1. Image by Eric Veach and Leonidas Guibas
Ray Tracing Lecture 10

2. Introduction
OpenGL is based on a pipeline model in which primitives are rendered one at time
No shadows (except by tricks or multiple renderings)
No reflections (though we can use multiple renderings)
No refraction – thus no caustics
Global approaches
Ray tracing – student project
Rendering equation – many approaches, such as
Radiosity – approach to solve rendering equation

3. Currently We have rasterization with Phong lighting
Strictly local illumination model:
Ambient
Diffuse
Specular
Shadows can be accomplished – we will discuss
Shadow maps
Shadow volumes
Can generate reflections using environment mapping but:
Approximate
Expensive for scenes with many reflective objects
Hard to capture complex inter-reflections

4. Example
Gilles Tran

5. Global Illumination Direct Illumination
A surface point receives light from all visible lights
Global Illumination
Even if point is in shadow,
Rays may pass through translucent material
Light rays will bounce off other material in scene
Want to sum all light that hits an object

6. Timeline
[Appel 68]
“Some techniques for shading machine renderings of solids”
Basic Ray Casting – designed for pen plotters: provides
Perspective
Accurate shadows
Hidden Surface Removal
[Whitted 80]
“An Improved Illumination Model for Shaded Display”
Reflection
Refraction

7. Perspective Ray Tracing provides natural perspective
The schemes used to develop perspective were tracing rays
Albrecht Durer

8. Two Alternatives
Albrecht Durer
Albrecht Durer

9. Ray Tracing
Track the path of light between light source and the eye

10. Ray Tracing Track the path of light between light source and the eye
Where do we start – at light or at eye?

11. Ray Casting Shot a ray from the eye through each pixel of the screen
Can cast more than one ray through a pixel
Calculate which objects in the scene the ray hits
If it hits an object, we know basic color
Images from SIGGRAPH
Tutorial on Ray Tracing
G. Scott Owens, 1999

12. Ray Casting Shot a ray from the eye through each pixel of the screen
Calculate which objects in the scene the ray hits
Now check to see if surface is illuminated or in shade

13. Shadow Ray Shoot a "shadow" ray from intersection point towards light
If shadow ray hits an object before it hits light, point is in shadow
This is Appel's original Algorithm – called Ray Casting today
Only uses direct illumination – does not track light bouncing off intermediate surfaces

14. Ray Casting cast ray Intersect all objects: select minimal t
Color = ambient color of object
For every light
Cast shadow ray
color += local shading term

15. In practice Run the first few steps of the following applet

16. Whitted Raytracing (1980)
Turner Witted

17. Turner Witted
“An Improved Illumination Model for Shaded Display”
First global illumination model. An object’s color is influenced by lights and other objects in the scene
First to simulate specular reflection and refractive transmission

18. Amiga Juggler
Eric Graham – "In November 1966, a month after I learned to program, I had written a simple ray tracing program. I was then a grad student at Manchester…After getting the program to actually work, I only ran it once.
"I used ray tracing because that was the simplest and most obvious way to render an image. I can't claim priority over Turner Whitted … because it was never published, but I suspect lots of other people also figured it out over the years.
"In the fall of 1986, … I re-implemented the ray tracer. It took about a week to get running, and then another week or two to make a few models and put the compression scheme together. That was November and the Juggler was born.
"The reaction amongst the Amiga community in Albuquerque encouraged me to send it to Commodore. Their legal department thought it was a hoax, and that I'd done the image generation on a mainframe..."
home.comcast.net/~erniew/juggler.html#avi

19. Reflected Ray If the object is shiny, we send a third "reflection" ray
If this hits an object, the color of the object will be reflected in the original screen point
To see if the new point is in the shade, send shadow ray
The new object may be shiny…

20. Ray Tree

21. Full Algorithm We have sketched the first steps. Will need to recurse…
Need to modify the path of refracted light
May wish to add specular highlights …

22. What does Ray Tracing Offer?
Hidden Surface Removal
Shading due to direct Illumination
Global specular interaction effects
Reflections
Refraction of light
Shadow Computation
Though the shadows are hard-edged

23. Diffuse Surfaces
Theoretically scattering at each point of intersection generates an infinite number of new rays to trace
We only trace the transmitted and reflected rays and use Phong model to compute shade at intersection
To get full simulation of indirect lighting, we will need to turn to the Rendering Equations
But first look at Ray Tracing in greater detail

24. Match point in image
with its construction

25. In practice Run more steps of the following applet Now move objects

26. Raytracing Is Simple
Paul Heckbert wrote a raytracer that fits on a business card
Prints something like a 32x32 P3 .ppm file to standard out.
typedef struct{double x,y,z}vec;vec U,black,amb={.02,.02,.02};struct sphere{ vec cen,color;double rad,kd,ks,kt,kl,ir}*s,*best,sph[]={0.,6.,.5,1.,1.,1.,.9, .05,.2,.85,0.,1.7,-1.,8.,-.5,1.,.5,.2,1.,.7,.3,0.,.05,1.2,1.,8.,-.5,.1,.8,.8, 1.,.3,.7,0.,0.,1.2,3.,-6.,15.,1.,.8,1.,7.,0.,0.,0.,.6,1.5,-3.,-3.,12.,.8,1., 1.,5.,0.,0.,0.,.5,1.5,};yx;double u,b,tmin,sqrt(),tan();double vdot(A,B)vec A ,B;{return A.x*B.x+A.y*B.y+A.z*B.z;}vec vcomb(a,A,B)double a;vec A,B;{B.x+=a* A.x;B.y+=a*A.y;B.z+=a*A.z;return B;}vec vunit(A)vec A;{return vcomb(1./sqrt( vdot(A,A)),A,black);}struct sphere*intersect(P,D)vec P,D;{best=0;tmin=1e30;s= sph+5;while(s-->sph)b=vdot(D,U=vcomb(-1.,P,s->cen)),u=b*b-vdot(U,U)+s->rad*s ->rad,u=u>0?sqrt(u):1e31,u=b-u>1e-7?b-u:b+u,tmin=u>=1e-7&&u<tmin?best=s,u: tmin;return best;}vec trace(level,P,D)vec P,D;{double d,eta,e;vec N,color; struct sphere*s,*l;if(!level--)return black;if(s=intersect(P,D));else return amb;color=amb;eta=s->ir;d= -vdot(D,N=vunit(vcomb(-1.,P=vcomb(tmin,D,P),s->cen )));if(d<0)N=vcomb(-1.,N,black),eta=1/eta,d= -d;l=sph+5;while(l-->sph)if((e=l ->kl*vdot(N,U=vunit(vcomb(-1.,P,l->cen))))>0&&intersect(P,U)==l)color=vcomb(e ,l->color,color);U=s->color;color.x*=U.x;color.y*=U.y;color.z*=U.z;e=1-eta* eta*(1-d*d);return vcomb(s->kt,e>0?trace(level,P,vcomb(eta,D,vcomb(eta*d-sqrt (e),N,black))):black,vcomb(s->ks,trace(level,P,vcomb(2*d,N,D)),vcomb(s->kd, color,vcomb(s->kl,U,black))));}main(){printf("%d %d\n",32,32);while(yx<32*32) U.x=yx%32-32/2,U.z=32/2-yx++/32,U.y=32/2/tan(25/ ),U=vcomb(255., trace(3,black,vunit(U)),black),printf("%.0f %.0f %.0f\n",U);}/*minray!*/
See Heckbert's work on ray tracing Jello

27. Minimal Ray Tracing Contest
Andrew Kensler's contribution
Walkthrough…

28. Minimal Ray Tracing Contest

29. Raytracing trace ray Intersect all objects color = ambient term
For every light
cast shadow ray
color += local shading term
If mirror
color += colorrefl * trace reflected ray
If transparent
color += colortrans * trace transmitted ray

30. Examples
Persistence of Vision Raytracer
POV Ray
Gilles Tran

31. POV Ray
Gilles Tran

32. POV Ray
Gilles Tran

33. POV Ray
Steve Davis, CS 234

34. POV Ray Start Steve Davis, CS 234 //Steve Davis POV Ray
#include "colors.inc"
#include "textures.inc"
#include "shapes.inc"
#include "metals.inc"
#include "glass.inc"
#include "woods.inc"
global_settings {max_trace_level 5}
camera {
location <-5, 5, -10>
direction <0, 0, 2.25>
right x*1.33
look_at <0,0,0> }
Steve Davis, CS 234

35. POV Ray Lighting Steve Davis, CS 234 //Lighting #declare Dist=80.0;
light_source {< -50, 25, -50> color White
fade_distance Dist fade_power 2
area_light <-40, 0, -40>, <40, 0, 40>, 3, 3
adaptive 1
jitter }
light_source {< 50, 10, -4> color Gray30
area_light <-20, 0, -20>, <20, 0, 20>, 3, 3
...
Steve Davis, CS 234

36. POV Ray Surface Steve Davis, CS 234 // Water Surface
plane{<0,1,0>, 0
texture {
pigment { rgbf <0.2, 0.2, 0.9, > }
finish {
ambient 0.1
diffuse 0.7
brilliance 6.0
reflection 0.6
phong 0.8
phong_size 120 }
normal {bumps 0.2
scale <1,0.25,0.25>*1
turbulence 0.5 }
Steve Davis, CS 234

37. POV Ray Spheres Steve Davis, CS 234 //first row
texture { Shadow_Clouds }
finish { specular 0.25 roughness ambient 0.2 }
normal { bumps 0.4 scale 0.2 } }
//second row
sphere { <-0.4, 0.0, -0.5>, 0.4
texture { T_Copper_4A }
sphere { <0.4, 0.0, -0.5>, 0.4
texture { T_Glass4 }
Steve Davis, CS 234

38. Cover of Rendering with Radiance, Larson and Shakespeare

39. When does it end? Diffuse surface – will not transmit or bounce
Recursion depth: Stop after a number of bounces
Ray contribution: Stop if reflected / transmitted contribution becomes too small
No reflection
One reflection
Two reflections

40. When does it end?

41. Depth one and two

42. Adding levels

43. Adding…

44. Hard Work
One difficulty is computing the intersection of the ray and the object
Need to do this quickly: to find closest object, we may have to compute many intersections

45. Math: Ray-Plane Intersection
Equations of Ray and Plane. Ray is parameterized
Plug the values for P from parameterized ray into Plane, and solve for t

46. Ray Plane Intersection
Solve this again with explicit values.

47. Example: Ray Casting a Sphere
Sphere is quadric equation, and ray is parameterized again.
Resulting equation is a scalar quadratic equation
It gives entry and exit points of ray

48. Ray intersects Sphere Equations of Sphere and Ray
Plug the values for (x, y, z) from parameterized ray into Sphere, and solve for t

49. What does it mean? There are three cases:
Equations of Sphere and Ray
There are three cases:
no roots (discriminant is negative), 2 roots, one root
What does each mean?

50. Finer Points If we have a solution with a negative t, we reject it
Why? What does a negative t mean?
If we have a solution with a very small positive t, we reject it (why?)

51. Finer Points If we have a solution with a negative t, we reject it
Why? What does a negative t mean?
If we have a solution with a very small positive t, we reject it (why?)
Hint – this can arise with a reflection ray

52. Epsilon
"Acne" - Move intersection by epsilon along surface normal

53. What does it mean?
Intersections with general quadratic equation is similar

54. Ray Tracing Quadrics Constructive Solid Geometry (CSG)
Primitives are solids
Build objects with set operations
Union, Intersection, Set Difference
Wikipedia

55. Tree of objects
Wikipedia

56. CSG Intersections Spheres Polygons Boxes Quadrics
Parameteric patches – we will see some later on…
Subdivision surfaces
Surfaces of revolution
Fractals …

57. Polyhedra
Generally we will be intersecting with closed objects such as polygons and polyhedra rather than planes
Hence we have to worry about inside/outside testing
For convex objects such as polyhedra there are some fast tests

58. Ray Tracing Polyhedra
Polyhedron is formed by intersection of half-planes
If ray enters an object, it must enter a front facing polygon and leave a back facing polygon
If entry is further away than exit, ray must miss the polyhedron

59. Problems for Ray Tracing
Aliasing
Finding all the intersections
The technique we outlined had you intersect ray with all objects – time consuming
Generates Hard Shadows
Ambient Light is not uniform in real life

60. Aliasing

61. Aliasing

62. Aliasing Three types of solutions Super sampling Many samples
Stochastic Sampling
Random points
Adaptive Sampling
Stop if no change seen
How to pick?

63. Finding All Intersections
The technique we outlined intersects each ray with all objects
Time consuming
Major alternatives
Try to bundle rays
Describe scene as collection of objects
Carve space up into regions

64. Intersections with Objects
To reduce the number of computations, we may put bounding boxes (or spheres) around complex objects in the scene
Easier to compute intersection with simpler object (box)
If we miss the bounding box, we miss its contents
Early Reject

65. Dividing Space Alternatives
Do we work perpendicular to axis or to objects or to ray?
Carve space up into regions – two approaches
Volumetric Pixels (voxels)
Keep track of which voxels each object hits
Split space via half planes: Two alternatives
Binary Space Partitioning (BSP) Trees
k-Dimensional (kd) Trees

66. Voxel Grid

67. Voxel Grid Figure out which voxels the ray hits
See which shapes are in those voxels

68. Grid
Divide scene
deltaX need not be
deltaY

69. Precompute In advance, figure out which grid boxes each object hits
Each grid box keeps list of objects

70. For each box along a ray Ray hits this box Box hits these objects
Test ray against each object in boxes it hits
Return the closest intersection, if any

71. Multiple Visits An object may hit multiple boxes
We may encounter the same object multiple times
Twice here
Only test for intersections once: mark the object

72. Grid If intersection is not within the grid box boundaries, continue
There may be a better intersection in a later box

73. Grid Binary Space Partitioning (BSP) Trees
Build a binary tree: each node tests against a plane – is point left or right?

74. Grid Binary Space Partitioning (BSP) Trees
In this example, we check almost all objects, but we always check in order
This makes it much easier to find the first intersection

75. KD Trees All planes are parallel to standard axes
Use lines through points to decompose space
Alternate dimensions. x, then y, then z, then x…

76. KD Trees

77. Speed up Computation Would like to handle all physical interactions
Most secondary rays do not affect what we see
Scattering produces many (infinite) additional rays
Alternative:
Ray Casting
Various techniques to speed up Ray Tracing

78. Light Map One technique that was introduced in Quake was light map
Precompute lighting: paste it on like a texture

79. Global Illumination Direct Illumination
A surface point receives light from all visible lights
Global Illumination
Even if point is in shadow,
Rays may pass through translucent material
Light rays will bounce off other material in scene
Want to sum all light that hits an object
Beyond Radiosity: The Light of Mies van der Rohe

80. Rendering Equation
Ray tracing is best with many highly specular surfaces
Not characteristic of real scenes
Rendering equation describes general shading problem
Radiosity solves rendering equation for perfectly diffuse surfaces

81. Rendering Equation

82. Rendering Equation Based on Heat Equation. 1986 paper by James Kajiya
As explained by Sir Paul McCartney in 1969
"The light you take is proportional to the light you make"
Keep track of the light exchange between points
A red wall should cause color to bleed into white floor
Look at sides of the boxes in this figure
Matt surface with lots of incident light
will reflect lots of light

83. Rendering Equation Does not have a general closed form solution
Solved in practice by iterative methods

84. Problems
We have replaced a difficult problem with one even more difficult
Rather than trace rays from the eye, we need to trace rays from light to the eye
Many techniques are used to address this
Radiance models
Finite element method: break world into patches
Monte-Carlo Methods
Metropolis Light Transport
Photon Mapping
Cornell Box is widely used as a benchmark

85. Cornell Box

86. Joseph Cornell Box

87. Radiosity
E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley 2012

88. Rendering Equation Cornell Box
Cornell Box is a Sample Scene
Compare output of algorithm with picture of the scene

89. Split model into Patches
E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley 2012

90. Computing Form Factors
Consider two flat patches
E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley 2012

91. Using Differential Patches
Must take angles into account
foreshortening
E. Angel and D. Shreiner: Interactive Computer Graphics 6E © Addison-Wesley 2012

92. Use illuminating hemisphere
Center hemisphere on patch with normal pointing up
Must shift hemisphere for each point on patch
In practice, use hemicube

93. Bidirectional Path Tracing
Track the path of light between light source and the eye
Start at both ends, and look at how they meet

94. Example Metropolis Light Transport Eric Veach and Leonidas Guibas
Image by Eric Veach and Leonidas Guibas

95. Metropolis Light Transport
Speeds up Monte-Carlo simulation
Uses technique attributed to Metropolis et. al.
Start with a sample path that may not be representative
Using probability model, mutate path
Track the contributions of representative paths

96. Compare to Monte-Carlo
Metropolis Light Transport

97. Photon Mapping

98. Photon Mapping

99. Voxel Based Solutions
Recent paper just added:
Thiedemann - Voxel Based Global Illumination
Combination of techniques
Textures to store global atlas of voxels
Maps to hold direct illumination per pixel
Integrate over the pixels you can see…
Focus on near term effects: distant sources are dim

100. The Cathedral Won award as Best Animated Short at Siggraph 2002

101. Image Credits Ed Angel - University of New Mexico
Allan Watt's text, 3D Computer Graphics
Brian Salomon, UNC
G. Scott Owen
Paul Bourke
Gilles Tran
and many others

102. Resources
Persistence of Vision (POV-RAY) – free ray tracer. Descriptions in HLL
Radiance – another package
radsite.lbl.gov/radiance/HOME.html
Internet Ray Tracing Competition – many winners used POV-RAY
SIGGRAPH Educational Material
Applet to Ray Trace a simple scene
Efficiency Issues for Ray Tracing, Brian Smits,, 1999
Also see the series of papers "State of the Art in Ray Tracing…"
Tutorials

103. Links See Lecture Links page for more…
Demo
Applet
Amiga Juggler
Dark Ridge Twin Engine example
Beyond Radiosity: The Light of Mies van der Rohe
Metropolis Light Transport
Photon Mapping
Sample Programs
The Cathedral
See Lecture Links page for more…

104. Summary Ray Tracing provides a way to generate rich images
Was too expensive for many uses
Faster machines
Better Data Structures and algorithms
Offline processing
Much work has gone into speeding up Ray Tracing
Machines are fast enough for us to consider the Rendering Equation for illumination
Compute the background illumination in advance
Much work underway to speed up the evaluation or RE