1. The Rendering Equation
CS 655 – Computer Graphics
The Rendering Equation

2. The Rendering Equation
Developed by Kajiya in 1986
An attempt to unify rendering so that all rendering has a basic model as a basis
Accounts for all light interactions in an environment

3. Energy Balance Outgoing – Incoming = Emitted – Absorbed
The total light energy put into the system must equal the energy leaving the system (usually, via heat).
Outgoing = Emitted + Reflected + Transmitted

4. Outgoing Energy wo H+ : wi · n(x) < 0 N wir N = surface normal
wo = outgoing energy
wir = incoming reflected energy
wit = incoming transmitted energy
H- : wi · n(x) > 0
H+ = half space above object
wit
H- = half space below object

5. Steady State. How come we never notice?
How long does it take lighting to stabilize in a 20x20x10 room with a point light in the center of the room?
suppose the room has semi-gloss paint which reflects let's say t
turn on light
diagonal distance from light (center of ceiling) to corner
E-08
light hits far corner
speed of light (ft/sec)
E-08
light travels bounces corner to corner 30
diagonal distance from corner to corner
7.8612E-08
light travels corner to corner again
E-07
E-07
E-07
E-07
E-07
E-07
2.9212E-07
Single bounce time as freq.
32,785,701.87
32.8 gigahertz
TV runs at about 60 hz
Ears can hear upto 16,000 Hz

6. Radiosity vs. Radiance vs. Irradiance vs. Radiant Power
Radiant Power: aka flux, energy flowing to/from a surface per unit time (Watt, aka Joule/sec)
Radiance: exiting power per unit area (Watts/m^2)
Irradiance: incoming power per unit area (Watts/m^2)
Radiosity: exiting power per unit projected area per unit solid angle
Varies with position and direction.
Captures the notion of appearance (for a given location and direction).

7. Radiance
Unit is:

8. Basic Equations Radiant power Incident radiant power at x
Exitant radiant power at x

9. BRDF Bidirectional Reflectance Distribution Function
A measurement of the amount of energy being distributed about all directions from a point.

10. Diffuse BRDF
Light is reflected equally in all directions

11. Specular BRDF
Light is reflected only in one direction

12. Glossy BRDF Light is reflected unequally in many directions
Several models exist that attempt to represent glossy BRDFs.

13. Phong Model

14. Blinn - Phong Model

15. Modified Blinn - Phong Model

16. Outgoing Energy wo H+ : wi · n(x) < 0 N wir H- : wi · n(x) > 0
Outgoing = Emitted + Reflected + Transmitted
H- : wi · n(x) > 0
wit

17. Outgoing Energy
Outgoing = Emitted + Reflected + Transmitted

18. Outgoing Energy
Outgoing = Emitted + Reflected + Transmitted

19. Outgoing Energy wo N H+
wir H-
wit

20. The Rendering Equation
Unoccluded two point transfer
No participating media
If occluded, this is 0
If not occluded, this is the inverse square
of the distance between x and x’
Energy emitted from point x’ that reaches point x

21. The Rendering Equation
The intensity of energy originating from x’’,
coming through point x’, and
terminating at point x
The BRDF

22. The Rendering Equation
In other words, the transport intensity from x’ to x is the sum of the emitted light from x’ that reaches x, plus all of the light from x’’ that eventually gets to x through x’
We can rewrite the equation as:
Where M is the linear integral operator

23. Breaking Down the Rendering Equation
Rearranging terms gives:

24. Breaking Down the Rendering Equation
Light to
x directly
from x’
Light
from
light
source
to x’,
then to x
Light to
x via x’
scattered
twice
Light to
x via x’
scattered
three
times
etc.

25. Applying the Rendering Equation
Local reflection models:
only the first two terms are used
the ge term is non-zero only for light sources
M operates on e rather than g, so shadows are not computed

26. Applying the Rendering Equation
Basic ray tracing:
Mo is the sum of the reflection and refraction terms
the geo gives shadows, but only for point light sources
Ambient lighting is accounted for in e
M is generally approximated by a small sum