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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

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), 344-386. doi: doi:10.1016/j.jfa.2007.03.022.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

P. Funk, Über Flächen mit lauter geschlossenen geodätischen Linien, Math. Ann., 74 (1913), 278-300. doi: doi:10.1007/BF01456044.  Google Scholar

[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.  Google Scholar

[6]

L. Hörmander, "The Analysis of Linear Partial Differential Operators IV. Fourier Integral Operators," Springer, 1985.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

F. Natterer, "The Mathematics of Computerized Tomography," B.G.Teubner, John Wiley & Sons, Stuttgart, 1986.  Google Scholar

[10]

S. K. Patch, Moment conditions indirectly improve image quality, in "Radon Transform and Tomography," 193-205, Amer. Math. Soc., Providence, RI, 2001.  Google Scholar

[11]

S. K. Patch and O. Scherzer, Photo- and thermo-acoustic imaging, Inverse Problems, 23 (2007), S1-S10.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

E. T. Quinto, The dependence of the generalized Radon transform on defining measures, Trans. Amer. Math. Soc., 257 (1980), 331-346.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

P. Funk, Über Flächen mit lauter geschlossenen geodätischen Linien, Math. Ann., 74 (1913), 278-300. doi: doi:10.1007/BF01456044.  Google Scholar

[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.  Google Scholar

[6]

L. Hörmander, "The Analysis of Linear Partial Differential Operators IV. Fourier Integral Operators," Springer, 1985.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

F. Natterer, "The Mathematics of Computerized Tomography," B.G.Teubner, John Wiley & Sons, Stuttgart, 1986.  Google Scholar

[10]

S. K. Patch, Moment conditions indirectly improve image quality, in "Radon Transform and Tomography," 193-205, Amer. Math. Soc., Providence, RI, 2001.  Google Scholar

[11]

S. K. Patch and O. Scherzer, Photo- and thermo-acoustic imaging, Inverse Problems, 23 (2007), S1-S10.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

E. T. Quinto, The dependence of the generalized Radon transform on defining measures, Trans. Amer. Math. Soc., 257 (1980), 331-346.  Google Scholar

[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.  Google Scholar

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