This document summarizes a class on acceleration structures for ray tracing. It discusses building bounding volume hierarchies and using them to accelerate ray intersection tests. Uniform grids, kd-trees, and binary space partitioning trees are covered as approaches for spatial subdivision. The role of acceleration structures in speeding up global illumination calculations is also discussed.

1. CS 354
Acceleration
Structures
Mark Kilgard
University of Texas
April 24, 2012

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Today’s material
 In-class quiz
 On global illumination lecture
 Lecture topic
 Project 4
 Acceleration structures

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My Office Hours
 Tuesday, before class
 Painter (PAI) 5.35
 8:45 a.m. to 9:15
 Thursday, after class
 ACE 6.302
 11:00 a.m. to 12
 Randy’s office hours
 Monday & Wednesday
 11 a.m. to 12:00
 Painter (PAI) 5.33

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Last time, this time
 Last lecture, we discussed
 Global illumination
 This lecture
 Acceleration structures
 Projects
 Project 4 on ray tracing on Piazza
 Due May 2, 2012
 Get started!

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On a sheet of paper
Daily Quiz • Write your EID, name, and date
• Write #1, #2, #3, #4 followed by its answer
 Multiple choice: In the Russian
Roulette approach to termination of  Multiple choice: Modeling the
tracing recursive rays, termination influence of participating media
occurs simulates
a) after a fixed number of ray a) motion blur
traces
b) fog
b) after a random number of ray
casts between 1 and a fixed c) smoke
constant
d) a., b., and c.
c) when an analytic solution can
be reached e) b. and c.
d) every trace has a random  True of False: Classic Radiosity
chance of being terminated assume the Bidirectional Reflectance
Distribution Function of all surfaces in
 True or False: A bidirectional the scene are Lambertian.
reflectance distribution function
returns a negative value
approximately 50% of the time.

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Project 4
 Provides ray tracing framework
 Use FLTK toolkit for user interface
 Includes sample scenes—.ray files
dragon.ray

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Sample File: sphere.ray
SBT-raytracer 1.0 point_light {
position = (-2, 2, -2);
camera { colour = (1, 0.3, 0.3);
position = (0,0,-4); constant_attenuation_coeff= 0.25;
viewdir = (0,0,1); linear_attenuation_coeff =
aspectratio = 1; 0.003372407;
updir = (0,1,0); quadratic_attenuation_coeff =
0.000045492;
}
}
directional_light {
material = {
direction = (0, 0, 1);
diffuse = (0.4,0.8,0.2);
colour = (0.2, 0.2, 0.2);
specular = (1,1,0);
}
shininess = 64;
}
scale(1.5,
sphere {
})
scale(2,sphere{});

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What You Must Implement
 Blinn-Phong lighting  Implement shadow ray
model  Modify most
 Ambient, diffuse, specular RayTracer.cpp
 Modify scene/material.cpp  Implement triangle mesh
 Point light distance intersection
attenuation  Including normal
 Inverse squared distance interpolation
fall-off  Modify
 Modify scene/light.cpp SceneObjects/trimesh.cpp
 Implement reflection and
refraction rays
 Modify most
RayTracer.cpp

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Extra Credit Embellishments
 Spatial data structures
 Speed up the ray traces
 Texture mapping
 Anti-aliasing
 Cast multiple rays per pixel
 Lighting effects
 Normal mapping, bump mapping,
environment mapping

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Other Project 4 Scenes
recursive_depth.ray turtle.ray hitchcook.ray
cone.ray spheres.ray

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Debugging Display

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Two Versions of
Rendering Equation
Occlusion (G) is zero
o
Le (x, ω , λ , t ) +
Le (x, ω , λ , t ) +
Lo (x, ω , λ , t ) =
∫
Ω
f r (x, ω ′, ω , λ , t ) Li (x, ω ′, λ , t ) (−ω ′ • n) dω ′ Lo (x, ω , λ , t ) =
∫ f r (x, ωyx , ω , λ , t ) L(y, ωyx , λ , t ) G (x, y ) dy
y∈Γ
Integrate over hemisphere Integrate over all surface points

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Photon Mapping
 Two-pass global illumination algorithm
 Developed by Henrick Jensen (1996)
 Two passes
 Randomly distribute photons around the scene
 Called “photon map construction”
 Render treating photons as mini-light sources
 Capable of efficiently generating otherwise very
expensive effects
 Caustics
 Diffuse inter-reflections, such as color bleed
 Sub-surface scattering

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Light Tracing
 Trace rays from the light
 Contribution rays accumulate image samples

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Photon Mapping Examples
without diffuse photo map
caustics interreflection visualization
sub-surface scattering
diffuse interreflection

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Global Illumination Often Gated
by Ray Tracing Speed
 Shooting rays tends to be the bottleneck
 Why?
 Lots of rays cast for quality
 Shadow rays, lots for soft shadows
 Reflection and refraction rays
 Speeding up global illumination generally
means speeding up tracing of rays
 Shading operations can be expensive too
 But rays involve data structure traversal

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More bounces
Recursive Rays
 Reflections and
refractions can spawn
lots of rays
Fewer bounces 17

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Sufficient Shadow Ray Sampling

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Distribution Ray Tracing
 Soft shadows
 Distribute shadow rays over light source region
All shadow rays No shadow rays Some shadow rays
hit light source, hit light source, hit light source,
fully illuminated fully shadowed partially shadowed

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Distribution Ray Tracing
 Motion blur
 Distribute rays over time
Pool Balls
Tom Porter
RenderMan

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Distribution Ray Tracing
 Depth of field
 Distribute rays across a discrete camera aperture
No depth-of-field More rays
Jittered
depth-of-field
More images for depth-of-field Even more rays

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Acceleration Techniques
Ray Tracing Acceleration Techniques
Fast Fewer Generalized
Intersections Rays Rays
Statistical Beam tracing
Faster Fewer optimizations
ray-object ray-object for illumination Cone tracing
convergence
intersections intersections Pencil tracing
Object bounding Bounding volume
volumes hierarchies

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Accelerating Ray Trace
Intersection Operations
 Two key optimizations
3. Exploit binary searching
 Rather than linear searches
4. Group objects spatially
 Discard hierarchically
 Use quick-and-coarse tests…
 …to avoid slow-and-exact intersection tests

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Acceleration Structures:
Bounding Volume Hierarchies
 Build hierarchy of bounding volumes
 Bounding volume of interior node has its children

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Accelerate Ray Intersections
 Traverse hierarchy to accelerate ray
intersections
 Intersect node content
only if ray hits the
bounding volume
Skip intersection A, D, E, and F

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Accelerate Ray Intersection
Algorithm
 Sort hits and detect early termination
FindIntersection( Ray ray, Node node )
{
// Find intersections with child node bounding volumes
…
// Sort intersections closest to farthest
…
// Process intersections, checking for early termination
min_t = infinity;
for each intersected child i {
if (min_t < bv_t[i]) break;
shape_t = FindIntersection(ray, child);
if (shape_t < min_t) { min_t = shape_t; }
}
return min_t; // closest intersection
}

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Bounding Volumes
 Axis-Aligned Bounding Boxes (AABB)
 min (x,y,z) & max(x,y,z)
 Trivial plane equations
 Bounding spheres
 Point and radius
 Ray and sphere intersection is easy
 Solving a quadratic equation
 Oriented Bounding Box
 Might have tighter bounds than AABB
 Convex Polyhedron (Polytope)

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Spatial Hierarchy
 Uniform grid
 Quadtree (2D) and Octree (3D)
 Exactly four or eight children
 Equal area/volume for each children
 KD Tree
 Two children, splitting in X, Y, or Z
 Axis aligned splitting planes
 Not necessarily equal area/volume
 Binary Space Partitioning (BSP) Tree
 Arbitrary splitting planes
 Just two children

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Space Subdivision
Approaches
Uniform grid Octree
Quadtree
Binary
KD Tree Space
Partitioning
Tree

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Uniform Grid Construction
Preprocess scene
2. Find bounding box
3. Determine grid
resolution

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Uniform Grid Construction
Preprocess scene
2. Find bounding box
3. Determine grid
resolution
4. Place object in cell if
its bounding box
overlaps the cell

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Uniform Grid Construction
Preprocess scene
2. Find bounding box
3. Determine grid
resolution
4. Place object in cell if
its bounding box
overlaps the cell
5. Check that object
overlaps cell

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Uniform Grid Traversal
After processing…
Traverse grid
3D line = 3D-DDA
Digital Differential Analyzer
Advantages
Simple construction
Simple traversal
Disadvantage
Poor at sparse or huge
scenes

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Binary Space Partitioning Tree
 2D view of BSP

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Binary Space Partitioning Trees
 Recursive search
 Partitioning plane has two nodes
FindIntersection( Ray ray, Node node )
{
if node is leaf {
intersect ray with each object in node
return closest object (or nil)
}
near = child of node in half space containing ray’s origin
far = the other child
hit = FindIntersection( ray, near )
if hit is null and ray intersections plane defined by node {
hit = FindIntersection( ray, far )
}
return hit;
}

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BSP Intersection with a Ray

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Optimizing Bounding Hierarchies
 Complex meshes need to be partitioned
into bounding hierarchy
[Saut, Sidobre, 2012]

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Octree Building
 Building octree from boundary representation

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KD-tree
 Like an Octree
 But dividing planes aren’t necessarily in even
octo squares
 Tighter bounds
[Mahmoud Zidan]

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Close Cousin of Ray Tracing:
Volume Rendering
 Common task: visualization of volumetric data
 Data arranged in 3D “voxel” grid
 Voxel = volume element
 Applications: medical, oil & gas exploration

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Simple Case of Ray Casting
 Rays are all coherent
 GPU-oriented Volume rendering
 Draw 3D textured polygons slicing through a 3D texture
 Apply transfer function
 Blend in ray order—use framebuffer blending

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Transfer Function
 Give viewer control of how volumetric data
maps to color & opacity
 Transfer functions enable visualization of
otherwise difficult-to-understand mass of data

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Volume Rendering Examples
Head with clip planes
Liver tumor

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Next Class
 Next lecture
 Performance analysis
 Considerations for tuning interactive graphics
applications
 Reading
 Chapter 8, 455-460
 Chapter 11, 578-601
 Project 4
 Project 4 is a simple ray tracer
 Due Wednesday, May 2, 2012