The Bas-Relief Ambiguity

When an unknown object with Lambertian reflectance is viewed orthographically, there is an implicit ambiguity in determining its 3-d structure: we show that the object's visible surface f(x, y) is indistinguishable from a “generalized bas-relief” transformation of the object's geometry, $ \bar f $ (x, y) = λf(x, y) + μx + νy, and a corresponding transformation on the object's albedo. For each image of the object illuminated by an arbitrary number of distant light sources, there exists an identical image of the transformed object illuminated by similarly transformed light sources. This result holds both for the illuminated regions of the object as well as those in cast and attached shadows. Furthermore, neither small motion of the object, nor of the viewer will resolve the ambiguity in determining the flattening (or scaling) λ of the object's surface. Implications of this ambiguity on structure recovery and shape representation are discussed.