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