1. Path Tracing (some material from University of Wisconsin)
CS 655 – Computer Graphics
Path Tracing
(some material from University of Wisconsin)

2. Path Tracing An extension to ray tracing Simulates global illumination
Can handle all possible light bounces
L(S|D)*E (this is a reg exp, not a product)
Introduced by Kajiya as a solution to the rendering equation
Stochastically samples all light paths
Handles area light sources, diffuse reflections
Approximates the integral of illumination

3. Path Tracing Algorithms
Determine the intensity of each pixel by tracing light transport paths
Paths that start at light sources and carry energy
A path of length k is a sequence of vertices
<x0, … , xk-1> where every xi and xi+1 is mutually
visible and x0 is on a light

4. “Important” Paths We are most interested in the “important” paths
Paths that go from a light source to the eye
Paths that carry the most energy
The rendering equation can be written as a sum of integrals, each one integrating over a different path length

5. The Rendering Equation - reordered
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.

6. Sampling Important Paths
We want to evaluate the integral using importance sampling
i.e., attempt to sample the most important paths
How do we find those paths?
Several approaches have been tried

7. Naïve Path Tracing (version 1)
Start at a light
Build a path by randomly choosing a direction at each bounce, send the ray in that direction, and add the point hit to the path vertex list
Join the last point to the eye
Problems? What path is achieved by this approach?

8. Naïve Path Tracing (version 2)
Start at eye
Build a path by randomly choosing a direction at each bounce, follow the path in that direction, add the point hit to the path vertex list
Optionally join the last point to a light
Problems? What paths are generated?

9. Path Tracing (Kajiya) Start at eye
At each bounce, send a ray out in the direction determined according to some distribution
At each point on the path, cast a shadow ray and add direct lighting contribution at that point
Send multiple paths per pixel, average the contributions to get the final intensity

10. Path Tracing (Kajiya) – Sampling Strategies
The way in which the direction of each bounce is determined makes a big difference in image quality.
Stratified Sampling
Break the possible directions into sub-regions and cast one sample per sub-region.
Importance Sampling
Sample according to the BRDF

12. Importance sampling

13. Path Tracing (Kajiya) - Summary
Pros:
Focuses on objects that are visible – doesn’t waste time on objects that can’t be seen
Spends equal time on all path lengths
ray tracing spends more time on longer paths
Cons:
Little information gain for each ray cast
Not easy to get good (important) samples
Spends equal time on slow-varying diffuse components and fast varying specular components

14. Path Tracing Algorithm
Send a ray through a pixel
Trace the ray to its first intersected object
From the intersected object, send out
One ray to each light source
One additional ray
a diffusely reflected ray,
a specularly reflected ray, or
a transmissive ray
Trace the ray and recursively follow it, as above
Produces a ray “path” – not a ray tree

15. Light ray
Light
Eye
Image plane

16. Illuminance ray
Light
Diffuse?
Specular?
Transmission?
Eye
Image plane

17. Path Tracing
This provides a Monte Carlo approach to global illumination
Stochastic samples are taken that should represent the actual surface properties
The lighting distribution is sampled by tracing rays stochastically along all possible light paths
Averaging a large number of sample rays gives an estimate of the integral of all light paths through the pixel

18. Selecting the ray to trace
How do we select which ray to trace?
Each material has a kd, ks, and kt
Let ktot = kd+ ks + kt
Select a random number R in the range (0, ktot )
if (R < kd) then send diffuse ray
else if (R < kd + ks) then send specular ray
else send transmission ray

19. Tracing a Diffuse Reflection
Ideal diffuse reflection:
Light is scattered equally in all directions

20. Tracing a Diffuse Reflection
Tracing a ray from the eye, the light I see could have come from any direction (direct from the light, or indirect from another object)
Randomly pick a direction and trace it

21. Computing the Diffuse reflection
We can compute a random direction as follows:
Given two random numbers x1in [0,1] and x2 in [0,1], the randomly reflected direction wd is given by
wd = (q, f) = (cos-1(sqrt(x1)), 2px2 )

22. Tracing a Specular Reflection
Ideal specular reflection:
Light is scattered according to the reflection direction

23. Tracing a Specular Reflection
As with diffuse lighting, tracing a ray from the eye, the light I see could have come from any direction (direct from the light, or indirect from another object)
Light is scattered according to the reflection direction

24. Problems Need to trace a lot of rays to get an accurate image
Typically trace 100 – 1000 rays per pixel

25. Problems Need to trace a lot of rays to get an accurate image
Typically trace 100 – 1000 rays per pixel

26. 10,000 rays per pixel (Henrik Wann Jensen)

28. Bi-directional Path Tracing
What if we were to combine rays from both directions?
Send a ray from the eye and trace it into the scene
Send a ray from the light and trace it into the scene
Combine them by sending a ray from the end of one path to the end of the other path

29. Bi-directional Path Tracing
Pros:
Each ray cast contributes to many paths
Building from both ends can catch difficult cases
All specular paths
Caustics
The idea extends to participating media
Cons:
Still spends a lot of time in slow varying diffuse areas
May not sample some of the more difficult paths

30. Example: Mirror Eye ray tracing: ES*DL Light ray tracing: LS*DE
Problematic: ES*DS*L
Even Worse: LS*DS*DS*E