Reconstruction of convex bodies from brightness functions
The brightness function of a convex body in \(\mathbb{R}^{n}\) gives the \((n-1)\)-dimensional volume of its orthogonal projections onto hyperplanes. By Aleksandrov's projection theorem any two origin-symmetric convex bodies which have same brightness function must be equal. The authors provide algorithms for reconstructing an approximation to an unknown origin-symmetric convex body from finitely many values of its brightness function. Different variants, including an oracle-polynomial time algorithm, are discussed, and all of these algorithms proceed in two phases. First the values of the brightness function are used to find the surface area measure of an approximating polytope, then the polytope itself is reconstructed from its area measure. The convergence of the polytope to the body is proved under certain assumptions on the body, and the proof also provides estimates on the rate of convergence. Implementations of the presented algorithms suggest that they can work even with noisy data.
scientific article; zbMATH DE number 1969088