fast and simple agglomerative lbvh construction

FAST AND SIMPLE AGGLOMERATIVE LBVH

CONSTRUCTION

Ciprian Apetrei

Computer Graphics & Visual Computing (CGVC) 2014

Introduction

Hierarchies What are they for?

Introduction

Hierarchies What are they for? Fast Construction or Better Quality?

Introduction

Hierarchies What are they for? Fast Construction or Better Quality?

When do we prefer speed? View-frustum culling Collision detection Particle simulation Voxel-based global illumination

Background

Previous GPU methods constructed the hierarchy in 4 steps: Morton code calculation Sorting the primitives Hierarchy generation Bottom-ul traversal to fit bounding-boxes

Binary radix tree

The binary representations of each key are in lexicographical order.

The keys are partitioned according to their first differing bit.

Because the keys are ordered, each node covers a linear range of keys.

A radix tree with n keys has n-1 internal nodes.

Binary radix tree

The binary representations of each key are in lexicographical order.

The keys are partitioned according to their first differing bit.

Because the keys are ordered, each node covers a linear range of keys.

A radix tree with n keys has n-1 internal nodes.

The parent of a node splits the hierarchy immediately before the first key of its right child and after the last key of its left child.

Binary radix tree

0010

0101

1000

1100

0100

0011

1101

1111

0 1 2 3 4 5 6 7

Algorithm Overview

Define a numbering scheme for the nodes Establish a connection with the keys Gain some knowledge about the parent

Algorithm Overview

Define a numbering scheme for the nodes Establish a connection with the keys Gain some knowledge about the parent

Each internal node i splits the hierarchy between keys i and i + 1

i

i

i+1

Algorithm Overview

We can find the parent of a node by knowing the range of keys covered by it.

The parent splits the hierarchy either immediately before the first key or after the last key.

Last key of the left child First key of the right child

i

i

i+1

Algorithm Overview

To determine the parent we have to analyze the split point of the two nodes. The one that splits the hierarchy between two more similar subtrees is the direct parent.

i

The metric distance between two keys indicates the dissimilarity between the left child and the right child of node i.

i i+1

Algorithm Overview

Algorithm steps: Start from each leaf node.

Algorithm Overview

Algorithm steps: Start from each leaf node. Choose the parent between the two internal

nodes that split the hierarchy at the ends of the range of keys covered by the current node.

Pass the range of keys to the parent.

Algorithm Overview

Algorithm steps: Start from each leaf node. Choose the parent between the two internal

nodes that split the hierarchy at the ends of the range of keys covered by the current node.

Pass the range of keys to the parent. Calculate the bounding box of the node. Advance towards the root. Only process a node if it has both its

children set.

Binary Radix Tree Construction

20 1 43 5 6

0 1 2 3 4 5 6 7

0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 70010

0101

1000

1100

0100

0011

1101

1111

0 1 2 3 4 5 6 7

Binary Radix Tree Construction

20

1

4

3

5

6

0 1 2 3

4

5 6

7

0 1 2 3 4 5 6 7

0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 70010

0101

1000

1100

0100

0011

1101

1111

0 1 2 3 4 5 6 7

Binary Radix Tree Construction

20

1 4

3

5

6

0 1 2 3

4

5 6

7

0 3

5

0 1 2 3 4 5 6 7

0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 70010

0101

1000

1100

0100

0011

1101

1111

0 1 2 3 4 5 6 7

Binary Radix Tree Construction

20

1 4

3

5

6

1 3 6

7

3 7

0

0 2 5

5

0 4

0 1 2 3 4 5 6 7

0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 70010

0101

1000

1100

0100

0011

1101

1111

0 1 2 3 4 5 6 7

Binary Radix Tree Construction

20

1 4

3

5

6

1 3 6

7

3 7

70

0 2 5

5

0 4

0 1 2 3 4 5 6 7

0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 70010

0101

1000

1100

0100

0011

1101

1111

0 1 2 3 4 5 6 7

Pseudocode

Pseudocode for choosing the parent: 1: def ChooseParent(Left, Right, currentNode)

2: If ( Left = 0 or ( Right != n and δ(Right) < δ(Left-1) ) )

3: then

4: parent à Right

5: InternalNodesparent.childA Ã currentNode

6: RangeOfKeysparent.left à Left

7: else

8: parent à Left - 1

9: InternalNodesparent.childB Ã currentNode

10: RangeOfKeysparent.right à Right

Outline

General aspects about our algorithm: Bottom-up construction Finds the parent at each step O(n) time complexity Simple to implement Can be used for constructing different types

of trees Allows an user-defined distance metric for

choosing the parent.

Results

We used the bottom-up reduction algorithm presented by Karras[2012] for implementation.

Compare against Karras binary radix tree construction and bounding-box calculation.

Evaluate performance on GeForce GT 745M CUDA, 30-bit Morton Codes Used Thrust library radix sort

Results

Scene Sort Previous Bottom-up Radix tree

Squared Distance

Stanford Bunny (69K tris)

14.9 

1.78 4.53 5.56 0.85 4.74

Armadillo(345K tris)

32.1 

5.01 10.0 12.03 2.4 11.9

Skeleton Hand(654k tris)

77.8 

14.1 28.3 32.5 6.54 31.8

Stanford Dragon(871K tris)

102 19.6 37.1 42.9 8.61 42.2

Happy Buddha(1087K tris)

125 

23.2 46.8 53.4 10.7 52.7

Turbine Blade(1765K tris)

210 

37.3 73.9 85.9 17.3 85.3

Thank You

Questions