1. Advanced Graphics Algorithms Ying Zhu Georgia State University
Ray Tracing and
Radiosity

2. Outline Basics Recursive ray tracing Intersection Shading
Ray tracing vs. rasterization

3. By Giles Tran (www.povray.org)
What is ray tracing?
Ray-tracing is a rendering technique that calculates an image of a scene by simulating the way rays of light travel in the real world.
Calculating the light rays between light source and camera.
However it does its job backwards.
By Giles Tran (

4. Follow the Photon
Follow each photon through its chain of bounces, till it either:
Runs out of energy
Departs the predefined space, or
Strikes the image, and then we add its contribution into the appropriate pixel.
The trajectories of photons look like rays so this process is called ray tracing.

5. Forward Ray Tracing
The process mentioned above is called forward ray tracing.
Problems with forward ray tracing:
Many rays never reach the image
We’ll waste a lot of time following the photons that make no contribution to the image.
The solution: backward ray tracing.
This is what we use in computer graphics.

6. Backward Ray Tracing
A ray of light is traced starting from the camera into the scene, and back through interactions with geometry, to see if it ends up back at a light source .
At least one ray per pixel.

7. Backward Ray Tracing Backward recursive ray tracing:
Draw a line (primary ray) through each image pixel and the eye point and follow that line outward.
If the primary ray happens to hit a light source, then we are done. Then the pixel color is the same as light source color.
If the primary ray hits nothing and flies straight through to the darkness of the space, we are done. The pixel color is the same as background color.
If the primary ray hits a surface, we apply local illumination model to this point to calculate its color. Then we shoot out more secondary rays (reflection and/or refraction) and recursively apply step 2, 3, and 4.
The color of the primary ray is the weighted sum of colors of all of its secondary rays.
The color of each secondary ray is the weighted sum of colors of all of its subsequent secondary rays, and so on …

8. Ray Casting Non-recursive ray tracing is called ray casting.
For every pixel (x,y) Construct a ray from the eye; color[x,y] = castRay(ray);
plot(x, y, color[x, y]);
Complexity?
O(n * m)
n: number of objects, m: number of pixels

9. Why use ray tracing? More elegant than polygon scan conversion
Testbed for numerous techniques and effects
modeling
rendering
texturing
Relatively easy to implement

10. Ray Tracing Example
More pictures at

11. Ray Tracing Example

12. Ray Tracing Example

13. Ray Tracing Example

14. Where does ray tracing come from?
Ray tracing technique has been used for lens design since the 1900s.
Geometric ray tracing is used to describe the propagation of light rays through a lens system or optical instrument
This is used to optimize the design of the instrument before it is built.
Gamma ray tracing software was developed (by MAG Inc.) in late 1960s for military applications
Help design radiation-proof military vehicles

15. Where does ray tracing come from?
A. Appel, “On Calculating the Illusion of Reality”, IFIP Congress 1968
First to introduce ray casting
T. Whitted, “An Improved Illumination Model for Shaded Display”, Communications of ACM 1980
First global illumination model
First to simulate specular reflection and refractive transmission.

16. Ray Tracing Overview
For each pixel, trace a primary ray to the first visible surface
For each intersection, trace secondary rays:
Shadow ray (in direction to light source)
Reflection ray
Transmission ray (refraction)

17. Ray Tracing Overview
Primary rays: the ones that bring light directly to the image
Shadow rays: that bring light (or shadow if blocked by other objects) directly from a light source to an object source.
Reflected rays: carrying light reflected from a surface.
Transmission rays: carrying light passing through an object.

18. Ray Tracing Overview
Refration
rays
Primary
ray
Reflection
rays

19. Recursive Ray Tracing Steps (1)
Construct a primary ray (eye-pixel ray)
Backward tracking of photons that could have arrived along primary ray
Intersect with all object in scene
Determine nearest object and intersection point
Generate secondary rays
To light sources (shadow ray)
In reflection-associated directions
In refraction-associated directions

20. Recursive Ray Tracing Steps (2)
Continue recursively for each secondary ray
Terminate after certain levels
Collect color information from secondary rays and accumulate suitably averaged information for primary ray
Assign primary ray color to the pixel

21. Ray Trees Eye Ni surface normal Ri reflected ray Li shadow ray
Ti transmitted (refracted) ray
Eye R2 R3 T3

22. Ray Casting
Similar to ray tracing, but stop before generating secondary rays
Apply illumination model at nearest object intersection with no regard to light occlusion
Only a local illumination model
Only consider light that is directly emitted from the source and the light ray’s intersection with an isolated surface

23. Intersection A typical ray tracing algorithm has three basic elements:
Ray-object intersection
Shading
Secondary ray generation
Intersection determines the performance of ray tracing.
How do we represent rays and objects?
Object representation is covered in the lectures about modeling.
Definition of a ray: E + t(P – E)
E is the eyepoint and P is the pixel point

24. Construct Primary Rays
We can use a convenient camera model to compute primary eye rays.
For example, the OpenGL look-at model:
Camera position
Look-at point
Up vector
Field-of-view angle
Width/height
Given pixel coordinates (i, j), we can compute the ray direction d.

25. Construct Primary Rays
Direction of the ray

26. Object Representation
Scene object: direct implicit form
Express as f(Q)=0 where Q is a surface point, and f is given formula
Intersection computation is an equation to solve: find t such hat f(E + t(P – E)) = 0

27. Quadratic Surfaces Surfaces given by Ellipsoid
Hyperboloid of one sheet
Cone
Elliptic cylinder

28. Quadric Surfaces Ray is given by
Substitute ray x, y, z into surface formula
Quadratic equation results for t and then we have the intersection point

29. Polygons
The plane of the polygon should be known as Ax + By + Cz + D = 0, where (A, B, C, 0) is the normal vector.
How to come up with the plane formula?
Pick three successive vertices
Normal vector is the cross product
Once you have the normal vector, you have values for A, B, C.
Substitute ray x, y, z into surface formula
Linear equation results for t and then we have the intersection point

30. The shading process
If the primary ray hits a light, then the pixel color is the light’s color.
If the ray hits a surface, then the pixel color I(P, v) as seen from point P along direction v is I(P, v) = Id + Kr*Ir + Kt*It,
where Id is computed from shadow rays using local illumination model,
Ir is the reflection ray color
It is refraction ray color.

31. Shadow Rays We can easily render shadow in recursive ray tracing.
We cast shadow rays to determine if a particular light source is visible to a ray-object intersection point.
If a light source is visible, we use local illumination model to calculate its contribution to the pixel color
Can use a similar lighting equation as in OpenGL
If it is not visible, which means this point is in the shadow of this light source, then we omit the contribution of this light source when applying the local illumination model.

32. Shadow Rays ob->intersect(ray2, hit2, 0); // find intersection
Color castRay(ray) For every object ob
ob->intersect(ray, hit); // find intersection
// calculate direct component
Color col=ambient*hit->getMaterial()->getDiffuse();
For every light L
// cast shadow ray
Ray ray2(hitPoint, L->getDir());
For every object ob
ob->intersect(ray2, hit2, 0); // find intersection
// if light is closer than the shadow ray intersection
If (hit->getT > L->getDist())
// include light source in the shading calculation
col=col+hit->getMaterial()->shade (ray, hit, L->getDir(), L->getColor());
Return col;

33. Reflection Rays q q Compute mirror contribution
Multiply by transparency coefficient (Kr)
Cast ray
In direction symmetric with regards to normal N V R q q R V

34. Refraction Rays Compute transmitted contribution Cast ray
In refracted direction
Multiply by transparency coefficient (Kt)

35. Refraction Don’t forget to normalize Snell-Descartes Law
Note that I is the negative of the incoming ray
Don’t forget to normalize
Total internal reflection when the square root is imaginary

36. Putting It Together I(P, v) = Id + Kr*Ir + Kt*It
Id can be calculated using Phong illumination model (depending on whether the intersection point is in shadow)
Kr controls global mirror reflections
Kt controls global specular transmission
Ir and It are the global contributions computed using recursive ray tracing.

37. Recap: Recursive Ray Tracing
traceRay
Intersect all objects;
Ambient shading;
For every light
Shadow ray;
Shading (local illumination);
If mirror
Trace reflected ray;
If transparent
Trace transmitted ray;
Add local and global illumination contributions;

38. Rasterization vs. Ray Tracing
What’s the difference between ray tracing and rasterization?
What are the benefits and drawbacks?
Will ray tracing eventually replace rasterization?

39. Rasterization for each global illumination pass { for each object {
for covered pixels {
shade
update framebuffer }
} // Objects are processed one at a time
Loop through all objects for each illumination effect
reflections
shadows (map, or to draw shadow volumes)
all pixels conceptually done for one object before the next one
alpha sorting is a problem

40. Rasterization Interactive rendering is possible
Most global effects will be sacrificed, but some can be approximated:
Reflections, refractions, bump-mapping
Various lighting models, shadows
Motion blur, anti-aliasing, depth-of-field
Technically, can reproduce all effects ray-tracing has through pre-rendering and texture mapping

41. Ray-tracing for all pixels { for each ray in path {
find intersected object {
shade }
update framebuffer
} // Pixels are processed one at a time

42. Ray-tracing
Each pixel rendered fully and independently

43. Ray-tracing Complex effects easy to implement
Can render anything that can be intersected with a ray
But may require significant math per intersection
Secondary interactions need help
e.g. Caustics, color bleeding
Solutions exist - Photon mapping

44. Ray-tracing Time complexity
Acceleration is necessary
Usually too slow for interactive application
Hard to implement in hardware
Must fit entire scene in memory

45. Ray-tracing vs. Rasterization
Rasterization is FAST
> 100 million polygons per second
> 1 billion pixels per second
Pipelined, parallelized hardware
Ray-tracing is SLOW (on CPU)
tens of millions of raw triangle intersections/sec
Not including texture accesses, shading...

46. Ray-tracing vs. Rasterization
Ray tracing is actually easier to implement.
Raytracing can rely on Moore's law
CPU will become faster and faster
Quality of rasterization depends on cleverness
Vertex shaders
Pixel shaders
Multipass rendering
Rasterization is so fast it’s almost fast enough! It will soon matter less how fast the program runs and more how high the quality is.
In my mind, RT Is easier to implement and understand, but a whole generation of programmers are growing up with high-speed depth buffer cards in their PCs with programmable pixel processing - will they care about ray-tracing?

47. Ray-tracing vs. Rasterization
Will ray tracing replace rasterization?
Not likely, but we will see more and more ray-tracing implementations on graphics hardware

48. Is real-time ray tracing possible?
Ray tracing on graphics hardware
“Ray tracing on a stream processor”, TJ Purcell, PhD dissertation, Stanford U, March 2004 (
“Ray tracing on programmable graphics hardware”, TJ Purcell, et al. SIGGRAPH 2002 paper (

49. Is real-time ray tracing possible?
“Realtime ray tracing of dynamic scenes on an FPGA chip”, J Schmittler, et al. Graphics Hardware 2004
The ray engine, Nathan Carr, et al., UIUC (

50. Ray Tracing Resources
Persistence of Vision Raytracer at
A free ray tracer
Use a scene description language to define scene
The user specifies the location of the camera, light sources, and objects as well as the surface texture properties of objects, their interiors (if transparent) and any atmospheric media such as fog, haze, or fire.
Include a lot of pre-defined scenes

51. Example of a POV-Ray Scene File
#include "colors.inc"
background { color Cyan }
camera {
location <0, 2, -3>
look_at <0, 1, 2> }
sphere {
<0, 1, 2>, 2
texture {
pigment { color Yellow }
light_source { <2, 4, -3> color White}

52. Summary What is ray tracing and why use it? Recursive ray tracing
Intersection
Shading and secondary rays
Shadow rays, reflection rays, and refraction rays
Ray tracing vs. rasterization

53. Radiosity

54. Outline What is radiosity? Radiosity process
Form factor
Solve radiosity
Advantages and limitations of radiosity

55. Background
Radiosity method is based on the theory of thermal heat transfer.
Heat transfer theory describes radiation as the transfer of energy from a surface when that surface has been thermally excited.
This thermal radiation theory can be used to describe the transfer of many kinds of energy between surfaces, including light energy.

56. What is radiosity?
The radiosity of a surface is the rate at which energy leaves that surface (energy per unit time per unit area).
It includes the energy emitted by a surface as well as the energy reflected from other surfaces.
Radiosity methods allow the intensity of radiant energy arriving at a surface to be computed.
These intensities can then be used to determine the shading of the surface.
In graphics, the term “radiosity” refers to the rendering algorithm that models the lighting process as energy transfer

57. Radiosity Process Process Group mesh surface into patches
Calculate form factors for patches
Solve radiosity
Display patches

58. The Radiosity Equation
The "radiosity equation" describes the amount of energy which can be emitted from a surface, as the sum of the energy inherent in the surface (a light source, for example) and the energy which strikes the surface, being emitted from some other surface.
The energy which leaves a surface (surface "j") and strikes another surface (surface "i") is attenuated by two factors:
the "form factor" between surfaces "i" and "j", which accounts for the physical relationship between the two surfaces
the reflectivity of surface "i", which will absorb a certain percentage of light energy which strikes the surface.

59. The Radiosity Equation

60. Form Factor
The "form factor" describes the fraction of energy which leaves one surface and arrives at a second surface.
It takes into account the distance between the surfaces, computed as the distance between the center of each of the surfaces.
It also takes into account their orientation in space relative to each other, computed as the angle between each surface's normal vector and a vector drawn from the center of one surface to the center of the other surface.

61. Form Factor
To use this form factor with surfaces which have a positive area, the equation must be integrated over one or both surface areas.
The form factor between a point on one surface and another surface with positive area can be used if the assumption is made that the single point is representative of all of the points on the surface.

62. The Form Factor

63. The Nusselt Analog
Differentiation of the basic form factor equation is difficult even for simple surfaces.
Nusselt developed a geometric analog which allows the simple and accurate calculation of the form factor between a surface and a point on a second surface.
The "Nusselt analog" involves placing a hemispherical projection body, with unit radius, at a point on a surface.
The second surface is spherically projected onto the projection body, then cylindrically projected onto the base of the hemisphere.
The form factor is, then, the area projected on the base of the hemisphere divided by the area of the base of the hemisphere.

64. The Nusselt Analog

65. The Hemicube Approximation
In the "hemicube" form factor calculation, the center of a cube is placed at a point on a surface
The upper half of the cube is used as a projection body as defined by the "Nusselt analog."
Each face of the hemicube is subdivided into a set of small, usually square ("discrete") areas, each of which has a pre-computed form factor value.
When a surface is projected onto the hemicube, the sum of the form factor values of the discrete areas of the hemicube faces which are covered by the projection of the surface is the form factor between the point on the first surface (about which the cube is placed) and the second surface (the one which was projected).

66. The Hemicube Approximation
The speed and accuracy of this method of form factor calculation can be affected by changing the size and number of discrete areas on the faces of the hemicube.

67. The Hemicube Anolog
This illustration demonstrates the calculation of form factors between a particular surface on the wall of a room and several surfaces of objects in the room.

68. The Hemicube Anolog
A radiosity algorithm will compute the form factors from a point on a surface to all other surfaces, by projecting all other surfaces onto the hemicube and storing, at each discrete area, the identifying index of the surface that is closest to the point.
When all surfaces have been projected onto the hemicube, the discrete areas contain the indices of the surfaces which are ultimately visible to the point.
From there the form factors between the point and the surfaces are calculated.
For greater accuracy, a large surface would typically be broken into a set of small surfaces before any form factor calculation is performed.

69. Radiosity form factor learniing tools
Brown University Exploratories project:
Radiosity Form Factor:
This applet shows how the form factor between two patches varies in magnitude depending on the relative geometry of the patches.

70. The Full Matrix Radiosity Algorithm
Two classes of radiosity algorithms have been developed which will calculate the energy equilibrium solution in an environment.
The "full matrix" radiosity solution calculates the form factors between each pair of surfaces in the environment, then forms a series of simultaneous linear equations.
This matrix equation is solved for the "B" values, which can be used as the final intensity (or color) value of each surface.

71. The Full Matrix Radiosity Algorithm

72. The Full Matrix Radiosity Algorithm
This method produces a complete solution, at the substantial cost of first calculating form factors between each pair of surfaces and then the solution of the matrix equation.
Each of these steps can be quite expensive if the number of surfaces is large
complex environments typically have upwards of ten thousand surfaces, and environments with one million surfaces are not uncommon.
This leads to substantial costs not only in computation time but in storage.

73. The Progressive Radiosity Algorithm
The "progressive" radiosity solution is an incremental method, yielding intermediate results at much lower computation and storage costs.
Each iteration of the algorithm requires the calculation of form factors between a point on a single surface and all other surfaces, rather than all N-squared form factors (where "N" is the number of surfaces in the environment).
After the form factor calculation, radiosity values for the surfaces of the environment are updated.

74. The Progressive Radiosity Algorithm
This method will eventually produce the same complete solution as the "full matrix" method, though, unlike the "full matrix" method, it will also produce intermediate results, each more accurate than the last.
It can be halted when the desired approximation is reached.
It also exacts no large (again, N-squared) storage cost.

75. The Progressive Radiosity Algorithm

76. Progressive Radiosity Example

77. Progressive Radiosity Variants
Several variations on the basic progressive radiosity algorithm have been developed.
“Gathering” variant
“Shooting” variant
The "gathering" variant updates one surface by collecting light energy from all other surfaces.
In this variant, as well as the "shooting" variant, the "base" surface is arbitrarily chosen.

78. Progressive Radiosity Variants
The "shooting" variant updates all surfaces by distributing light energy from one surface
The "shooting and sorting" variant chooses the surface with the greatest unshot light energy and distributes its light energy to the surfaces in the environment.
The "shooting and sorting" method is the most desirable, as it finds the surface with the greatest potential contribution to the intensity solution and updates all other surfaces in the environment with its energy.
In addition to these, an initial "ambient" term can be approximated for the environment and adjusted at each iteration, gradually replaced by the true ambient contribution to the rendered image.

79. Radiosity Learning Tool
Brown University Exploratories project:
Radiosity Shooting vs Gathering
This applet shows the graphic difference between two radiosity algorithms: shooting and gathering.
The algorithms are animated for each iteration of the algorithms and the resulting visualization of a discretized scene and of its form factor matrix can be viewed as the animation progresses.

80. Comparison of Progressive Variants
shooting
gathering
Shooting & sorting & ambient
Shooting & sorting

81. The Two-Pass Radiosity Solution
View independent, global diffuse illumination computed with radiosity pre-process.
View dependent, global specular illumination computed with ray tracing post-process.
Combining the strengths of radiosity and ray tracing achieves a more accurate and efficient solution to the problem of light transport.

82. Participating Media
Another variation on the basic diffuse radiosity solution adds the contribution of light passing through a participating medium, such as smoke, fog, or water vapor in the air
Creating volumetric lighting effect
In this algorithm, light energy is sent through a three-dimensional volume representing a participating medium, which both attenuates the light energy and adds to the intensity solution through illumination of the participating medium.

83. Participating Media

84. Advantages of Radiosity
Photorealistic image quality
Accurate simulation of energy transfer
Can simulate color bleeding.
Light reflected from a surface is attenuated by the reflectivity of the surface, which is closely associated with the color of the surface.
The reflected light energy often is colored, to some small extent, by the color of the surface from which it was reflected.

85. Advantages of Radiosity
The reflection of light energy in an environment often produces a phenomenon known as "color bleeding," where a brightly colored surface's color will "bleed" onto adjacent surfaces. The image in this slide illustrates this phenomenon, as both the red and blue walls "bleed" their color onto the white walls, ceiling and floor.

86. Radiosity vs. Ray Tracing
Simulate soft shadows: shadows with soft edges
Ray tracing often creates hard-edged shadows
Viewpoint independent: the solution will be the same regardless of the viewpoint of the image
Ray tracing is viewpoint dependent
Radiosity is an object space algorithm while ray tracing is an image space algorithm

87. Limitations of Radiosity
Large computational and storage costs
Must preprocess polygonal environments
Difficult to compute form factor
Does not consider transparent/translucent surfaces
Non-diffuse components of light not represented
Specular highlights hard to achieve

88. Radiosity Examples

89. Radiosity Examples
Over 1 million polygons

90. Radiosity Examples

91. Radiosity tools Pixar’s PRMan Radiosity is supported in POV-Ray

92. Summary What is radiosity? Radiosity process
Form factor
Solve radiosity
Advantages and limitations of radiosity