Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Inversion formula and singularities of the solution
for the backprojection operator in tomography


Author: A. G. Ramm
Journal: Proc. Amer. Math. Soc. 124 (1996), 567-577
MSC (1991): Primary 44A15, 45P05
DOI: https://doi.org/10.1090/S0002-9939-96-03155-3
MathSciNet review: 1301044
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $R^\ast \mu := \int _{S^2} \mu (\alpha , \alpha \cdot x) d\alpha $, $x \in {\mathbb{R}}^n$, be the backprojection operator. The range of this operator as an operator on non-smooth functions $R^\ast : X:=L^\infty _0 (S^{n-1} \times {\mathbb{R}}) \to L_{\mathrm{loc}}^2 ({\mathbb{R}}^n)$ is described and formulas for $(R^\ast )^{-1}$ are derived. It is proved that the operator $R^\ast $ is not injective on $X $ but is injective on the subspace $X_e$ of $X$ which consists of even functions $\mu (\alpha , p) = \mu (-\alpha , -p)$. Singularities of the function $(R^\ast )^{-1} h$ are studied. Here $h$ is a piecewise-smooth compactly supported function. Conditions for $\mu $ to have compact support are given. Some applications are considered.


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

  • BE H. Bateman and A. Erdelyi, Tables of integral transforms, McGraw-Hill, New York, 1954, MR 15:868a.
  • FWZ V. Faber, M. Wing, and J. Zahrt, Pseudotomography, LANL (1993 manuscript).
  • G F. Gakhov, Boundary value problems, Pergamon Press, Oxford, 1966, MR 33:6311.
  • GGV I. Gelfand, M. Graev, and N. Vilenkin, Integral geometry and representation theory, Academic Press, New York, 1965, MR 34:7726.
  • H L. Hörmander, Analysis of linear partial differential operators, Springer-Verlag, New York, 1983--1985, MR 85g:35002b; MR 85g:35002a.
  • He A. Hertle, On the range of the Radon transform and its dual, Math. Ann. 267 (1984), 91--99, MR 86e:44004b.
  • N F. Natterer, Mathematics of computerized tomography, Wiley, New York, 1986, MR 88m:44008.
  • R A. G. Ramm, Multidimensional inverse scattering problems, Longman/Wiley, New York, 1992; expanded Russian edition: MIR, Moscow, 1994, MR 94e:35004.
  • R1 ------, Finding discontinuities from tomographic data, Proc. Amer. Math. Soc. 123 (1995), 2499--2505.
  • R2 ------, The Radon transform is an isomorphism between $L^2(B_a) $ and $H_e(Z_a) $, Appl. Math. Lett. 8 (1995), 25--29.
  • R3 ------, Opimal local tomography formula, preprint, Pan Amer. Math. J. 4 (1994), 125--127.
  • RK A. G. Ramm and A. I. Katsevich, The Radon transform and local tomography, CRC, Boca Raton (to appear).
  • RZ1 A. G. Ramm and A. I. Zaslavsky, Reconstructing singularities of a function given its Radon transform, Math. Comput. Modeling 18 (1993), 109--138, MR 94j:44006.
  • RZ2 ------, Singularities of the Radon transform, Bull. Amer. Math. Soc. (N.S.) 25 (1993), 109--115, MR 93i:44003.
  • RZ3 ------, X-ray transform, the Legendre transform and envelopes, J. Math. Anal. Appl. 183 (1994), 528--546, CMP 94:12.
  • RZ4 ------, Asymptotics of the Fourier transform of piecewise-smooth functions, C. R. Acad. Sci. Paris Sér. I Math. 316 (1993), 541--545, MR 94d:42019.
  • Ru W. Rudin, Real and complex analysis, McGraw-Hill, New York, 1974, MR 49:8783.
  • S D. Solmon, Asymptotic formulas of the dual Radon transform and applications, Math. Z. 195 (1987), 321--343, MR 88i:44006.
  • Z L. Zalcman, Uniqueness and nonuniqueness for the Radon transform, Bull. London Math. Soc. 14 (1982), 241--245, MR 83h:42020.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 44A15, 45P05

Retrieve articles in all journals with MSC (1991): 44A15, 45P05


Additional Information

A. G. Ramm
Affiliation: Department of Mathematics, Kansas State University, Manhattan, Kansas 66506-2602; Los Alamos National Laboratory, Los Alamos, New Mexico 87545
Email: ramm@math.ksu.edu

DOI: https://doi.org/10.1090/S0002-9939-96-03155-3
Keywords: Tomography, range, inversion formulas, backprojection
Received by editor(s): May 10, 1994
Received by editor(s) in revised form: September 12, 1994
Additional Notes: The author thanks NSF and LANL for support, Complutense University, Madrid, for hospitality, and A. Katsevich for discussions.
Communicated by: David Sharp
Article copyright: © Copyright 1996 American Mathematical Society

American Mathematical Society