Picking directions on the upper hemisphere

[vchizhov]

This is more of a suggestion that should provide a better intuition for people learning ray tracing for the first time. As I see it, currently it is not quite clear why the scattering directions need to be picked on a unit sphere offset along the normal (even though that provides a cosine weighted hemisphere distribution) - the geometric and algebraic derivations are both non-trivial, and possibly non-intuitive (this is clearly illustrated by the fact that picking directions inside the ball yields a cos^3 distribution, while picking points on the sphere yields the desired cos distribution).
My suggestion is to directly generate points in or on the upper hemisphere, which is a lot more intuitive and can be directly related to the rendering equation and light scattering (even informally). This can be done without inverse transform sampling - by generating point in the unit ball (or on the unit sphere) as was already done, and reflecting all vectors in the lower hemisphere to be in the upper hemisphere, with respect to the normal (requires a dot check per vector). Note that this will induce a uniform distribution (so higher variance, and a normalization factor of 2pi and not pi), and will require a cos(theta) multiplication in the recursion (which will be a good point to introduce Lambert's law which will help motivate the name of the material too).

[hollasch]

Thoughts, @petershirley ?

[petershirley]

I've been pondering this and not converging. Here are some issues:

1. "diffuse" does not need to be Lambertian
2. But Lambertian has some advantages
3. The methods of generating rays on the unit shell is indeed unintuitive

Position 1: the simplest way to generate Lambertian directions should be used
Position 2: the simplest way to generate any kind of diffuse rays should be used as VC suggests

I am pretty comfortable with both those arguments-- they both have a point. Anyone have a strong opinion?

[vchizhov]

I wouldn't say that what I propose is necessarily simpler to implement. Sampling points within the unit ball, normalizing those to get points on the unit sphere, and offsetting those along the scattering normal doesn't seem more complex, than sampling points in the unit ball, and selectively reflecting those around the scattering plane (using a dot product).

Theoretically, however, motivating the former approach is a lot harder, and possibly not very intuitive. Algebraically it requires notions of probability theory and knowledge on the properties of the Dirac delta: https://github.com/vchizhov/Derivations/blob/master/Probability%20density%20functions%20of%20the%20projected%20offset%20disk%2C%20circle%2C%20ball%2C%20and%20sphere.pdf

Geometrically it requires some creativity and knowledge of Archimedes' Hat-Box theorem: http://amietia.com/lambertnotangent.html

[trevordblack]

There's a compromise possible here. I think that having random scattering above the horizon is intuitive. But I also think that we should eventually turn diffuse into cosine scattering. We could use horizon scattering in InOneWeekend. And go back through in NextWeek or RestOfYourLife to change to cos scattering. I unfortunately cannot think of where in either of those texts I'd happy putting the change.

…

On Sun, Oct 27, 2019, 06:35 Vassillen Chizhov ***@***.***> wrote: I wouldn't say that what I propose is necessarily simpler to implement. Sampling points within the unit ball, normalizing those to get points on the unit sphere, and offsetting those along the scattering normal doesn't seem more complex, than sampling points in the unit ball, and selectively reflecting those around the scattering plane (using a dot product). Theoretically, however, motivating the former approach is a lot harder, and possibly not very intuitive. Algebraically it requires notions of probability theory and knowledge on the properties of the Dirac delta: https://github.com/vchizhov/Derivations/blob/master/Probability%20density%20functions%20of%20the%20projected%20offset%20disk%2C%20circle%2C%20ball%2C%20and%20sphere.pdf Geometrically it requires some creativity and knowledge of Archimedes' Hat-Box theorem: http://amietia.com/lambertnotangent.html — You are receiving this because you were assigned. Reply to this email directly, view it on GitHub <#155?email_source=notifications&email_token=ACNZVKAUJ5BOTIQ3LPWTYGTQQSZYTA5CNFSM4ISKCHG2YY3PNVWWK3TUL52HS4DFVREXG43VMVBW63LNMVXHJKTDN5WW2ZLOORPWSZGOECKRNAQ#issuecomment-546641538>, or unsubscribe <https://github.com/notifications/unsubscribe-auth/ACNZVKDWLMBP7TELGKS6XRLQQSZYTANCNFSM4ISKCHGQ> .

[petershirley]

So thematically the three books are:

1. path of least resistance get results fast
2. oh so you want some better pictures here is some stuff
3. oh so you are hard core? Here is your kamikaze headband

Does that make it more clear (not to me yet)?

[trevordblack]

I guess we have three options for implemention in InOneWeekend 1. Scattering within the unit sphere offset by normal (previous, simplest to implement) 2. Uniform scattering above the horizon (most complicated to implement, intuitive) 3. Cos scattering about the normal (least intuitive, is correct) Looking at it from those terms, I could genuinely be convinced that any of the above is the right solution for the text.

…

On Sun, Oct 27, 2019, 11:54 petershirley ***@***.***> wrote: So thematically the three books are: 1. path of least resistance get results fast 2. oh so you want some better pictures here is some stuff 3. oh so you are hard core? Here is your kamikaze headband Does that make it more clear (not to me yet)? — You are receiving this because you were assigned. Reply to this email directly, view it on GitHub <#155?email_source=notifications&email_token=ACNZVKGUDQZB75TZTIR6KJ3QQT7HJA5CNFSM4ISKCHG2YY3PNVWWK3TUL52HS4DFVREXG43VMVBW63LNMVXHJKTDN5WW2ZLOORPWSZGOECKVFDY#issuecomment-546656911>, or unsubscribe <https://github.com/notifications/unsubscribe-auth/ACNZVKEGWE2KQAAKOVTLBHLQQT7HJANCNFSM4ISKCHGQ> .

[vchizhov]

@trevordblack

I think the implementation complexities of the three are comparable:

//1) Generates normal offset points in the unit ball - cos^3 distributed on the hemisphere
vec3 generate_cosine_cube_point_hemisphere(Sampler* sampler, const vec3& normal)
{
	vec3 in_ball = generate_uniform_point_ball(sampler);
	return normal + in_ball;
}

//2) Generates uniform points in the unit ball, with reflection - uniform distriubted on the hemisphere
vec3 generate_uniform_point_hemisphere(Sampler* sampler, const vec3& normal)
{
	vec3 in_ball = generate_uniform_point_ball(sampler);
	return dot(in_ball, normal) < 0.0f ? -in_ball : in_ball;
}

//3) Generates normal offset points on the unit ball - cos distributed on the hemisphere
vec3 generate_cosine_point_hemisphere(Sampler* sampler, const vec3& normal)
{
	vec3 on_ball = normalize(generate_uniform_point_ball(sampler));
	return normal + on_ball;
}

Theoretically, however, 1) and 3) are considerably harder to prove and intuitively reason about (beyond just blind trust that this is the case). Actually, the proof that they are correct requires further analysis knowledge than for generating the points through the cdf inversion method, so in a sense this is harder to theoretically show than what is presented in the third book. Anyways, if 1) is preferred, the estimator should be divided by a factor of 2/cos^2 if we want to get the target brdf.

The cosine factor that will show up in the estimator, if using uniform sampling, should also be taken into account. I would argue that is a good thing however, since it is currently implicit, because it cancels out with the pdf.

[petershirley]

I don't see any big advantage of 1 over 3.

[vchizhov]

The first variant was what was initially in the book, and the reason for the discrepancy between the first and third book (essentially it makes it so that the Lambertian material is actually a cos^2 material). The third variant is the correction of 1) which produces the desired cosine distribution over the hemisphere (as opposed to cos^3). I proposed 2) since it is trivial to reason about without requiring a university level mathematical background, while 1) and 3) may require some further mathematical knowledge (or at least I cannot think of a way to derive the results from intuition).