Radon’s inversion formulas

Madych, W.

doi:10.1090/S0002-9947-04-03404-X

Radon’s inversion formulas
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by W. R. Madych

Trans. Amer. Math. Soc. 356 (2004), 4475-4491

DOI: https://doi.org/10.1090/S0002-9947-04-03404-X

Published electronically: January 16, 2004

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Abstract:

Radon showed the pointwise validity of his celebrated inversion formulas for the Radon transform of a function $f$ of two real variables (formulas (III) and (III$’$) in J. Radon, Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten, Ber. Verh. Sächs. Akad. Wiss. Leipzig, Math.-Nat. kl. 69 (1917), 262-277) under the assumption that $f$ is continuous and satisfies two other technical conditions. In this work, using the methods of modern analysis, we show that these technical conditions can be relaxed. For example, the assumption that $f$ be in $L^p(\mathbb {R}^2)$ for some $p$ satisfying $4/3<p<2$ suffices to guarantee the almost everywhere existence of its Radon transform and the almost everywhere validity of Radon’s inversion formulas.

References

J. Bourgain, Averages in the plane over convex curves and maximal operators, J. Analyse Math. 47 (1986), 69–85. MR 874045, DOI 10.1007/BF02792533
Stanley R. Deans, The Radon transform and some of its applications, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1983. MR 709591
S. Helgason, The Radon Transform, Birkhauser, Boston, 1980.
W. R. Madych, Summability and approximate reconstruction from Radon transform data, Integral geometry and tomography (Arcata, CA, 1989) Contemp. Math., vol. 113, Amer. Math. Soc., Providence, RI, 1990, pp. 189–219. MR 1108655, DOI 10.1090/conm/113/1108655
W. R. Madych, Tomography, approximate reconstruction, and continuous wavelet transforms, Appl. Comput. Harmon. Anal. 7 (1999), no. 1, 54–100. MR 1699598, DOI 10.1006/acha.1998.0258
F. Natterer, The mathematics of computerized tomography, B. G. Teubner, Stuttgart; John Wiley & Sons, Ltd., Chichester, 1986. MR 856916
J. Radon, Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten, Ber. Verh. Sächs. Akad. Wiss. Leipzig, Math. Nat. kl. 69 (1917), 262-277.
L. A. Shepp and J. B. Kruskal, Computerized tomography, the new medical X-ray technology, Amer. Math. Monthly 85, (1978), 420-439.
Kennan T. Smith, Donald C. Solmon, and Sheldon L. Wagner, Practical and mathematical aspects of the problem of reconstructing objects from radiographs, Bull. Amer. Math. Soc. 83 (1977), no. 6, 1227–1270. MR 490032, DOI 10.1090/S0002-9904-1977-14406-6
Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
A. Zygmund, Trigonometric series: Vols. I, II, Cambridge University Press, London-New York, 1968. Second edition, reprinted with corrections and some additions. MR 0236587