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X-ray transform on Damek-Ricci spaces
Local Sobolev estimates of a function by means of its Radon transform
1. | Department of Mathematics, Stockholm University, 10691 Stockholm, Sweden |
2. | Department of Mathematics, Tufts University, Medford, MA 02155, United States |
References:
[1] |
M. A. Anastasio, Y. Zou, E. Y. Sidky and X. Pan, Local cone-beam tomography image reconstruction on chords,, Journal of the Optical Society of America A, 24 (2007), 1569.
doi: doi:10.1364/JOSAA.24.001569. |
[2] |
E. Candès and L. Demanet, Curvelets and Fourier Integral Operators,, C. R. Math. Acad. Sci. Paris. Serie I, 336 (2003), 395.
|
[3] |
E. J. Candès and D. L. Donoho, Curvelets and Reconstruction of Images from Noisy Radon Data,, in, 4119 (2000). Google Scholar |
[4] |
D. V. Finch, I.-R. Lan and G. Uhlmann, Microlocal Analysis of the restricted X-ray transform with sources on a curve,, in, 47 (2003), 193. Google Scholar |
[5] |
A. Greenleaf and G. Uhlmann, Non-local inversion formulas for the X-ray transform,, Duke Math. J., 58 (1989), 205.
doi: doi:10.1215/S0012-7094-89-05811-0. |
[6] |
A. Greenleaf and G. Uhlmann, Estimates for singular Radon transforms and pseudodifferential operators with singular symbols,, J. Funct. Anal., 89 (1990), 202.
doi: doi:10.1016/0022-1236(90)90011-9. |
[7] |
A. Greenleaf and G. Uhlmann, Microlocal techniques in integral geometry,, Contemp. Math., 113 (1990), 121.
|
[8] |
V. Guillemin and D. Schaeffer, Fourier integral operators from the Radon transform point of view,, Proc. Sympos. Pure Math., 27 (1975), 297.
|
[9] |
V. Guillemin and S. Sternberg, "Geometric Asymptotics,'', American Mathematical Society, (1977).
|
[10] |
M. G. Hahn and E. T. Quinto, Distances between measures from 1-dimensional projections as implied by continuity of the inverse Radon transform,, Zeit. Wahr., 70 (1985), 361.
doi: doi:10.1007/BF00534869. |
[11] |
A. Hertle, Continuity of the Radon transform and its inverse on Euclidean space,, Math. Z., 184 (1983), 165.
doi: doi:10.1007/BF01252856. |
[12] |
A. Katsevich, Improved cone beam local tomography,, Inverse Problems, 22 (2006), 627.
doi: doi:10.1088/0266-5611/22/2/015. |
[13] |
A. I. Katsevich, Cone beam local tomography,, SIAM J. Appl. Math., 59 (1999), 2224.
doi: doi:10.1137/S0036139998336043. |
[14] |
A. K. Louis, "Analytische Methoden in der Computer Tomographie,", Habilitationsschrift, (1981). Google Scholar |
[15] |
F. Natterer, The mathematics of computerized tomography,, in, (2001).
|
[16] |
F. Natterer and F. Wübbeling, Mathematical methods in image reconstruction,, in, (2001).
|
[17] |
B. Petersen, "Introduction to the Fourier Transform and Pseudo-Differential Operators,", Pittman, (1983).
|
[18] |
E. T. Quinto, Singularities of the X-ray transform and limited data tomography in $R^2$ and $R^3$,, SIAM J. Math. Anal., 24 (1993), 1215.
doi: doi:10.1137/0524069. |
[19] |
E. T. Quinto, T. Bakhos and S. Chung, A local algorithm for Slant Hole SPECT,, in, (2008), 321.
|
[20] |
E. T. Quinto and O. Öktem, Local tomography in electron microscopy,, SIAM J. Appl. Math., 68 (2008), 1282.
doi: doi:10.1137/07068326X. |
[21] |
A. G. Ramm and A. I. Zaslavsky, Singularities of the Radon transform,, Bull. Amer. Math. Soc., 25 (1993), 109.
doi: doi:10.1090/S0273-0979-1993-00350-1. |
[22] |
M. Beals and M. Reed, Propagation of singularities for hyperbolic pseudodifferential operators with nonsmooth coefficients,, Comm. Pure Appl. Math., 35 (1982), 169.
doi: doi:10.1002/cpa.3160350203. |
[23] |
M. Beals and M. Reed, Microlocal regularity theorems for nonsmooth pseudodifferential operators and applications to nonlinear problems,, Trans. Amer. Math. Soc. 285 (1984), 285 (1984), 159.
|
[24] |
H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators,'', Second edition. Johann Ambrosius Barth, (1995).
|
[25] |
Y. Ye, H. Yu and G. Wang, Cone beam pseudo-lambda tomography,, Inverse Problems, 23 (2007), 203.
doi: doi:10.1088/0266-5611/23/1/010. |
show all references
References:
[1] |
M. A. Anastasio, Y. Zou, E. Y. Sidky and X. Pan, Local cone-beam tomography image reconstruction on chords,, Journal of the Optical Society of America A, 24 (2007), 1569.
doi: doi:10.1364/JOSAA.24.001569. |
[2] |
E. Candès and L. Demanet, Curvelets and Fourier Integral Operators,, C. R. Math. Acad. Sci. Paris. Serie I, 336 (2003), 395.
|
[3] |
E. J. Candès and D. L. Donoho, Curvelets and Reconstruction of Images from Noisy Radon Data,, in, 4119 (2000). Google Scholar |
[4] |
D. V. Finch, I.-R. Lan and G. Uhlmann, Microlocal Analysis of the restricted X-ray transform with sources on a curve,, in, 47 (2003), 193. Google Scholar |
[5] |
A. Greenleaf and G. Uhlmann, Non-local inversion formulas for the X-ray transform,, Duke Math. J., 58 (1989), 205.
doi: doi:10.1215/S0012-7094-89-05811-0. |
[6] |
A. Greenleaf and G. Uhlmann, Estimates for singular Radon transforms and pseudodifferential operators with singular symbols,, J. Funct. Anal., 89 (1990), 202.
doi: doi:10.1016/0022-1236(90)90011-9. |
[7] |
A. Greenleaf and G. Uhlmann, Microlocal techniques in integral geometry,, Contemp. Math., 113 (1990), 121.
|
[8] |
V. Guillemin and D. Schaeffer, Fourier integral operators from the Radon transform point of view,, Proc. Sympos. Pure Math., 27 (1975), 297.
|
[9] |
V. Guillemin and S. Sternberg, "Geometric Asymptotics,'', American Mathematical Society, (1977).
|
[10] |
M. G. Hahn and E. T. Quinto, Distances between measures from 1-dimensional projections as implied by continuity of the inverse Radon transform,, Zeit. Wahr., 70 (1985), 361.
doi: doi:10.1007/BF00534869. |
[11] |
A. Hertle, Continuity of the Radon transform and its inverse on Euclidean space,, Math. Z., 184 (1983), 165.
doi: doi:10.1007/BF01252856. |
[12] |
A. Katsevich, Improved cone beam local tomography,, Inverse Problems, 22 (2006), 627.
doi: doi:10.1088/0266-5611/22/2/015. |
[13] |
A. I. Katsevich, Cone beam local tomography,, SIAM J. Appl. Math., 59 (1999), 2224.
doi: doi:10.1137/S0036139998336043. |
[14] |
A. K. Louis, "Analytische Methoden in der Computer Tomographie,", Habilitationsschrift, (1981). Google Scholar |
[15] |
F. Natterer, The mathematics of computerized tomography,, in, (2001).
|
[16] |
F. Natterer and F. Wübbeling, Mathematical methods in image reconstruction,, in, (2001).
|
[17] |
B. Petersen, "Introduction to the Fourier Transform and Pseudo-Differential Operators,", Pittman, (1983).
|
[18] |
E. T. Quinto, Singularities of the X-ray transform and limited data tomography in $R^2$ and $R^3$,, SIAM J. Math. Anal., 24 (1993), 1215.
doi: doi:10.1137/0524069. |
[19] |
E. T. Quinto, T. Bakhos and S. Chung, A local algorithm for Slant Hole SPECT,, in, (2008), 321.
|
[20] |
E. T. Quinto and O. Öktem, Local tomography in electron microscopy,, SIAM J. Appl. Math., 68 (2008), 1282.
doi: doi:10.1137/07068326X. |
[21] |
A. G. Ramm and A. I. Zaslavsky, Singularities of the Radon transform,, Bull. Amer. Math. Soc., 25 (1993), 109.
doi: doi:10.1090/S0273-0979-1993-00350-1. |
[22] |
M. Beals and M. Reed, Propagation of singularities for hyperbolic pseudodifferential operators with nonsmooth coefficients,, Comm. Pure Appl. Math., 35 (1982), 169.
doi: doi:10.1002/cpa.3160350203. |
[23] |
M. Beals and M. Reed, Microlocal regularity theorems for nonsmooth pseudodifferential operators and applications to nonlinear problems,, Trans. Amer. Math. Soc. 285 (1984), 285 (1984), 159.
|
[24] |
H. Triebel, "Interpolation Theory, Function Spaces, Differential Operators,'', Second edition. Johann Ambrosius Barth, (1995).
|
[25] |
Y. Ye, H. Yu and G. Wang, Cone beam pseudo-lambda tomography,, Inverse Problems, 23 (2007), 203.
doi: doi:10.1088/0266-5611/23/1/010. |
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