1. Monte Carlo Path Tracing
Today
Path tracing
Random walks and Markov chains
Eye vs. light ray tracing
Bidirectional ray tracing
Next
Irradiance caching
Photon mapping

2. Light Path
Should we show the directional form of this equation?

3. Light Transport Integrate over all paths Questions
How to sample space of paths

4. Path Tracing

5. Penumbra: Trees vs. Paths
Path tracing vs. recursive ray tracing using trees
Use less samples for this
Superimpose paths and trees over these images
4 eye rays per pixel
16 shadow rays per eye ray
64 eye rays per pixel
1 shadow ray per eye ray

6. Path Tracing: From Camera
Step 1. Choose a camera ray r given the (x,y,u,v,t) sample
weight = 1;
Step 2. Find ray-surface intersection
Step 3.
if light
return weight * Le();
else
weight *= reflectance(r)
Choose new ray r’ ~ BRDF pdf(r)
Go to Step 2.
Should illustrate this algorithm using the Cornell Box
Confusion about reflectance
The BRDF is not normalized to be a pdf
The directional hemispherical reflectance is the correct normalization factor
How to choose between Le and Lr

7. M. Fajardo Arnold Path Tracer
Return to this when discussing the adjoint formulation

8. Cornell Box: Path Tracing
Check that I am referring to these images in the correct way
10 rays per pixel
100 rays per pixel
From Jensen, Realistic Image Synthesis Using Photon Maps

9. Path Tracing: Include Direct Lighting
Step 1. Choose a camera ray r given the (x,y,u,v,t) sample
weight = 1;
Step 2. Find ray-surface intersection
Step 3.
weight += Lr(light sources)
Choose new ray r’ ~ BRDF pdf(r)
Go to Step 2.
Should illustrate this algorithm using the Cornell Box
Confusion about reflectance
The BRDF is not normalized to be a pdf
The directional hemispherical reflectance is the correct normalization factor
How to choose between Le and Lr

10. Discrete Random Walk

11. Discrete Random Process
States
Creation
Termination
Transition
Assign probabilities to each process

12. Discrete Random Process
States
Creation
Termination
Transition
Equilibrium number of particles in each state

13. Discrete Random Walk States Creation Termination
Generate random particles from sources.
Undertake a discrete random walk.
Count how many terminate in state i
[von Neumann and Ulam; Forsythe and Leibler; 1950s]
States
Creation
Termination
Transition

14. Monte Carlo Algorithm Define a random variable on the space of paths
Probability:
Estimator:
Expectation:

15. Monte Carlo Algorithm Define a random variable on the space of paths
Probability:
Estimator:

16. Estimator Count the number of particles terminating in state j
Do I have I,j reversed in Mi,j?

17. Equilibrium Distribution of States
Total probability of being in states P
Note that this is the solution of the equation
Thus, the discrete random walk is an unbiased estimate of the equilibrium number of particles in each state

18. Light Ray Tracing

19. Backward ray tracing, Arvo 1986
Examples
Backward ray tracing, Arvo 1986

20. Path Tracing: From Lights
Step 1. Choose a light ray
Choose a light source according to the light source power distribution function.
Choose a ray from the light source radiance (area) or intensity (point) distribution function
w = 1;
Step 2. Trace ray to find surface intersect
Step 3. Interaction

21. Path Tracing: From Lights
Step 1. Choose a light ray
Step 2. Find ray-surface intersection
Step 3. Interaction
u = rand()
if u < reflectance
Choose new ray r ~ BRDF
goto Step 2
else
terminate on surface; deposit energy

22. Bidirectional Path Tracing

23. Bidirectional Ray Tracing
Need to make notation clear
K is the number of vertices on a path
K-2 is the number of bounces on the path
Should probably show an example with k=4, not k=3

24. Path Pyramid From Veach and Guibas 25 samples per pixel
Combined using the power heuristic
Middle image in the k=4 row
Rightmost image in the bottom row
5 light-path vertices
5 steps from the floor-lamp to the table
Everyone really liked this picture
Understand the phenomenology behind each subimage
Is there a version of this picture that contains the weights for each pixel
Review multiple importance sampling: need to keep the weights and values
If there are multiple samples for each pixel, Kurt suggests premultiplying by the weights and keeping a running sum of the weights.
From Veach and Guibas

25. Comparison Bidirectional path tracing Path tracing
From Veach and Guibas

26. Generating All Paths

27. Adjoint Formulation

28. Symmetric Light Path All this adjoint stuff is confusing
What are the key points
Can always formulate adjoint version of the problem
Rendering operators are self-adjoint
Solution of the adjoint equation can be used as an importance function
Clear explanation with examples of why you want to reverse sources and receivers

29. Symmetric Light Path

30. Symmetric Light Path

31. Three Consequences Forward estimate equal backward estimate
- May use forward or backward ray tracing
Adjoint solution
- Importance sampling paths
Solve for small subset of the answer

32. Example: Linear Equations
Solve a linear system
Solve for a single xi?
Solve the adjoint equation
Source
Estimator
More efficient than solving for all the unknowns
[von Neumann and Ulam]