Document Type

Article

Publication Date

6-2006

Keywords

Convex body, support function, brightness function, surface area measure, least squares, set-valued estimator, cosine transform, algorithm, geometric tomography, stereology, fiber process, directional measure, rose of intersections

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, . . . , uk. The key idea of these algorithms is to construct a convex polytope Pk whose support function (or brightness function) best approximates the given measurements in the directions u1, . . . , uk (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, Pk 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, . . . , uk. 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.

Publication Title

The Annals of Statistics

Volume

34

Issue

3

First Page

1331

Last Page

1374

Required Publisher's Statement

© Institute of Mathematical Statistics, 2006

Stable URL: http://www.jstor.org/stable/25463460

Subjects - Topical (LCSH)

Convex bodies; Convex polytopes; Surfaces--Areas and volumes; Dimension theory (Topology)

Genre/Form

articles

Type

Text

Rights

Copying of this document in whole or in part is allowable only for scholarly purposes. It is understood, however, that any copying or publication of this document for commercial purposes, or for financial gain, shall not be allowed without the author’s written permission.

Language

English

Format

application/pdf

Included in

Mathematics Commons

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