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

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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

Abstract
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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.

Keywords: tomography, circular-arc, Radon, spherical, transform., Compton.

Mathematics Subject Classification: Primary: 44A12, 65R10; Secondary: 92C5.

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.
[2]	A. M. Cormack, Representation of a function by its line integrals, with some radiological applications,, \emph{Journal of Applied Physics}, 34 (1963), 2722.
[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.
[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.
[5]	J. Fawcett, Inversion of $n$-dimensional spherical averages,, \emph{SIAM Journal on Applied Mathematics}, 45 (1985), 336. doi: 10.1137/0145018.
[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.
[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.
[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).
[9]	S. Gindikin, J. Reeds and L. Shepp, Spherical tomography and spherical integral geometry,, In \emph{Tomography, (1993), 7.
[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.
[11]	S. Helgason, A duality in integral geometry: some generalizations of the Radon transform,, \emph{Bulletin of the American Mathematical Society}, 70 (1964), 435.
[12]	H. Hellsten and L. E. Andersson, An inverse method for the processing of synthetic aperture radar data,, \emph{Inverse Problems}, 3 (1987).
[13]	F. John, Plane Waves and Spherical Means Applied to Partial Differential Equations,, Dover Books on Mathematics Series. Dover Publications, (2004).
[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.
[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.
[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.
[17]	D. Ludwig, The Radon transform on Euclidean space,, \emph{Communications on Pure and Applied Mathematics}, 19 (1966), 49.
[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.
[19]	F. Natterer, The Mathematics of Computerized Tomography,, Classics in Applied Mathematics. Society for Industrial and Applied Mathematics, (2001). doi: 10.1137/1.9780898719284.
[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.
[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.
[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).
[23]	C. J. Nolan and M. Cheney, Synthetic aperture inversion,, \emph{Inverse Problems}, 18 (2002). doi: 10.1088/0266-5611/18/1/315.
[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.
[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.
[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.
[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.
[28]	N. T. Redding and G. N. Newsam, Inverting the circular Radon transform,, \emph{DTSO Research Report DTSO-Ru-0211}, (2001).
[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.
[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.
[31]	A. E. Yagle, Inversion of spherical means using geometric inversion and Radon transforms,, \emph{Inverse Problems}, 8 (1992).
[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.
[33]	L. Zalcman, Offbeat integral geometry,, \emph{The American Mathematical Monthly}, 87 (1980), 161. doi: 10.2307/2321600.

show all references

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.
[2]	A. M. Cormack, Representation of a function by its line integrals, with some radiological applications,, \emph{Journal of Applied Physics}, 34 (1963), 2722.
[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.
[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.
[5]	J. Fawcett, Inversion of $n$-dimensional spherical averages,, \emph{SIAM Journal on Applied Mathematics}, 45 (1985), 336. doi: 10.1137/0145018.
[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.
[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.
[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).
[9]	S. Gindikin, J. Reeds and L. Shepp, Spherical tomography and spherical integral geometry,, In \emph{Tomography, (1993), 7.
[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.
[11]	S. Helgason, A duality in integral geometry: some generalizations of the Radon transform,, \emph{Bulletin of the American Mathematical Society}, 70 (1964), 435.
[12]	H. Hellsten and L. E. Andersson, An inverse method for the processing of synthetic aperture radar data,, \emph{Inverse Problems}, 3 (1987).
[13]	F. John, Plane Waves and Spherical Means Applied to Partial Differential Equations,, Dover Books on Mathematics Series. Dover Publications, (2004).
[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.
[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.
[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.
[17]	D. Ludwig, The Radon transform on Euclidean space,, \emph{Communications on Pure and Applied Mathematics}, 19 (1966), 49.
[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.
[19]	F. Natterer, The Mathematics of Computerized Tomography,, Classics in Applied Mathematics. Society for Industrial and Applied Mathematics, (2001). doi: 10.1137/1.9780898719284.
[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.
[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.
[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).
[23]	C. J. Nolan and M. Cheney, Synthetic aperture inversion,, \emph{Inverse Problems}, 18 (2002). doi: 10.1088/0266-5611/18/1/315.
[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.
[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.
[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.
[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.
[28]	N. T. Redding and G. N. Newsam, Inverting the circular Radon transform,, \emph{DTSO Research Report DTSO-Ru-0211}, (2001).
[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.
[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.
[31]	A. E. Yagle, Inversion of spherical means using geometric inversion and Radon transforms,, \emph{Inverse Problems}, 8 (1992).
[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.
[33]	L. Zalcman, Offbeat integral geometry,, \emph{The American Mathematical Monthly}, 87 (1980), 161. doi: 10.2307/2321600.

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