Fast reconstruction algorithms for the thermoacoustic tomography in certain domains with cylindrical or spherical symmetries

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February 2012, 6(1): 111-131. doi: 10.3934/ipi.2012.6.111

Fast reconstruction algorithms for the thermoacoustic tomography in certain domains with cylindrical or spherical symmetries

Leonid Kunyansky ^1,

1.	Department of Mathematics, University of Arizona, 617 N Santa Rita Ave, Tucson, AZ 85721, United States

Received February 2011 Revised August 2011 Published February 2012

Abstract
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We propose three fast algorithms for solving the inverse problem of the thermoacoustic tomography corresponding to certain acquisition geometries. Two of these methods are designed to process the measurements done with point-like detectors placed on a circle (in 2D) or a sphere (in 3D) surrounding the object of interest. The third inversion algorithm works with the data measured by the integrating line detectors arranged in a cylindrical assembly rotating around the object. The number of operations required by these techniques is equal to $\mathcal{O}(n^{3} \log n)$ and $\mathcal{O}(n^{3} \log^2 n)$ for the 3D techniques (assuming the reconstruction grid with $n^3$ nodes) and to $\mathcal{O}(n^{2} \log n)$ for the 2D problem with $n \times n$ discretizetion grid. Numerical simulations show that on large computational grids our methods are at least two orders of magnitude faster than the finite-difference time reversal techniques. The results of reconstructions from real measurements done by the integrating line detectors are also presented, to demonstrate the practicality of our algorithms.

Keywords: integrating detectors., fast algorithms, thermoacoustic tomography, Radon transform, spherical means.

Mathematics Subject Classification: Primary: 44A12, 65M32; Secondary: 92C5.

Citation: Leonid Kunyansky. Fast reconstruction algorithms for the thermoacoustic tomography in certain domains with cylindrical or spherical symmetries. Inverse Problems and Imaging, 2012, 6 (1) : 111-131. doi: 10.3934/ipi.2012.6.111

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]	G. Ambartsoumian and P. Kuchment, A range description for the planar circular Radon transform, SIAM J. Math. Anal., 38 (2006), 681-692. doi: 10.1137/050637492.
[3]	L.-E. Andersson, On the determination of a function from spherical averages, SIAM J. Math. Anal., 19 (1988), 214-232. doi: 10.1137/0519016.
[4]	P. Burgholzer, J. Bauer-Marschallinger, H. Grün, M. Haltmeier and G. Paltauf, Temporal back-projection algorithms for photoacoustic tomography with integrating line detectors, Inverse Problems, 23 (2007), S65-S80. doi: 10.1088/0266-5611/23/6/S06.
[5]	P. Burgholzer, C. Hofer, G. Paltauf, M. Haltmeier and O. Scherzer, Thermoacoustic tomography with integrating area and line detectors, IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 52 (2005), 1577-1583. doi: 10.1109/TUFFC.2005.1516030.
[6]	P. Burgholzer, C. Hofer, G. J. Matt, G. Paltauf, M. Haltmeier and O. Scherzer, Thermoacoustic tomography using a fiber-based Fabry-Perot interferometer as an integrating line detector, Proc. SPIE, 6086 (2006), 434-442.
[7]	P. Burgholzer, G. J. Matt, M. Haltmeier and G. Paltauf, Exact and approximative imaging methods for photoacoustic tomography using an arbitrary detection surface, Phys. Review E, 75 (2007), 046706. doi: 10.1103/PhysRevE.75.046706.
[8]	D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory," Applied Mathematical Sciences, 93, Springer-Verlag, Berlin, 1992.
[9]	A. Dutt and V. Rokhlin, Fast fourier transforms for nonequispaced data, SIAM J. Sci. Comput., 14 (1993), 1368-1393. doi: 10.1137/0914081.
[10]	J. A. Fawcett, Inversion of $n$-dimensional spherical averages, SIAM J. Appl. Math., 45 (1985), 336-341. doi: 10.1137/0145018.
[11]	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.
[12]	D. Finch, S. Patch and Rakesh, Determining a function from its mean values over a family of spheres, SIAM J. Math. Anal., 35 (2004), 1213-1240. doi: 10.1137/S0036141002417814.
[13]	M. Haltmeier, O. Scherzer, P. Burgholzer, R. Nuster and G. Paltauf, Thermoacoustic tomography and the circular radon transform: Exact inversion formula, Mathematical Models and Methods in Applied Sciences, 17 (2007), 635-655. doi: 10.1142/S0218202507002054.
[14]	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.
[15]	D. M. Healy, Jr., D. N. Rockmore, P. J. Kostelec and S. Moore, FFTs for the 2-sphere-Improvements and variations, J. Fourier Anal. and Appl., 9 (2003), 341-385. doi: 10.1007/s00041-003-0018-9.
[16]	Y. Hristova, P. Kuchment and L. Nguyen, On reconstruction and time reversal in thermoacoustic tomography in acoustically homogeneous and inhomogeneous media, Inverse Problems, 24 (2008), 055006, 25 pp.
[17]	R. A. Kruger, P. Liu, Y. R. Fang and C. R. Appledorn, Photoacoustic ultrasound (PAUS) reconstruction tomography, Med. Phys., 22 (1995), 1605-1609. doi: 10.1118/1.597429.
[18]	P. Kuchment and L. Kunyansky, Mathematics of photoacoustic and thermoacoustic tomography, in "Handbook of Mathematical Methods in Imaging" (ed. Otmar Scherzer), Springer, 2011. doi: 10.1007/978-0-387-92920-0_19.
[19]	L. Kunyansky, Explicit inversion formulae for the spherical mean Radon transform, Inverse Problems, 23 (2007), 373-383. doi: 10.1088/0266-5611/23/1/021.
[20]	L. 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.
[21]	M. J. Mohlenkamp, A fast transform for spherical harmonics, J. Fourier Anal. Appl., 5 (1999), 159-184. doi: 10.1007/BF01261607.
[22]	F. Natterer, "The Mathematics of Computerized Tomography," B. G. Teubner, Stuttgart, John Wiley & Sons, Ltd., Chichester, 1986.
[23]	F. Natterer and F. Wübbeling, "Mathematical Methods in Image Reconstruction," SIAM Monographs on Mathematical Modeling and Computation, SIAM, Philadelphia, PA, 2001.
[24]	L. Nguyen, A family of inversion formulas in thermoacoustic tomography, Inverse Problems and Imaging, 3 (2009), 649-675. doi: 10.3934/ipi.2009.3.649.
[25]	S. J. Norton, Reconstruction of a two-dimensional reflecting medium over a circular domain: Exact solution, J. Acoust. Soc. Am., 67 (1980), 1266-1273. doi: 10.1121/1.384168.
[26]	S. J. Norton and M. Linzer, Ultrasonic reflectivity imaging in three dimensions: Exact inverse scattering solutions for plane, cylindrical, and spherical apertures, IEEE Transactions on Biomedical Engineering, 28 (1981), 200-202.
[27]	A. A. Oraevsky, S. L. Jacques, R. O. Esenaliev and F. K. Tittel, Laser-based ptoacoustic imaging in biological tissues, Proc. SPIE, 2134 (1994), 122-128.
[28]	G. Paltauf, R. Nuster, M. Haltmeier and P. Burgholzer, Thermoacoustic computed tomography using a Mach-Zehnder interferometer as acoustic line detector, Appl. Opt., 46 (2007), 3352-3358. doi: 10.1364/AO.46.003352.
[29]	G. Paltauf, R. Nuster, M. Haltmeier and P. Burgholzer, Experimental evaluation of reconstruction algorithms for limited view photoacoustic tomography with line detectors, Inverse Problems, 23 (2007), S81-S94. doi: 10.1088/0266-5611/23/6/S07.
[30]	G. Paltauf, R. Nuster and P. Burgholzer, Weight factors for limited angle photoacoustic tomography, Phys. Med. Biol., 54 (2009), 3303-3314. doi: 10.1088/0031-9155/54/11/002.
[31]	D. Potts, G. Steidl and M. Tasche, Fast and stable algorithms for discrete spherical Fourier transforms, Proceedings of the Sixth Conference of the International Linear Algebra Society (Chemnitz, 1996), Linear Algebra Appl., 275/276 (1998), 433-450. doi: 10.1016/S0024-3795(97)10013-1.
[32]	A. G. Ramm, Injectivity of the spherical means operator, C. R. Math. Acad. Sci. Paris, 335 (2002), 1033-1038.
[33]	R. Suda and M. Takami, A fast spherical harmonics transform algorithm, Mathematics of Computation, 71 (2002), 703-715. doi: 10.1090/S0025-5718-01-01386-2.
[34]	V. S. Vladimirov, "Equations of Mathematical Physics," Translated from the Russian by Audrey Littlewood, Edited by Alan Jeffrey, Pure and Applied Mathematics, 3, Marcel Dekker, Inc., New York, 1971.
[35]	L. Wang, ed., "Photoacoustic Imaging and Spectroscopy," CRC Press, Boca Raton, FL, 2009.
[36]	L. V. Wang and H. Wu, "Biomedical Optics. Principles and Imaging," Wiley-Interscience, 2007.
[37]	M. Xu and L.-H. V. Wang, Time-domain reconstruction for thermoacoustic tomography in a spherical geometry, IEEE Trans. Med. Imag., 21 (2002), 814-822. doi: 10.1109/TMI.2002.801176.
[38]	M. Xu and L.-H. V. Wang, Universal back-projection algorithm for photoacoustic computed tomography, Phys. Rev. E, 71 (2005), 016706. doi: 10.1103/PhysRevE.71.016706.

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]	G. Ambartsoumian and P. Kuchment, A range description for the planar circular Radon transform, SIAM J. Math. Anal., 38 (2006), 681-692. doi: 10.1137/050637492.
[3]	L.-E. Andersson, On the determination of a function from spherical averages, SIAM J. Math. Anal., 19 (1988), 214-232. doi: 10.1137/0519016.
[4]	P. Burgholzer, J. Bauer-Marschallinger, H. Grün, M. Haltmeier and G. Paltauf, Temporal back-projection algorithms for photoacoustic tomography with integrating line detectors, Inverse Problems, 23 (2007), S65-S80. doi: 10.1088/0266-5611/23/6/S06.
[5]	P. Burgholzer, C. Hofer, G. Paltauf, M. Haltmeier and O. Scherzer, Thermoacoustic tomography with integrating area and line detectors, IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 52 (2005), 1577-1583. doi: 10.1109/TUFFC.2005.1516030.
[6]	P. Burgholzer, C. Hofer, G. J. Matt, G. Paltauf, M. Haltmeier and O. Scherzer, Thermoacoustic tomography using a fiber-based Fabry-Perot interferometer as an integrating line detector, Proc. SPIE, 6086 (2006), 434-442.
[7]	P. Burgholzer, G. J. Matt, M. Haltmeier and G. Paltauf, Exact and approximative imaging methods for photoacoustic tomography using an arbitrary detection surface, Phys. Review E, 75 (2007), 046706. doi: 10.1103/PhysRevE.75.046706.
[8]	D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory," Applied Mathematical Sciences, 93, Springer-Verlag, Berlin, 1992.
[9]	A. Dutt and V. Rokhlin, Fast fourier transforms for nonequispaced data, SIAM J. Sci. Comput., 14 (1993), 1368-1393. doi: 10.1137/0914081.
[10]	J. A. Fawcett, Inversion of $n$-dimensional spherical averages, SIAM J. Appl. Math., 45 (1985), 336-341. doi: 10.1137/0145018.
[11]	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.
[12]	D. Finch, S. Patch and Rakesh, Determining a function from its mean values over a family of spheres, SIAM J. Math. Anal., 35 (2004), 1213-1240. doi: 10.1137/S0036141002417814.
[13]	M. Haltmeier, O. Scherzer, P. Burgholzer, R. Nuster and G. Paltauf, Thermoacoustic tomography and the circular radon transform: Exact inversion formula, Mathematical Models and Methods in Applied Sciences, 17 (2007), 635-655. doi: 10.1142/S0218202507002054.
[14]	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.
[15]	D. M. Healy, Jr., D. N. Rockmore, P. J. Kostelec and S. Moore, FFTs for the 2-sphere-Improvements and variations, J. Fourier Anal. and Appl., 9 (2003), 341-385. doi: 10.1007/s00041-003-0018-9.
[16]	Y. Hristova, P. Kuchment and L. Nguyen, On reconstruction and time reversal in thermoacoustic tomography in acoustically homogeneous and inhomogeneous media, Inverse Problems, 24 (2008), 055006, 25 pp.
[17]	R. A. Kruger, P. Liu, Y. R. Fang and C. R. Appledorn, Photoacoustic ultrasound (PAUS) reconstruction tomography, Med. Phys., 22 (1995), 1605-1609. doi: 10.1118/1.597429.
[18]	P. Kuchment and L. Kunyansky, Mathematics of photoacoustic and thermoacoustic tomography, in "Handbook of Mathematical Methods in Imaging" (ed. Otmar Scherzer), Springer, 2011. doi: 10.1007/978-0-387-92920-0_19.
[19]	L. Kunyansky, Explicit inversion formulae for the spherical mean Radon transform, Inverse Problems, 23 (2007), 373-383. doi: 10.1088/0266-5611/23/1/021.
[20]	L. 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.
[21]	M. J. Mohlenkamp, A fast transform for spherical harmonics, J. Fourier Anal. Appl., 5 (1999), 159-184. doi: 10.1007/BF01261607.
[22]	F. Natterer, "The Mathematics of Computerized Tomography," B. G. Teubner, Stuttgart, John Wiley & Sons, Ltd., Chichester, 1986.
[23]	F. Natterer and F. Wübbeling, "Mathematical Methods in Image Reconstruction," SIAM Monographs on Mathematical Modeling and Computation, SIAM, Philadelphia, PA, 2001.
[24]	L. Nguyen, A family of inversion formulas in thermoacoustic tomography, Inverse Problems and Imaging, 3 (2009), 649-675. doi: 10.3934/ipi.2009.3.649.
[25]	S. J. Norton, Reconstruction of a two-dimensional reflecting medium over a circular domain: Exact solution, J. Acoust. Soc. Am., 67 (1980), 1266-1273. doi: 10.1121/1.384168.
[26]	S. J. Norton and M. Linzer, Ultrasonic reflectivity imaging in three dimensions: Exact inverse scattering solutions for plane, cylindrical, and spherical apertures, IEEE Transactions on Biomedical Engineering, 28 (1981), 200-202.
[27]	A. A. Oraevsky, S. L. Jacques, R. O. Esenaliev and F. K. Tittel, Laser-based ptoacoustic imaging in biological tissues, Proc. SPIE, 2134 (1994), 122-128.
[28]	G. Paltauf, R. Nuster, M. Haltmeier and P. Burgholzer, Thermoacoustic computed tomography using a Mach-Zehnder interferometer as acoustic line detector, Appl. Opt., 46 (2007), 3352-3358. doi: 10.1364/AO.46.003352.
[29]	G. Paltauf, R. Nuster, M. Haltmeier and P. Burgholzer, Experimental evaluation of reconstruction algorithms for limited view photoacoustic tomography with line detectors, Inverse Problems, 23 (2007), S81-S94. doi: 10.1088/0266-5611/23/6/S07.
[30]	G. Paltauf, R. Nuster and P. Burgholzer, Weight factors for limited angle photoacoustic tomography, Phys. Med. Biol., 54 (2009), 3303-3314. doi: 10.1088/0031-9155/54/11/002.
[31]	D. Potts, G. Steidl and M. Tasche, Fast and stable algorithms for discrete spherical Fourier transforms, Proceedings of the Sixth Conference of the International Linear Algebra Society (Chemnitz, 1996), Linear Algebra Appl., 275/276 (1998), 433-450. doi: 10.1016/S0024-3795(97)10013-1.
[32]	A. G. Ramm, Injectivity of the spherical means operator, C. R. Math. Acad. Sci. Paris, 335 (2002), 1033-1038.
[33]	R. Suda and M. Takami, A fast spherical harmonics transform algorithm, Mathematics of Computation, 71 (2002), 703-715. doi: 10.1090/S0025-5718-01-01386-2.
[34]	V. S. Vladimirov, "Equations of Mathematical Physics," Translated from the Russian by Audrey Littlewood, Edited by Alan Jeffrey, Pure and Applied Mathematics, 3, Marcel Dekker, Inc., New York, 1971.
[35]	L. Wang, ed., "Photoacoustic Imaging and Spectroscopy," CRC Press, Boca Raton, FL, 2009.
[36]	L. V. Wang and H. Wu, "Biomedical Optics. Principles and Imaging," Wiley-Interscience, 2007.
[37]	M. Xu and L.-H. V. Wang, Time-domain reconstruction for thermoacoustic tomography in a spherical geometry, IEEE Trans. Med. Imag., 21 (2002), 814-822. doi: 10.1109/TMI.2002.801176.
[38]	M. Xu and L.-H. V. Wang, Universal back-projection algorithm for photoacoustic computed tomography, Phys. Rev. E, 71 (2005), 016706. doi: 10.1103/PhysRevE.71.016706.

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