Bidirectional reflectance distribution function (BRDF) encodes the behavior of light that “bounces off” surface. In particular, given incoming direction , how much light gets scattered in any outgoing direction .

Definition

In the most general case, light can enter some surface at a point and incident direction vector . and can leave the surface at some other point and exitant direction . The function define this relationship is called the BSSRDF (bidirectional surface scattering reflectance distribution function).

We can make additional assumption that and ignore subsurface scattering, in this case we have bidirectional reflectance distribution function (BRDF).

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 , that is something can be considered in a texture map.

Microfacet BRDF

See microfacet theory

See Also

Reference