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 When do we prefer speed? 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 1 2 3 4 5 6 7 1 1 01 1 1 1 1101 1111

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. i i i+1 Last key of the left child First key of the right child

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 i i+1 The metric distance between two keys indicates the dissimilarity between the left child and the right child of node i.

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 1 2 3 4 5 6 1 2 3 4 5 6 7 1 1 2 2 3 3 4 4 5 5 6 6 7 7 1 1 01 1 1 1 1101 1111 1 2 3 4 5 6 7

Binary Radix Tree Construction 4 4 6 7 2 5 1 2 3 5 6 1 3 1 2 3 4 5 6 7 1 1 2 2 3 3 4 4 5 5 6 6 7 7 1 1 01 1 1 1 1101 1111 1 2 3 4 5 6 7

Binary Radix Tree Construction 1 4 4 3 6 5 7 2 5 1 2 3 5 6 3 1 2 3 4 5 6 7 1 1 2 2 3 3 4 4 5 5 6 6 7 7 1 1 01 1 1 1 1101 1111 1 2 3 4 5 6 7

Binary Radix Tree Construction 3 1 4 4 7 3 6 5 7 2 5 1 2 3 5 6 1 2 3 4 5 6 7 1 1 2 2 3 3 4 4 5 5 6 6 7 7 1 1 01 1 1 1 1101 1111 1 2 3 4 5 6 7

Binary Radix Tree Construction 3 7 1 4 4 7 3 6 5 7 2 5 1 2 3 5 6 1 2 3 4 5 6 7 1 1 2 2 3 3 4 4 5 5 6 6 7 7 1 1 01 1 1 1 1101 1111 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 à currentNode 6: RangeOfKeysparent.left à Left 7: else 8: parent à Left - 1 9: InternalNodesparent.childB à 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

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