# 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), 344-386. 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
 [1] Mark Agranovsky, David Finch, Peter Kuchment. Range conditions for a spherical mean transform. Inverse Problems & Imaging, 2009, 3 (3) : 373-382. doi: 10.3934/ipi.2009.3.373 [2] Alberto Ibort, Alberto López-Yela. Quantum tomography and the quantum Radon transform. Inverse Problems & Imaging, 2021, 15 (5) : 893-928. doi: 10.3934/ipi.2021021 [3] Figen Özpinar, Fethi Bin Muhammad Belgacem. The discrete homotopy perturbation Sumudu transform method for solving partial difference equations. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 615-624. doi: 10.3934/dcdss.2019039 [4] Gareth Ainsworth, Yernat M. Assylbekov. On the range of the attenuated magnetic ray transform for connections and Higgs fields. Inverse Problems & Imaging, 2015, 9 (2) : 317-335. doi: 10.3934/ipi.2015.9.317 [5] Linh V. Nguyen. A family of inversion formulas in thermoacoustic tomography. Inverse Problems & Imaging, 2009, 3 (4) : 649-675. doi: 10.3934/ipi.2009.3.649 [6] Aleksander Denisiuk. On range condition of the tensor x-ray transform in $\mathbb R^n$. Inverse Problems & Imaging, 2020, 14 (3) : 423-435. doi: 10.3934/ipi.2020020 [7] Kaifang Liu, Lunji Song, Shan Zhao. A new over-penalized weak galerkin method. Part Ⅰ: Second-order elliptic problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2411-2428. doi: 10.3934/dcdsb.2020184 [8] Lunji Song, Wenya Qi, Kaifang Liu, Qingxian Gu. A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2581-2598. doi: 10.3934/dcdsb.2020196 [9] William Guo. The Laplace transform as an alternative general method for solving linear ordinary differential equations. STEM Education, 2021, 1 (4) : 309-329. doi: 10.3934/steme.2021020 [10] Hiroshi Fujiwara, Kamran Sadiq, Alexandru Tamasan. Partial inversion of the 2D attenuated $X$-ray transform with data on an arc. Inverse Problems & Imaging, 2022, 16 (1) : 215-228. doi: 10.3934/ipi.2021047 [11] Daniel Fusca. The Madelung transform as a momentum map. Journal of Geometric Mechanics, 2017, 9 (2) : 157-165. doi: 10.3934/jgm.2017006 [12] James W. Webber, Sean Holman. Microlocal analysis of a spindle transform. Inverse Problems & Imaging, 2019, 13 (2) : 231-261. doi: 10.3934/ipi.2019013 [13] Doǧan Çömez. The modulated ergodic Hilbert transform. Discrete & Continuous Dynamical Systems - S, 2009, 2 (2) : 325-336. doi: 10.3934/dcdss.2009.2.325 [14] Sean Holman, Plamen Stefanov. The weighted Doppler transform. Inverse Problems & Imaging, 2010, 4 (1) : 111-130. doi: 10.3934/ipi.2010.4.111 [15] Leonid Kunyansky. Fast reconstruction algorithms for the thermoacoustic tomography in certain domains with cylindrical or spherical symmetries. Inverse Problems & Imaging, 2012, 6 (1) : 111-131. doi: 10.3934/ipi.2012.6.111 [16] Chase Mathison. Thermoacoustic Tomography with circular integrating detectors and variable wave speed. Inverse Problems & Imaging, 2020, 14 (4) : 665-682. doi: 10.3934/ipi.2020030 [17] C E Yarman, B Yazıcı. A new exact inversion method for exponential Radon transform using the harmonic analysis of the Euclidean motion group. Inverse Problems & Imaging, 2007, 1 (3) : 457-479. doi: 10.3934/ipi.2007.1.457 [18] Muhammad Arfan, Kamal Shah, Aman Ullah, Soheil Salahshour, Ali Ahmadian, Massimiliano Ferrara. A novel semi-analytical method for solutions of two dimensional fuzzy fractional wave equation using natural transform. Discrete & Continuous Dynamical Systems - S, 2022, 15 (2) : 315-338. doi: 10.3934/dcdss.2021011 [19] Simon Gindikin. A remark on the weighted Radon transform on the plane. Inverse Problems & Imaging, 2010, 4 (4) : 649-653. doi: 10.3934/ipi.2010.4.649 [20] Sebastian Reich, Seoleun Shin. On the consistency of ensemble transform filter formulations. Journal of Computational Dynamics, 2014, 1 (1) : 177-189. doi: 10.3934/jcd.2014.1.177

2020 Impact Factor: 1.639