Abstract
Let f be a function in a Euclidean plane with compact support in a half disc H. The problem of reconstruction of the function from the data of its integrals over half circles A ⊂ H with centers at the diameter of H is studied. An explicit formula and a microlocal analysis of stability of the reconstruction are given.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Berenstein, C.A. and Tarabusi, E.C. (1992). On the Radon and Riesz transforms in the real hyperbolic space,Geometric Analysis, Contemp. Math.,140, 1–22.
Berenstein, C.A. and Tarabusi, E.C. (1993). Range of the k-dimensional Radon transform in real hyperbolic spaces,Forum Math.,5, 603–616.
Cormack, A.M. and Quinto, E.T. (1980). A Radon transform on spheres through the origin in ℝn and applications to the Darboux equation,Trans. Am. Math. Soc.,260, 575–581.
Courant, R. and Hilbert, D. (1962).Methods of Mathematical Physics II. Interscience Publ., New York.
Denisjuk, A.S. and Palamodov, V.P. (1988). Inversion de transformation de Radon d'apres les donée non-complétes,C.R. Acad. Sci. Paris,307, Ser. I, 181–183.
Fawcett, J.A. (1985). Inversion of N-dimensional spherical averages,SIAM J. Appl. Math.,45, 336–341.
Firbas, P. (1987). Tomography from seismic profiles, in:Seismic Tomography. Nolet, G., Ed., Reidel, 189–202/
Harger, R.O. (1970).Synthetic Aperture Radar Systems, Academic Press, New York.
Hellsten, H. and Andersson, L.E. (1987). An inverse method for the processing of synthetic aperture radar data,Inverse Problems. 3, 111–124.
John, F. (1960). Continuous dependence on data for solutions of partial differential equations with a prescribed bound,Comm. Pure Appl. Math.,13, 551–585.
Kostelyanets, P.O. and Reshetnyak, Yu.G. (1954). Determination of an additive function from its values on halfspaces,Uspekhi Mat. Nauk,9, 135.
Natterer, F. (1986).The Mathematics of Computerized Tomography, Teubner-Wiley, Stuttgart.
Palamodov, V.P. (1990). Some singular problems in Tomography, in:Mathematical Problems of Tomography, Translations of Mathematical Monographs,81,AMS, Rhode Island, 123–140.
Palamodov, V.P. (1991). Inversion formulas for the three-dimensional ray transform, inLecture Notes in Mathematics,1497, 53–61.
Palamodov, V.P. (1998). Reconstruction from line integrals in spaces of constant curvature,Math. Nachr.,196, 167–188.
Quinto, E.T. (1993). Singularities of the X-ray transform and limited data tomography in ℝ2 and ℝ3,SIAM J. Math. Anal.,24(5), 1215–1225.
Romanov, V.G. (1969). A problem of integral geometry and a linearized inverse problem for a hyperbolic equation,Siberian Math. J.,10(6), 1011–1018.
Romanov, V.G. (1969).Nekotorye obratnye zadachi dlja uravnenij giperbolicheskogo tipa. Nauka, Novosibirsk (Russian); English translationIntegra Geometry and Inverse Problems for Hyperbolic Equations. (1974), Springer-Verlag, Berlin.
Tarantola, A. (1987). Inversion of travel times and seismic waveforms, inSeismic Tomography. Nolet, G., Ed., Reidel, 135–157.
Trampert, J. (1998). Global seismic tomography: the inverse problem and beyond,Inverse Problems,14, 371–385.
Author information
Authors and Affiliations
Additional information
Communicated by Eric Todd Quinto
Rights and permissions
About this article
Cite this article
Palamodov, V.P. Reconstruction from limited data of arc means. The Journal of Fourier Analysis and Applications 6, 25–42 (2000). https://doi.org/10.1007/BF02510116
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02510116