# American Institute of Mathematical Sciences

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

 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

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.
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. 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. 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. 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). 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, 52 (2005), 1577. 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. 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). doi: 10.1103/PhysRevE.75.046706. Google Scholar [8] D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory,", Applied Mathematical Sciences, 93 (1992). Google Scholar [9] A. Dutt and V. Rokhlin, Fast fourier transforms for nonequispaced data,, SIAM J. Sci. Comput., 14 (1993), 1368. doi: 10.1137/0914081. Google Scholar [10] J. A. Fawcett, Inversion of $n$-dimensional spherical averages,, SIAM J. Appl. Math., 45 (1985), 336. 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. 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. 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. 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. 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. 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). 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. doi: 10.1118/1.597429. Google Scholar [18] P. Kuchment and L. Kunyansky, Mathematics of photoacoustic and thermoacoustic tomography,, in, (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. 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). 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. doi: 10.1007/BF01261607. Google Scholar [22] F. Natterer, "The Mathematics of Computerized Tomography,", B. G. Teubner, (1986). Google Scholar [23] F. Natterer and F. Wübbeling, "Mathematical Methods in Image Reconstruction,", SIAM Monographs on Mathematical Modeling and Computation, (2001). Google Scholar [24] L. Nguyen, A family of inversion formulas in thermoacoustic tomography,, Inverse Problems and Imaging, 3 (2009), 649. 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. 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. 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. 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. 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). 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. 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, 275/276 (1998), 433. 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. Google Scholar [33] R. Suda and M. Takami, A fast spherical harmonics transform algorithm,, Mathematics of Computation, 71 (2002), 703. 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, 3 (1971). Google Scholar [35] L. Wang, ed., "Photoacoustic Imaging and Spectroscopy,", CRC Press, (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. 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). 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. 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. 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. 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). 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, 52 (2005), 1577. 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. 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). doi: 10.1103/PhysRevE.75.046706. Google Scholar [8] D. Colton and R. Kress, "Inverse Acoustic and Electromagnetic Scattering Theory,", Applied Mathematical Sciences, 93 (1992). Google Scholar [9] A. Dutt and V. Rokhlin, Fast fourier transforms for nonequispaced data,, SIAM J. Sci. Comput., 14 (1993), 1368. doi: 10.1137/0914081. Google Scholar [10] J. A. Fawcett, Inversion of $n$-dimensional spherical averages,, SIAM J. Appl. Math., 45 (1985), 336. 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. 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. 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. 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. 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. 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). 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. doi: 10.1118/1.597429. Google Scholar [18] P. Kuchment and L. Kunyansky, Mathematics of photoacoustic and thermoacoustic tomography,, in, (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. 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). 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. doi: 10.1007/BF01261607. Google Scholar [22] F. Natterer, "The Mathematics of Computerized Tomography,", B. G. Teubner, (1986). Google Scholar [23] F. Natterer and F. Wübbeling, "Mathematical Methods in Image Reconstruction,", SIAM Monographs on Mathematical Modeling and Computation, (2001). Google Scholar [24] L. Nguyen, A family of inversion formulas in thermoacoustic tomography,, Inverse Problems and Imaging, 3 (2009), 649. 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. 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. 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. 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. 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). 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. 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, 275/276 (1998), 433. 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. Google Scholar [33] R. Suda and M. Takami, A fast spherical harmonics transform algorithm,, Mathematics of Computation, 71 (2002), 703. 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, 3 (1971). Google Scholar [35] L. Wang, ed., "Photoacoustic Imaging and Spectroscopy,", CRC Press, (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. 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). doi: 10.1103/PhysRevE.71.016706. Google Scholar
 [1] 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 [2] 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 [3] 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 [4] 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 [5] 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 [6] 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 [7] Linh V. Nguyen. Spherical mean transform: A PDE approach. Inverse Problems & Imaging, 2013, 7 (1) : 243-252. doi: 10.3934/ipi.2013.7.243 [8] 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 [9] 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 [10] 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 [11] 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 [12] 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 [13] 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 [14] Gemma Huguet, Rafael de la Llave, Yannick Sire. Computation of whiskered invariant tori and their associated manifolds: New fast algorithms. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1309-1353. doi: 10.3934/dcds.2012.32.1309 [15] 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 [16] Antoni Ferragut, Jaume Llibre, Adam Mahdi. Polynomial inverse integrating factors for polynomial vector fields. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 387-395. doi: 10.3934/dcds.2007.17.387 [17] Dieter Armbruster, Michael Herty, Xinping Wang, Lindu Zhao. Integrating release and dispatch policies in production models. Networks & Heterogeneous Media, 2015, 10 (3) : 511-526. doi: 10.3934/nhm.2015.10.511 [18] Alexander Barg, Oleg R. Musin. Codes in spherical caps. Advances in Mathematics of Communications, 2007, 1 (1) : 131-149. doi: 10.3934/amc.2007.1.131 [19] Jan Boman. A local uniqueness theorem for weighted Radon transforms. Inverse Problems & Imaging, 2010, 4 (4) : 631-637. doi: 10.3934/ipi.2010.4.631 [20] Marianna Euler, Norbert Euler. Integrating factors and conservation laws for some Camassa-Holm type equations. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1421-1430. doi: 10.3934/cpaa.2012.11.1421

2018 Impact Factor: 1.469

## Metrics

• PDF downloads (8)
• HTML views (0)
• Cited by (13)

## Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]