**Foundations**

**Path tracing is great for simulating global illumination by tracing rays, and of course, these rays have to follow "realistic paths". As observed a long time ago (1760) by Johann Lambert, intensity of light from an ideal diffuse surface is directly proportional to the cosine of the angle $\theta$ between the direction of the incident light and the surface normal. This is called Lambert's cosine law in optics.**

In path tracing, we mainly use the cosine law to calculate the contribution of light from the next ray. When tracing from a diffuse surface, the direction of the next ray is usually calculated by generating a random direction in the hemisphere of the surface. When tracing from a mirror, you can imagine there is only one direction on the hemisphere that is correct: the perfect reflection. Metal-like surfaces contain both elements, called glossy reflections. This image shows what I'm talking about in 2D:

If you would order them by calculation cost for the path tracing algorithm, the reflection would be the easiest, there is only one solution, one ray to calculate. The glossy one is harder to calculate, since you have to accumulate rays with different weights, but still simpler to fully calculate than the diffuse option. Besides these simple examples, materials can contain many functions to bend incoming rays. The collection of these functions describe the Bidirectional Reflectance Distribution Function (BRDF). Which in turn, you can even extend to allow for rays extending through the surface with a Bidirectional Scattering Distribution Function (BSDF).

**Cosine weighted sampling**

The intensity of light from an ideal diffuse surface is directly proportional to the cosine of the angle $\theta$ between the direction of the incident light and the surface normal.This leads to an interesting optimization: cosine weighted sampling. Since the incoming and outgoing light is measured by a cosine, the outgoing rays nearly parallel to the surface don't add any (useful) information to the final solution. I could write this whole post about this optimization, but that would be a waste of time, since there are already many good examples out there. This post from Rory Driscoll shows results as well as the maths behind it.

This image shows the cosine weighted sampling on the top and the uniform sampling on the bottom. This was taken from the pathtracing blog |

**Sampling optimization**

**As mentioned above, when you've found a ray-triangle collision, the direction of the new ray is usually 'bruteforce' calculated by taking a random direction on the hemisphere. For diffuse surfaces, this is no problem, since every outgoing direction weights the same. If you would apply the same to perfectly reflecting surfaces, chances are you will most likely never find the correct outgoing ray.**

By knowing the material properties you can determine how to sample the outgoing ray. This sampling strategy can be a very easy optimization if you're willing to end up with an approximation. If you look at the 3 examples from the image above: one variable for the degree of reflectiveness can produce these examples.

The problems with this reflectiveness variable for path tracing are the weights shown in the glossy example. Solving the Monte Carlo problem of path tracing requires a probability distribution function (PDF) over these samples. The outer rays in the glossy examples are shown smaller, because they affect the outcome less than the rays from the perfect reflection.

This optimization is as simple or as hard as you want to make it. The simpler you keep the sampling, the more approximate your solution is going to be. The best solution would be to sample your chosen BRDF. The cosine weighted sampling is shown to be perfect for diffuse only materials (diffuse materials have perfect lambertian reflectance). I found a sampling method for the Cook-Torrance BRDF in this paper: Microfacet Models for Refraction through Rough Surfaces.

**Direct and indirect lighting**

Direct lighting is easy to calculate. This is shown in almost all released 3D games out there. Simply calculating a dot product of the surface normal and light direction would result in perfect lambertian shading.

In path tracing we can apply an optimization to allow for direct light sampling. It's called next event estimation. This is where the math and statistics get nasty. To allow for direct light sampling, we have to apply a different probability distribution function. We have to sample the light and divide the contribution of direct light by the area of the light:

Area sampling. Image taken from [2] |

Whereas the indirect illumination should be sampled by the original method using hemispheres. With one exception, all of the rays hitting the light should be discarded (since the light gets sampled by the other PDF from above).

Hemisphere sampling without direct light sampling (dashed lines). Image taken from [2] |

**Multiple importance sampling**

**In larger applications, thousands of material types and BRDFs are going to be used. Having a different sampling strategy for each of them is going to be time consuming and frankly undo-able. We already have two different PDFs to sample all of these materials, the direct and indirect sampling. The problem that arises from all of this shown in images:**

These are the two sampling methods shown from above, the BSDF sampling is the hemisphere indirect sampling and the light source sampling is the direct sampling using the area of the light. The problem is the glossy reflection on the four plates. You can see that indirect sampling works if the light is large enough to have some collisions with rays, and the direct light source sampling works fine for small area lights.

Multiple Importance Sampling is a technique that can combine different sampling techniques to find a low variance estimate of the integral. By combining both sampling methods we can get the following result:

Images taken from [4] |

**Bidirectional path tracing**

**This leads us to more recent techniques in path tracing, namely bidirectional path tracing. This technique is based on the inverse of path tracing, instead of finding the light by tracing rays from the camera, find the camera by tracing from the light. These are the shooting rays, and the original camera rays the gathering rays. The bidirectional part indicates that these two types of rays can be combined.**

This technique wasn't meant for real time path tracing applications which require responsiveness. Since for every gathering ray you shoot, the amount of calculation is doubled by also tracing shooting rays. This nearly doubles the time per frame, and thus decreases responsiveness. While you lose responsiveness, this algorithm increases the convergence rate for path tracing a lot, so in the end it's worth it.

As extension to bidirectional path tracing, there is an algorithm called Metropolis light transport. This algorithm allows finding more difficult light paths by extending existing paths. An example: A shooting and gathering ray have been found by the bidirectional raytracing method. Every bounce in these rays is recorded as a node. The metropolis light transport can add extra nodes on these rays to modify the existing ray to a new one. These nodes are placed in areas with high gradients. You can find more on this technique in this paper: Metropolis Light Transport.

Another extension I found, but have not fully explored is: Light Transport Simulation with Vertex Connection and Merging. The basic idea is to reuse existing paths by connecting vertices. This can increase the amount of generated paths per pixel:

Vertex merging: reusing paths to increase paths per pixel. Image taken from [6] |

**Further reading**

Most recent techniques include gradient domain tracing. It first originated from metropolis light transport in: Gradient-domain Metropolis Light Transport. The paper on Gradient Domain Bidirectional Path Tracing applies this to a bidirectional path tracer, and Gradient-Domain Path Tracing applies the same principle for normal path tracers. The path tracer also contains source code. You can see the path tracing algorithm is converging faster already with only one sample per pixel:

Showing the difference between Gradient domain path tracer (G-PT) and normal path tracer (PT). Image taken from [9]. |

http://blog.hvidtfeldts.net/index.php/2015/01/path-tracing-3d-fractals/

Path tracing 3D fractals. Also explains cosine weighted sampling, importance sampling, next event estimation, and image based lighting. Implementation in GLSL.

http://raytracey.blogspot.com

Everything about ray tracing, and is currently working on a GPU path tracer.

https://www.reddit.com/r/pathtracing/

https://www.reddit.com/r/raytracing/

Community for path- and ray tracing.

**References**

**Presentations**

[1] http://cg.informatik.uni-freiburg.de/course_notes/graphics2_09_pathTracing.pdf

[2] http://www.cs.dartmouth.edu/~cs77/slides/18_PathTracing.pdf

Papers

[3] Walter, Bruce, et al. "Microfacet models for refraction through rough surfaces."Proceedings of the 18th Eurographics conference on Rendering Techniques. Eurographics Association, 2007.

[4] Veach, Eric. Robust monte carlo methods for light transport simulation. Diss. Stanford University, 1997.

[5] Veach, Eric, and Leonidas J. Guibas. "Metropolis light transport." Proceedings of the 24th annual conference on Computer graphics and interactive techniques. ACM Press/Addison-Wesley Publishing Co., 1997.
[6] Georgiev, Iliyan, et al. "Light transport simulation with vertex connection and merging." ACM Trans. Graph. 31.6 (2012): 192.

[7] Lehtinen, Jaakko, et al. "Gradient-domain metropolis light transport." ACM Transactions on Graphics (TOG) 32.4 (2013): 95.
[8] Manzi, Marco, et al. "Gradient-Domain Bidirectional Path Tracing." (2015).

[9] Kettunen, Markus, et al. "Gradient-domain path tracing." ACM Transactions on Graphics (TOG) 34.4 (2015): 123.

[9] Kettunen, Markus, et al. "Gradient-domain path tracing." ACM Transactions on Graphics (TOG) 34.4 (2015): 123.

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