Journal of Mathematical Imaging and Vision

, Volume 61, Issue 7, pp 1007–1021 | Cite as

Reconstruction Algorithms for Photoacoustic Tomography in Heterogeneous Damping Media

  • Markus Haltmeier
  • Linh V. NguyenEmail author


In this article, we study several reconstruction methods for the inverse source problem of photoacoustic tomography with spatially variable sound speed and damping. The backbone of these methods is the adjoint operators, which we thoroughly analyze in both the \(L^2\)- and \(H^1\)-settings. They are casted in the form of a nonstandard wave equation. We derive the well posedness of the aforementioned wave equation in a natural functional space and also prove the finite speed of propagation. Under the uniqueness and visibility condition, our formulations of the standard iterative reconstruction methods, such as Landweber’s and conjugate gradients (CG), achieve a linear rate of convergence in either \(L^2\)- or \(H^1\)-norm. When the visibility condition is not satisfied, the problem is severely ill posed and one must apply a regularization technique to stabilize the solutions. To that end, we study two classes of regularization methods: (i) iterative and (ii) variational regularization. In the case of full data, our simulations show that the CG method works best; it is very fast and robust. In the ill-posed case, the CG method behaves unstably. Total variation regularization method (TV), in this case, significantly improves the reconstruction quality.


Photoacoustic tomography Tikhonov regularization Total variation Attenuation Visibility condition Adjoint operator Finite speed of propagation 



Linh Nguyen’s research is partially supported by the NSF grants DMS 1212125 and DMS 1616904. Markus Haltmeier acknowledges the support of the Austrian Science Fund (FWF), project P 30747.


  1. 1.
    Acosta, S., Palacios, B.: Thermoacoustic tomography for an integro-differential wave equation modeling attenuation. J. Differ. Equ. 5, 1984–2010 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Agranovsky, M., Kuchment, P.: Uniqueness of reconstruction and an inversion procedure for thermoacoustic and photoacoustic tomography with variable sound speed. Inverse Probl. 23, 2089 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ammari, H., Bretin, E., Garnier, J., Wahab, A.: Time reversal in attenuating acoustic media. Contemp. Math. 548, 151–163 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Ammari, H., Bretin, E., Jugnon, V., Wahab, A.: Photoacoustic imaging for attenuating acoustic media. In: Ammari, H. (ed.) Mathematical Modeling in Biomedical Imaging II, pp. 57–84. Springer (2012)Google Scholar
  5. 5.
    Arridge, S., Beard, P., Betcke, M., Cox, B., Huynh, N., Lucka, F., Ogunlade, O., Zhang, E.: Accelerated high-resolution photoacoustic tomography via compressed sensing. Phys. Med. Biol. 61, 8908 (2016)CrossRefGoogle Scholar
  6. 6.
    Arridge, S.R., Betcke, M.M., Cox, B.T., Lucka, F., Treeby, B.E.: On the adjoint operator in photoacoustic tomography. Inverse Probl. 32, 115012 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Barannyk, L.L., Frikel, J., Nguyen, L.V.: On artifacts in limited data spherical radon transform: curved observation surface. Inverse Probl. 32, 015012 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Belhachmi, Z., Glatz, T., Scherzer, O.: A direct method for photoacoustic tomography with inhomogeneous sound speed. Inverse Probl. 32, 045005 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Burgholzer, P., Grün, H., Haltmeier, M., Nuster, R., Paltauf, G.: Compensation of acoustic attenuation for high-resolution photoacoustic imaging with line detectors. Proc. SPIE 6437, 643724 (2007)CrossRefGoogle Scholar
  10. 10.
    Burgholzer, P., Matt, G.J., Haltmeier, M., Paltauf, G.: Exact and approximate imaging methods for photoacoustic tomography using an arbitrary detection surface. Phys. Rev. E 75, 046706 (2007)CrossRefGoogle Scholar
  11. 11.
    Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 40, 120–145 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Clason, C., Klibanov, M.V.: The quasi-reversibility method for thermoacoustic tomography in a heterogeneous medium. SIAM J. Sci. Comput. 30, 1–23 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Compani-Tabrizi, B.: K-space scattering formulation of the absorptive full fluid elastic scalar wave equation in the time domain. J. Acoust. Soc. Am. 79, 901–905 (1986)CrossRefGoogle Scholar
  14. 14.
    Cox, B., Kara, S., Arridge, S., Beard, P.: k-space propagation models for acoustically heterogeneous media: application to biomedical photoacoustics. J. Acoust. Soc. Am. 121, 3453–3464 (2007)CrossRefGoogle Scholar
  15. 15.
    Chen, W., Holm, S.: Fractional laplacian time-space models for linear and nonlinear lossy media exhibiting arbitrary frequency power-law dependency. J. Acoust. Soc. Am. 115, 1424–1430 (2004)CrossRefGoogle Scholar
  16. 16.
    Dean-Ben, X.L., Buehler, A., Ntziachristos, V., Razansky, D.: Accurate model-based reconstruction algorithm for three-dimensional optoacoustic tomography. IEEE Trans. Med. Imaging 31, 1922–1928 (2012)CrossRefGoogle Scholar
  17. 17.
    Elbau, P., Scherzer, O., Shi, C.: Singular values of the attenuated photoacoustic imaging operator. J. Differ. Equ. 263, 5330–5376 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems, vol. 375. Springer, Berlin (1996)CrossRefzbMATHGoogle Scholar
  19. 19.
    Finch, D., Haltmeier, M., Rakesh: Inversion of spherical means and the wave equation in even dimensions. SIAM J. Appl. Math. 68, 392–412 (2007)Google Scholar
  20. 20.
    Finch, D., Rakesh, Patch, S.K.: Determining a function from its mean values over a family of spheres. SIAM J. Math. Anal. 35, 1213–1240 (2004). (electronic)Google Scholar
  21. 21.
    Frikel, J., Quinto, E.T.: Artifacts in incomplete data tomography with applications to photoacoustic tomography and sonar. SIAM J. Math. Anal. 75, 703–725 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Haltmeier, M.: Inversion of circular means and the wave equation on convex planar domains. Comput. Math. Appl. 65, 1025–1036 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Haltmeier, M.: Universal inversion formulas for recovering a function from spherical means. SIAM J. Math. Anal. 46, 214–232 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Haltmeier, M., Nguyen, L.V.: Analysis of iterative methods in photoacoustic tomography with variable sound speed. SIAM J. Imaging Sci. 10, 751–781 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Hanke, M.: Conjugate Gradient Type Methods for Ill-Posed Problems, vol. 327. CRC Press, Boca Raton (1995)zbMATHGoogle Scholar
  26. 26.
    Homan, A.: Multi-wave imaging in attenuating media. Inverse Probl. Imaging 7, 1235–1250 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Hristova, Y.: Time reversal in thermoacoustic tomography—an error estimate. Inverse Probl. 25, 055008, 14 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Hristova, Y., Kuchment, P., Nguyen, L.: Reconstruction and time reversal in thermoacoustic tomography in acoustically homogeneous and inhomogeneous media. Inverse Probl. 24, 055006, 25 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Huang, C., Wang, K., Nie, L., Wang, L.V., Anastasio, M.A.: Full-wave iterative image reconstruction in photoacoustic tomography with acoustically inhomogeneous media. IEEE Trans. Med. Imaging 32, 1097–1110 (2013)CrossRefGoogle Scholar
  30. 30.
    Javaherian, A., Holman, S.: A multi-grid iterative method for photoacoustic tomography. IEEE Trans. Med. Imaging 36, 696–706 (2017)CrossRefGoogle Scholar
  31. 31.
    Kaltenbacher, B., Neubauer, A., Scherzer, O.: Iterative Regularization Methods for Nonlinear Ill-Posed Problems, Vol. 6 of Radon Series on Computational and Applied Mathematics. Walter de Gruyter GmbH & Co. KG, Berlin (2008)CrossRefzbMATHGoogle Scholar
  32. 32.
    Kowar, R., Scherzer, O., Bonnefond, X.: Causality analysis of frequency-dependent wave attenuation. Math. Methods Appl. Sci. 34, 108–124 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Kowar, R.: On time reversal in photoacoustic tomography for tissue similar to water. SIAM J. Imaging Sci. 7, 509–527 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Kowar, R., Scherzer,O.: Photoacoustic imaging taking into account attenuation. In: Ammari, H. (ed.) Mathematics and Algorithms in Tomography II, Lecture Notes in Mathematics 2035, pp. 85–130. Springer (2012)Google Scholar
  35. 35.
    Kuchment, P.: The Radon Transform and Medical Imaging, vol. 85. SIAM, Philadelphia (2014)zbMATHGoogle Scholar
  36. 36.
    Kuchment, P., Kunyansky, L.: Mathematics of thermoacoustic tomography. Eur. J. Appl. Math. 19, 191–224 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Kunyansky, L.A.: Explicit inversion formulae for the spherical mean Radon transform. Inverse Probl. 23, 373–383 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Kunyansky, L.A.: A series solution and a fast algorithm for the inversion of the spherical mean radon transform. Inverse Probl. 23, S11 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Leeman, S., Hutchins, L., Jones, J.P.: Bounded pulse propagation. In: Alais, P., Metbefell, A.E. (eds.) Acoustical Imaging, vol. 10, pp. 427–435. Plenum, Oxford (1982)CrossRefGoogle Scholar
  40. 40.
    La Riviere, P. J., Zhang, J., Anastasio, M. A.: Image reconstruction in optoacoustic tomography accounting for frequency-dependent attenuation. In: IEEE Nuclear Science Symposium Conference Record, p. 5 (2005)Google Scholar
  41. 41.
    La Riviére, P.J., Zhang, J., Anastasio, M.A.: Image reconstruction in optoacoustic tomography for dispersive acoustic media. Opt. Lett. 31, 781–783 (2006)CrossRefGoogle Scholar
  42. 42.
    Liebler, M., Ginter, S., Dreyer, T., Riedlinger, R.E.: Full wave modeling of therapeutic ultrasound: efficient time-domain implementation of the frequency power-law attenuation. J. Acoust. Soc. Am. 116, 2742–2750 (2004)CrossRefGoogle Scholar
  43. 43.
    Nachman, A.I., Smith III, J.F., Waag, R.C.: An equation for acoustic propagation in inhomogeneous media with relaxation losses. J. Acoust. Soc. Am. 88, 1584–1595 (1990)CrossRefGoogle Scholar
  44. 44.
    Natterer, F.: Photo-acoustic inversion in convex domains. Inverse Probl. Imaging 6, 315–320 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Nguyen, L.V.: A family of inversion formulas in thermoacoustic tomography. Inverse Probl. Imaging 3, 649–675 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Nguyen, L.V.: On artifacts in limited data spherical radon transform: flat observation surfaces. SIAM J. Math. Anal. 47, 2984–3004 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Nguyen, L.V., Kunyansky, L.A.: A dissipative time reversal technique for photoacoustic tomography in a cavity. SIAM J. Imaging Sci. 9, 748–769 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Palacios, B.: Reconstruction for multi-wave imaging in attenuating media with large damping coefficient. Inverse Probl. 32, 125008, 15 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Palamodov, V.P.: A uniform reconstruction formula in integral geometry. Inverse Probl. 28, 065014 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Paltauf, G., Nuster, R., Haltmeier, M., Burgholzer, P.: Experimental evaluation of reconstruction algorithms for limited view photoacoustic tomography with line detectors. Inverse Probl. 23, S81–S94 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Paltauf, G., Viator, J.A., Prahl, S.A., Jacques, S.L.: Iterative reconstruction algorithm for optoacoustic imaging. J. Opt. Soc. Am. 112, 1536–1544 (2002)Google Scholar
  52. 52.
    Rosenthal, A., Ntziachristos, V., Razansky, D.: Acoustic inversion in optoacoustic tomography: a review. Curr. Med. Imaging Rev. 9, 318 (2013)CrossRefGoogle Scholar
  53. 53.
    Scherzer, O., Grasmair, M., Grossauer, H., Haltmeier, M., Lenzen, F.: Variational methods in imaging, volume 167 of applied mathematical sciences. Springer Science+Business Media LLC, Berlin/Heidelberg (2009)zbMATHGoogle Scholar
  54. 54.
    Sidky, E.Y., Jørgensen, J.H., Pan, X.: Convex optimization problem prototyping for image reconstruction in computed tomography with the Chambolle–Pock algorithm. Phys. Med. Biol. 57, 3065 (2012)CrossRefGoogle Scholar
  55. 55.
    Stefanov, P., Uhlmann, G.: Thermoacoustic tomography with variable sound speed. Inverse Probl. 25, 075011, 16 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  56. 56.
    Stefanov, P., Uhlmann, G.: Thermoacoustic tomography arising in brain imaging. Inverse Probl. 27, 045004 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Stefanov, P., Yang, Y.: Multiwave tomography with reflectors: Landweber’s iteration. Inverse Probl. Imaging 11, 373–401 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Szabo, T.L.: Time domain wave equations for lossy media obeying a frequency power law. J. Acoust. Soc. Am. 96, 491–500 (1994)CrossRefGoogle Scholar
  59. 59.
    Taylor, M.E.: Pseudodifferential Operators, volume 34 of Princeton Mathematical Series. Princeton, NJ (1981)Google Scholar
  60. 60.
    Treeby, B.E., Cox, B.T.: k-wave: Matlab toolbox for the simulation and reconstruction of photoacoustic wave fields. J. Biomed. Opt. 15, 021314 (2010)CrossRefGoogle Scholar
  61. 61.
    Treeby, B.E., Cox, B.T.: Modeling power law absorption and dispersion for acoustic propagation using the fractional laplacian. J. Acoust. Soc. Am. 127, 2741–2748 (2010)CrossRefGoogle Scholar
  62. 62.
    Treeby, B.E., Zhang, E.Z., Cox, B.: Photoacoustic tomography in absorbing acoustic media using time reversal. Inverse Probl. 26, 115003 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  63. 63.
    Wang, K., Schoonover, R.W., Su, R., Oraevsky, A., Anastasio, M.A.: Discrete imaging models for three-dimensional optoacoustic tomography using radially symmetric expansion functions. IEEE Trans. Med. Imaging 33, 1180–1193 (2014)CrossRefGoogle Scholar
  64. 64.
    Wang, K., Su, R., Oraevsky, A.A., Anastasio, M.A.: Investigation of iterative image reconstruction in three-dimensional optoacoustic tomography. Phys. Med. Biol. 57, 5399 (2012)CrossRefGoogle Scholar
  65. 65.
    Wells, P.N.T.: Biomedical Ultrasonics. Academic Press, New York (1977)Google Scholar
  66. 66.
    Xu, M., Wang, L.V.: Universal back-projection algorithm for photoacoustic computed tomography. Phys. Rev. E 71, 016706 (2005)CrossRefGoogle Scholar
  67. 67.
    Zhang, J., Anastasio, M.A., La Rivière, P.J., Wang, L.V.: Effects of different imaging models on least-squares image reconstruction accuracy in photoacoustic tomography. IEEE Trans. Med. Imaging 28, 1781–1790 (2009)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of InnsbruckInnsbruckAustria
  2. 2.Department of MathematicsUniversity of IdahoMoscowUSA
  3. 3.Faculty of Information TechnologyIndustrial University of Ho Chi Minh CityHo Chi MinhVietnam

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