Generalizing the definition of the bi-directional reflectance distribution function

The fundamental quantity that characterizes the reflectance property of a surface is the bi-directional reflectance distribution function (BRDF), defined as the ratio of the radiance scattered by a surface into a specified direction to the unidirectional irradiance incident on a surface. Its standard definition was derived under very restrictive conditions and it has only the angles of illumination and measurement as dependent variables. Several recent papers have attempted to generalize the BRDF to include the spatial attributes of illumination and measurement in order to make it applicable to heterogeneous media. The various BRDF definitions proposed by these papers are shown to be special cases of a generalized form of the BRDF derived herein. The spatial attributes of illumination and measurement are included as part of the nomenclature of the generalized BRDF. It is also shown that the generalized BRDF obeys reciprocity when properly weighted by the areas of illumination and measurement.