Extended Gaussian images, mixed volumes, shape reconstruction

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Abstract
The Extended Gaussian Image (EGI) of an object records the variation of surface area with surface orientation. The EGI is a unique representation for convex objects. For a polyhedron, each face is represented by its normal and its area. The inversion problem (from an EGI to a description in terms of vertices and faces) is solved for convex polyhedra, by providing an algorithm giving an iterative solution by a minimization[Little,1983]. The algorithm employs a geometric construction, the mixed volume, which was used in Minkowski's proof [1897] of the existence and uniqueness of an inverse. The mixed volume measures similarity of shape for convex objects.

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