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

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 & 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. doi: doi:10.1016/j.jfa.2007.03.022.
[2]	J. Boman, On stable inversion of the attenuated Radon transform with half data,, in, (2006), 19.
[3]	D. Finch and Rakesh, The range of the spherical mean value operator for functions supported in a ball,, Inverse Problems, 22 (2006), 923. doi: doi:10.1088/0266-5611/22/3/012.
[4]	P. Funk, Über Flächen mit lauter geschlossenen geodätischen Linien,, Math. Ann., 74 (1913), 278. doi: doi:10.1007/BF01456044.
[5]	V. Guillemin, On some results of Gelfand in integral geometry,, in, 43 (1985), 149.
[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.
[8]	R. G. Mukhometov, On a problem of integral geometry on the plane,, in, 180 (1978), 30.
[9]	F. Natterer, "The Mathematics of Computerized Tomography,", B.G.Teubner, (1986).
[10]	S. K. Patch, Moment conditions indirectly improve image quality,, in, (2001), 193.
[11]	S. K. Patch and O. Scherzer, Photo- and thermo-acoustic imaging,, Inverse Problems, 23 (2007).
[12]	D. A. Popov, The generalized Radon transform on the plane, its inversion, and the Cavalieri conditions,, Funct. Anal. Appl., 35 (2001), 270. 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. 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.
[15]	H. Rullgård, Stability of the inverse problem for the attenuated Radon transform with 180 $^\circ$ data,, Inverse Problems, 20 (2004), 781. 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. doi: doi:10.1016/j.jfa.2007.03.022.
[2]	J. Boman, On stable inversion of the attenuated Radon transform with half data,, in, (2006), 19.
[3]	D. Finch and Rakesh, The range of the spherical mean value operator for functions supported in a ball,, Inverse Problems, 22 (2006), 923. doi: doi:10.1088/0266-5611/22/3/012.
[4]	P. Funk, Über Flächen mit lauter geschlossenen geodätischen Linien,, Math. Ann., 74 (1913), 278. doi: doi:10.1007/BF01456044.
[5]	V. Guillemin, On some results of Gelfand in integral geometry,, in, 43 (1985), 149.
[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.
[8]	R. G. Mukhometov, On a problem of integral geometry on the plane,, in, 180 (1978), 30.
[9]	F. Natterer, "The Mathematics of Computerized Tomography,", B.G.Teubner, (1986).
[10]	S. K. Patch, Moment conditions indirectly improve image quality,, in, (2001), 193.
[11]	S. K. Patch and O. Scherzer, Photo- and thermo-acoustic imaging,, Inverse Problems, 23 (2007).
[12]	D. A. Popov, The generalized Radon transform on the plane, its inversion, and the Cavalieri conditions,, Funct. Anal. Appl., 35 (2001), 270. 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. 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.
[15]	H. Rullgård, Stability of the inverse problem for the attenuated Radon transform with 180 $^\circ$ data,, Inverse Problems, 20 (2004), 781. doi: doi:10.1088/0266-5611/20/3/008.

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