Remarks on the general Funk transform and thermoacoustic tomography

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November 2010, 4(4): 693-702. doi: 10.3934/ipi.2010.4.693

Remarks on the general Funk transform and thermoacoustic tomography

Victor Palamodov ^1,

1.	School of Mathematical Sciences, Tel Aviv University, Ramat Aviv Tel Aviv 69978, Israel

Received June 2009 Published September 2010

Abstract
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We discuss properties of a generalized Minkowski-Funk transform defined for a family of hypersurfaces. We prove two-side estimates and show that the range conditions can be written in terms of the reciprocal Funk transform. Some applications to the spherical mean transform are considered.

Keywords: Kaczmarz method., elliptic PDO, Funk transform of a density, reciprocal Funk transform, partial scan, conjugate points, thermoacoustic tomography, range conditions.

Mathematics Subject Classification: Primary: 53C65; Secondary: 65R32, 78A4.

Citation: Victor Palamodov. Remarks on the general Funk transform and thermoacoustic tomography. Inverse Problems and Imaging, 2010, 4 (4) : 693-702. doi: 10.3934/ipi.2010.4.693

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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