Singularities of the Radon transform

Ramm, A. G.; Zaslavsky, A. I.

doi:10.1090/S0273-0979-1993-00350-1

Singularities of the Radon transform
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by A. G. Ramm and A. I. Zaslavsky PDF

Bull. Amer. Math. Soc. 28 (1993), 109-115 Request permission

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

I. M. Gel′fand, M. I. Graev, and N. Ya. Vilenkin, Generalized functions. Vol. 5: Integral geometry and representation theory, Academic Press, New York-London, 1966. Translated from the Russian by Eugene Saletan. MR 0207913
F. Natterer, The mathematics of computerized tomography, B. G. Teubner, Stuttgart; John Wiley & Sons, Ltd., Chichester, 1986. MR 856916, DOI 10.1007/978-3-663-01409-6
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, DOI 10.1090/mmono/081/05
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, DOI 10.1088/0266-5611/4/3/019
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
A. G. Ramm, Inversion of limited-angle tomographic data, Comput. Math. Appl. 22 (1991), no. 4-5, 101–111. Multidimensional inverse problems. MR 1127218, DOI 10.1016/0898-1221(91)90135-Q
A. G. Ramm, Numerical differentiation, Izv. Vysš. Učebn. Zaved. Matematika 1968 (1968), no. 11 (78), 131–134 (Russian). MR 0251904

Estimates of the derivatives of random functions

102

110

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
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, DOI 10.1016/0895-7177(93)90083-B

Asymptotic behavior of the Fourier transform of a piecewise smooth function

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, DOI 10.1006/jmaa.1993.1328
René Thom, Sur la théorie des enveloppes, J. Math. Pures Appl. (9) 41 (1962), 177–192 (French). MR 141041

Dual varieties and Legendre transforms

V. A. Zalgaller, Teoriya ogibayushchikh, Izdat. “Nauka”, Moscow, 1975 (Russian). MR 0394450, DOI 10.1016/0022-247X(84)90217-8