Overview

The Bi-directional Reflectance Distribution Function (BRDF) is described by Jensen (2001, p. 19) as a tool for describing the reflection of light at a surface. Fundamentally, it is a simplification of the Bi-directional Surface Scattering Reflectance Distribution Function (BSSRDF) model that reduces the BSSRDF model from 8 to 6 dimensions and as such significantly lowers the computation time required to evaluate.

Essentially, while the BRDF accounts for light that enters and leaves a material at the same point, the BSSRDF model accounts for the incoming direction of light at one position and the outgoing direction of light from a different position i.e. light enters at one point before scattering below the surface and being radiated from a separate point (or points). This allows for the calculation of the transport of light below the surface of a material. The BRDF model results in a surface looking distinctly computer-generated due to it not blending surface features such as colour and geometry (Jensen et al. 2001, p. 511). It is relatively easy to notice the difference as a surface calculated using BSSRDF has a much softer look. This can be seen Figure 1:

Figure 1: The image on the top has skin generated using the BRDF model while the one on the bottom uses the BSSRDF model.

Theory

BRDF is defined by the differential reflected radiance at a point over the outgoing differential irradiance from the same point. Where radiance is defined as radiant energy emitted per unit time in a specified direction by a unit area of an emitting surface and irradiance is defined as the density of radiation incident on a given surface.

So the formula as taken from Jensen (2001, p. 20):

fr(x, w’, w) = dLr(x,w)/dEi(x,w’)

Relates to the change in the amount of light being reflected (dLr) from point x in direction w to the change in the amount of light being received (dEi) at point x arriving from direction w’. Note: Both w and w’ are facing away from the surface.

BSSRDF on the other hand is defined by the differential reflected radiance at a point over the differential incident radiant flux at a point providing secondary illumination. Where radiant flux is the rate of flow of radiant energy over time (the raw stream of photons)

So the formula as taken from Jensen (2001, p. 19):

fr(x, w, x’, w’) = dLr(x,w)/d Fi(x’,w’)

Relates to change in the amount of light being reflected (dLr) from point x in direction w over the change in the raw stream of photons arriving at point x’ from direction w’ (where w’ is pointing towards the source). The BSSRDF consts of both a single scattering (where light only bounces once before leaving the surface) and a multiple scattering term derived using a statistical dipole point source diffusion approximation.