How is the maximum value for alpha (roughness == 1) decided for microfacet models?

Question

Recently I have been looking at pbrt, and was looking at their remapping from roughness to $\alpha$ values: https://github.com/mmp/pbrt-v3/blob/master/src/core/microfacet.h#L86

I understand that it is quite common to use some non-linear remapping between roughness values and $\alpha$ values to make the changes in roughness perceptually linear and in the range $[0, 1]$. To me this function looks like it could be some kind of polynomial approximation of the function:

$\alpha = 1.62142\cdot\sqrt{roughness}$.

(Here is a Desmos link to show the match - if anyone knows if this is correct and also knows how they came up with this approximation I would love to find out, as it doesn't seem like a standard Taylor series expansion). If this is correct, it would seem to be a pretty standard remapping, using the square of $\alpha$ to get roughness values, and ensuring that the maximum value of $\alpha$ (which in this case is $1.62142$) goes to $1$ in roughness.

What I'm really curious about is how this maximum value of $1.62142$ is chosen. In pbrt, $\alpha = \sqrt{2}\cdot(RMS slope)$:

https://pbr-book.org/3ed-2018/Reflection_Models/Microfacet_Models#eq:beckmann-isotropic ,

meaning the maximum RMS slope is $\approx 1.14652$. This seems like a very specific value for maximum roughness - where does this limit come from? RMS slope has infinite bounds as far as I'm aware - is there some physical reason for the limit or some standard that they are conforming to?

Answer 1

In truth, there is no mathematical maximum value for $\alpha$. As you noted, microfacet slope is unbounded, so in principle you could have arbitrarily large slope values and hence arbitrarily large $\alpha$. There's nothing wrong with that—the mathematics of the microfacet model keeps working fine.

As a practical matter, beyond a certain point you don't really see a lot of perceptual change in the appearance of the surface with increasing $\alpha$. As the roughness increases, the specular highlights become more broad and soft until the surface looks essentially Lambertian.

I don't know what is the origin of the value 1.62142 or that particular roughness remapping curve, but I'd guess that the maximum $\alpha$ is mainly chosen to give a visually useful range of roughness values, with the maximum being about at the point where it becomes indistinguishable to a diffuse surface. As this is a fuzzy aesthetic judgement, there may be no significance to the exact value.

Also, from a PBR standpoint, typical BSDFs only model single scattering (no inter-reflection between microfacets), but as roughness increases, multiple scattering becomes more important to the surface appearance. So at some point with high $\alpha$, the accuracy of the model breaks down and it ceases to be really physically based, unless you incorporate multiple scattering in the BSDF. Materials with complex micro-geometry may also be better modeled by volumetric microflake distributions rather than (or in addition to) surface BSDFs.