After tasks 4 and 5, Pathtracer::sample_direct_lighting is no different than the indirect case: it simply samples a ray from the BSDF scattering function and traces it into the scene to gather direct lighting. In this task, you will modify the sampling algorithm by splitting samples between BSDF scatters and the surface of area lights, a procedure commonly known as next event estimation.

First consider why sampling lights is useful. Currently, we are only importance sampling the BSDF term of the rendering equation (in which we have included the cosine term). However, each sample we take will also be multiplied by incoming radiance. If we could somehow sample the full product, our monte carlo estimator would exhibit far lower variance. Sampling lights is one way to importance sample incoming radiance, but there are some caveats.

• Importance sampling indirect lighting is hard. Doing so in an un-biased fashion requires more advanced integration schemes like bidirectional path tracing or metropolis light transport. In Scotty3D, we will only be adding sampling for direct lighting, because we know a priori what directions it can come from: those that point at emissive objects. Although doing so will not importance sample the full distribution of incoming radiance, it will be a better approximation.

• Specular BSDFs, such as mirror and glass, will not be improved by directly sampling lights. This is because any ray that is not a perfect reflection or refraction has zero contribution. However, scenes using continuous distributions like the Lambertian, Blinn-Phong, or GGX BSDFs, can benefit tremendously, especially in the presence of small and/or intense area lights.

### How do we do it?

It’s tempting to simply compute an estimate of direct lighting from BSDF samples, another from sampling the surface of lights, and average the results. However, variance is additive: averaging two high-variance estimators does not give back a low-variance one. Instead, we can create a single new distribution that has the average PDF of the two inputs.

To do so, simply uniformly randomly choose which strategy to use before sampling from it as usual. Any given sample could then have been generated from either distribution, so the PDF at the sample is the average over each individual strategy’s PDF. This is called mixture sampling, or more properly single-sample multiple importance sampling.

Intuitively, consider that the average of multiple PDFs is itself a PDF: it is non-negative and integrates to one. The average PDF will assign higher weight to any region that one of its components did, producing a distribution following the strengths of each. We can then use the improved distribution to sample and accumulate monte carlo estimates as usual.

Previously, if we chose a low-weight BSDF sample that just happened to hit a bright light, we would get a huge (high-variance) result after dividing (large) incoming light by the (small) PDF. Now, regardless of whether that sample came from BSDF::scatter or Pathtracer::sample_area_lights, the PDF of the sample cannot be small, because its Pathtracer::area_lights_pdf component is not small.

For a more rigorous explanation of multiple importance sampling, refer to Physically Based Rendering chapters 13.10 and 14.3.

• Note: because the mixture PDF now depends on both strategies, we need to separate the concepts of sampling a distribution and evaluating its PDF. This is why we separate BSDF::scatter/BSDF::pdf and Pathtracer::sample_area_lights/Pathtracer::area_lights_pdf.

Lastly, note that when the true light distribution heavily favors only one strategy (e.g. a highly specular but still continuous BSDF, or an intense but small area light), we will end up wasting time on samples we got from the wrong strategy. Ideally, we could adaptively choose how much contribution we take from each option, which is known as multi-sample multiple importance sampling (see Extra Credit).

Finally, let’s upgrade Pathtracer::sample_direct_lighting. Start by reading the following functions:

• Pathtracer::sample_area_lights takes a world-space position, returning a world-space direction pointing towards an area light.
• Pathtracer::area_lights_pdf takes a world-space position and direction, returning the PDF for generating the direction at the point from the area lights in the scene. Note that these area light functions operate in world space, while BSDF functions operate in local space, relative to the surface at the ray intersection point. Pay close attention to the inputs and outputs of each of these functions, and make sure to look at what attributes you have available to you as part of the Pathtracer::Shading_Info struct. The direct lighting procedure should now follow these steps:

• If the BSDF is discrete, we don’t need to bother sampling lights: the behavior should be the same as task 4.

• Otherwise, we should randomly choose whether we get our sample from BSDF::scatter or Pathtracer::sample_area_lights. Choose between the strategies with equal probability.

• Create a new world-space ray (the “shadow ray”) and call Pathtracer::trace to get incoming light. You should modify Ray::dist_bounds so that the ray does not intersect at time = 0. We are still only interested in the emissive component, so the ray depth can be zero.

• Add estimate of incoming light scaled by BSDF attenuation. Given a sample, we don’t know whether it came from the BSDF or the light, so you should use BSDF::evaluate, BSDF::pdf, and Pathtracer::area_lights_pdf to compute the proper weighting. What is the PDF of our sample, given it comes from the combined distribution?

## Tips

• The converged output of all scenes should not change with the addition of task 6. If it does, you’ve done something wrong.
• We do not provide much in the way of reference images: make your own scene demonstrating what situations area light sampling is or isn’t well suited for.
• Use the ray log to visually debug what proportion of your shadow rays are being directed at lights and where they are going.

## Reference Results

You will now be able to render scenes featuring area lights using far fewer samples and still get good results. The effect will be particularly pronounced when small and/or intense area lights are used with Lambertian materials (e.g. see task 7, grace.exr).

Here are cbox.dae and cbox_lambertian.dae without (left) and with (right) area light sampling (32 samples, max depth = 8):

## Extra Credit

• Upgrade the mixture sampling procedure to use proper multiple importance sampling. This will involve always generating both a light sample and BSDF sample, then weighting their direct light contributions by the balance or power heuristic. You may also want to re-combine the direct and indirect sampling procedures. Refer to Physically Based Rendering chapter 14.3 for more information.
• Currently, computing the area light PDF involves checking whether the sampled ray intersects each individual emissive triangle. Improve the brute-force approach by building and querying a BVH containing only emissive triangles. You will need a traversal algorithm that returns all triangles intersected by a ray, not just the closest.
• (Advanced) Implement a more powerful integration scheme such as bidirectional path tracing or photon mapping. Refer to Physically Based Rendering chapter 16.
• (Advanced) Implement specular next event estimation for the mirror and glass materials.