The bas-relief ambiguity

Since antiquity, artisans have created flattened 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 confined 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 z_true =f(x, y), then we define the “generalized bas-relief transformation” as z=λf(x, y)+μx+νy, with a corresponding transformation of the albedo. For each image of a Lambertian surface f(x, y) produced by a point light source at infinity, 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 μ=ν=0 (e.g. a classical bas-relief sculpture), we show that the set of possible motion fields 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