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

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

Jon,

It is inarguable that H produces much more realistic results - a simple
observation of light streaks on wet roads and similar scenes with a
comparison to renderings of the two formulations proves this without a
doubt.

However, there are also good, fundamental, theoretical reasons to prefer
H. I seem to be explaining this poorly - I will give it another try, but
first, here are some pointers to other explanations:

There is a good diagram illustrating the difference in behavior of the two
vectors in Figure 7 of this paper:
http://people.csail.mit.edu/addy/research/ngan05_brdf_eval.pdf.

There is also some discussion about it in Appendix A of this paper:
http://graphics.stanford.edu/courses/cs448-05-winter/Schilling-1997-TechRep.pdf

"Real-Time Rendering, 3rd edition" also has some discussion of it on pages
249-251. If you don't have a copy of the book, you can "look inside" at
Amazon:
http://www.amazon.com/Real-Time-Rendering-Third-Tomas-Akenine-Moller/dp/1568814240,
click on "search inside this book", look for "half vector" (in quotes) -
you will get  a link to page 249.

OK, now I'll have another go As we both agree, the reflection vector is
fundamental for a perfectly flat mirror. Imagine a directional or point
light shining on the mirror. There are only visible reflections when V ==
R(L, N) (view vector is equal to the reflection of the light vector around
the surface normal).

Now how should we treat a surface which is not perfectly flat? A good
model (which comes to us from fields outside graphics but has been very
successful in graphics) is to treat such a surface as a statistical
collection of stochastically-oriented perfect mirrors, each one too tiny
to be individually visible. A useful description of such a surface for
purposes of rendering is a normal distribution function, or NDF, which
gives the statistical distribution of the microfacet normals relative to
the overall macroscopic normal.

Given a light direction L and a view direction V, how bright will we
observe the surface to be? Let's assume for simplicity that each of these
mirrors is 100% reflective at all angles (silver comes close to that).
Then it is clear that the brightness is proportional to the percentage of
microfacets from which there are visible reflections, in other words those
for which V == R(L, N_u) (here I use N_u for the microfacet normal to
distinguish from the overall surface normal N).

It is simple to demonstrate that this is equivalent to N_u == H. Therefore
we should "plug" H into the microfacet distribution function, which yields
the (N dot H) formulation for isotropic surfaces.

I can think of no similarly-principled way to derive the reflection vector
formulation, and none has appeared in the literature.

I hope this has convinced you that the H formulation is superior to R both
in terms of realism and theoretical soundness.

Thanks,

Naty Hoffman

> But that's equally true for the reflection vector! If all the
> micro-mirrors were perfectly flat, then an infinite specular power
> would be applied, and you'd get a perfect reflection of the lighting
> environment -- in fact, this is what environment mapping gives you.
>
> As the mirrors start deviating from the perfectly flat state, the
> specular power would decrease, and the specular reflection area would
> grow in size. I don't see how you can say that the half-angle
> formulation is more meaningful. We're still talking about reflected
> light. In the perfectly reflected case, clearly the reflection vector
> is 100% meaningful and accurate, and any other formulation would be
> less meaningful. I don't see how "meaningfulness" would change as
> smoothness goes from 100% to 99.9% or 95% or 50%.
>
> I do agree that the math gives you a different assumed microfacet
> distribution in the case of the reflection formulation versus the
> half-angle formulation. Both are approximations, of course. However,
> what I don't get, is why the reflection vector approximation is
> considered so inferior to the more expensive half-angle vector
> approximation. Does it have anything to do with the space integral of
> the reflection cone formed by the vector in question spread out by the
> power function? If so, how?
>
> Sincerely,
>
> jw
>
>
> On Sun, Nov 8, 2009 at 9:41 AM, Nathaniel Hoffman <na...@io...> wrote:
>
>> The half-angle formulation is not just more physically correct than the
>> reflection-vector formulation, it is fundamentally more meaningful.
> ...
>> The half-vector comes from microfacet theory. Imagine that the surface
>> is
>> actually a large collection of tiny flat mirrors when viewed under
>> magnification. Recall that a mirror only reflects light in the
>> reflection
>> direction. For given light vector L and view vector V, only mirrors
>> which
>
>
> --
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> standards for the benefit of people in the rest of the world.
> Nevertheless, whether we get there willingly or not, we shall soon
> have lower consumption rates, because our present rates are
> unsustainable.
>
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