Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

 

 

Singularities of the Radon transform


Authors: A. G. Ramm and A. I. Zaslavsky
Journal: Bull. Amer. Math. Soc. 28 (1993), 109-115
MSC: Primary 44A12; Secondary 53C65, 92C55
MathSciNet review: 1168516
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Singularities of the Radon transform of a piecewise smooth function $ f(x)$, $ x \in {R^n}$, $ n \geq 2$, are calculated. If the singularities of the Radon transform are known, then the equations of the surfaces of discontinuity of $ f(x)$ are calculated by applying the Legendre transform to the functions, which appear in the equations of the discontinuity surfaces of the Radon transform of $ f(x)$; examples are given. Numerical aspects of the problem of finding discontinuities of $ f(x)$, given the discontinuities of its Radon transform, are discussed.


References [Enhancements On Off] (What's this?)

  • [GGV] I. M. Gel′fand, M. I. Graev, and N. Ya. Vilenkin, Generalized functions. Vol. 5: Integral geometry and representation theory, Translated from the Russian by Eugene Saletan, Academic Press, New York-London, 1966. MR 0207913
  • [N] F. Natterer, The mathematics of computerized tomography, B. G. Teubner, Stuttgart; John Wiley & Sons, Ltd., Chichester, 1986. MR 856916
  • [P] V. P. Palamodov, Some singular problems in tomography, Mathematical problems of tomography, Transl. Math. Monogr., vol. 81, Amer. Math. Soc., Providence, RI, 1990, pp. 123–140. MR 1104016
  • [Q] Eric Todd Quinto, Tomographic reconstructions from incomplete data—numerical inversion of the exterior Radon transform, Inverse Problems 4 (1988), no. 3, 867–876. MR 965651
  • [R1] A. G. Ramm, Random fields estimation theory, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 48, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1990. MR 1103995
  • [R2] A. G. Ramm, Inversion of limited-angle tomographic data, Comput. Math. Appl. 22 (1991), no. 4-5, 101–111. Multidimensional inverse problems. MR 1127218, 10.1016/0898-1221(91)90135-Q
  • [R3] A. G. Ramm, Numerical differentiation, Izv. Vysš. Učebn. Zaved. Matematika 1968 (1968), no. 11 (78), 131–134 (Russian). MR 0251904
  • [R4] -, Estimates of the derivatives of random functions, J. Math. Anal. Appl. 102 (1984), 244-250; (with T. Miller) 110 (1985), 429-435.
  • [R5] A. G. Ramm, Multidimensional inverse scattering problems, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 51, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1992. MR 1211417
  • [RZ1] A. G. Ramm and A. I. Zaslavsky, Reconstructing singularities of a function from its Radon transform, Math. Comput. Modelling 18 (1993), no. 1, 109–138. MR 1245197, 10.1016/0895-7177(93)90083-B
  • [RZ2] -, Asymptotic behavior of the Fourier transform of a piecewise smooth function, Technion, preprint 1992.
  • [RSZ] A. G. Ramm, A. M. Steinberg, and A. I. Zaslavsky, Stable calculation of the Legendre transform, J. Math. Anal. Appl. 178 (1993), no. 2, 592–602. MR 1238898, 10.1006/jmaa.1993.1328
  • [T] René Thom, Sur la théorie des enveloppes, J. Math. Pures Appl. (9) 41 (1962), 177–192 (French). MR 0141041
  • [Z] A. I. Zaslavsky, Dual varieties and Legendre transforms, submitted.
  • [Z1] V. A. Zalgaller, Teoriya ogibayushchikh, Izdat. “Nauka”, Moscow, 1975 (Russian). MR 0394450

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC: 44A12, 53C65, 92C55

Retrieve articles in all journals with MSC: 44A12, 53C65, 92C55


Additional Information

DOI: https://doi.org/10.1090/S0273-0979-1993-00350-1
Article copyright: © Copyright 1993 American Mathematical Society