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Local tomography with nonsmooth attenuation

Author: A. I. Katsevich
Journal: Trans. Amer. Math. Soc. 351 (1999), 1947-1974
MSC (1991): Primary 35S99, 44A12, 65R10, 92C55
Published electronically: January 27, 1999
MathSciNet review: 1466950
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Abstract: Local tomography for the Radon transform with nonsmooth attenuation is proposed and justified. The main theoretical tool is analysis of singularities of pseudodifferential operators with nonsmooth symbols. Results of numerical testing of local tomography are presented.

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  • [BR] M. Beals and M. Reed, Microlocal regularity theorems for nonsmooth pseudodifferential operators and applications to nonlinear problems, Trans. Amer. Math. Soc. 285 (1) (1984), 159-184. MR 86a:35156
  • [BW] C.A. Berenstein and D.F. Walnut, Local inversion of the Radon transform in even dimensions using wavelets, 75 Years of Radon Transform (S. Gindikin and P. Michor, eds.), Conference Proceedings and Lecture Notes in Mathematical Physics, Vol. 4, International Press, Boston, 1994, pp. 45-69. MR 95m:42037
  • [Be] G. Beylkin, The inversion problem and applications of the generalized Radon transform, Comm. on Pure and Appl. Math. 37 (1984), 579-599. MR 86a:44002
  • [BH] N. Bleistein and R. Handelsman, Asymptotic Expansions of Integrals, Dover, Mineola, N.Y., 1986. MR 89d:41049
  • [De] S. Deans, The Radon Transform and Some of Its Applications, Wiley, New York, 1983. MR 86a:44003
  • [FRS] A. Faridani, E.L. Ritman, and K.T. Smith, Local tomography, SIAM J. Appl. Math. 52 (2) (1992), 459 - 484, 1193 - 1198. MR 93b:92008; CMP 92:15
  • [FFRS] A. Faridani, D. Finch, E.L. Ritman, and K.T. Smith, Local tomography II, SIAM J. Appl. Math. 57 (1997), 1095-1127. CMP 97:16
  • [GS] I.M. Gelfand and G.E. Shilov, Generalized Functions, Volume I, Academic Press, New York, 1964. MR 29:3869
  • [GuSt] V. Guillemin and S. Sternberg, Geometric Asymptotics, Amer. Math. Soc., Providence, RI, 1977. MR 58:24404
  • [GU] V. Guillemin and G. Uhlmann, Oscillatory integrals with singular symbols, Duke Mathematical Journal 48 (1981), 251-267. MR 28d:58065
  • [Hor] L. Hörmander, The Analysis of Linear Partial Differential Operators. III, Springer-Verlag, Berlin, 1985. MR 87d:35002a
  • [K1] A.I. Katsevich, Local tomography for the generalized Radon transform, SIAM J. Appl. Math. 57 (4) (1997), 1128-1162. CMP 97:16
  • [K2] -, Local tomography for the limited-angle problem, J. Math. Anal. Appl. 213 (1997), 160-182. CMP 98:01
  • [KR1] A.I. Katsevich and A.G. Ramm, Asymptotics of PDO on discontinuous functions near singular support, Applicable Analysis 58 (3-4) (1995), 383-390. MR 97d:35259
  • [KR2] -, Finding jumps of a function using local tomography, PanAmerican Mathematical Journal 6 (2) (1996), 1-21. MR 97e:44004
  • [KR3] -, New methods for finding values of the jumps of a function from its local tomographic data, Inverse Problems 11 (5) (1995), 1005-1023. CMP 96:02
  • [KLM] P. Kuchment, K. Lancaster, and L. Mogilevskaya, On local tomography, Inverse Problems 11 (1995), 571-589. MR 96i:44007
  • [KS] P. Kuchment and I. Shneiberg, Some inversion formulas in the single photon emission computed tomography, Applicable Analysis 53 (1994), 221-231. MR 96m:44003
  • [Kun] L.A. Kunyansky, Generalized and attenuated Radon transforms: restorative approach to the numerical inversion, Inverse Problems 8 (1992), 809-819. MR 93k:65103
  • [Ma1] J. Marschall, Parametrices for nonregular elliptic pseudodifferential operators, Math. Nachr. 159 (1992), 175-188. MR 94h:35287
  • [Ma2] -, On the boundedness and compactness of nonregular pseudo-differential operators, Math. Nachr. 175 (1995), 231-262. MR 96k:47090
  • [MU] R. Melrose and G. Uhlmann, Lagrangian intersection and the Cauchy problem, Comm. Pure Appl. Math. 32 (1979), 483-519. MR 81b:58052
  • [R1] A.G. Ramm, Optimal local tomography formulas, PanAmer. Math. J. 4 (4) (1994), 125-127. MR 95h:44008
  • [R2] -, Finding discontinuities from tomographic data, Proc. Amer. Math. Soc. 123 (8) (1995), 2499-2505. MR 95j:44001
  • [R3] -, Necessary and sufficient conditions for a PDO to be a local tomography operator, Comptes Rend. Acad. Sci, Paris, Sér. I Math. 332 (7) (1996), 613-618. MR 97g:47046
  • [RK] A.G. Ramm and A.I. Katsevich, The Radon Transform and Local Tomography, CRC Press, Boca Raton, FL, 1996. MR 97g:44009
  • [RZ1] A.G. Ramm and A.I. Zaslavsky, Singularities of the Radon transform, Bull. Amer. Math. Soc. 25 (1) (1993), 109-115. MR 93i:44003
  • [RZ2] -, Reconstructing singularities of a function its Radon transform, Math. and Comput. Modelling 18 (1) (1993), 109 - 138. MR 94j:44006
  • [Shu] M.A. Shubin, Pseudodifferential Operators and Spectral Theory, Springer-Verlag, Berlin, 1987. MR 88c:47105
  • [SK] K.T. Smith and F. Keinert, Mathematical foundations of computed tomography, Appl. Optics (1985), 3950 - 3957.
  • [Ta] M.E. Taylor, Pseudodifferential Operators and Nonlinear PDE, Progress in Math., Vol. 100, Birkhauser, Boston, MA, 1991. MR 92j:35193
  • [TM] O.J. Tretiak and C.L. Metz, The exponential Radon transform, SIAM J. Appl. Math. 39 (1980), 341 - 354. MR 82a:44004
  • [VKK] E.I. Vainberg, I.A. Kazak, and V.P. Kurczaev, Reconstruction of the internal three-dimensional structure of objects based on real-time integral projections, Soviet J. Nondest. Test. 17 (1981), 415-423.
  • [W] R. Wong, Asymptotic Approximations of Integrals, Academic Press, Boston, 1989. MR 90j:41061

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Additional Information

A. I. Katsevich
Affiliation: Los Alamos National Laboratory, MS K-990, Los Alamos, New Mexico 87545
Address at time of publication: Department of Mathematics, University of Central Florida, Orlando, Florida 32816-1364

Received by editor(s): May 30, 1996
Received by editor(s) in revised form: November 13, 1996
Published electronically: January 27, 1999
Article copyright: © Copyright 1999 American Mathematical Society

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