One of the implicit assumptions in the definition of the BRDF is that it is measured and used at a scale larger than the scale of the largest variation in the underlying material. Thus, it makes some sense to measure the BRDF of granite for use in aerial sensing applications, where a single pixel sensor may record light reflected from many square meters of granite surface, but it does not make sense to use that same BRDF in trying to predict the appearance of a microscopic picture of granite. Suppose that we have modeled some object with local variation in appearance—a piece of paper with printing on it, or a flat metal tray with fingerprints around the rim—and we wish to make a picture of it from a distance so that the entire object will occupy just a few pixels on the imaging sensor. It’s natural to use MIP mapping for this.

(a) Argue why it is reasonable to average the spatially varying BRDF over a region of the surface to estimate a BRDF for the larger surface region, at least in the case of the paper and the flat metal tray.

(b) Argue that even in the case of a flat surface, it’s not generally reasonable to average the model parameters (such as the Phong exponent, or the CookTorrance specular color, or the index of refraction), and then use these averaged values to estimate the BRDF of the larger surface region.

(c) Suppose that your surface has a fairly constant BRDF (like the curved tile on a Spanish tile roof), but the underlying surface has substantial curvature at a smaller scale than one imager pixel (i.e., a Spanish tile rooftop that projects to just a few imager pixels). How would you compute a BRDF for the larger surface?