I think Helmholtz reciprocity doesn't apply to diffuse light at all
That's where I get confused, because I've read in many sources (wikipedia is the easiest to cite
) that a physically plausible BRDF must obey reciprocity...
My Lambertian diffuse surface is
physically plausible by this definition, until I try maintain energy conservation by splitting the energy between diffuse/specular using Fresnel's law. This article
points out the same thing -- by maintaining energy conservation (making the diffuse darker when the spec is brighter), then it ruins the reciprocity.
So either everyone teaching that physically plausible BRDF's have to obey Helmholtz is wrong, or (Occam says: more likely
) my method of conserving energy is just a rough approximation...
The halfway vector is the same whether you calculate it from (L+V)/length(L+V) or (V+L)/length(V+L), and thus the microfacet distribution function returns the same value, since it relies on NDotH. Fresnel relies on LDotH...
In the case of my perfectly flat surface, the distribution term will always be 0 for all cases except where H==N, in which case the distribution will be 100%. I think this example makes a few edge cases more visible.
In all cases where N!=H, the fresnel term calculated from LDotH is meaningless as it ends up being multiplied by 0; these microfacets don't exist. But, say they did exist, this fresnel term tells us how much energy is reflected and refracted (then diffused/re-emitted
) for the sub-set of the total microfacets that are oriented towards H. I guess this means that to find out the total amount of refracted energy (energy available to the diffuse term
), we'd have to evaluate the fresnel term for every possible microfacet orientation weighted by probability.
In my example case, the flat plane, this is simple; there is only one microfacet orientation, so we don't have to bother doing any integration! We just use the fresnel term for LDotN, as 100% of the microfacets are oriented towards N. Suddenly, we've got a part of the BRDF that relies on L but not V... The calculation to find the total reflected vs refracted energy balance only depends on L, N and F(0º) and the surface roughness. Hence my dilemma -- how do we implement physically correct energy conservation in any BRDF without upsetting Helmholtz? Or, is Helmholtz more of a guideline than a rule in the realm of BRDFs? Or is there some important detail elsewhere in the rendering equation, outside of the BRDF, that I'm missing?
Helmholtz reciprocity isn't supposed to be correct for this process, because it's not just a single reflection
Are you saying it only breaks down because we're dealing with a composition of many different waves/particles instead of a single one? e.g. if we could track the path of one photon, it would obey the law, but when we end up with multiple overlapping probability distributions, we're no longer tracking individual rays so reciprocity has become irrelevant?
Edited by Hodgman, 26 February 2013 - 07:48 AM.