1. Monte Carlo Path Tracing and Caching Illumination
Part II – Bidirectional Path Tracing and Photon Mapping

2. Path Notation A path is written as a regular expression. Examples:
Ray tracing: LD[S*]E
Radiosity: LD*E
Complete global illumination: L(D|S)*E

3. Path Tracing
See Pharr’s PBRT 2nd Ed. 15.3

4. Bi-direction Path Tracing
From Pharr’s PBRT 2nd Ed., Section
Subpath from camera: p1, p2, …pi
Subpath from light: q1, q2, …qj
The whole path is p1, …, pi, qj, …, q1
Check if pi can see qj
Some call this “vertex connection”

5. Bi-direction Path Tracing
Many open questions:
Where does pi or qj end? Does it end on specular or diffuse surfaces?
How to calculate the contribution of qj to pi?

6. Photon Mapping
See Pharr’s PBRT 2nd Ed., Section 15.6 or SIGGRAPH 2008 Course, Advanced Global Illumination using Photon Mapping

7. Step 1: Photon Emmision
From: SIGGRAPH 2008 Course, Advanced Global Illumination using Photon Mapping Figure 4.1

8. Step 2: Photon Tracing
From: SIGGRAPH 2008 Course, Advanced Global Illumination using Photon Mapping Figure 4.3

9. Step 3: Photon Storing
From: SIGGRAPH 2008 Course, Advanced Global Illumination using Photon Mapping Figure 4.4

10. Caustic vs. Global Photon Maps
From: SIGGRAPH 2008 Course, Advanced Global Illumination using Photon Mapping Figure 4.6

11. Step 4: Radiance Estimate
Some call this “vertex merging”
From: SIGGRAPH 2008 Course, Advanced Global Illumination using Photon Mapping Figure 4.8

12. Typical Distributed Ray Path

13. Integrals In rendering equation: In image formation (camera)
Reflection and transmission.
Visibility
Light source
In image formation (camera)
Pixel
Aperture
Time
Wavelength

14. Integral of BRDF and Light
Rendering Equation (revisted)
Ignoring emitted light and occlusion, we still have an expensive integral:
Let f(Xi)= (…) I (…) and evaluate its integral with Monte Carlo methods.

15. Integral of BRDF and Light
Let f(Xi)= (…) I (…) and evaluate its integral.
Case1: a diffuse surface and a few area lights
Case2: a specular surface and environment lighting
Uniform sampling isn’t efficient in both cases. Why?
 (…)
I (…)

16. Can Importance Sampling Cure Them All?
Consider these two example:
How to handle diffuse reflection?
How to handle large area light source?
More in Veach’s thesis (especially Figure 9.2)
Sampling BRDF vs. sampling light sources

17. Multiple Importance Sampling
See Pharr’s PBRT 2nd Ed. 14.4
(a) sampling the surface reflectance distribution
(b) sampling the area light source

18. Source: Eric Veach, “Robust Monte Carlo Methods for Light Transport Simulation” Page 255, Figure 9.2.
Ph.D. Thesis, Stanford University

19. References
SIGGRAPH 2008 Course, Advanced Global Illumination using Photon Mapping
SIGGRAPH 2012 Courses, Advanced (quasi) Monte Carlo methods for image synthesis