Abstract.
We consider the problem of reconstructing a planar convex set from noisy observations of its moments. An estimation method based on pointwise recovering of the support function of the set is developed. We study intrinsic accuracy limitations in the shape–from–moments estimation problem by establishing a lower bound on the rate of convergence of the mean squared error. It is shown that the proposed estimator is near–optimal in the sense of the order. An application to tomographic reconstruction is discussed, and it is indicated how the proposed estimation method can be used for recovering edges from noisy Radon data.
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Ang, D.D., Gorenflo, R., Le, V.K., Trong, D.D.: Moment Theory and Some Inverse Problems in Potential Theory and Heat Conduction. Lecture Notes in Mathematics 1792, Springer, Berlin, 2002
Akhiezer, N.I.: The Classical Moment Problem and Some Related Questions in Analysis. Oliver and Boyd, Edinburgh, 1965
Brandolini, L., Rigoli, M., Travaglini, G.: Average decay of Fourier transforms and geometry of convex sets. Rev. Mat. Iberoamericana 14, 519–560 (1998)
Candés, E.J., Donoho, D.L.: Recovering edges in ill–posed inverse problems: optimality of curvlet frames. Ann. Statist. 30, 784–842 (2002)
Erdélyi, A., Magnus, W., Oberhettinger, F., Tricomi F.G.: Higher transcendental functions. Vol. II. Based, in part, on notes left by Harry Bateman. McGraw-Hill, New York, 1953
Fisher, N.I., Hall, P., Turlach, B.A., Watson, G.S.: On the estimation of a convex set from noisy data on its support function. J. Am. Statist. Assoc. 92, 84–91 (1997)
Gardner, R.: Geometric Tomography. Cambridge University Press, Cambridge, 1995
Golub, G.H., Milanfar, P., Varah, J.: A stable numerical method for inverting shape from moments. SIAM J. Sci. Comput. 21, 1222–1243 (1999)
Gustafsson, B., He, C., Milanfar, P., Putinar, M.: Reconstructing planar domains from their moments. Inverse Problems 16, 1053–1070 (2000)
Johnstone, I.M., Silverman, B.: Speed of estimation in positron emission tomography and related inverse problems. Ann. Statist. 18, 251–280 (1990)
Korostelev, A., Tsybakov, A.: Minimax Theory of Image Reconstruction. Lecture Notes in Statist. 82, New York, Springer, 1993
Logan, B., Shepp, L.: Optimal reconstruction of a function from its projections. Duke Math. J. 42, 645–659 (1975)
Milanfar, P., Karl, W.C., Willsky, A.S.: A moment–based variational approach to tomographic reconstruction. IEEE Trans. Image Processing 5, 459–470 (1996)
Milanfar, P., Verghese, G.C., Karl, W.C., Willsky, A.S.: Reconstructing polygons from moments with connections to array processing. IEEE Trans. Signal Processing 43, 432–443 (1995)
Natanson, I.P.: Constructive Function Theory. Gostechizdat, Moscow (in Russian), 1949
Pawlak, M.: On the reconstruction aspects of moments descriptors. IEEE Trans. Inform. Theory 38, 1698–1708 (1992)
Prince, J.L., Willsky, A.S.: Reconstructing convex sets from support line measurements. IEEE Trans. Pattern Anal. Machine Intell. 12, 377–389 (1990)
Quinto, E.T.: Singularities of the X-ray transform and limited data tomography in ℝ2 and ℝ3. SIAM J. Math. Anal. 24, 1215–1225 (1993)
Ramm, A.G., Katsevich, A.I.: The Radon Transform and Local Tomography. CRC Press, Boca Raton, 1996
Talagrand, M.: Sharper bounds for Gaussian and empirical processes. Ann. Probab. 22, 28–76 (1994)
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Mathematics Subject Classification (2000): 62C20, 62G20, 94A12
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Goldenshluger, A., Spokoiny, V. On the shape–from–moments problem and recovering edges from noisy Radon data. Probab. Theory Relat. Fields 128, 123–140 (2004). https://doi.org/10.1007/s00440-003-0303-1
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DOI: https://doi.org/10.1007/s00440-003-0303-1