Ray Tracing 2 Trace Harder COMP 575/770 Spring 2013

Ray-Triangle Intersection  Core operation while ray tracing triangle meshes  Given:  Ray with origin 𝑜 and direction 𝑑  Triangle with vertices 𝑣0 , 𝑣1 , and 𝑣2

 Figure out the intersection point

Ray-Triangle Intersection  Any point on the ray is given by: 𝑝 = 𝑜 + 𝑡𝑑  Any point on the triangle is given by: 𝑝 = 𝑣0 + 𝛽 𝑣1 − 𝑣0 + 𝛾 𝑣2 − 𝑣0  Barycentric coordinates!

Ray-Triangle Intersection  One equation for each coordinate: 𝑥𝑜 + 𝑡𝑥𝑑 = 𝑥𝑎 + 𝛽 𝑥𝑏 − 𝑥𝑎 + 𝛾 𝑥𝑐 − 𝑥𝑎 𝑦𝑜 + 𝑡𝑦𝑑 = 𝑦𝑎 + 𝛽 𝑦𝑏 − 𝑦𝑎 + 𝛾 𝑦𝑐 − 𝑦𝑎 𝑧𝑜 + 𝑡𝑧𝑑 = 𝑧𝑎 + 𝛽 𝑧𝑏 − 𝑧𝑎 + 𝛾(𝑧𝑐 − 𝑧𝑎 )

 In matrix form: 𝑥𝑎 − 𝑥𝑏 𝑦𝑎 − 𝑦𝑏 𝑧𝑎 − 𝑧𝑏

𝑥𝑎 − 𝑥𝑐 𝑦𝑎 − 𝑦𝑐 𝑧𝑎 − 𝑧𝑐

𝑥𝑑 𝑦𝑑 𝑧𝑑

 Solve using Cramer’s rule

𝑥𝑎 − 𝑥𝑒 𝛽 𝛾 = 𝑦𝑎 − 𝑦𝑒 𝑧𝑎 − 𝑧𝑒 𝑡

Ray-Triangle Intersection  A faster approach: Moller-Trumbore intersection algorithm

http://www.cs.virginia.edu/~gfx/Courses/2003/ImageSynthesis/papers/Acc eleration/Fast%20MinimumStorage%20RayTriangle%20Intersection.pdf

Ray-Triangle Intersection  In general, two parts to any intersection test:  Determining 𝑡 (ray-plane test)  Determining 𝛽, 𝛾 (barycentric test)

 Ray-plane test finds a potential hit point  Barycentric test checks that a hit point exists  Does the ordering matter?

Ray-Triangle Intersection  Eye Rays  Need to find the closest hit  If 𝑡 is greater than the current closest hit, bail  Ray-plane test before barycentric test

 Shadow Rays  Need to find any hit in an interval  If the ray can’t possible intersect, bail  Barycentric test before ray-plane test

Triangle Meshes  Test each ray with each triangle?  Too slow!  1M pixels (rays), 1000 triangles = 1 billion tests!

 Rasterization is much better at this  Why?

Triangle Meshes  Ray Tracing: for each pixel for each triangle test (ray, triangle)  Rasterization: for each triangle for each pixel test (pixel, triangle)

Triangle Meshes  Ray Tracing: for each pixel for each triangle test (ray, triangle)  Rasterization: for each triangle for each pixel potentially covered test (pixel, triangle)

Triangle Meshes  Ray Tracing: for each pixel for each triangle potentially intersecting test (ray, triangle)  Rasterization: for each triangle for each pixel potentially covered test (pixel, triangle)

Acceleration Structures  A special data structure to accelerate ray tracing  Used to quickly determine which triangles might intersect a ray

 Usually a hierarchical tree structure  Object hierarchies  Spatial hierarchies

Outline  Ray-Triangle Intersection  Bounding Volume Hierarchy  Ray-Box Intersection

 kD Trees  Efficient Ray Tracing

Bounding Volume Hierarchies  Hierarchical clustering of objects (triangles)  Each cluster represented as a bounding volume

Bounding Volume Hierarchies

Bounding Volume Hierarchies  Typically binary trees  Each internal node splits a set of triangles into two

 Each node stores its bounding volume

 Each leaf typically contains one triangle  Variations are possible

Bounding Volumes  A BVH can be built using any kind of bounding volume  Typical choices:    

Spheres Axis-Aligned Bounding Boxes (AABBs) Oriented Bounding Boxes (OBBs) Discrete Oriented Polytopes (DOPs)

 Need to test whether a ray passes through a bounding volume

Ray-AABB Intersection  An AABB is a region bounded by 3 slabs  2D example shown here for convenience

Ray-AABB Intersection  AABB intersection expressed as 3 slab intersection tests  2 tests in 2D

 Need to find interval of 𝑡 in which ray is in AABB

Ray-AABB Intersection  Consider a box with min. coordinates (𝑥0 , 𝑦0 , 𝑧0 ) and max. coordinates 𝑥1 , 𝑦1 , 𝑧1  The 𝑥-slab goes from 𝑥 = 𝑥0 to 𝑥 = 𝑥1

 Which will the ray encounter first?  Depends on which way the ray is traveling

 If 𝑑𝑥 > 0, 𝑥𝑚𝑖𝑛 = 𝑥0 , 𝑥𝑚𝑎𝑥 = 𝑥1 , otherwise the other way around

Ray-AABB Intersection  Ray intersection with 𝑥 = 𝑥𝑚𝑖𝑛 : 𝑜𝑥 + 𝑡𝑑𝑥 = 𝑥𝑚𝑖𝑛  Similarly for intersection with 𝑥 = 𝑥𝑚𝑎𝑥  Ray passes through the 𝑥-slab in the interval [𝑡𝑥𝑚𝑖𝑛 , 𝑡𝑥𝑚𝑎𝑥 ] 𝑥𝑚𝑖𝑛 − 𝑜𝑥 𝑑𝑥 𝑥𝑚𝑎𝑥 − 𝑜𝑥 = 𝑑𝑥

𝑡𝑥𝑚𝑖𝑛 = 𝑡𝑥𝑚𝑎𝑥

Ray-AABB Intersection  Repeat process for 𝑦-slab (and 𝑧-slab)  Ray is in AABB when it’s in both (all three) slabs  Need to find the intersection of intervals

Ray-AABB Intersection Ray intersects AABB if 𝑡𝑚𝑖𝑛 ≤ 𝑡𝑚𝑎𝑥

BVH Traversal  Recursive algorithm: at current node: if ray doesn’t intersect current node’s bounding volume, bail if current node is leaf: check for intersection with stored triangle else (if current node is internal node): recurse on children

 Start with root node

 Still have to test against multiple triangles  Hopefully, bounding volume tests eliminate most triangles

Outline  Ray-Triangle Intersection  Bounding Volume Hierarchy  Ray-Box Intersection

 kD Trees  Efficient Ray Tracing

kD Trees  Or, binary spatial partitioning (BSP) trees  Instead of a hierarchy of objects, a hierarchy of spatial regions

 Allows front-to-back spatial sorting  Stop as soon as a ray hits something

kD Trees  Binary tree of spatial regions  At each node, the spatial region is split using an axis-aligned plane x y

y

kD Trees

kD Trees

kD Trees

kD Trees

kD Tree Traversal  Recursive algorithm: at current node: if current node is leaf: intersect with all stored triangles if ray hits a stored triangle, stop else (if current node is internal node): find intersection of ray with splitting plane if ray intersects both children: recurse on near side, then far side otherwise: recurse on side it intersects

kD Tree Construction  Given:  An AABB describing a region of space (“cell”)  List of triangles in the cell

 Core operation:  Pick an axis-aligned plane to split the cell  Assign triangles to each sub-cell  Some triangles may be in both

 Recurse  Subject to some termination criteria

kD Tree Construction  What axis to split along?

 Where to place splitting plane?

 When to terminate recursion?

kD Tree Construction  What axis to split along? Round-robin? Largest extent?

 Where to place splitting plane? Middle of extent? Median of geometry?

 When to terminate recursion? 1 triangle left in cell? Max. tree depth?

Outline  Ray-Triangle Intersection  Bounding Volume Hierarchy  Ray-Box Intersection

 kD Trees  Efficient Ray Tracing

Efficient Ray Tracing  Many techniques developed to speed up ray tracing       

Sophisticated tree construction algorithms Compact, cache-friendly memory layout Parallel programming on multi-core CPUs SIMD instructions for exploiting coherence Replace recursion with iteration Optimize inner loops …

BVH vs kD Tree  BVH:  Pros: Can be updated for dynamic geometry  Cons: Slower traversal, takes more memory

 kD Tree:  Pros: Faster traversal, takes less memory  Cons: Static scenes only