Advanced Graphics – Real-time Ray Tracing 37

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Top-level BVH

Advanced Graphics – Real-time Ray Tracing 38

Combining BVHs

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Top-level BVH

Advanced Graphics – Real-time Ray Tracing 39

Combining BVHs

Two BVHs can be combined into a single BVH, by simply adding a new root node pointing to the two BVHs. β–ͺ This works regardless of the method used to build each BVH β–ͺ This can be applied repeatedly to combine many BVHs

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Advanced Graphics – Real-time Ray Tracing 40

Scene Graph

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Top-level BVH

Advanced Graphics – Real-time Ray Tracing 41

Scene Graph

world car wheel wheel wheel wheel turret plane plane car wheel wheel wheel wheel turret buggy wheel wheel wheel wheel dude dude dude

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Top-level BVH

Advanced Graphics – Real-time Ray Tracing 42

Scene Graph

If our application uses a scene graph, we can construct a BVH for each scene graph node. The BVH for each node is built using an appropriate construction algorithm: β–ͺ High-quality SBVH for static scenery (offline) β–ͺ Fast binned SAH BVHs for dynamic scenery The extra nodes used to combine these BVHs into a single BVH are known as the Top-level BVH .

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Top-level BVH

Advanced Graphics – Real-time Ray Tracing 43

Rigid Motion

Applying rigid motion to a BVH:

  • 1. Refit the top-level BVH
  • 2. Refit the affected BVH
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Top-level BVH

Advanced Graphics – Real-time Ray Tracing 44

Rigid Motion

Applying rigid motion to a BVH:

  • 1. Refit the top-level BVH
  • 2. Refit the affected BVH
  • r:
  • 2. Transform the ray, not the node

Rigid motion is achieved by transforming the rays by the inverse transform upon entering the sub-BVH.

(this obviously does not only apply to translation)

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Top-level BVH

Advanced Graphics – Real-time Ray Tracing 45

The Top-level BVH - Construction

Input: list of axis aligned bounding boxes for transformed scene graph nodes Algorithm:

  • 1. Find the two elements in the list for which the AABB has the smallest

surface area

  • 2. Create a parent node for these elements
  • 3. Replace the two elements in the list by the parent node
  • 4. Repeat until one element remains in the list.

Note: algorithmic complexity is 𝑃(𝑂3).

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Top-level BVH

Advanced Graphics – Real-time Ray Tracing 46

The Top-level BVH – Faster Construction*

Algorithm:

Node A = list.GetFirst(); Node B = list.FindBestMatch( A ); while (list.size() > 1) { Node C = list.FindBestMatch( B ); if (A == C) { list.Remove( A ); list.Remove( B ); A = new Node( A, B ); list.Add( A ); B = list.FindBestMatch( A ); } else A = B, B = C; }

*: Fast Agglomerative Clustering for Rendering, Walter et al., 2008

A B C A B A B

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SLIDE 47

Top-level BVH

Advanced Graphics – Real-time Ray Tracing 47

The Top-level BVH – Traversal

The leafs of the top-level BVH contain the sub-BVHs. When a ray intersects such a leaf, it is transformed by the inverted transform matrix of the sub-BVH. After this, it traverses the sub-BVH. Once the sub-BVH has been traversed, we transform the ray again, this time by the transform matrix of the sub-BVH. For efficiency, we store the inverted matrix with the sub-BVH root.

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Top-level BVH

Advanced Graphics – Real-time Ray Tracing 48

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Top-level BVH

Advanced Graphics – Real-time Ray Tracing 49

The Top-level BVH – Summary

The top-level BVH enables complex animated scenes: β–ͺ for static objects, it contains high-quality sub-BVHs; β–ͺ for objects undergoing rigid motion, it also contains high-quality sub-BVHs, with a transform matrix and its inverse; β–ͺ for deforming objects, it contains sub-BVHs that can be refitted; β–ͺ for arbitrary animations, it contains lower quality sub-BVHs. Combined, this allows for efficient maintenance of a global BVH.

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INFOMAGR – Advanced Graphics

Jacco Bikker - November 2019 - February 2020

END of β€œThe Perfect BVH”

next lecture: β€œPath Tracing”

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SLIDE 51

Practical:

  • 1. Converging
  • 2. Handling materials and textures