Accelerating Ray Tracing using Constrained Tetrahedralizations

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Accelerating Ray Tracing usingConstrained
Tetrahedralizations
19th Eurographics Symposium on Rendering

• Ares Lagae Philip Dutré

EGSR 2008
Wednesday, June 25th
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Introduction

• Acceleration structures for ray tracing
• Computer graphics
• BVH, kd-tree, grid
• ? Mostly practical (complexity? dynamic
  geometry?)
• Computational geometry
• Delaunay triangulation
• ? Mostly theoretical (theorems, proofs,
  implementations?)
• ? Constrained tetrahedralizations

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Introduction

• ? Constrained tetrahedralizations

Construct constrained tetrahedralization as a
preprocess
Use constrained tetrahedralization during ray
traversal
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Constrained Triangulation

• 2D triangulation constraints (edges)

constraints
constrained Delaunay triangulation
conforming Delaunay triangulation
quality Delaunay triangulation
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Constrained Tetrahedralization

• 3D tetrahedralization constraints (faces)

constrained Delaunay tetrahedralization
quality Delaunay tetrahedralization
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Construction

• Piecewise linear complex (PLC)
• Very general geometry representation
• Arbitrary polygons, holes, non-manifold geometry,
  
• Polygons must properly intersect
• Tetrahedralizations cannot have intersecting
  faces
• 1. Triangle soup ? PLC
• Eliminate all self-intersections
• 2. PLC ? constrained tetrahedralization
• TetGen, CGAL

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Ray Traversal

• Ray traversal
• Locate ray origin
• Traverse tetrahedralization one tetrahedron at a
  time
• Stop at constrained face

locate ray origin
traverse triangulation
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Ray Traversal

• Locate ray origin
• Potentially costly
• Accelerate
• Linear search ? grid, monotone subdivision
• Avoid by exploiting ray connectivity
• Rays start at camera position or where previous
  ray ended

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Ray Traversal

• Traverse tetrahedralization
• One tetrahedron at a time
• Given entry face, determine exit face
• Several methods

plane intersections
half space classification
scalar triple products
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Ray Traversal

• Example

ray hitting scene geometry
ray just missing scene geometry
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Ray Tracing Cost

• Comparison with kd-tree
• Ray tracing cost number of tetrahedra / nodes
  visited

scene
quality Delaunay tetrahedralization
constrained Delaunay tetrahedralization
kd-tree
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Ray Tracing Cost

• Teapot-in-a-stadium problem

scene
quality Delaunay tetrahedralization
constrained Delaunay tetrahedralization
kd-tree
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Advantages

• Deforming and dynamic geometry
• Deforming ? theoretical guarantee
• Dynamic ? efficient update

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Advantages

• Time complexity of ray traversal
• Constrained tetrahedralization
• Linear in local geometric complexity
• Hierarchical acceleration structures (kd-tree,
  bvh)
• Logarithmic at best in global geometric
  complexity
• No practical results yet
• Effect might only show up for large scenes

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Advantages

• Optimal constrained tetrahedralizations
• Weight tetrahedralization SAH for kd-trees
• Unified data structure for global illumination
• Associate data with vertices, edges, faces,
  tetrahedra
• Level-of-detail
• Meshes and triangulations use similar data
  structures

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Disadvantages

• Constructing constrained tetrahedralizations
• TetGen, CGAL
• Geometry preconditioning
• Eliminating all self-intersections from triangle
  soup
• Absolute performance
• Limited testing, limited optimization

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Conclusion Future Work

• Conclusion
• Constrained tetrahedralizations
• have a number of unique and interesting
  properties and
• offer several new perspectives on acceleration
  structures
• Future work
• Geometry preconditioning
• More elaborate testing
• Further explore advantages

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Thanks!

• Questions?

Acknowledgments Ares Lagae is a Postdoctoral
Fellow of the Research Foundation Flanders
(FWO) Peter Vangorp and Jurgen Laurijssen Jan
Welkenhuyzen from Materialize Tim Volodine
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Numerical Robustness

• Construction
• Adaptive precision floating point arithmetic
• Robust geometric predicates
• ? Common practice in computational geometry
• Traversal
• Ignore robustness errors and degenerate cases
• ? Common practice in computer graphic
• Detection and correction is possible
• ray parameters of plane intersections should be
  increasing
• temporarily move points