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

Leonid Kunyansky

doi:10.3934/ipi.2012.6.111

Previous Article
The order of convergence for Landweber Scheme with $\alpha,\beta$-rule
IPI Home
This Issue
Next Article
A multiphase logic framework for multichannel image segmentation

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
Related Papers

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 & 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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[8]	D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory," Applied Mathematical Sciences, 93, Springer-Verlag, Berlin, 1992. Google Scholar
[9]	A. Dutt and V. Rokhlin, Fast fourier transforms for nonequispaced data, SIAM J. Sci. Comput., 14 (1993), 1368-1393. doi: 10.1137/0914081. Google Scholar
[10]	J. A. Fawcett, Inversion of $n$-dimensional spherical averages, SIAM J. Appl. Math., 45 (1985), 336-341. doi: 10.1137/0145018. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[21]	M. J. Mohlenkamp, A fast transform for spherical harmonics, J. Fourier Anal. Appl., 5 (1999), 159-184. doi: 10.1007/BF01261607. Google Scholar
[22]	F. Natterer, "The Mathematics of Computerized Tomography," B. G. Teubner, Stuttgart, John Wiley & Sons, Ltd., Chichester, 1986. Google Scholar
[23]	F. Natterer and F. Wübbeling, "Mathematical Methods in Image Reconstruction," SIAM Monographs on Mathematical Modeling and Computation, SIAM, Philadelphia, PA, 2001. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[32]	A. G. Ramm, Injectivity of the spherical means operator, C. R. Math. Acad. Sci. Paris, 335 (2002), 1033-1038. Google Scholar
[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. Google Scholar
[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. Google Scholar
[35]	L. Wang, ed., "Photoacoustic Imaging and Spectroscopy," CRC Press, Boca Raton, FL, 2009. Google Scholar
[36]	L. V. Wang and H. Wu, "Biomedical Optics. Principles and Imaging," Wiley-Interscience, 2007. Google Scholar
[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. Google Scholar
[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. Google Scholar

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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[8]	D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory," Applied Mathematical Sciences, 93, Springer-Verlag, Berlin, 1992. Google Scholar
[9]	A. Dutt and V. Rokhlin, Fast fourier transforms for nonequispaced data, SIAM J. Sci. Comput., 14 (1993), 1368-1393. doi: 10.1137/0914081. Google Scholar
[10]	J. A. Fawcett, Inversion of $n$-dimensional spherical averages, SIAM J. Appl. Math., 45 (1985), 336-341. doi: 10.1137/0145018. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[21]	M. J. Mohlenkamp, A fast transform for spherical harmonics, J. Fourier Anal. Appl., 5 (1999), 159-184. doi: 10.1007/BF01261607. Google Scholar
[22]	F. Natterer, "The Mathematics of Computerized Tomography," B. G. Teubner, Stuttgart, John Wiley & Sons, Ltd., Chichester, 1986. Google Scholar
[23]	F. Natterer and F. Wübbeling, "Mathematical Methods in Image Reconstruction," SIAM Monographs on Mathematical Modeling and Computation, SIAM, Philadelphia, PA, 2001. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[32]	A. G. Ramm, Injectivity of the spherical means operator, C. R. Math. Acad. Sci. Paris, 335 (2002), 1033-1038. Google Scholar
[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. Google Scholar
[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. Google Scholar
[35]	L. Wang, ed., "Photoacoustic Imaging and Spectroscopy," CRC Press, Boca Raton, FL, 2009. Google Scholar
[36]	L. V. Wang and H. Wu, "Biomedical Optics. Principles and Imaging," Wiley-Interscience, 2007. Google Scholar
[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. Google Scholar
[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. Google Scholar

[1]	Chase Mathison. Thermoacoustic Tomography with circular integrating detectors and variable wave speed. Inverse Problems & Imaging, 2020, 14 (4) : 665-682. doi: 10.3934/ipi.2020030
[2]	Alberto Ibort, Alberto López-Yela. Quantum tomography and the quantum Radon transform. Inverse Problems & Imaging, 2021, 15 (5) : 893-928. doi: 10.3934/ipi.2021021
[3]	Victor Palamodov. Remarks on the general Funk transform and thermoacoustic tomography. Inverse Problems & Imaging, 2010, 4 (4) : 693-702. doi: 10.3934/ipi.2010.4.693
[4]	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
[5]	Hans Rullgård, Eric Todd Quinto. Local Sobolev estimates of a function by means of its Radon transform. Inverse Problems & Imaging, 2010, 4 (4) : 721-734. doi: 10.3934/ipi.2010.4.721
[6]	Linh V. Nguyen. A family of inversion formulas in thermoacoustic tomography. Inverse Problems & Imaging, 2009, 3 (4) : 649-675. doi: 10.3934/ipi.2009.3.649
[7]	Simon Gindikin. A remark on the weighted Radon transform on the plane. Inverse Problems & Imaging, 2010, 4 (4) : 649-653. doi: 10.3934/ipi.2010.4.649
[8]	Michael Krause, Jan Marcel Hausherr, Walter Krenkel. Computing the fibre orientation from Radon data using local Radon transform. Inverse Problems & Imaging, 2011, 5 (4) : 879-891. doi: 10.3934/ipi.2011.5.879
[9]	Linh V. Nguyen. Spherical mean transform: A PDE approach. Inverse Problems & Imaging, 2013, 7 (1) : 243-252. doi: 10.3934/ipi.2013.7.243
[10]	Mark Agranovsky, David Finch, Peter Kuchment. Range conditions for a spherical mean transform. Inverse Problems & Imaging, 2009, 3 (3) : 373-382. doi: 10.3934/ipi.2009.3.373
[11]	Ali Gholami, Mauricio D. Sacchi. Time-invariant radon transform by generalized Fourier slice theorem. Inverse Problems & Imaging, 2017, 11 (3) : 501-519. doi: 10.3934/ipi.2017023
[12]	Gaik Ambartsoumian, Leonid Kunyansky. Exterior/interior problem for the circular means transform with applications to intravascular imaging. Inverse Problems & Imaging, 2014, 8 (2) : 339-359. doi: 10.3934/ipi.2014.8.339
[13]	Gabriella Bretti, Roberto Natalini, Benedetto Piccoli. Fast algorithms for the approximation of a traffic flow model on networks. Discrete & Continuous Dynamical Systems - B, 2006, 6 (3) : 427-448. doi: 10.3934/dcdsb.2006.6.427
[14]	Chenchen Wu, Wei Lv, Yujie Wang, Dachuan Xu. Approximation algorithm for spherical $ k $-means problem with penalty. Journal of Industrial & Management Optimization, 2021 doi: 10.3934/jimo.2021067
[15]	C E Yarman, B Yazıcı. A new exact inversion method for exponential Radon transform using the harmonic analysis of the Euclidean motion group. Inverse Problems & Imaging, 2007, 1 (3) : 457-479. doi: 10.3934/ipi.2007.1.457
[16]	Jean-François Crouzet. 3D coded aperture imaging, ill-posedness and link with incomplete data radon transform. Inverse Problems & Imaging, 2011, 5 (2) : 341-353. doi: 10.3934/ipi.2011.5.341
[17]	Lu Tan, Ling Li, Senjian An, Zhenkuan Pan. Nonlinear diffusion based image segmentation using two fast algorithms. Mathematical Foundations of Computing, 2019, 2 (2) : 149-168. doi: 10.3934/mfc.2019011
[18]	Ningyu Sha, Lei Shi, Ming Yan. Fast algorithms for robust principal component analysis with an upper bound on the rank. Inverse Problems & Imaging, 2021, 15 (1) : 109-128. doi: 10.3934/ipi.2020067
[19]	Gemma Huguet, Rafael de la Llave, Yannick Sire. Computation of whiskered invariant tori and their associated manifolds: New fast algorithms. Discrete & Continuous Dynamical Systems, 2012, 32 (4) : 1309-1353. doi: 10.3934/dcds.2012.32.1309
[20]	Antoni Ferragut, Jaume Llibre, Adam Mahdi. Polynomial inverse integrating factors for polynomial vector fields. Discrete & Continuous Dynamical Systems, 2007, 17 (2) : 387-395. doi: 10.3934/dcds.2007.17.387

2020 Impact Factor: 1.639

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors