## When do Fresnel Equations Apply/Fresnel BRDFs

Practical and theoretical implementation discussion.
Geometrian
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### When do Fresnel Equations Apply/Fresnel BRDFs

As I understand it, when light hits a surface, the Fresnel equations apply. These equations depend on the complex-valued refractive index of the material, which itself depends on the incident light's wavelength.

However, for purely-diffuse surfaces, the results are also generally bad. Paper, wood, and various paints, for example, tend to have very low specular, yet this cannot be accounted for by the refractive index alone (since it is not large enough). Wikipedia states that the equations assume the interface is flat, planar, and homogeneous, and that the light is a plane wave. My suspicion is that the key word here is "flat". Surfaces have microgeometry. For these surfaces is the Right Thing to blend the Fresnel factor with zero empirically? From looking at commercial raytracing package tutorials, I think this solution might be what they do.

For metals, the refractive index varies significantly with wavelength, and the imaginary component is nontrivial. Would computing R, G, and B separately and using the full complex index of refraction help enough?

tarlack
Posts: 27
Joined: Mon Feb 10, 2014 7:48 am

### Re: When do Fresnel Equations Apply/Fresnel BRDFs

For the "flat" aspect : for pure metals and dielectrics, with beams with very small cross-sections such as the ones implicitly defined by rays, a planar square of 10*10 wavelengths^2 is in practice good enough for the "flat" condition to apply: in this case, when considering the results from wave optics, the main specular lobe is largely dominant compared to the secondary lobes (these lobes appear because of interferences, nothing in common with the lobes from BRDF-like models which model the macro effect of specular reflection on small disturbed surfaces). The maximum wavelength of the visible spectrum is roughly 800nm, so as long as your microgeometry is larger than 8 microns * 8 microns, Fresnel equations apply relatively well for each microgeometry.

For metals and the computations of separate values, even if I did not experience with it directly, that cannot harm to take into account the different refraction indices, as long as you've a clear definition of the R, G and B : these must be bands of the visible spectrum, not coordinates in a color space. In which case you will have to convert the input textures to this spectrum representation (with some deconvolution of the color-space values with respect to the response function of the color space they are encoded in, and hoping that metamerism will not cause too many problems), and the final colors will be obtained by convolving your 3-values spectrum with the specific response functions of the color space you want to use as output.

ypoissant
Posts: 97
Joined: Wed Nov 30, 2011 12:44 pm

### Re: When do Fresnel Equations Apply/Fresnel BRDFs

Geometrian wrote:However, for purely-diffuse surfaces, the results are also generally bad. Paper, wood, and various paints, for example, tend to have very low specular, yet this cannot be accounted for by the refractive index alone (since it is not large enough). Wikipedia states that the equations assume the interface is flat, planar, and homogeneous, and that the light is a plane wave. My suspicion is that the key word here is "flat". Surfaces have microgeometry. For these surfaces is the Right Thing to blend the Fresnel factor with zero empirically? From looking at commercial raytracing package tutorials, I think this solution might be what they do.
Within the microfacet-based BRDF theory, the Fresnel factor applies to each properly oriented microfacet. Here the microfacet is a facet of the micogeometry. Given this definition, the Fresnel factor can only be applied this way within a path tracing context.

In a ray tracing context, I found correctly applying the Fresnel factor is not trivial. The Fresnel factor has to be multiplied by the proportion of properly oriented microfacet surfaces within a differential area. And this function is not available AFAIK (although it is computable using Monte-Carlo techniques). Not doing that results in a Fresnel effect that becomes way too strong as surfaces get rougher. And to complicate matter a little bit more, real eroded rough surfaces tend to have their normal distribution vary with incident angle. So it is not even a matter of computing the proportion of properly oriented microfacet surfaces within a differential area but the proportion of properly oriented visible microfacet surfaces within a viewing volume within a differential cross-section area.

tarlack
Posts: 27
Joined: Mon Feb 10, 2014 7:48 am

### Re: When do Fresnel Equations Apply/Fresnel BRDFs

ypoissant wrote:The Fresnel factor has to be multiplied by the proportion of properly oriented microfacet surfaces within a differential area. And this function is not available AFAIK (although it is computable using Monte-Carlo techniques)
Isn't that given by the pdf associated with the microfacet distribution ? I must have missed something...

macnihilist
Posts: 8
Joined: Thu Mar 22, 2012 9:36 am

### Re: When do Fresnel Equations Apply/Fresnel BRDFs

I don't have much to add to the discussion, but I thought I'd throw in two recent papers on 'visible vs total' microfacet distribution that I found quite illuminating:
http://hal.inria.fr/hal-00942452
http://hal.inria.fr/hal-00996995/en
The second one proposes to importance sample according to the visible distribution instead of the 'total' D, which seems like a very good idea.

Maybe of interest for some people in this thread.

ypoissant
Posts: 97
Joined: Wed Nov 30, 2011 12:44 pm

### Re: When do Fresnel Equations Apply/Fresnel BRDFs

tarlack wrote:
ypoissant wrote:The Fresnel factor has to be multiplied by the proportion of properly oriented microfacet surfaces within a differential area. And this function is not available AFAIK (although it is computable using Monte-Carlo techniques)
Isn't that given by the pdf associated with the microfacet distribution ? I must have missed something...
It's a pdf, which can go from zero to infinity. What is needed is a probability that goes from 0 to 1.

tarlack
Posts: 27
Joined: Mon Feb 10, 2014 7:48 am

### Re: When do Fresnel Equations Apply/Fresnel BRDFs

Ah, ok. But does it vary so wildly that the proportion distribution can not be discretized in a relatively accurate way, with more direct methods than MC (gauss-legendre, ...) ?

ypoissant
Posts: 97
Joined: Wed Nov 30, 2011 12:44 pm

### Re: When do Fresnel Equations Apply/Fresnel BRDFs

I never attempted quadrature methods on BRDF normal distribution functions. I already have a set of Monte-Carlo functions for testing different properties of BRDFs in a BRDF unit testing file. Those functions output Excel spreadsheets. For me, it is easier to just add another one so I tend to think along this line. It is on my todo list to implement such function. But for now, I just very empirically scale the Fresnel term with the complement of the BRDF roughness.

junketsu
Posts: 2
Joined: Thu May 15, 2014 5:43 am

### Re: When do Fresnel Equations Apply/Fresnel BRDFs

To get nearly proper fresnel reflection of glossy surface is enough to multiply it by masking term of the Smiths Geometry Attenuation Factor (G).

An application of G is described in the original paper of Cook and Torrance and more particularly in Walters "Microfacet models for refraction through rough surfaces".