Reconstruction of Convex Bodies from Brightness Functions

Abstract. Algorithms are given for reconstructing an approximation to an unknown convex body from finitely many values of its brightness function, the function giving the volumes of its projections onto hyperplanes. One of these algorithms constructs a convex polytope with less than a prescribed number of facets, while the others do not restrict the number of facets. Convergence of the polytopes to the body is proved under certain essential assumptions including origin symmetry of the body. Also described is an oracle-polynomial-time algorithm for reconstructing an approximation to an origin-symmetric rational convex polytope of fixed and full dimension that is only accessible via its brightness function. Some of the algorithms have been implemented, and sample reconstructions are provided.

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