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Incomplete data problems in wave equation imaging
Remarks on the general Funk transform and thermoacoustic tomography
1. | School of Mathematical Sciences, Tel Aviv University, Ramat Aviv Tel Aviv 69978, Israel |
References:
[1] |
M. Agranovsky, P. Kuchment and E. T. Quinto, Range descriptions for the spherical mean Radon transform, J. Funct. Anal., 248 (2007), 344-386.
doi: doi:10.1016/j.jfa.2007.03.022. |
[2] |
J. Boman, On stable inversion of the attenuated Radon transform with half data, in "Integral Geometry and Tomography," 19-26, Amer. Math. Soc., Providence, RI, 2006. |
[3] |
D. Finch and Rakesh, The range of the spherical mean value operator for functions supported in a ball, Inverse Problems, 22 (2006), 923-938.
doi: doi:10.1088/0266-5611/22/3/012. |
[4] |
P. Funk, Über Flächen mit lauter geschlossenen geodätischen Linien, Math. Ann., 74 (1913), 278-300.
doi: doi:10.1007/BF01456044. |
[5] |
V. Guillemin, On some results of Gelfand in integral geometry, in "Pseudodifferential Operators and Applications," 149-155, Proc. Sympos.Pure Math., 43, Amer. Math. Soc., Provindence, RI, 1985. |
[6] |
L. Hörmander, "The Analysis of Linear Partial Differential Operators IV. Fourier Integral Operators," Springer, 1985. |
[7] |
M. M. Lavrent'ev and A. L. Buhgeim, A certain class of problems of integral geometry, Dokl. Akad. Nauk SSSR, 211 (1973), 38-39. |
[8] |
R. G. Mukhometov, On a problem of integral geometry on the plane, in "Methods of Functional Analysis in Problems of Mathematical Physics (Russian)," 30-37, Akad. Nauk Ukrain. SSR, 180, Inst. Mat., Kiev, 1978. |
[9] |
F. Natterer, "The Mathematics of Computerized Tomography," B.G.Teubner, John Wiley & Sons, Stuttgart, 1986. |
[10] |
S. K. Patch, Moment conditions indirectly improve image quality, in "Radon Transform and Tomography," 193-205, Amer. Math. Soc., Providence, RI, 2001. |
[11] |
S. K. Patch and O. Scherzer, Photo- and thermo-acoustic imaging, Inverse Problems, 23 (2007), S1-S10. |
[12] |
D. A. Popov, The generalized Radon transform on the plane, its inversion, and the Cavalieri conditions, Funct. Anal. Appl., 35 (2001), 270-283.
doi: doi:10.1023/A:1013126507543. |
[13] |
D. A. Popov and D. V. Sushko, Image restoration in optical-acoustic tomography, Probl. Inf. Transm., 40 (2004), 254-278.
doi: doi:10.1023/B:PRIT.0000044261.87490.05. |
[14] |
E. T. Quinto, The dependence of the generalized Radon transform on defining measures, Trans. Amer. Math. Soc., 257 (1980), 331-346. |
[15] |
H. Rullgård, Stability of the inverse problem for the attenuated Radon transform with 180 $^\circ$ data, Inverse Problems, 20 (2004), 781-797.
doi: doi:10.1088/0266-5611/20/3/008. |
show all references
References:
[1] |
M. Agranovsky, P. Kuchment and E. T. Quinto, Range descriptions for the spherical mean Radon transform, J. Funct. Anal., 248 (2007), 344-386.
doi: doi:10.1016/j.jfa.2007.03.022. |
[2] |
J. Boman, On stable inversion of the attenuated Radon transform with half data, in "Integral Geometry and Tomography," 19-26, Amer. Math. Soc., Providence, RI, 2006. |
[3] |
D. Finch and Rakesh, The range of the spherical mean value operator for functions supported in a ball, Inverse Problems, 22 (2006), 923-938.
doi: doi:10.1088/0266-5611/22/3/012. |
[4] |
P. Funk, Über Flächen mit lauter geschlossenen geodätischen Linien, Math. Ann., 74 (1913), 278-300.
doi: doi:10.1007/BF01456044. |
[5] |
V. Guillemin, On some results of Gelfand in integral geometry, in "Pseudodifferential Operators and Applications," 149-155, Proc. Sympos.Pure Math., 43, Amer. Math. Soc., Provindence, RI, 1985. |
[6] |
L. Hörmander, "The Analysis of Linear Partial Differential Operators IV. Fourier Integral Operators," Springer, 1985. |
[7] |
M. M. Lavrent'ev and A. L. Buhgeim, A certain class of problems of integral geometry, Dokl. Akad. Nauk SSSR, 211 (1973), 38-39. |
[8] |
R. G. Mukhometov, On a problem of integral geometry on the plane, in "Methods of Functional Analysis in Problems of Mathematical Physics (Russian)," 30-37, Akad. Nauk Ukrain. SSR, 180, Inst. Mat., Kiev, 1978. |
[9] |
F. Natterer, "The Mathematics of Computerized Tomography," B.G.Teubner, John Wiley & Sons, Stuttgart, 1986. |
[10] |
S. K. Patch, Moment conditions indirectly improve image quality, in "Radon Transform and Tomography," 193-205, Amer. Math. Soc., Providence, RI, 2001. |
[11] |
S. K. Patch and O. Scherzer, Photo- and thermo-acoustic imaging, Inverse Problems, 23 (2007), S1-S10. |
[12] |
D. A. Popov, The generalized Radon transform on the plane, its inversion, and the Cavalieri conditions, Funct. Anal. Appl., 35 (2001), 270-283.
doi: doi:10.1023/A:1013126507543. |
[13] |
D. A. Popov and D. V. Sushko, Image restoration in optical-acoustic tomography, Probl. Inf. Transm., 40 (2004), 254-278.
doi: doi:10.1023/B:PRIT.0000044261.87490.05. |
[14] |
E. T. Quinto, The dependence of the generalized Radon transform on defining measures, Trans. Amer. Math. Soc., 257 (1980), 331-346. |
[15] |
H. Rullgård, Stability of the inverse problem for the attenuated Radon transform with 180 $^\circ$ data, Inverse Problems, 20 (2004), 781-797.
doi: doi:10.1088/0266-5611/20/3/008. |
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