Re: approximation to pow(n,x)?

The progression Jon mentioned between a rough surface and a smooth one is
interesting to me though. The use of the half-vector does break down for
very smooth surfaces, it seems. It could be worth considering what that
means exactly.

I just finished a game where we used cube maps for specular lighting
contributions from the sun & sky. Several cube images were used, each
"pre-blurred" to a set specular power around the reflection vector.
Half-vector was not an option I think (every pixel in the map is a light
source). Is there a way to get this "half-vector-like" behavior out of a
cube lookup? Feels like the answer is no...

On Wed, Nov 11, 2009 at 12:31 PM, Nathaniel Hoffman <[email protected]> wrote:

> 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 <[email protected]> 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
> >
> >
> > --
> > Americans might object: there is no way we would sacrifice our living
> > 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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-- 
Jeff Russell
Engineer, 8monkey Labs
www.8monkeylabs.com

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