Re: [Algorithms] Lighting (HDR)

[Matt P. <ma...@ph...>]

(below)

On Jul 25, 2009, at 12:00 PM, Fabian Giesen wrote:
> Joe Meenaghan wrote:
>>
>> 1. I have differing information about the normalization term for the
>> Blinn-Phong BRDF and I'd like to know which is correct. In Real-Time
>> Rendering the authors suggest (m + 8) / (8 * pi) based on a
>> derivation from Sloan and Hoffman. However, in Pharr and Humphrey
>> (Physically Based Rendering, p.446) they calculate (m + 2) / (2 *  
>> pi).
>> The definite integral solution demonstrated in the
>> latter appears reasonable to me, so I'm uncertain. Normally I've
>> associated the m+2 version as the normalization term for plain Phong,
>> yet, Pharr and Humphrey are very explicitly referring to the Blinn  
>> BRDF
>> and are using the half vector rather than the reflection vector in  
>> their
>> discussion, and they are convincing. Is there something I'm missing?

Unfortunately RTR doesn't seem to have a derivation, so it's hard to  
see precisely where the difference comes from.  One general note is  
that the PBR book is normalizing a microfacet distribution, but RTR is  
(from a quick skim) normalizing a BRDF.  So I think that the 2pi vs  
8pi difference in the denominator comes from the fact that 2pi works  
out to be the right denominator for a normalized microfacet  
distribution, but if you fold in the 1/4 term that comes in the  
Torrance-Sparrow BRDF (discussed on p442 of the PBR book), then that  
covers that difference.

The (m+2) vs (m+8) stuff in the numerator I don't have any insight  
on.  As far as I know the PBR derivation is correct and I just wrote a  
short program to verify the result numerically and it all came out as  
expected.  Hopefully Naty can chime in?

>
> I went through the same thing a few months ago. See the discussion in
> the comments here:
>
>   http://www.rorydriscoll.com/2009/01/25/energy-conservation-in-games/
>
> (Including a comment from Naty Hoffman who did the derivation  
> mentioned
> in RTR). The exact normalization factor obviously depends on what
> variant of the BRDF you use. I did the computation for typical  
> variants
> of Phong and Blinn-Phong here:
>
>   http://www.farbrausch.de/~fg/articles/phong.pdf

I think there is a bug in the Blinn-Phong normalization there.  In  
particular, I don't think that the first step of going from an  
integral over cos theta h to an integral of cos theta/2 is right (or  
needed--the microfacet distribution can be normalized fine in theta_h  
land.)

A related note is that the extra cos theta factor doesn't come from it  
being a BRDF, but comes from the geometry of microfacets--i.e. the  
projection of a microfacet with angle theta_h from the actual surface  
normal projects to a differential area scaled by cos theta_h on the  
actual surface.

>
> (also referenced in the discussion above). Hope that helps!
>
> Cheers,
> -Fabian "ryg" Giesen
>
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