Convergence of algorithms for reconstructing convex bodies and directional measures

Abstract

We investigate algorithms for reconstructing a convex body K in ℝⁿ from noisy measurements of its support function or its brightness function in k directions u₁,…,u_k. The key idea of these algorithms is to construct a convex polytope P_k whose support function (or brightness function) best approximates the given measurements in the directions u₁,…,u_k (in the least squares sense). The measurement errors are assumed to be stochastically independent and Gaussian.

It is shown that this procedure is (strongly) consistent, meaning that, almost surely, P_k tends to K in the Hausdorff metric as k→∞. Here some mild assumptions on the sequence (u_i) of directions are needed. Using results from the theory of empirical processes, estimates of rates of convergence are derived, which are first obtained in the L₂ metric and then transferred to the Hausdorff metric. Along the way, a new estimate is obtained for the metric entropy of the class of origin-symmetric zonoids contained in the unit ball.

Similar results are obtained for the convergence of an algorithm that reconstructs an approximating measure to the directional measure of a stationary fiber process from noisy measurements of its rose of intersections in k directions u₁,…,u_k. Here the Dudley and Prohorov metrics are used. The methods are linked to those employed for the support and brightness function algorithms via the fact that the rose of intersections is the support function of a projection body.