Re: [Algorithms] approximation to pow(n,x)?

[Nathaniel H. <na...@io...>]

> What are you using this for? If it's lighting or shading, there's no
> real reason to use a power function on the cosine term to get tighter
> specular highlights. It's just a shaping function that people use
> because it's easy to control and AFAICT it has no physical basis in
> defining a BRDF from the micropolygon point of view.
>
> - Robin Green.

It's true that there is no reason to _exactly_ match a cosine power, and I
agree with Robin that for this purpose, any curve which vaguely resembles
cosine power will do.

As a side note, I just wanted to say a few words in the defense of the
lowly cosine power - it's not completely physically meaningless. If you
are using Blinn-Phong (N dot H, which is much to be preferred over
original Phong - R dot L), then using a cosine power is equivalent to
assuming that the microfacet normal distribution follows a cosine power
curve.

Now, there is no physical reason to assume that microfacet distributions
necessarily follow a cosine power curve, but (except for very low powers)
this curve very closely matches one that _does_ have a physical basis -
the Beckmann distribution (the one used in the Cook-Torrance BRDF). The
match is amazingly close considering that Bui-Tong Phong just eyeballed
the function; he didn't do any curve fitting. BTW, Beckmann's behavior for
very low powers is interesting - it stops behaving like a Gaussianish blob
and starts turning inside-out (which makes sense when you look at the
definition of the "m" parameter).

Beckmann isn't the last word on microfacet distributions; the EGSR 2007
paper "Microfacet Models for Refraction through Rough Surfaces" (which is
a great paper overall and well worth reading for anyone interested in
microfacet BRDFs) makes a good case for a different curve, with a more
gradual falloff. Which again supports Robin's original point.

Naty Hoffman