Brown Applied Mathematics Pattern Theory and Vision Seminar
Abstract: Since antiquity, artisans have created flattened forms called bas-reliefs which give an exaggerated perception of depth when viewed from a particular vantage point. This talk gives 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. The fact that a surface and a generalized bas-relief transformation of the surface produce the same set of images arises from an implicit duality. For each image of a surface f(x, y) produced by a light source S, there exists an identical image of the bas-relief produced by a transformed light source \bar{S}. This equality holds for both shaded and shadowed regions. Thus, the set of possible images (illumination cone) is invariant over generalized bas-relief transformations. The generalized bas-relief transformation is the only transformation of the surface for which this ambiguity exists. Furthermore, 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 will be discussed.
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