Contents
Characteristics of a BRDF
So, what's a BRDF?
As I see it, it's an abstract tool that helps us to describe the macroscopic behavior of a material when photons hit this material. It's a convenient black box, a huge multi-dimensional lookup table (3, 4, or sometimes even 5, 6 dimensions when including spatial variations) that somehow encodes the amount of photons bouncing off the surface in a specific (outgoing) direction when coming from another specific (incoming) direction (and potentially, from another location).
It comes in many flavours
A Bidrectional Reflectance Distribution Function or BRDF is only a subset of the phenomena that happen when photons hit a material but there are plenty of other kinds of BxDFs:
- The BRDF only deals about reflection, so we're talking about photons coming from outside the material and scattered back to the outside the material as well.
- The BTDF (Transmittance) only deals about transmission of photons coming from outside the material and scattering inside the material (i.e. refraction).
- Note that the BRDF and BTDF only need to consider the upper or lower hemispheres of directions (which we call <math>\Omega</math>, or sometimes <math>\Omega_+</math> and <math>\Omega_-</math> if the distinction is required)
- The BSDF (Scattering) is the general term that encompasses both the BRDF and BTDF. This time, it considers the entire sphere of directions.
- Anyway, the BSDF, BRDF and BTDF are generally 4-dimensional as they make the (usually correct) assumption that both the incoming and outgoing rays interact with the material at a unique and same location.
- Also, the BSDF could be viewed as an incorrect term since it not only accounts for scattering but also for the other phenomenon happening to photons when they hit a material: absorption. This is because of absorption that the total integral of the BRDF for any outgoing direction is less than 1.
- The BSSRDF (Surface Scattering Reflectance) is a much larger model that also accounts for different locations for the incoming and outgoing rays. It thus becomes 5- or even 6-dimensional.
- This is an expensive but really important model when dealing with translucent materials (e.g. skin, marble, wax, milk, even plastic) where light diffuses through the surface to reappear at some other place.
- For skin rendering, it's an essential model otherwise your character will look dull and plastic, as was the case for a very long time in real-time computer graphics. Fortunately, there are many simplifications that one can use to remove 3 of the 6 original dimensions of the BSSRDF, but it's only recently that real-time methods were devised [1].
First, what's the color of a pixel?
Well, a pixel encodes what the eye or a CCD sensor is sensitive to: it's called radiance.
Radiance is the radiant flux of photons per unit area per unit solid angle and is written as <math>L(x,\omega)</math>. Its unit is the Watt per square meter per steradian (<math>W.m^{-2}.sr^{-1}</math>).
- <math>x</math> is the location where the radiance is evaluated, it's a 3D vector!
- <math>\omega</math> is the direction in which the radiance is evaluated, it's also a 3D vector but it's normalized so it can be written as a couple of spherical coordinates <math>\langle \phi,\theta \rangle</math>.
The radiant flux of photons –or simply flux– is basically the amount of photons/energy per amount of time.
And since we're considering a single CCD sensor element or a single photo-receptor in the back of the eye (e.g. cone):
- We only perceive that flux in a single location, hence the "per square meter". We need the flux flowing through an infinitesimal piece of surface (at least, the area of a rod or a cone, or the area of a single CCD sensor element).
- We only perceive that flux in a single direction, hence the "per steradian". We need the flux flowing through an infinitesimal piece of the whole sphere of directions (or at least the solid angle <math>d\omega</math> covered by the cone or single CCD sensor element as shown in the figure below).
So the radiance is this: the amount of photons per seconds flowing along a ray of solid angle <math>d\omega</math> and reaching a small surface <math>dA</math>. And that's what is stored in the pixels of an image.
A good source of radiance is one of those HDR cube maps used for Image Based Lighting (IBL): each texel of the cube map represents a piece of the photon flux reaching the point at the center of the cube map. It encodes the entire light field around an object and if you use the cube map well, your object can seamlessly integrate into the real environment where the cube map photograph was taken (thanks to our dear Paul Debevec) (ever noticed how movies before 1999 had poor CGI? And since his paper on HDR probes, it's a real orgy! 
).
But IBL is also very expensive: ideally, you would need to integrate each texel of the cube map and dot it with your normal and multiply it by some special function to obtain the perceived color of your surface in the view direction.
And guess what this special function is?
Well, yes! It's the BRDF and it's used to completely describe the behavior of radiance when it interacts with a material. Any material...
Mathematically
We're going to use <math>\omega_i</math> and <math>\omega_o</math> to denote the incoming and outgoing directions respectively. Each of these 2 directions are encoded in spherical coordinates by a couple of angles <math>\langle \phi_i,\theta_i \rangle</math> and <math>\langle \phi_o,\theta_o \rangle</math>. These only represent generic directions, we don't care if it's a view direction or light direction.
For example, for radiance estimates, the outgoing direction is usually the view direction while the incoming direction is the light direction. For importance estimates, it's the opposite.
Also note that we use vectors pointing toward the view or the light.
Irradiance
The integration of radiance arriving to a surface element <math>dA</math>, times <math>n.\omega_i</math> yields the irradiance (<math>W.m^{-2}</math>):
<math>E_r(x) = \int_\Omega dE_i(x,\omega_i) = \int_\Omega L_i(x,\omega_i) (n.\omega_i) \, d\omega_i</math>
It means that by summing the radiance (<math>W.m^{-2}.sr^{-1]</math>) coming from all possible directions, we get rid of the angular component (the <math>sr^{-1]</math> part).
Irradiance is the energy per unit surface. It's not very useful because what we need for our pixels is radiance, as we saw earlier.
Intuitively, we can imagine that we need to multiply that quantity by a value that will yield back a radiance. This value has the units of per steradian (<math>sr^{-1}</math>).
First try
So, perhaps we could include the BRDF in front of the irradiance integral and obtain a radiance like this:
<math>L_r(x,\omega_o) = f_r(x,\omega_o) \int_\Omega L_i(\omega_i) (n.\omega_i) \, d\omega_i</math>
Well, it can work for a few cases. For example, in the case of a perfectly diffuse reflector (Lambert model) then the BRDF is a simple constant <math>f_r(x,\omega_o) = \frac{\rho(x)}{\pi}</math> where <math>\rho(x)</math> is called the reflectance (or albedo) of the surface. The division by <math>\pi</math> is here to account for our "per steradian" need.
This is okay as long as we don't want to model materials that behave in a more complex manner. More complex materials certainly don't handle incoming radiance in the same way whatever the incoming direction! They must redistribute radiance in special and strange ways sometimes...
For example:
- Many materials imply a specular peak: a strong reflection of photons that tend to bounce off the surface almost in the direction perfectly symmetrical to the incoming direction (your typical mirror is the ultimate example).
- Also, many rough materials imply a Fresnel peak: a strong reflection of photons that arrive to the surface with glancing angles (fabrics are a good example of Fresnel effect)
That makes us come to a realization: the BRDF needs to be inside the integral and be dependent on the incoming direction <math>\omega_i</math>!
First try
But when we inject the BRDF into the integral, we obtain a new radiance :
<math>L_r(x,\omega_o) = \int_\Omega f_r(x,\omega_o,\omega_i) L_i(\omega_i) (n.\omega_i) \, d\omega_i</math>
Indeed, the expression of the BRDF (<math>sr^{-1}</math>) is:
<math>f_r(x,\omega_o,\omega_i) = \frac{dL_r(x,\omega_o)}{dE_i(x,\omega_i)}</math>
From the first equation of the irradiance, we can deduce that:
<math>dE_i(x,\omega_i) = L_i(x,\omega_i) (n.\omega_i) d\omega_i</math> (note that we simply removed the integral signs to get this)
We can then rewrite the BRDF as:
<math>f_r(x,\omega_o,\omega_i) = \frac{dL_r(x,\omega_o)}{L_i(x,\omega_i) (n.\omega_i) d\omega_i}</math>
It can be seen as the infinitesimal amount of reflected radiance by the infinitesimal amount of incoming irradiance.
We thus realize it's defined as an infinitesimal quantity (so we need to integrate it to obtain the final result) and it's expressed in <math>sr^{-1}</math> units.
The fundamental characteristics of a real material BRDf are:
- Reciprocity (a.k.a. Helmoltz principle), guaranteeing the BRDF returns the same value if <math>\omega_o</math> and <math>\omega_i</math> are reversed (i.e. view is swapped with light).
- Energy conservation, guaranteeing the total amount of reflected light is less or equal to the amount of incoming light.
Simplifications
Change of variables: http://www.cs.princeton.edu/~smr/papers/brdf_change_of_variables/
When the materials are isotropic, this change in variables allows to account for only 3 dimensions instead of 4.
MERL database.
Sources
I've been reading interesting papers from the Siggraph 2012 talk about Practical Physically Based Shading in Film and Game Production which is available here
Some ideas and generalizations/ideas/explanations are worth mentioning.
From "Background: Physics and Math of Shading (Naty Hoffman)" I learned that:
- metals completely absorb photons if they are not reflected specularly: metals have no diffuse components.
- Moreover, metals usually have a colored specular (due to RGB variations in the Fresnel reflections) while dielectric materials have a uniform specular and need only luminance encoding.
References
[1] "Screen-Space Perceptual Rendering of Human Skin" Jimenez et al. (2009) http://www.iryoku.com/sssss/