Formulating a scene probability equation to differentiate the effects of shape and albedo on image brightness

One goal of scene understanding is to find the shape and color of surfaces in a scene given a single image. This goal is complicated by the fact that given a particular distribution of illumination sources, local variations of image brightness may be due to local change in surface orientation with respect to the light source (shading), variation of the natural color (albedo) of the surface, or a combination of both. If we make a priori assumptions about the distribution of surface normals (the generic surface assumption) and also define the criterion for object segmentation as regions of similar albedo, and then maintain that shape and albedo are statistically independent then we should be able to determine exactly the pdf of albedos in the scene given the pdf for image brightness. However, this may prove difficult since image brightness is a complex function dependant upon distribution of illumination and derived from surface shape and surface albedo. A new method is presented to find the pdf of a derived random variable regardless of the complexity of the derivation function, subject only to its inverse being differentiable. Because this formulation is also linear, the process can be inverted to find an estimate for the pdf of albedo that is optimal in a least squares error sense. This information should in turn be helpful in attacking such tasks as object segmentation, shadow cancellation and albedo normalization.