1. CS 551 / 645: Introductory Computer Graphics
Ray Tracing
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2. Administrivia
Assignment 5: Intel okay
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3. Realism What is not realistic about the following images?
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4. Realism?
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5. Realism?
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6. Realism?
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7. Realism?
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8. Realism?
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9. Realism… A big part of realism is realistic lighting
Is Phong’s lighting model realistic?
Empirical specular term
“Ambient” term
No shadows
No surface interreflection
Crude light-surface interaction model
Strictly local illumination model
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10. Global Illumination Realistic lighting is global, not local
Surfaces shadow each other
Surfaces illuminate each other
Two long-time approaches
Ray-tracing: simulate light-ray optics
Radiosity: simulate physics of light transfer
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11. Ray Tracing Overview To determine visible surfaces at each pixel:
Cast a ray from the eyepoint through the center of the pixel
Intersect the ray with all objects in the scene
Whatever it hits first is the visible object
This is called ray casting
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12. Ray Casting
An example:
Eyepoint
Screen
Scene
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13. Recursive Ray Tracing Obvious extension:
Spawn additional rays off reflective surfaces
Spawn transmitted rays through transparent surfaces
Leads to recursive ray tracing
Secondary Rays
Primary Ray
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14. Recursive Ray Tracing Slightly less obvious extension:
Trace a ray from point of intersection to light source
If ray hits anything before light source, object is in shadow
These are called shadow rays
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15. Ray Tracing Overview Ray tracing is simple Ray tracing is powerful
No clipping, perspective projection matrices, scan conversion of polygons
Ray tracing is powerful
Hidden surface problem
Shadow computation
Reflection/refraction
Ray tracing is slow
Complexity proportional to # of pixels
Typical screen ~ 1,000,000 pixels
Typical scene « 1,000,000 polygons
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16. Recursive Ray Tracing
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17. Recursive Ray Tracing
Pixel
Image Plane
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18. Recursive Ray Tracing
“Direct” Ray To Eye
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19. Recursive Ray Tracing
“Indirect” Ray To Eye
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20. Recursive Ray Tracing
Physically, we’re interested in path of light rays from light source to eye.
In practice, we trace rays backwards from eye to source (Why?)
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21. Recursive Ray Tracing
Physically, we’re interested in path of light rays from light source to eye.
In practice, we trace rays backwards from eye to source (Why?)
Computational efficiency: we want the finite subset of rays that leave source, bounce around, and pass through eye
Can’t predict where a ray will go, so start with rays we know reach eye
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22. Basic Algorithm
Function TraceRay() Recursively trace ray R and return resulting color C
if R intersects any objects then
Find nearest object O
Find local color contribution
Spawn reflected and transmitted rays, using TraceRay() to find resulting colors
Combine colors; return result
else
return background color
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23. Basic Algorithm: Code Object allObs[]; Color image[]; RayTraceScene()
allObs = initObjects();
for (Y  all rows in image)
for (X  all pixels in row)
Ray R = calcPrimaryRay(X,Y);
image[X,Y] = TraceRay(R); display(image);
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24. Basic Algorithm: Code Color TraceRay(Ray R) if rayHitsObjects(R) then
Color localC, reflectC, refractC;
Object O = findNearestObject(R);
localC = shade(O,R);
Ray reflectedRay = calcReflect(O,R)
Ray refractedRay = calcRefract(O,R)
reflectC = TraceRay(reflectedRay);
refractC = TraceRay(refractedRay);
return localC  reflectC  refractC
else return backgroundColor
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25. Refining the Basic Algorithm
Color TraceRay(Ray R)
if rayHitsObjects(R) then
Color localC, reflectC, refractC;
Object O = findNearestObject(R);
localC = shade(O,R);
Ray reflectedRay = calcReflect(O,R)
Ray refractedRay = calcRefract(O,R)
reflectC = TraceRay(reflectedRay);
refractC = TraceRay(refractedRay);
return localC  reflectC  refractC
else return backgroundColor
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26. Ray-Object Intersection
Given a ray and a list of objects, what objects (if any) intersect the ray?
Query: Does ray R intersect object O?
How to represent ray?
What kind of object?
Sphere
Polygon
Box
General quadric
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27. R = O + tD Representing Rays O D How might we represent rays?
We represent a ray parametrically:
A starting point O
A direction vector D
A scalar t
R = O + tD O
t < 0
t = 1
t > 1 D
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28. Ray-Sphere Intersection
Ray R = O + tD
x = Ox + t Dx
y = Oy + t Dy
z = Oz + t Dz
Sphere at (l, m, n) of radius r is:
(x - l)2 + (y - m)2 + (z - n)2 = r 2
Substitute for x,y,z and solve for t…
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29. Ray-Sphere Intersection
Works out as a quadratic equation:
at2 + bt + c = 0
where
a = Dx2 + Dy2 + Dz2
b = 2Dx (Ox - l) + 2Dy (Oy - m) + 2Dz (Oz - n)
c = l2 + m2 + n2 + Ox2 + Oy2 + Oz (l Ox + m Oy + n Oz + r2)
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30. Ray-Sphere Intersection
If solving for t gives no real roots: ray does not intersect sphere
If solving gives 1 real root r, ray grazes sphere where t = r
If solving gives 2 real roots (r1, r2), ray intersects sphere at t = r1 & t = r2
Ignore negative values
Smallest value is first intersection
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31. Ray-Sphere Intersection
Find intersection point Pi = (xi, yi, zi) by plugging t back into ray equation
Find normal at intersection point by subtracting sphere center from Pi and normalizing:
When will we need the normal? When not?
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32. Ray-Polygon Intersection
Polygons are the most common model representation
Can render in hardware
Lowest common denominator
Basic approach:
Find plane equation of polygon
Find point of intersection between ray and plane
Does polygon contain intersection point?
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33. Ray-Polygon Intersection
Find plane equation of polygon: ax + by + cz + d = 0
Remember how?
N = [a, b, c]
d = N  P1 y N P2 P1 d x
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34. Ray-Polygon Intersection
Find intersection of ray and plane:
t = -(aOx + bOy + cOz + d) / (aDx + bDy + cDz)
Does polygon contain intersection point Pi ?
One simple algorithm:
Draw line from Pi to each polygon vertex
Measure angles between lines (how?)
If sum of angles between lines is 360°, polygon contains Pi
Slow — better algorithms available
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35. Ray-Box Intersection
Often want to find whether a ray hits an axis-aligned box (Why?)
One way:
Intersect ray with pairs of parallel planes that form box
If intervals of intersection overlap, the ray intersects the volume.
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36. Shadow Rays Simple idea: Q: how much extra work is involved?
Where a ray intersects a surface, send a shadow ray to each light source
If the shadow ray hits any surface before the light source, ignore light
Q: how much extra work is involved?
A: each ray-surface intersection now spawns n + 2 rays, n = # light sources
Remember: intersections 95% of work
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37. Shadow Rays Some problems with using shadow rays as described:
Lots of computation
Infinitely sharp shadows
No semitransparent object shadows
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38. Shadow Rays Some problems with using shadow rays as described:
Lots of computation
Infinitely sharp shadows
No semitransparent object shadows
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39. Shadow Ray Problems: Too Much Computation
Light buffer (Haines/Greenberg, 86)
Precompute lists of polygons surrounding light source in all directions
Sort each list by distance to light source
Now shadow ray need only be intersected with appropriate list!
Light Buffer
Shadow Ray
Occluding Polys
Current Intersection Point
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40. Shadow Rays Some problems with using shadow rays as described:
Lots of computation
Infinitely sharp shadows
No semitransparent object shadows
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41. Shadow Ray Problems: Sharp Shadows
Why are the shadows sharp?
A: Infinitely small point light sources
What can we do about it?
A: Implement area light sources
How?
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42. Shadow Ray Problems: Area Light Sources
Could trace a conical beam from point of intersection to light source:
Track portion of beam blocked by occluding polygons:
30% blockage
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43. Shadow Ray Problems: Area Light Sources
Too hard! Approximate instead:
Sample the light source over its area and take weighted average:
50% blockage
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44. Shadow Ray Problems: Area Light Sources
Disadvantages:
Less accurate (50% vs. 30% blockage)
Oops! Just quadrupled (at least) number of shadow rays
Moral of the story:
Soft shadows are very expensive in ray tracing
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45. Shadow Rays Some problems with using shadow rays as described:
Lots of computation
Infinitely sharp shadows
No semitransparent object shadows
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46. Shadow Ray Problems: Semitransparent Objects
In principle:
Translucent colored objects should cast colored shadows
Translucent curved objects should create refractive caustics
In practice:
Can fake colored shadows by attenuating color with distance
Caustics need backward ray tracing
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47. Speedup Techniques Intersect rays faster Shoot fewer rays
Shoot “smarter” rays
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48. Speedup Techniques Intersect rays faster Shoot fewer rays
Shoot “smarter” rays
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49. Intersect Rays Faster Bounding volumes Spatial partitions
Reordering ray intersection tests
Optimizing intersection tests
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50. 7 7 9 9 5 5 8 8 Bounding Volumes 4 4 3 3 2 2 Bounding volumes
Idea: before intersecting a ray with a collection of objects, test it against one simple object that bounds the collection 7 7 5 5 3 3 1 1 8 8 6 6 9 9 2 4 2 4
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51. Bounding Volumes Hierarchical bounding volumes
Group nearby volumes hierarchically
Test rays against hierarchy top-down
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52. Bounding Volumes Different bounding volumes Spheres
Cheap intersection test
Poor fit
Tough to calculate optimal clustering
Axis-aligned bounding boxes (AABBs)
Relatively cheap intersection test
Usually better fit
Trivial to calculate clustering
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53. Bounding Volumes More bounding volumes Oriented bounding boxes (OBBs)
Medium-expensive intersection test
Very good fit (asymptotically better)
Medium-difficult to calculate clustering
Slabs (parallel planes)
Comparatively expensive
Very good fit
Difficult to calculate good clustering
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54. Spatial Partitioning
Hierarchical bounding volumes surround objects in the scene with (possibly overlapping) volumes
Spatial partitioning techniques classify all space into non-overlapping portions
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55. Spatial Partitioning Example spatial partitions:
Uniform grid (2-D or 3-D)
Octree
k-D tree
BSP-tree
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56. Uniform Grid Uniform grid pros: Uniform grid cons:
Very simple and fast to generate
Very simple and fast to trace rays across (How?)
Uniform grid cons:
Not adaptive
Wastes storage on empty space
Assumes uniform spread of data
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57. Octree Octree pros: Octree cons: Simple to generate Adaptive
Nontrivial to trace rays across
Adaptive only in scale
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58. k-D Trees k-D tree pros: k-D tree cons: Moderately simple to generate
More adaptive than octrees
k-D tree cons:
Less efficient to trace rays across
Moderately complex data structure
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59. BSP Trees BSP tree pros: BSP tree cons: Extremely adaptive
Simple & elegant data structure
BSP tree cons:
Very hard to create optimum BSP
Splitting planes can explode storage
Simple but slow to trace rays across
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60. Reordering Ray Intersection Tests
Caching ray hits (esp. shadow rays)
Memory-coherent ray tracing
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61. Optimizing Ray Intersection Tests
Fine-tune the math!
Share subexpressions
Precompute everything possible
Code with care
Even use assembly, if necessary
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