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June 2006 Convergence of algorithms for reconstructing convex bodies and directional measures
Richard J. Gardner, Markus Kiderlen, Peyman Milanfar
Ann. Statist. 34(3): 1331-1374 (June 2006). DOI: 10.1214/009053606000000335


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.


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Richard J. Gardner. Markus Kiderlen. Peyman Milanfar. "Convergence of algorithms for reconstructing convex bodies and directional measures." Ann. Statist. 34 (3) 1331 - 1374, June 2006.


Published: June 2006
First available in Project Euclid: 10 July 2006

zbMATH: 1097.52503
MathSciNet: MR2278360
Digital Object Identifier: 10.1214/009053606000000335

Primary: 52A20 , 62M30 , 65D15
Secondary: 52A21 , 60D05 , 60G10

Keywords: algorithm , brightness function , convex body , convex polytope , Cosine transform , directional measure , fiber process , geometric tomography , least squares , rose of intersections , set-valued estimator , stereology , support function , surface area measure

Rights: Copyright © 2006 Institute of Mathematical Statistics


Vol.34 • No. 3 • June 2006
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