# 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,, \emph{SIAM Journal on Mathematical Analysis}, 19 (1988), 214. doi: 10.1137/0519016. Google Scholar [2] A. M. Cormack, Representation of a function by its line integrals, with some radiological applications,, \emph{Journal of Applied Physics}, 34 (1963), 2722. Google Scholar [3] A. M. Cormack, Representation of a function by its line integrals, with some radiological applications. II,, \emph{Journal of Applied Physics}, 35 (1964), 2908. 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,, \emph{Transactions of the American Mathematical Society}, 260 (1980), 575. doi: 10.2307/1998023. Google Scholar [5] J. Fawcett, Inversion of $n$-dimensional spherical averages,, \emph{SIAM Journal on Applied Mathematics}, 45 (1985), 336. doi: 10.1137/0145018. Google Scholar [6] D. Finch, M. Haltmeier and Rakesh, Inversion of spherical means and the wave equation in even dimensions,, \emph{SIAM Journal on Applied Mathematics}, 68 (2007), 392. 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,, \emph{SIAM Journal on Mathematical Analysis}, 35 (2004), 1213. 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 \emph{Photoacoustic Imaging and Spectroscopy} (L. Wang ed.), (2009). Google Scholar [9] S. Gindikin, J. Reeds and L. Shepp, Spherical tomography and spherical integral geometry,, In \emph{Tomography, (1993), 7. Google Scholar [10] M. Haltmeier, Exact reconstruction formula for the spherical mean Radon transform on ellipsoids,, \emph{Inverse Problems}, 30 (2014). doi: 10.1088/0266-5611/30/10/105006. Google Scholar [11] S. Helgason, A duality in integral geometry: some generalizations of the Radon transform,, \emph{Bulletin of the American Mathematical Society}, 70 (1964), 435. Google Scholar [12] H. Hellsten and L. E. Andersson, An inverse method for the processing of synthetic aperture radar data,, \emph{Inverse Problems}, 3 (1987). 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,, \emph{Inverse Problems}, 23 (2007). 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,, \emph{Inverse Problems}, 23 (2007). 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,, \emph{Inverse Problems and Imaging}, 6 (2012), 111. doi: 10.3934/ipi.2012.6.111. Google Scholar [17] D. Ludwig, The Radon transform on Euclidean space,, \emph{Communications on Pure and Applied Mathematics}, 19 (1966), 49. Google Scholar [18] E. K. Narayanan and Rakesh, Spherical means with centers on a hyperplane in even dimensions,, \emph{Inverse Problems}, 26 (2010). 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, (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, (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,, \emph{Inverse Problems}, 26 (2010). 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 \emph{Aip Conference Proceedings}, (1281). Google Scholar [23] C. J. Nolan and M. Cheney, Synthetic aperture inversion,, \emph{Inverse Problems}, 18 (2002). 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,, \emph{The Journal of the Acoustical Society of America}, 67 (1980), 853. doi: 10.1121/1.384168. Google Scholar [25] E. T. Quinto, Null spaces and ranges for the classical and spherical Radon transforms,, \emph{Journal of Mathematical Analysis and Applications}, 90 (1982), 408. 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,, \emph{Journal of Mathematical Analysis and Applications}, 95 (1983), 437. 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 $\mathbbR^3$,, \emph{SIAM Journal on Mathematical Analysis}, 24 (1993), 1215. doi: 10.1137/0524069. Google Scholar [28] N. T. Redding and G. N. Newsam, Inverting the circular Radon transform,, \emph{DTSO Research Report DTSO-Ru-0211}, (2001). Google Scholar [29] H. Rhee, A representation of the solutions of the Darboux equation in odd-dimensional spaces,, \emph{Transactions of the American Mathematical Society}, 150 (1970), 491. 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,, \emph{Bulletin of the American Mathematical Society}, 82 (1977), 1227. Google Scholar [31] A. E. Yagle, Inversion of spherical means using geometric inversion and Radon transforms,, \emph{Inverse Problems}, 8 (1992). Google Scholar [32] C. E. Yarman and B. Yazici, Inversion of the circular averages transform using the Funk transform,, \emph{Inverse Problems}, 27 (2011). doi: 10.1088/0266-5611/27/6/065001. Google Scholar [33] L. Zalcman, Offbeat integral geometry,, \emph{The American Mathematical Monthly}, 87 (1980), 161. doi: 10.2307/2321600. Google Scholar

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##### References:
 [1] L. Andersson, On the determination of a function from spherical averages,, \emph{SIAM Journal on Mathematical Analysis}, 19 (1988), 214. doi: 10.1137/0519016. Google Scholar [2] A. M. Cormack, Representation of a function by its line integrals, with some radiological applications,, \emph{Journal of Applied Physics}, 34 (1963), 2722. Google Scholar [3] A. M. Cormack, Representation of a function by its line integrals, with some radiological applications. II,, \emph{Journal of Applied Physics}, 35 (1964), 2908. 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,, \emph{Transactions of the American Mathematical Society}, 260 (1980), 575. doi: 10.2307/1998023. Google Scholar [5] J. Fawcett, Inversion of $n$-dimensional spherical averages,, \emph{SIAM Journal on Applied Mathematics}, 45 (1985), 336. doi: 10.1137/0145018. Google Scholar [6] D. Finch, M. Haltmeier and Rakesh, Inversion of spherical means and the wave equation in even dimensions,, \emph{SIAM Journal on Applied Mathematics}, 68 (2007), 392. 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,, \emph{SIAM Journal on Mathematical Analysis}, 35 (2004), 1213. 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 \emph{Photoacoustic Imaging and Spectroscopy} (L. Wang ed.), (2009). Google Scholar [9] S. Gindikin, J. Reeds and L. Shepp, Spherical tomography and spherical integral geometry,, In \emph{Tomography, (1993), 7. Google Scholar [10] M. Haltmeier, Exact reconstruction formula for the spherical mean Radon transform on ellipsoids,, \emph{Inverse Problems}, 30 (2014). doi: 10.1088/0266-5611/30/10/105006. Google Scholar [11] S. Helgason, A duality in integral geometry: some generalizations of the Radon transform,, \emph{Bulletin of the American Mathematical Society}, 70 (1964), 435. Google Scholar [12] H. Hellsten and L. E. Andersson, An inverse method for the processing of synthetic aperture radar data,, \emph{Inverse Problems}, 3 (1987). 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,, \emph{Inverse Problems}, 23 (2007). 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,, \emph{Inverse Problems}, 23 (2007). 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,, \emph{Inverse Problems and Imaging}, 6 (2012), 111. doi: 10.3934/ipi.2012.6.111. Google Scholar [17] D. Ludwig, The Radon transform on Euclidean space,, \emph{Communications on Pure and Applied Mathematics}, 19 (1966), 49. Google Scholar [18] E. K. Narayanan and Rakesh, Spherical means with centers on a hyperplane in even dimensions,, \emph{Inverse Problems}, 26 (2010). 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, (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, (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,, \emph{Inverse Problems}, 26 (2010). 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 \emph{Aip Conference Proceedings}, (1281). Google Scholar [23] C. J. Nolan and M. Cheney, Synthetic aperture inversion,, \emph{Inverse Problems}, 18 (2002). 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,, \emph{The Journal of the Acoustical Society of America}, 67 (1980), 853. doi: 10.1121/1.384168. Google Scholar [25] E. T. Quinto, Null spaces and ranges for the classical and spherical Radon transforms,, \emph{Journal of Mathematical Analysis and Applications}, 90 (1982), 408. 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,, \emph{Journal of Mathematical Analysis and Applications}, 95 (1983), 437. 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 $\mathbbR^3$,, \emph{SIAM Journal on Mathematical Analysis}, 24 (1993), 1215. doi: 10.1137/0524069. Google Scholar [28] N. T. Redding and G. N. Newsam, Inverting the circular Radon transform,, \emph{DTSO Research Report DTSO-Ru-0211}, (2001). Google Scholar [29] H. Rhee, A representation of the solutions of the Darboux equation in odd-dimensional spaces,, \emph{Transactions of the American Mathematical Society}, 150 (1970), 491. 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,, \emph{Bulletin of the American Mathematical Society}, 82 (1977), 1227. Google Scholar [31] A. E. Yagle, Inversion of spherical means using geometric inversion and Radon transforms,, \emph{Inverse Problems}, 8 (1992). Google Scholar [32] C. E. Yarman and B. Yazici, Inversion of the circular averages transform using the Funk transform,, \emph{Inverse Problems}, 27 (2011). doi: 10.1088/0266-5611/27/6/065001. Google Scholar [33] L. Zalcman, Offbeat integral geometry,, \emph{The American Mathematical Monthly}, 87 (1980), 161. doi: 10.2307/2321600. Google Scholar
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