The Bas-Relief Ambiguity

• Authors: Peter N. Belhumeur;David J. Kriegman;Alan L. Yuille
• Venue: CVPR '97 Proceedings of the 1997 Conference on Computer Vision and Pattern Recognition (CVPR '97)
• Year: 1997

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=\lambda f(x,y) +\mu x +\nu y with a corresponding transformation of the albedo. For each image of a surface f(x,y) produced by a light source, there exists an identical image of the 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 \mu=\nu=0 (e.g.\ a classical bas-relief sculpture), we show that the set of possible motion fields are also identical. Thus, neither small motions nor changes of illumination can resolve the bas-relief ambiguity. Implications of this ambiguity on structure recovery and shape representation are discussed.