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A multiphase logic framework for multichannel image segmentation
Fast reconstruction algorithms for the thermoacoustic tomography in certain domains with cylindrical or spherical symmetries
1. | Department of Mathematics, University of Arizona, 617 N Santa Rita Ave, Tucson, AZ 85721, United States |
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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