bidirectional reflectance distribution function (BRDF)

[7.3.3.17] f_r^*
The ratio of the differential luminance of a ray dL_r(θ_r,φ_r) reflected in a given direction (θ_r,φ_r) to the differential luminous flux density dE_i(θ_i,φ_i) incident from a given direction (θ_i,φ_i) that produces it.
Units: sr^-1

f_{r}\left (\theta _{i},\phi _{i}; \theta _{r},\phi _{r} \right ) \equiv \frac{dL_{r}\left ( \theta _{r},\phi _{r} \right )}{dE_{i}\left ( \theta _{i},\phi _{i} \right )} = \frac{dL_{r}\left ( \theta _{r},\phi _{r} \right )}{L_{i}\left ( \theta _{i},\phi _{i} \right )d\Omega _{i}} ,

where dΩ ≡ dω · cos θ

Notes:

(i) This distribution function is the basic parameter for describing (geometrically) the reflecting properties of an opaque surface element (negligible internal scattering).

(ii) It may have any positive value and will approach infinity in the specular direction for ideally specular reflectors.

(iii) The spectral and polarization aspects must be defined for complete specification, since the BRDF as given above only defines the geometric aspects.

^{* Nicodemus FE, et al. Geometrical Considerations and Nomenclature for Reflectance, NBS Monograph 160. Washington, DC: U.S. Department of Commerce; Oct 1977.}