# American Institute of Mathematical Sciences

November  2012, 6(4): 645-661. doi: 10.3934/ipi.2012.6.645

## Efficient and accurate computation of spherical mean values at scattered center points

 1 University Osnabrück, Institute of Mathematics, 49069 Osnabrück, Germany 2 University Chemnitz, Department of Mathematics, 09107 Chemnitz, Germany 3 University Osnabrück, Institute of Mathematics, 49069 Osnabrück, and, Helmholtz Zentrum München, Institute for Biomathematics and Biometry, 85764 Neuherberg, Germany

Received  December 2011 Revised  September 2012 Published  November 2012

Spherical means are a widespread model in modern imaging modalities like photoacoustic tomography. Besides direct inversion methods for specific geometries, iterative methods are often used as reconstruction scheme such that each iteration asks for the efficient and accurate computation of spherical means. We consider a spectral discretization via trigonometric polynomials such that the computation can be done via nonequispaced fast Fourier transforms. Moreover, a recently developed sparse fast Fourier transform is used in the three dimensional case and gives optimal arithmetic complexity. All theoretical results are illustrated by numerical experiments.
Citation: Torsten Görner, Ralf Hielscher, Stefan Kunis. Efficient and accurate computation of spherical mean values at scattered center points. Inverse Problems and Imaging, 2012, 6 (4) : 645-661. doi: 10.3934/ipi.2012.6.645
##### References:
 [1] M. Agranovsky and P. Kuchment, Uniqueness of reconstruction and an inversion procedure for thermoacoustic and photoacoustic tomography with variable sound speed, Inverse Problems, 23 (2007), 2089-2102. doi: 10.1088/0266-5611/23/5/016. [2] M. Agranovsky, P. Kuchment and L. Kunyansky, On reconstruction formulas and algorithms for the thermoacoustic tomography, in "Photoacoustic Imaging and Spectroscopy" (ed. L. V. Wang), chapter 8, CRC Press, Boca Raton, FL, (2009), 89-101. doi: 10.1201/9781420059922.ch8. [3] M. Agranovsky, P. Kuchment and E. T. Quinto, Range descriptions for the spherical mean Radon transform, J. Funct. Anal., 248 (2007), 344-386. doi: 10.1016/j.jfa.2007.03.022. [4] A. Buehler, A. Rosenthal, T. Jetzfellner, A. Dima, D. Razansky and V. Ntziachristos, Model-based optoacoustic inversions with incomplete projection data, Med. Phys., 38 (1694), 2011. [5] P. Burgholzer, G. J. Matt, M. Haltmeier and G. Paltauf, Exact and approximate imaging methods for photoacoustic tomography using an arbitrary detection surface, Phys. Rev. E, 75 (2007), 046706. doi: 10.1103/PhysRevE.75.046706. [6] Y. Dong, T. Görner and S. Kunis, An iterative reconstruction scheme for photoacoustic imaging, preprint, 2011. [7] F. Filbir, R. Hielscher and W. R. Madych, Reconstruction from circular and spherical mean data, Appl. Comput. Harmon. Anal., 29 (2010), 111-120. doi: 10.1016/j.acha.2009.10.001. [8] D. Finch, M. Haltmeier and Rakesh, Inversion of spherical means and the wave equation in even dimensions, SIAM J. Appl. Math., 68 (2007), 392-412. doi: 10.1137/070682137. [9] M. Haltmeier, A mollification approach for inverting the spherical mean Radon transform, SIAM J. Appl. Math., 71 (2011), 1637-1652. doi: 10.1137/110821561. [10] M. Haltmeier, Inversion of circular means and the wave equation on convex planar domains, arXiv:1206.1246, 2012. [11] M. Haltmeier, Universal inversion formulas for recovering a function from spherical means, arXiv:1206.3424, 2012. [12] M. Haltmeier, O. Scherzer, P. Burgholzer and G. Paltauf, Thermoacoustic computed tomography with large planar receivers, Inverse Problems, 20 (2004), 1663-1673. doi: 10.1088/0266-5611/20/5/021. [13] M. Haltmeier, O. Scherzer and G. Zangerl, A reconstruction algorithm for photoacoustic imaging based on the nonuniform FFT, IEEE Trans. Med. Imag., 28 (2009), 1727-1735. doi: 10.1109/TMI.2009.2022623. [14] M. Haltmeier, T. Schuster and O. Scherzer, Filtered backprojection for thermoacoustic computed tomography in spherical geometry, Math. Methods Appl. Sci., 28 (2005), 1919-1937. doi: 10.1002/mma.648. [15] M. Haltmeier and G. Zangerl, Spatial resolution in photoacoustic tomography: Effects of detector size and detector bandwidth, Inverse Problems, 26 (2010), 125002, 14 pp. [16] F. John, "Plane Waves and Spherical Means Applied to Partial Differential Equations," Reprint of the 1955 original, Dover Publications, Inc., Mineola, NY, 2004. [17] J. Keiner, S. Kunis and D. Potts, Using {NFFT 3-a software library for various nonequispaced fast Fourier transforms}, ACM Trans. Math. Software, 36 (2009), Art. 19, 30 pp. [18] P. Kuchment and L. Kunyansky, Mathematics of thermoacoustic tomography, European J. Appl. Math., 19 (2008), 191-224. [19] S. Kunis and I. Melzer, A stable and accurate butterfly sparse Fourier transform, SIAM J. Numer. Anal., 50 (2012), 1777-1800. doi: 10.1137/110839825. [20] L. Kunyansky, Reconstruction of a function from its spherical (circular) means with the centers lying on the surface of certain polygons and polyhedra, Inverse Problems, 27 (2011), 025012, 22 pp. [21] L. A. Kunyansky, Explicit inversion formulae for the spherical mean Radon transform, Inverse Problems, 23 (2007), 373-383. [22] L. A. Kunyansky, A series solution and a fast algorithm for the inversion of the spherical mean Radon transform, Inverse Problems, 23 (2007), S11-S20. doi: 10.1088/0266-5611/23/6/S02. [23] F. Natterer, Photo-acoustic inversion in convex domains, Inverse Probl. Imaging, 6 (2012), 1-6. doi: 10.3934/ipi.2012.6.315. [24] G. Paltauf, R. Nuster, M. Haltmeier and P. Burgholzer, Photoacoustic tomography with integrating area and line detectors, in "Photoacoustic Imaging and Spectroscopy" (ed. L. V. Wang), Optical Science and Engineering, chapter 20, CRC Press, Boca Raton, FL, (2009), 251-263. [25] E. T. Quinto, Helgason's support theorem and spherical Radon transforms, in "Radon Transforms, Geometry, and Wavelets," Contemp. Math., 464, Amer. Math. Soc., Providence, RI, (2008), 249-264. [26] L. V. Wang and H. Wu, "Biomedical Optics - Principles and Imaging," John Wiley & Sons Inc., Hoboken, NJ, 2007. [27] G. N. Watson, "A Treatise on the Theory of Bessel Functions," Second edition, Cambridge University Press, Cambridge, 1966. [28] L. Ying, Sparse Fourier transform via butterfly algorithm, SIAM J. Sci. Comput., 31 (2009), 1678-1694. doi: 10.1137/08071291X. [29] G. Zangerl and O. Scherzer, Exact reconstruction in photoacoustic tomography with circular integrating detectors II: Spherical geometry, Math. Methods Appl. Sci., 33 (2010), 1771-1782. doi: 10.1002/mma.1266. [30] G. Zangerl, O. Scherzer and M. Haltmeier, Exact series reconstruction in photoacoustic tomography with circular integrating detectors, Commun. Math. Sci., 7 (2009), 665-678. [31] A. Zygmund, "Trigonometric Series. Vol. I, II," Third edition, With a foreword by Robert A. Fefferman, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2002.

show all references

##### References:
 [1] M. Agranovsky and P. Kuchment, Uniqueness of reconstruction and an inversion procedure for thermoacoustic and photoacoustic tomography with variable sound speed, Inverse Problems, 23 (2007), 2089-2102. doi: 10.1088/0266-5611/23/5/016. [2] M. Agranovsky, P. Kuchment and L. Kunyansky, On reconstruction formulas and algorithms for the thermoacoustic tomography, in "Photoacoustic Imaging and Spectroscopy" (ed. L. V. Wang), chapter 8, CRC Press, Boca Raton, FL, (2009), 89-101. doi: 10.1201/9781420059922.ch8. [3] M. Agranovsky, P. Kuchment and E. T. Quinto, Range descriptions for the spherical mean Radon transform, J. Funct. Anal., 248 (2007), 344-386. doi: 10.1016/j.jfa.2007.03.022. [4] A. Buehler, A. Rosenthal, T. Jetzfellner, A. Dima, D. Razansky and V. Ntziachristos, Model-based optoacoustic inversions with incomplete projection data, Med. Phys., 38 (1694), 2011. [5] P. Burgholzer, G. J. Matt, M. Haltmeier and G. Paltauf, Exact and approximate imaging methods for photoacoustic tomography using an arbitrary detection surface, Phys. Rev. E, 75 (2007), 046706. doi: 10.1103/PhysRevE.75.046706. [6] Y. Dong, T. Görner and S. Kunis, An iterative reconstruction scheme for photoacoustic imaging, preprint, 2011. [7] F. Filbir, R. Hielscher and W. R. Madych, Reconstruction from circular and spherical mean data, Appl. Comput. Harmon. Anal., 29 (2010), 111-120. doi: 10.1016/j.acha.2009.10.001. [8] D. Finch, M. Haltmeier and Rakesh, Inversion of spherical means and the wave equation in even dimensions, SIAM J. Appl. Math., 68 (2007), 392-412. doi: 10.1137/070682137. [9] M. Haltmeier, A mollification approach for inverting the spherical mean Radon transform, SIAM J. Appl. Math., 71 (2011), 1637-1652. doi: 10.1137/110821561. [10] M. Haltmeier, Inversion of circular means and the wave equation on convex planar domains, arXiv:1206.1246, 2012. [11] M. Haltmeier, Universal inversion formulas for recovering a function from spherical means, arXiv:1206.3424, 2012. [12] M. Haltmeier, O. Scherzer, P. Burgholzer and G. Paltauf, Thermoacoustic computed tomography with large planar receivers, Inverse Problems, 20 (2004), 1663-1673. doi: 10.1088/0266-5611/20/5/021. [13] M. Haltmeier, O. Scherzer and G. Zangerl, A reconstruction algorithm for photoacoustic imaging based on the nonuniform FFT, IEEE Trans. Med. Imag., 28 (2009), 1727-1735. doi: 10.1109/TMI.2009.2022623. [14] M. Haltmeier, T. Schuster and O. Scherzer, Filtered backprojection for thermoacoustic computed tomography in spherical geometry, Math. Methods Appl. Sci., 28 (2005), 1919-1937. doi: 10.1002/mma.648. [15] M. Haltmeier and G. Zangerl, Spatial resolution in photoacoustic tomography: Effects of detector size and detector bandwidth, Inverse Problems, 26 (2010), 125002, 14 pp. [16] F. John, "Plane Waves and Spherical Means Applied to Partial Differential Equations," Reprint of the 1955 original, Dover Publications, Inc., Mineola, NY, 2004. [17] J. Keiner, S. Kunis and D. Potts, Using {NFFT 3-a software library for various nonequispaced fast Fourier transforms}, ACM Trans. Math. Software, 36 (2009), Art. 19, 30 pp. [18] P. Kuchment and L. Kunyansky, Mathematics of thermoacoustic tomography, European J. Appl. Math., 19 (2008), 191-224. [19] S. Kunis and I. Melzer, A stable and accurate butterfly sparse Fourier transform, SIAM J. Numer. Anal., 50 (2012), 1777-1800. doi: 10.1137/110839825. [20] L. Kunyansky, Reconstruction of a function from its spherical (circular) means with the centers lying on the surface of certain polygons and polyhedra, Inverse Problems, 27 (2011), 025012, 22 pp. [21] L. A. Kunyansky, Explicit inversion formulae for the spherical mean Radon transform, Inverse Problems, 23 (2007), 373-383. [22] L. A. Kunyansky, A series solution and a fast algorithm for the inversion of the spherical mean Radon transform, Inverse Problems, 23 (2007), S11-S20. doi: 10.1088/0266-5611/23/6/S02. [23] F. Natterer, Photo-acoustic inversion in convex domains, Inverse Probl. Imaging, 6 (2012), 1-6. doi: 10.3934/ipi.2012.6.315. [24] G. Paltauf, R. Nuster, M. Haltmeier and P. Burgholzer, Photoacoustic tomography with integrating area and line detectors, in "Photoacoustic Imaging and Spectroscopy" (ed. L. V. Wang), Optical Science and Engineering, chapter 20, CRC Press, Boca Raton, FL, (2009), 251-263. [25] E. T. Quinto, Helgason's support theorem and spherical Radon transforms, in "Radon Transforms, Geometry, and Wavelets," Contemp. Math., 464, Amer. Math. Soc., Providence, RI, (2008), 249-264. [26] L. V. Wang and H. Wu, "Biomedical Optics - Principles and Imaging," John Wiley & Sons Inc., Hoboken, NJ, 2007. [27] G. N. Watson, "A Treatise on the Theory of Bessel Functions," Second edition, Cambridge University Press, Cambridge, 1966. [28] L. Ying, Sparse Fourier transform via butterfly algorithm, SIAM J. Sci. Comput., 31 (2009), 1678-1694. doi: 10.1137/08071291X. [29] G. Zangerl and O. Scherzer, Exact reconstruction in photoacoustic tomography with circular integrating detectors II: Spherical geometry, Math. Methods Appl. Sci., 33 (2010), 1771-1782. doi: 10.1002/mma.1266. [30] G. Zangerl, O. Scherzer and M. Haltmeier, Exact series reconstruction in photoacoustic tomography with circular integrating detectors, Commun. Math. Sci., 7 (2009), 665-678. [31] A. Zygmund, "Trigonometric Series. Vol. I, II," Third edition, With a foreword by Robert A. Fefferman, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2002.
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