Convergence of algorithms for reconstructing convex bodies and directional measures

Abstract

We investigate algorithms for reconstructing a convex body K in R n 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 (ui) 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.While most of the paper is devoted to reconstruction of convex bodies, Section 9 focuses on a problem from stereology, that of reconstructing an unknown directional measure of a stationary fiber process from a finite number of noisy measurements of its rose of intersections. It turns out that the corresponding algorithm, Algorithm NoisyRoseLSQ, is very closely related to Algorithm NoisyBrightLSQ, due to the fact that the rose of intersections is the support function of a projection body. This fact was also used by Kiderlen [20], where an estimation method similar to Algorithm NoisyRoseLSQ was suggested and analyzed. Convergence of Algorithm NoisyRoseLSQ was proved by Männle [23], but also follows easily from our earlier results (see Proposition 9.1). With suitable extra assumptions, we can once again obtain estimates of rates of convergence of the approximating measures to the unknown directional measure. These are first given for the Dudley metric in Theorem 9.4, but can easily be converted to rates for the Prohorov metric. For example, for the Prohorov metric, the rate is of order k −1/20 in the three-dimensional case. 2. Definitions, notation and preliminaries. As usual, S n−1 denotes the unit sphere, B the unit ball, o the origin and · the norm in Euclidean nspace R n . It is assumed throughout that n ≥ 2. A direction is a unit vector, that is, an element of S n−1 . If u is a direction, then u ⊥ is the (n − 1)dimensional subspace orthogonal to u. If x, y ∈ R n , then x · y is the inner product of x and y and [x, y] denotes the line segment with ...

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Untitled

“…Algorithm NoisyCovBlaschke utilizes the known fact that −∂g K 0 (tu)/∂t, evaluated at t = 0, equals the brightness function value b K 0 (u), that is, the (n − 1)-dimensional volume of the orthogonal projection of K 0 in the direction u. This connection allows most of the work to be done by a very efficient algorithm, Algorithm NoisyBrightLSQ, designed earlier by Gardner and Milanfar (see [24]) for reconstructing an o-symmetric convex body from finitely many noisy measurements of its brightness function.…”