BRDF

BRDF stands for Bidirectional Reflectance Distribution Function. It describes how light is reflected at an opaque surface by providing a relationship between the incoming light direction wi and the outgoing light direction wo at a given point on the surface.

Formally, the BRDF fr(wi, wo) is defined so that the differential reflected radiance dLr(wo) equals fr(wi, wo) times the differential irradiance dEi from direction wi: dLr(wo) = fr(wi, wo) dEi. The incoming irradiance can be expressed as dEi = Li cos(theta_i) dA, where theta_i is the angle between wi and the surface normal. The BRDF has units of inverse steradians (1/sr) and is defined for all pairs of directions within the hemisphere above the surface. Important properties include reciprocity, fr(wi, wo) = fr(wo, wi), for stationary, return-symmetric materials, and energy conservation, which requires that the reflected energy does not exceed the incoming energy: integral over the hemisphere of fr(wi, wo) cos(theta_i) dwi ≤ 1.

Common models and use cases: the Lambertian model yields fr = ρ/π, representing ideal diffuse reflection with constant radiance in all directions. More realistic diffuse surfaces are described by models such as Oren-Nayar. Specular reflection is often captured by microfacet-based models like Cook-Torrance, which combine a distribution of microfacets, Fresnel terms, and geometric masking. The Phong model provides a simple specular highlight approximation but is not physically exact as a BRDF.

Measurement and applications: BRDFs are characterized with gonioreflectometers and are widely used in computer graphics, visual effects, remote sensing, and material science to simulate and analyze surface appearance. Limitations include the assumption of a surface point with fixed viewing and illumination directions and the exclusion of subsurface scattering and transmission.