Why is H (blinn) used instead of R (phong) in specular shading?

Question

I can't find a good reason for this anywhere. The reflection vector used in phong has a simple basis in physics. But the half vector used in blinn seemingly has no rational basis, and does not constitute a proper reflection. And yet it is used in every so-called "physically based" shading function. If there is a good physical basis for it, I'd like to know.

What I've been able to find are a few reasons:

It's faster - there's mixed information on this, but even so it would have been a great reason... in the year 1998.

It handles angles higher than 90 degrees better - as far as I can tell the only reason for this is because the phong term has been used improperly. The dot product of the reflection and the view gives an angle between -1 and +1. Usually this angle is clamped to 0 to 1, this is the direct cause of the 90 degree problem. Re-normalize the angle instead of clamping it and you get the full 180 degree coverage. I refuse to believe a simple x * 0.5 + 0.5 operation has eluded the graphics world for 40 years.

it handles edges better - The edge "problem" also exists in the blinn solution, just to a lesser degree. The main cause is improper simulation of area lighting at the terminator, which should be essential for any "physically based" shader. But even in simpler situations a sigmoid function can approximate a soft terminator line correctly. Multiplying into a lambert term is incorrect as it attenuates the specular term improperly, this could cancel out a fresnel term and lead to further errors.

It has long reflections at the edge - It seems to me that while anisotropic reflections may be realistic, blinn is not the correct way to implement them, as they only appear at the edge. It is merely a happy coincidence that an error in the H term happens to look realistic.

None of these reasons are satisfactory, I want to sort out this madness.

I want to clarify that I am not talking about blinn and phong specifically, but instead about the vector components H and R, which are used as the basis for these shaders as well as others.

Answer 1

For perfectly reflective surfaces Phong-model makes sense. However, where does the n in (R.V)^n of Phong-model for approximating rougher surfaces come from? Where is the theory that you have to raise the result of the dot product to the power except that it just appears to empirically give the proper result?

For Blinn-model there's physically based microfacet theory to support all the components in the equation and there's also empirical evidence that the model approximates real world surfaces more closely (though not perfectly). The half-vector in Blinn model is used as an input to normal distribution function (NDF), which is an approximation how microfacets are distributed about surface normal as the function of the surface roughness. I.e. when H-vector points to the normal direction the value is highest since most microfacets point to that direction, and the probability is decreased accordingly when the angle between normal and H-vector is increased.

Blinn-model isn't perfect by any means though and it doesn't for example take the geometry term of the microfacet model into account (i.e. shadowing and masking of microfacets whose importance increases in grazing angles).

Answer 2

Actually, I think you yourself listed the reasons why Blinn is the default over Phong.

Each reason you listed there is, in fact, an area where Blinn proves superior to Phong.

Taken as a whole, all of these lead to Blinn being a better default than Phong.

Is Blinn perfect? Is it better than Phong?

No.

But it is a reasonable default. Feel free to substitute Phong for Blinn in any renderer/shader you write.

Answer 3

I have discovered the reason for the H vector to be used. Unfortunately it is not the way it is used in most shading models, which can then be concluded to be incorrect.

For physically based shading reflected light must obey the fresnel equations. (Most "physically based" shaders don't) Microfacets must also obey the fresnel equations, which relies on the angle of incidence for the light as well as the index of refraction of the interface to produce a correct result.

According to the law of reflection the angle of incidence must be mirrored with the angle of reflection along the surface normal. For a ray of light to have hit the camera - which we know it did - it must have been reflected from the light - that we know the direction of. Thus the surface normal must by deduction be the mirror axis for these two directions. This gives us the half vector H which is in between them. Calculated by normalizing the sum of both.

Now by calculating the angle between the light direction L and the half vector H we acquire the angle of incidence for the specular reflection of a microfacet, and can correctly attenuate it using the fresnel term.

Note that the view direction is equal to R for that microfacet, H is not a reflection term. Blinn, Cook, Torrance, and Sparrow can suck it. Phong and Fresnel were right.