# American Institute of Mathematical Sciences

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

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.
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.  Google Scholar [2] J. Boman, On stable inversion of the attenuated Radon transform with half data,, in, (2006), 19.   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.  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.  doi: doi:10.1007/BF01456044.  Google Scholar [5] V. Guillemin, On some results of Gelfand in integral geometry,, in, 43 (1985), 149.   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.   Google Scholar [8] R. G. Mukhometov, On a problem of integral geometry on the plane,, in, 180 (1978), 30.   Google Scholar [9] F. Natterer, "The Mathematics of Computerized Tomography,", B.G.Teubner, (1986).   Google Scholar [10] S. K. Patch, Moment conditions indirectly improve image quality,, in, (2001), 193.   Google Scholar [11] S. K. Patch and O. Scherzer, Photo- and thermo-acoustic imaging,, Inverse Problems, 23 (2007).   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.  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.  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.   Google Scholar [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.  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.  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, (2006), 19.   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.  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.  doi: doi:10.1007/BF01456044.  Google Scholar [5] V. Guillemin, On some results of Gelfand in integral geometry,, in, 43 (1985), 149.   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.   Google Scholar [8] R. G. Mukhometov, On a problem of integral geometry on the plane,, in, 180 (1978), 30.   Google Scholar [9] F. Natterer, "The Mathematics of Computerized Tomography,", B.G.Teubner, (1986).   Google Scholar [10] S. K. Patch, Moment conditions indirectly improve image quality,, in, (2001), 193.   Google Scholar [11] S. K. Patch and O. Scherzer, Photo- and thermo-acoustic imaging,, Inverse Problems, 23 (2007).   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.  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.  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.   Google Scholar [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.  Google Scholar
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