TL;DR of the paper 'Stratified Sampling of Projected Spherical Caps'

I will in this post write down a TL;DR of the paper 'Stratified Sampling of Projected Spherical Caps'. Observe that all images are from the paper, and do not belong to me

The paper describes an importance sampling strategy for spherical light sources, useful for implementing a path tracer. When rendering a light source in a path tracer, we need a way of getting a random position on that light source. This is necessary for implementing the Monte Carlo method, which is an important part of speeding up the path tracing process.

The above image illustrates the situation. From every point $p$, we want to determine the final color. In order to do this, we must send out and trace many rays. These rays are all from the green hemisphere that covers the point. As part of the monte carlo method, we want a sampling strategy that generates rays that hit the yellow light source. Different sampling strategies give noise-free results faster or slower.

It is important to realize that not all points on the spherical light source are even visible. Observe the below image. Only the points on the blue arc are visible from the point $x$. So, one sampling strategy, is that we sample only points from this arc, since they are the only points on the light source that will actually contribute anything to the final image. A common strategy is that we uniformly sample a point from this arc. Which means that all points are equally likely to be chosen.

In the three dimensional case, the arc becomes a spherical cap, which is illustrated in the image below. The spherical cap is blue.

So, an easy solution would be that we sample points from the spherical cap, in order to generate rays that hit the light source. However, the paper describes a better solution: project the spherical cap onto the tangent plane of the point $x$:

As can be observed above, the projected shape will either be a regular ellipse(left), the combination of an ellipse and a shape called a lune(middle), or just a lune(right). At grazing angles, a lune is much more likely. Now, the authors suggest that we do our sampling from this projected 2D shape. So, we take a uniform sample from this 2D shape. In the middle image above, $w$ was sampled from the projected shape. $w$ is then projected onto the sphere, and we thus get our final ray $\omega$.

Now, why should we sample from the projected spherical cap, instead of directly sampling from the spherical cap? At a first glance, it seems like an awfully convoluted and roundabout approach. But there is a very intuitive explanation for this. Observe the below image

In the image, we have a green and blue spherical cap. The caps are equally wide, but are at different angles, with respect to the normal. The green cap is at a much more grazing angle. As can be observed, the blue spherical cap is projected to a much larger area than the green spherical cap. This is because the green spherical cap is at a much more grazing angle.

Now, what will happen if we sample from a projected spherical cap in our sampling strategy? The areas of the spherical cap that are at a non-grazing angle will be projected to a much larger area, and so they are more likely to be sampled. Because if our sampling strategy is correctly implemented, then all points on the projected spherical cap are equally likely to be chosen as samples. Ergo, points of the spherical cap that are at non-grazing angles are much more likely to be chosen as samples. And this is exactly what we want. Consider the rendering equation, as given in the paper:

Of particular importance is the cosine term. The cosine term means that samples that are at non-grazing angles, will be given more importance. However, the sampling method in the paper, actually is more likely to sample non-grazing rays(as we just described above), and this means that the Monte Carlo method will give a noise-free result faster! Basically, the integrand in the rendering equation will have a high value for certain samples, and if we are more likely to generate such samples, then the Monte Carlo method converges quicker.

To conclude, merely sampling the spherical cap is a worse sampling strategy, since this strategy doesn't take the cosine term into account. However, if we sample from the projected spherical cap, then the cosine term is actually taken into account, and this results in a better sampling strategy, that gives us noise free results in a less amount of time.

Finally, the authors provide easily usable source code that implements the method.