# American Institute of Mathematical Sciences

May  2016, 15(3): 1029-1039. doi: 10.3934/cpaa.2016.15.1029

## Inversion of the spherical Radon transform on spheres through the origin using the regular Radon transform

 1 Department of Mathematical Sciences, Ulsan National Institute of Science and Technology, Ulsan 689-798, South Korea

Received  July 2014 Revised  April 2015 Published  February 2016

A spherical Radon transform whose integral domain is a sphere has many applications in partial differential equations as well as tomography. This paper is devoted to the spherical Radon transform which assigns to a given function its integrals over the set of spheres passing through the origin. We present a relation between this spherical Radon transform and the regular Radon transform, and we provide a new inversion formula for the spherical Radon transform using this relation. Numerical simulations were performed to demonstrate the suggested algorithm in dimension 2.
Citation: Sunghwan Moon. Inversion of the spherical Radon transform on spheres through the origin using the regular Radon transform. Communications on Pure & Applied Analysis, 2016, 15 (3) : 1029-1039. doi: 10.3934/cpaa.2016.15.1029
##### References:
 [1] L. Andersson, On the determination of a function from spherical averages, SIAM Journal on Mathematical Analysis, 19 (1988), 214-232. doi: 10.1137/0519016.  Google Scholar [2] A. M. Cormack, Representation of a function by its line integrals, with some radiological applications, Journal of Applied Physics, 34 (1963), 2722-2727. Google Scholar [3] A. M. Cormack, Representation of a function by its line integrals, with some radiological applications. II, Journal of Applied Physics, 35 (1964), 2908-2913. Google Scholar [4] A. M. Cormack and E. T. Quinto, A Radon transform on spheres through the origin in $\mathbbR^n$ and applications to the Darboux equation, Transactions of the American Mathematical Society, 260 (1980), 575-581. doi: 10.2307/1998023.  Google Scholar [5] J. Fawcett, Inversion of $n$-dimensional spherical averages, SIAM Journal on Applied Mathematics, 45 (1985), 336-341. doi: 10.1137/0145018.  Google Scholar [6] D. Finch, M. Haltmeier and Rakesh, Inversion of spherical means and the wave equation in even dimensions, SIAM Journal on Applied Mathematics, 68 (2007), 392-412. doi: 10.1137/070682137.  Google Scholar [7] D. Finch, S. Patch and Rakesh, Determining a function from its mean values over a family of spheres, SIAM Journal on Mathematical Analysis, 35 (2004), 1213-1240. doi: 10.1137/S0036141002417814.  Google Scholar [8] D. Finch and Rakesh, Recovering a function from its spherical mean values in two and three dimensions, In Photoacoustic Imaging and Spectroscopy (L. Wang ed.), Optical Science and Engineering. Taylor & Francis, 2009. Google Scholar [9] S. Gindikin, J. Reeds and L. Shepp, Spherical tomography and spherical integral geometry, In Tomography, Impedance Imaging, and Integral Geometry: 1993 AMS-SIAM Summer Seminar on the Mathematics of Tomography, Impedance Imaging, and Integral Geometry, June 7-18, 1993, Mount Holyoke College, Massachusetts (E. T. Quinto, M. Cheney, P. Kuchment and American Mathematical Society eds.), Lectures in Applied Mathematics Series, pages 83-92. American Mathematical Society, 1994.  Google Scholar [10] M. Haltmeier, Exact reconstruction formula for the spherical mean Radon transform on ellipsoids, Inverse Problems, 30 (2014), 035001. doi: 10.1088/0266-5611/30/10/105006.  Google Scholar [11] S. Helgason, A duality in integral geometry: some generalizations of the Radon transform, Bulletin of the American Mathematical Society, 70 (1964), 435-446.  Google Scholar [12] H. Hellsten and L. E. Andersson, An inverse method for the processing of synthetic aperture radar data, Inverse Problems, 3 (1987), 111.  Google Scholar [13] F. John, Plane Waves and Spherical Means Applied to Partial Differential Equations, Dover Books on Mathematics Series. Dover Publications, 2004.  Google Scholar [14] L. A. Kunyansky, Explicit inversion formulae for the spherical mean Radon transform, Inverse Problems, 23 (2007), 373. doi: 10.1088/0266-5611/23/1/021.  Google Scholar [15] L. A. Kunyansky, A series solution and a fast algorithm for the inversion of the spherical mean Radon transform, Inverse Problems, 23 (2007), S11. doi: 10.1088/0266-5611/23/6/S02.  Google Scholar [16] L. A. Kunyansky, Fast reconstruction algorithms for the thermoacoustic tomography in certain domains with cylindrical or spherical symmetries, Inverse Problems and Imaging, 6 (2012), 111-131. doi: 10.3934/ipi.2012.6.111.  Google Scholar [17] D. Ludwig, The Radon transform on Euclidean space, Communications on Pure and Applied Mathematics, 19 (1966), 49-81.  Google Scholar [18] E. K. Narayanan and Rakesh, Spherical means with centers on a hyperplane in even dimensions, Inverse Problems, 26 (2010), 035014. doi: 10.1088/0266-5611/26/3/035014.  Google Scholar [19] F. Natterer, The Mathematics of Computerized Tomography, Classics in Applied Mathematics. Society for Industrial and Applied Mathematics, Philadelphia, 2001. doi: 10.1137/1.9780898719284.  Google Scholar [20] F. Natterer and F. Wübbeling, Mathematical methods in image reconstruction, SIAM Monographs on mathematical modeling and computation. SIAM, Society of industrial and applied mathematics, Philadelphia (Pa.), 2001. doi: 10.1137/1.9780898718324.  Google Scholar [21] M. K. Nguyen and T. T. Truong, Inversion of a new circular-arc Radon transform for Compton scattering tomography, Inverse Problems, 26 (2010), 065005. doi: 10.1088/0266-5611/26/6/065005.  Google Scholar [22] M. K. Nguyen, G Rigaud and T. T. Truong, A new circular-arc Radon transform and the numerical method for its inversion, In Aip Conference Proceedings, volume 1281, page 1064, 2010. Google Scholar [23] C. J. Nolan and M. Cheney, Synthetic aperture inversion, Inverse Problems, 18 (2002), 221. doi: 10.1088/0266-5611/18/1/315.  Google Scholar [24] S. J. Norton, Reconstruction of a reflectivity field from line integrals over circular paths, The Journal of the Acoustical Society of America, 67 (1980), 853-863. doi: 10.1121/1.384168.  Google Scholar [25] E. T. Quinto, Null spaces and ranges for the classical and spherical Radon transforms, Journal of Mathematical Analysis and Applications, 90 (1982), 408-420. doi: 10.1016/0022-247X(82)90069-5.  Google Scholar [26] E. T. Quinto, Singular value decompositions and inversion methods for the exterior radon transform and a spherical transform, Journal of Mathematical Analysis and Applications, 95 (1983), 437-448. doi: 10.1016/0022-247X(83)90118-X.  Google Scholar [27] E. Quinto, Singularities of the X-ray transform and limited data tomography in $\mathbbR^2$ and $\mathbbR^3$, SIAM Journal on Mathematical Analysis, 24 (1993), 1215-1225. doi: 10.1137/0524069.  Google Scholar [28] N. T. Redding and G. N. Newsam, Inverting the circular Radon transform, DTSO Research Report DTSO-Ru-0211, August 2001. Google Scholar [29] H. Rhee, A representation of the solutions of the Darboux equation in odd-dimensional spaces, Transactions of the American Mathematical Society, 150 (1970), 491-498.  Google Scholar [30] K. T. Smith, D. C. Solmon and S. L. Wagner, Practical and mathematical aspects of the problem of reconstructing a function from radiographs, Bulletin of the American Mathematical Society, 82 (1977), 1227-1270.  Google Scholar [31] A. E. Yagle, Inversion of spherical means using geometric inversion and Radon transforms, Inverse Problems, 8 (1992), 949.  Google Scholar [32] C. E. Yarman and B. Yazici, Inversion of the circular averages transform using the Funk transform, Inverse Problems, 27 (2011), 065001. doi: 10.1088/0266-5611/27/6/065001.  Google Scholar [33] L. Zalcman, Offbeat integral geometry, The American Mathematical Monthly, 87 (1980), 161-175. doi: 10.2307/2321600.  Google Scholar

show all references

##### References:
 [1] L. Andersson, On the determination of a function from spherical averages, SIAM Journal on Mathematical Analysis, 19 (1988), 214-232. doi: 10.1137/0519016.  Google Scholar [2] A. M. Cormack, Representation of a function by its line integrals, with some radiological applications, Journal of Applied Physics, 34 (1963), 2722-2727. Google Scholar [3] A. M. Cormack, Representation of a function by its line integrals, with some radiological applications. II, Journal of Applied Physics, 35 (1964), 2908-2913. Google Scholar [4] A. M. Cormack and E. T. Quinto, A Radon transform on spheres through the origin in $\mathbbR^n$ and applications to the Darboux equation, Transactions of the American Mathematical Society, 260 (1980), 575-581. doi: 10.2307/1998023.  Google Scholar [5] J. Fawcett, Inversion of $n$-dimensional spherical averages, SIAM Journal on Applied Mathematics, 45 (1985), 336-341. doi: 10.1137/0145018.  Google Scholar [6] D. Finch, M. Haltmeier and Rakesh, Inversion of spherical means and the wave equation in even dimensions, SIAM Journal on Applied Mathematics, 68 (2007), 392-412. doi: 10.1137/070682137.  Google Scholar [7] D. Finch, S. Patch and Rakesh, Determining a function from its mean values over a family of spheres, SIAM Journal on Mathematical Analysis, 35 (2004), 1213-1240. doi: 10.1137/S0036141002417814.  Google Scholar [8] D. Finch and Rakesh, Recovering a function from its spherical mean values in two and three dimensions, In Photoacoustic Imaging and Spectroscopy (L. Wang ed.), Optical Science and Engineering. Taylor & Francis, 2009. Google Scholar [9] S. Gindikin, J. Reeds and L. Shepp, Spherical tomography and spherical integral geometry, In Tomography, Impedance Imaging, and Integral Geometry: 1993 AMS-SIAM Summer Seminar on the Mathematics of Tomography, Impedance Imaging, and Integral Geometry, June 7-18, 1993, Mount Holyoke College, Massachusetts (E. T. Quinto, M. Cheney, P. Kuchment and American Mathematical Society eds.), Lectures in Applied Mathematics Series, pages 83-92. American Mathematical Society, 1994.  Google Scholar [10] M. Haltmeier, Exact reconstruction formula for the spherical mean Radon transform on ellipsoids, Inverse Problems, 30 (2014), 035001. doi: 10.1088/0266-5611/30/10/105006.  Google Scholar [11] S. Helgason, A duality in integral geometry: some generalizations of the Radon transform, Bulletin of the American Mathematical Society, 70 (1964), 435-446.  Google Scholar [12] H. Hellsten and L. E. Andersson, An inverse method for the processing of synthetic aperture radar data, Inverse Problems, 3 (1987), 111.  Google Scholar [13] F. John, Plane Waves and Spherical Means Applied to Partial Differential Equations, Dover Books on Mathematics Series. Dover Publications, 2004.  Google Scholar [14] L. A. Kunyansky, Explicit inversion formulae for the spherical mean Radon transform, Inverse Problems, 23 (2007), 373. doi: 10.1088/0266-5611/23/1/021.  Google Scholar [15] L. A. Kunyansky, A series solution and a fast algorithm for the inversion of the spherical mean Radon transform, Inverse Problems, 23 (2007), S11. doi: 10.1088/0266-5611/23/6/S02.  Google Scholar [16] L. A. Kunyansky, Fast reconstruction algorithms for the thermoacoustic tomography in certain domains with cylindrical or spherical symmetries, Inverse Problems and Imaging, 6 (2012), 111-131. doi: 10.3934/ipi.2012.6.111.  Google Scholar [17] D. Ludwig, The Radon transform on Euclidean space, Communications on Pure and Applied Mathematics, 19 (1966), 49-81.  Google Scholar [18] E. K. Narayanan and Rakesh, Spherical means with centers on a hyperplane in even dimensions, Inverse Problems, 26 (2010), 035014. doi: 10.1088/0266-5611/26/3/035014.  Google Scholar [19] F. Natterer, The Mathematics of Computerized Tomography, Classics in Applied Mathematics. Society for Industrial and Applied Mathematics, Philadelphia, 2001. doi: 10.1137/1.9780898719284.  Google Scholar [20] F. Natterer and F. Wübbeling, Mathematical methods in image reconstruction, SIAM Monographs on mathematical modeling and computation. SIAM, Society of industrial and applied mathematics, Philadelphia (Pa.), 2001. doi: 10.1137/1.9780898718324.  Google Scholar [21] M. K. Nguyen and T. T. Truong, Inversion of a new circular-arc Radon transform for Compton scattering tomography, Inverse Problems, 26 (2010), 065005. doi: 10.1088/0266-5611/26/6/065005.  Google Scholar [22] M. K. Nguyen, G Rigaud and T. T. Truong, A new circular-arc Radon transform and the numerical method for its inversion, In Aip Conference Proceedings, volume 1281, page 1064, 2010. Google Scholar [23] C. J. Nolan and M. Cheney, Synthetic aperture inversion, Inverse Problems, 18 (2002), 221. doi: 10.1088/0266-5611/18/1/315.  Google Scholar [24] S. J. Norton, Reconstruction of a reflectivity field from line integrals over circular paths, The Journal of the Acoustical Society of America, 67 (1980), 853-863. doi: 10.1121/1.384168.  Google Scholar [25] E. T. Quinto, Null spaces and ranges for the classical and spherical Radon transforms, Journal of Mathematical Analysis and Applications, 90 (1982), 408-420. doi: 10.1016/0022-247X(82)90069-5.  Google Scholar [26] E. T. Quinto, Singular value decompositions and inversion methods for the exterior radon transform and a spherical transform, Journal of Mathematical Analysis and Applications, 95 (1983), 437-448. doi: 10.1016/0022-247X(83)90118-X.  Google Scholar [27] E. Quinto, Singularities of the X-ray transform and limited data tomography in $\mathbbR^2$ and $\mathbbR^3$, SIAM Journal on Mathematical Analysis, 24 (1993), 1215-1225. doi: 10.1137/0524069.  Google Scholar [28] N. T. Redding and G. N. Newsam, Inverting the circular Radon transform, DTSO Research Report DTSO-Ru-0211, August 2001. Google Scholar [29] H. Rhee, A representation of the solutions of the Darboux equation in odd-dimensional spaces, Transactions of the American Mathematical Society, 150 (1970), 491-498.  Google Scholar [30] K. T. Smith, D. C. Solmon and S. L. Wagner, Practical and mathematical aspects of the problem of reconstructing a function from radiographs, Bulletin of the American Mathematical Society, 82 (1977), 1227-1270.  Google Scholar [31] A. E. Yagle, Inversion of spherical means using geometric inversion and Radon transforms, Inverse Problems, 8 (1992), 949.  Google Scholar [32] C. E. Yarman and B. Yazici, Inversion of the circular averages transform using the Funk transform, Inverse Problems, 27 (2011), 065001. doi: 10.1088/0266-5611/27/6/065001.  Google Scholar [33] L. Zalcman, Offbeat integral geometry, The American Mathematical Monthly, 87 (1980), 161-175. doi: 10.2307/2321600.  Google Scholar
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