Theoretical evaluation of the impact of finite intervals in the measurement of the bidirectional reflectance distribution function

The bidirectional reflectance distribution function (BRDF) is a well-defined quantity that allows the bidirectional reflectance of surfaces to be described. However, the light propagation in a specific direction cannot be perfectly realized in practice, because the physical apertures are not infinitesimal but finite. Consequently, a BRDF measurement cannot be considered fully bidirectional, although the measure coincides with the BRDF within a certain confidence level. To properly understand the impact of the finite apertures on the BRDF measures, the deviation between the real BRDF and the BRDF to be obtained using real experimental conditions with finite apertures was theoretically studied for surfaces with realistic BRDFs. The biconical reflectance factor was used to estimate these "measured BRDFs" in different geometrical conditions, and a family of negative exponential functions was defined to assess the impact on surfaces with different angular scattering distributions. An expression for estimating the relative error from finite apertures is given, which considers the angular variation of the BRDF and the different solid angles involved in the measurement.