Bidirectional reflectance distribution function (BRDF) encodes the behavior of light that “bounces off” surface. In particular, given incoming direction
Definition
In the most general case, light can enter some surface at a point
We can make additional assumption that
Definition
The BRDF at a point
is defined as the ratio of the differential radiance reflected in an exitant direction ( ), and the differential irradiance ( ) incident through a differential solid angle : where
is the cosine of the angle formed by the normal vector at the point , and the indecent direction vector .
Properties
- Range - The BRDF can take any positive value and can very with wavelength.
- Dimension - The BRDF is a four-dimensional function defined at each point on a surface; each two dimensions correspond to the incoming direction/outgoing direction
- Reciprocity - Helmholtz reciprocity
Because of the reciprocity property, the following notation is used for the BRDF: - Relation between incident and reflected radiance: TODO Advanced Global Illumination p33
- Energy conservation - The law of conservation of energy requires that the total amount of power reflected over all directions must be less than or equal to the total amount of power indecent on the surface
Example: Lambertian reflection
Lambertian reflection assume light is equally likely to be reflected in each output direction. In other words,
where is some constant. where
here is the irradiance. We often want to think in term between 0 and 1, so we sometimes use the albedo
where
Example: Perfect specular reflection
We will introduce some angles to describe specular reflection, where
and are the angles related to normal, and and (azimuthal angle) are the angles around the plane. And we can have the following distribution:
Where
is the Dirac delta function. In practice, we can’t find perfect specular reflection direction via random sampling and we simply pick the reflected direction.
Other Lighting Phenomenon to Consider
For refraction, we need to consider the Snell’s Law. Fresnel is also something to consider when doing reflection.
Another thing to consider is anisotropic reflection, where reflection depends on azimuthal angle
Microfacet BRDF
See Also
Reference
- “Advanced Global Illumination” 2.5
- Lecture 16: The Rendering Equation (CMU 15-462/662)