The Annals of Statistics

Convergence of algorithms for reconstructing convex bodies and directional measures

Richard J. Gardner, Markus Kiderlen, and Peyman Milanfar

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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.

Article information

Ann. Statist., Volume 34, Number 3 (2006), 1331-1374.

First available in Project Euclid: 10 July 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 52A20: Convex sets in n dimensions (including convex hypersurfaces) [See also 53A07, 53C45] 62M30: Spatial processes 65D15: Algorithms for functional approximation
Secondary: 52A21: Finite-dimensional Banach spaces (including special norms, zonoids, etc.) [See also 46Bxx] 60D05: Geometric probability and stochastic geometry [See also 52A22, 53C65] 60G10: Stationary processes

Convex body convex polytope 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


Gardner, Richard J.; Kiderlen, Markus; Milanfar, Peyman. Convergence of algorithms for reconstructing convex bodies and directional measures. Ann. Statist. 34 (2006), no. 3, 1331--1374. doi:10.1214/009053606000000335.

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  • Bourgain, J. and Lindenstrauss, J. (1988). Projection bodies. Geometric Aspects of Functional Analysis. Lecture Notes in Math. 1317 250--270. Springer, Berlin.
  • Bourgain, J. and Lindenstrauss, J. (1988). Distribution of points on spheres and approximation by zonotopes. Israel J. Math. 64 25--31.
  • Bronshtein, E. M. (1976). $\ee$-entropy of convex sets and functions. Siberian Math. J. 17 393--398.
  • Campi, S. (1988). Recovering a centred convex body from the areas of its shadows: A stability estimate. Ann. Mat. Pura Appl. (4) 151 289--302.
  • Campi, S., Colesanti, A. and Gronchi, P. (1995). Convex bodies with extremal volumes having prescribed brightness in finitely many directions. Geom. Dedicata 57 121--133.
  • Dudley, R. M. (1968). Distances of probability measures and random variables. Ann. Math. Statist. 39 1563--1572.
  • Dudley, R. M. (2002). Real Analysis and Probability, rev. reprint. Cambridge Univ. Press.
  • Fisher, N. I., Hall, P., Turlach, B. and Watson, G. S. (1997). On the estimation of a convex set from noisy data on its support function. J. Amer. Statist. Assoc. 92 84--91.
  • Fortet, R. and Mourier, E. (1953). Convergence de la répartition empirique vers la répartition théorique. Ann. Sci. École Norm. Sup. (3) 70 267--285.
  • Gardner, R. J. (1995). Geometric Tomography. Cambridge Univ. Press.
  • Gardner, R. J., Kiderlen, M. and Milanfar, P. (2005). Extended version of ``Convergence of algorithms for reconstructing convex bodies and directional measures.'' Available at
  • Gardner, R. J. and Milanfar, P. (2001). Shape reconstruction from brightness functions. In Proc. SPIE Conference on Advanced Signal Processing Algorithms Architectures and Implementations XI 4474 (F. T. Luk, ed.) 234--245. SPIE, Bellingham, WA.
  • Gardner, R. J. and Milanfar, P. (2003). Reconstruction of convex bodies from brightness functions. Discrete Comput. Geom. 29 279--303.
  • Gregor, J. and Rannou, F. R. (2002). Three-dimensional support function estimation and application for projection magnetic resonance imaging. Internat. J. Imaging Systems and Technology 12 43--50.
  • Groemer, H. (1996). Geometric Applications of Fourier Series and Spherical Harmonics. Cambridge Univ. Press.
  • Hall, P. and Turlach, B. (1999). On the estimation of a convex set with corners. IEEE Trans. Pattern Analysis and Machine Intelligence 21 225--234.
  • Hug, D. and Schneider, R. (2002). Stability results involving surface area measures of convex bodies. Rend. Circ. Mat. Palermo (2) Suppl. 70, part II 21--51.
  • Ikehata, M. and Ohe, T. (2002). A numerical method for finding the convex hull of polygonal cavities using the enclosure method. Inverse Problems 18 111--124.
  • Karl, W. C., Kulkarni, S. R., Verghese, G. C. and Willsky, A. S. (1996). Local tests for consistency of support hyperplane data. J. Math. Imaging Vision 6 249--267.
  • Kiderlen, M. (2001). Non-parametric estimation of the directional distribution of stationary line and fibre processes. Adv. in Appl. Probab. 33 6--24.
  • Lele, A. S., Kulkarni, S. R. and Willsky, A. S. (1992). Convex-polygon estimation from support-line measurements and applications to target reconstruction from laser-radar data. J. Opt. Soc. Amer. A 9 1693--1714.
  • Mammen, E., Marron, J. S., Turlach, B. A. and Wand, M. P. (2001). A general projection framework for constrained smoothing. Statist. Sci. 16 232--248.
  • Männle, S. (2002). Ein Kleinste--Quadrat--Schätzer des Richtungsmaßes für stationäre Geradenprozesse. Masters thesis, Univ. Karlsruhe, Germany.
  • Matoušek, J. (1996). Improved upper bounds for approximation by zonotopes. Acta Math. 177 55--73.
  • Pollard, D. (1984). Convergence of Stochastic Processes. Springer, New York.
  • Poonawala, A., Milanfar, P. and Gardner, R. J. (2006). Shape estimation from support and diameter functions. J. Math. Imaging Vision 24 229--244.
  • Prince, J. L. and Willsky, A. S. (1990). Reconstructing convex sets from support line measurements. IEEE Trans. Pattern Analysis and Machine Intelligence 12 377--389.
  • Schneider, R. (1993). Convex Bodies: The Brunn--Minkowski Theory. Cambridge Univ. Press.
  • Schneider, R. and Weil, W. (2000). Stochastische Geometrie. Teubner, Stuttgart.
  • Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and Its Applications, 2nd ed. Wiley, New York.
  • van de Geer, S. (1990). Estimating a regression function. Ann. Statist. 18 907--924.
  • van de Geer, S. (2000). Applications of Empirical Process Theory. Cambridge Univ. Press.