Accelerating Ray Tracing usingConstrained Tetrahedralizations

Ares Lagae & Philip Dutré

19th Eurographics Symposium on Rendering

EGSR 2008 Wednesday, June 25th

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

Introduction

Constrained tetrahedralizations

Construct constrained tetrahedralization

as a preprocess

Use constrained tetrahedralization

during ray traversal

Constrained Triangulation

• 2D triangulation + constraints (edges)

constraintsconstrained Delaunay

triangulation

conforming Delaunay

triangulation

quality Delaunay

triangulation

Constrained Tetrahedralization

• 3D tetrahedralization + constraints (faces)

constrained Delaunay tetrahedralization

quality Delaunay tetrahedralization

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

Ray Traversal

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

locate ray origintraverse

triangulation

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

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

Ray Traversal

ray hitting scene geometry

ray just missing scene geometry

• Example

Ray Tracing Cost

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

scenequality Delaunay tetrahedralization

constrained Delaunay tetrahedralization

kd-tree

low high

Ray Tracing Cost

• Teapot-in-a-stadium problem

scenequality Delaunay tetrahedralization

constrained Delaunay tetrahedralization

kd-tree

Advantages

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

(log )O G

( )O L

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

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

Disadvantages

• Constructing constrained tetrahedralizations– TetGen, CGAL

• Geometry preconditioning– Eliminating all self-intersections from triangle soup

• Absolute performance– Limited testing, limited optimization

Conclusion & Future Work

• ConclusionConstrained 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

Thanks!

• Questions?

AcknowledgmentsAres Lagae is a Postdoctoral Fellow of the Research Foundation Flanders (FWO)Peter Vangorp and Jurgen LaurijssenJan Welkenhuyzen from MaterializeTim Volodine

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