In the Proceedings of CVPR 97 . The Bas - Relief Ambiguity

Since antiquity, artisans have created attened forms, often called \bas-reliefs," which give an exaggerated perception of depth when viewed from a particular vantage point. This paper presents an explanation of this phenomena, showing that the ambiguity in determining the relief of an object is not con ned to bas-relief sculpture but is implicit in the determination of the structure of any object. Formally, if the object's true surface is denoted by ztrue = f(x; y), then we de ne the \generalized bas-relief transformation" as z = f(x; y) + x+ y; with a corresponding transformation of the albedo. For each image of a Lambertian surface f(x; y) produced by a point light source at in nity, there exists an identical image of a bas-relief produced by a transformed light source. This equality holds for both shaded and shadowed regions. Thus, the set of possible images (illumination cone) is invariant over generalized bas-relief transformations. When = = 0 (e.g. a classical bas-relief sculpture), we show that the set of possible motion elds are also identical. Thus, neither small unknown motions nor changes of illumination can resolve the bas-relief ambiguity. Implications of this ambiguity on structure recovery and shape representation are discussed.