Mathematical foundations for orientation based shape representation are reviewed. Basic tools include support function, mixed volume, vector addition, Blaschke addition, and the corresponding decompositions, as well as some basic facts about convex bodies, are presented. Results on several types of curvature measures such as spherical images, m-th order area functions are summarized. As a case study, the EGI approach is examined to see how the classical results on Minkowski's problem are utilized in computational vision. Finally, results on Christoffel's problem are surveyed, including constructive proofs.
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