Abstract

Photoacoustic tomography (PAT), also known as thermoacoustic or optoacoustic tomography, is a rapidly emerg ing biomedical imaging technique that combines optical image contrast with ultrasound detection principles. Most existing reconstruction algorithms for PAT assume the object of interest possesses homogeneous acoustic properties. The images produced by such algorithms can contain significant distortions and artifacts when the object’s acoustic properties are spatially variant. In this work, we establish an image reconstruction formula for PAT applications in which a planar detection surface is employed and the to-be-imaged optical absorber is embedded in a known planar layered acoustic medium. The reconstruction formula is exact in a mathematical sense and accounts for multiple acoustic reflections between the layers of the medium. Computer-simulation studies are conducted to demonstrate and investigate the proposed method.

© 2011 Optical Society of America

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  1. L. V. Wang, “Prospects of photoacoustic tomography,” Med. Phys. 35, 5758 (2008).
    [CrossRef]
  2. M. Xu and L. V. Wang, “Biomedical photoacoustics,” Rev. Sci. Instrum. 77, 041101 (2006).
    [CrossRef]
  3. A. A. Oraevsky and A. A. Karabutov, “Optoacoustic tomography,” in Biomedical Photonics Handbook, T.Vo-Dinh, ed. (CRC Press, 2003).
  4. L.Wang, ed., Photoacoustic Imaging and Spectroscopy (CRC Press, 2009).
    [CrossRef]
  5. V. Ntziachristos and D. Razansky, “Molecular imaging by means of multispectral optoacoustic tomography (MSOT),” Chem. Rev. 110, 2783–2794 (2010).
    [CrossRef] [PubMed]
  6. R. Kruger, D. Reinecke, and G. Kruger, “Thermoacoustic computed tomography—technical considerations,” Med. Phys. 26, 1832–1837 (1999).
    [CrossRef] [PubMed]
  7. M. Haltmeier, O. Scherzer, P. Burgholzer, and G. Paltauf, “Thermoacoustic computed tomography with large planar receivers,” Inverse Probl. 20, 1663–1673 (2004).
    [CrossRef]
  8. P. Ephrat, L. Keenliside, A. Seabrook, F. S. Prato, and J. J. L. Carson, “Three-dimensional photoacoustic imaging by sparse-array detection and iterative image reconstruction,” J. Biomed. Opt. 13, 054052 (2008).
    [CrossRef] [PubMed]
  9. B. T. Cox, S. R. Arridge, K. P. Köstli, and P. C. Beard, “Two-dimensional quantitative photoacoustic image reconstruction of absorption distributions in scattering media by use of a simple iterative method,” Appl. Opt. 45, 1866–1875 (2006).
    [CrossRef] [PubMed]
  10. K. Wang, S. Ermilov, R. Su, H. Brecht, A. Oraevsky, and M. Anastasio, “An imaging model incorporating ultrasonic transducer properties for three-dimensional optoacoustic tomography,” IEEE Trans. Med. Imag. 30, 203–214 (2010).
    [CrossRef]
  11. M. A. Anastasio, J. Zhang, D. Modgil, and P. J. L. Riviere, “Application of inverse source concepts to photoacoustic tomography,” Inverse Probl. 23, S21–S35 (2007).
    [CrossRef]
  12. L. A. Kunyansky, “Explicit inversion formulae for the spherical mean radon transform,” Inverse Probl. 23, 373–383 (2007).
    [CrossRef]
  13. D. Finch, M. Haltmeier, and Rakesh, “Inversion of spherical means and the wave equation in even dimensions,” SIAM J. Appl. Math. 68, 392–412 (2007).
    [CrossRef]
  14. M. Xu and L. V. Wang, “Universal back-projection algorithm for photoacoustic computed tomography,” Phys. Rev. E 71, 016706 (2005).
    [CrossRef]
  15. D. Finch, S. Patch, and Rakesh, “Determining a function from its mean values over a family of spheres,” SIAM J. Math. Anal. 35, 1213–1240 (2004).
    [CrossRef]
  16. Y. Xu, D. Feng, and L. V. Wang, “Exact frequency-domain reconstruction for thermoacoustic tomography: I. Planar geometry,” IEEE Trans. Med. Imag. 21, 823–828 (2002).
    [CrossRef]
  17. K. P. Köstli, M. Frenz, H. Bebie, and H. P. Weber, “Temporal backward projection of optoacoustic pressure transients using Fourier transform methods,” Phys. Med. Biol. 46, 1863–1872(2001).
    [CrossRef] [PubMed]
  18. R. A. Kruger, P. Liu, R. Fang, and C. Appledorn, “Photoacoustic ultrasound (PAUS) reconstruction tomography,” Med. Phys. 22, 1605–1609 (1995).
    [CrossRef] [PubMed]
  19. Y. Xu and L. V. Wang, “Effects of acoustic heterogeneity in breast thermoacoustic tomography,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 50, 1134–1146 (2003).
    [CrossRef] [PubMed]
  20. M. A. Anastasio, J. Zhang, X. Pan, Y. Zou, G. Keng, and L. V. Wang, “Half-time image reconstruction in thermoacoustic tomography,” IEEE Trans. Med. Imag. 24, 199–210 (2005).
    [CrossRef]
  21. D. Modgil, M. A. Anastasio, and P. J. L. Rivière, “Image reconstruction in photoacoustic tomography with variable speed of sound using a higher-order geometrical acoustics approximation,” J. Biomed. Opt. 15, 021308 (2010).
    [CrossRef] [PubMed]
  22. M. Agranovsky and P. Kuchment, “Uniqueness of reconstruction and an inversion procedure for thermoacoustic and photoacoustic tomography with variable sound speed,” Inverse Probl. 23, 2089–2102 (2007).
    [CrossRef]
  23. X. Jin and L. V. Wang, “Thermoacoustic tomography with correction for acoustic speed variations,” Phys. Med. Biol. 51, 6437–6448 (2006).
    [CrossRef] [PubMed]
  24. R. Willemink, S. Manohar, Y. Purwar, C. Slump, F. van der Heijden, and T. van Leeuwen, “Imaging of acoustic attenuation and speed of sound maps using photoacoustic measurements,” Proc. SPIE 6920, 692013 (2008).
    [CrossRef]
  25. Y. Hristova, P. Kuchment, and L. Nguyen, “Reconstruction and time reversal in thermoacoustic tomography in acoustically homogeneous and inhomogeneous media,” Inverse Probl. 24, 055006 (2008).
    [CrossRef]
  26. P. Stefanov and G. Uhlmann, “Thermoacoustic tomography with variable sound speed,” Inverse Probl. 25, 075011 (2009).
    [CrossRef]
  27. C. Tai, Dyadic Green Functions in Electromagnetic Theory (IEEE, 1994).
  28. M. Haltmeier, O. Scherzer, and G. Zangerl, “A reconstruction algorithm for photoacoustic imaging based on the nonuniform FFT,” IEEE Trans. Med. Imag. 28, 1727–1735 (2009).
    [CrossRef]
  29. P. J. L. Riviere, J. Zhang, and M. A. Anastasio, “Image reconstruction in optoacoustic tomography for dispersive acoustic media,” Opt. Lett. 31, 781–783 (2006).
    [CrossRef] [PubMed]
  30. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  31. M. Xu and L. V. Wang, “Analytic explanation of spatial resolution related to bandwidth and detector aperture size in thermoacoustic or photoacoustic reconstruction,” Phys. Rev. E 67, 056605 (2003).
    [CrossRef]
  32. P. Morse and K. Ingard, Theoretical Acoustics (Princeton University Press, 1986).
  33. M. A. Anastasio, M. Kupinski, and X. Pan, “Noise properties of reconstructed images in ultrasound diffraction tomography,” IEEE Trans. Nucl. Sci. 45, 2216–2223 (1998).
    [CrossRef]

2010 (3)

K. Wang, S. Ermilov, R. Su, H. Brecht, A. Oraevsky, and M. Anastasio, “An imaging model incorporating ultrasonic transducer properties for three-dimensional optoacoustic tomography,” IEEE Trans. Med. Imag. 30, 203–214 (2010).
[CrossRef]

D. Modgil, M. A. Anastasio, and P. J. L. Rivière, “Image reconstruction in photoacoustic tomography with variable speed of sound using a higher-order geometrical acoustics approximation,” J. Biomed. Opt. 15, 021308 (2010).
[CrossRef] [PubMed]

V. Ntziachristos and D. Razansky, “Molecular imaging by means of multispectral optoacoustic tomography (MSOT),” Chem. Rev. 110, 2783–2794 (2010).
[CrossRef] [PubMed]

2009 (2)

P. Stefanov and G. Uhlmann, “Thermoacoustic tomography with variable sound speed,” Inverse Probl. 25, 075011 (2009).
[CrossRef]

M. Haltmeier, O. Scherzer, and G. Zangerl, “A reconstruction algorithm for photoacoustic imaging based on the nonuniform FFT,” IEEE Trans. Med. Imag. 28, 1727–1735 (2009).
[CrossRef]

2008 (4)

L. V. Wang, “Prospects of photoacoustic tomography,” Med. Phys. 35, 5758 (2008).
[CrossRef]

R. Willemink, S. Manohar, Y. Purwar, C. Slump, F. van der Heijden, and T. van Leeuwen, “Imaging of acoustic attenuation and speed of sound maps using photoacoustic measurements,” Proc. SPIE 6920, 692013 (2008).
[CrossRef]

Y. Hristova, P. Kuchment, and L. Nguyen, “Reconstruction and time reversal in thermoacoustic tomography in acoustically homogeneous and inhomogeneous media,” Inverse Probl. 24, 055006 (2008).
[CrossRef]

P. Ephrat, L. Keenliside, A. Seabrook, F. S. Prato, and J. J. L. Carson, “Three-dimensional photoacoustic imaging by sparse-array detection and iterative image reconstruction,” J. Biomed. Opt. 13, 054052 (2008).
[CrossRef] [PubMed]

2007 (4)

M. Agranovsky and P. Kuchment, “Uniqueness of reconstruction and an inversion procedure for thermoacoustic and photoacoustic tomography with variable sound speed,” Inverse Probl. 23, 2089–2102 (2007).
[CrossRef]

M. A. Anastasio, J. Zhang, D. Modgil, and P. J. L. Riviere, “Application of inverse source concepts to photoacoustic tomography,” Inverse Probl. 23, S21–S35 (2007).
[CrossRef]

L. A. Kunyansky, “Explicit inversion formulae for the spherical mean radon transform,” Inverse Probl. 23, 373–383 (2007).
[CrossRef]

D. Finch, M. Haltmeier, and Rakesh, “Inversion of spherical means and the wave equation in even dimensions,” SIAM J. Appl. Math. 68, 392–412 (2007).
[CrossRef]

2006 (4)

2005 (2)

M. A. Anastasio, J. Zhang, X. Pan, Y. Zou, G. Keng, and L. V. Wang, “Half-time image reconstruction in thermoacoustic tomography,” IEEE Trans. Med. Imag. 24, 199–210 (2005).
[CrossRef]

M. Xu and L. V. Wang, “Universal back-projection algorithm for photoacoustic computed tomography,” Phys. Rev. E 71, 016706 (2005).
[CrossRef]

2004 (2)

D. Finch, S. Patch, and Rakesh, “Determining a function from its mean values over a family of spheres,” SIAM J. Math. Anal. 35, 1213–1240 (2004).
[CrossRef]

M. Haltmeier, O. Scherzer, P. Burgholzer, and G. Paltauf, “Thermoacoustic computed tomography with large planar receivers,” Inverse Probl. 20, 1663–1673 (2004).
[CrossRef]

2003 (2)

M. Xu and L. V. Wang, “Analytic explanation of spatial resolution related to bandwidth and detector aperture size in thermoacoustic or photoacoustic reconstruction,” Phys. Rev. E 67, 056605 (2003).
[CrossRef]

Y. Xu and L. V. Wang, “Effects of acoustic heterogeneity in breast thermoacoustic tomography,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 50, 1134–1146 (2003).
[CrossRef] [PubMed]

2002 (1)

Y. Xu, D. Feng, and L. V. Wang, “Exact frequency-domain reconstruction for thermoacoustic tomography: I. Planar geometry,” IEEE Trans. Med. Imag. 21, 823–828 (2002).
[CrossRef]

2001 (1)

K. P. Köstli, M. Frenz, H. Bebie, and H. P. Weber, “Temporal backward projection of optoacoustic pressure transients using Fourier transform methods,” Phys. Med. Biol. 46, 1863–1872(2001).
[CrossRef] [PubMed]

1999 (1)

R. Kruger, D. Reinecke, and G. Kruger, “Thermoacoustic computed tomography—technical considerations,” Med. Phys. 26, 1832–1837 (1999).
[CrossRef] [PubMed]

1998 (1)

M. A. Anastasio, M. Kupinski, and X. Pan, “Noise properties of reconstructed images in ultrasound diffraction tomography,” IEEE Trans. Nucl. Sci. 45, 2216–2223 (1998).
[CrossRef]

1995 (1)

R. A. Kruger, P. Liu, R. Fang, and C. Appledorn, “Photoacoustic ultrasound (PAUS) reconstruction tomography,” Med. Phys. 22, 1605–1609 (1995).
[CrossRef] [PubMed]

Agranovsky, M.

M. Agranovsky and P. Kuchment, “Uniqueness of reconstruction and an inversion procedure for thermoacoustic and photoacoustic tomography with variable sound speed,” Inverse Probl. 23, 2089–2102 (2007).
[CrossRef]

Anastasio, M.

K. Wang, S. Ermilov, R. Su, H. Brecht, A. Oraevsky, and M. Anastasio, “An imaging model incorporating ultrasonic transducer properties for three-dimensional optoacoustic tomography,” IEEE Trans. Med. Imag. 30, 203–214 (2010).
[CrossRef]

Anastasio, M. A.

D. Modgil, M. A. Anastasio, and P. J. L. Rivière, “Image reconstruction in photoacoustic tomography with variable speed of sound using a higher-order geometrical acoustics approximation,” J. Biomed. Opt. 15, 021308 (2010).
[CrossRef] [PubMed]

M. A. Anastasio, J. Zhang, D. Modgil, and P. J. L. Riviere, “Application of inverse source concepts to photoacoustic tomography,” Inverse Probl. 23, S21–S35 (2007).
[CrossRef]

P. J. L. Riviere, J. Zhang, and M. A. Anastasio, “Image reconstruction in optoacoustic tomography for dispersive acoustic media,” Opt. Lett. 31, 781–783 (2006).
[CrossRef] [PubMed]

M. A. Anastasio, J. Zhang, X. Pan, Y. Zou, G. Keng, and L. V. Wang, “Half-time image reconstruction in thermoacoustic tomography,” IEEE Trans. Med. Imag. 24, 199–210 (2005).
[CrossRef]

M. A. Anastasio, M. Kupinski, and X. Pan, “Noise properties of reconstructed images in ultrasound diffraction tomography,” IEEE Trans. Nucl. Sci. 45, 2216–2223 (1998).
[CrossRef]

Appledorn, C.

R. A. Kruger, P. Liu, R. Fang, and C. Appledorn, “Photoacoustic ultrasound (PAUS) reconstruction tomography,” Med. Phys. 22, 1605–1609 (1995).
[CrossRef] [PubMed]

Arridge, S. R.

Beard, P. C.

Bebie, H.

K. P. Köstli, M. Frenz, H. Bebie, and H. P. Weber, “Temporal backward projection of optoacoustic pressure transients using Fourier transform methods,” Phys. Med. Biol. 46, 1863–1872(2001).
[CrossRef] [PubMed]

Brecht, H.

K. Wang, S. Ermilov, R. Su, H. Brecht, A. Oraevsky, and M. Anastasio, “An imaging model incorporating ultrasonic transducer properties for three-dimensional optoacoustic tomography,” IEEE Trans. Med. Imag. 30, 203–214 (2010).
[CrossRef]

Burgholzer, P.

M. Haltmeier, O. Scherzer, P. Burgholzer, and G. Paltauf, “Thermoacoustic computed tomography with large planar receivers,” Inverse Probl. 20, 1663–1673 (2004).
[CrossRef]

Carson, J. J. L.

P. Ephrat, L. Keenliside, A. Seabrook, F. S. Prato, and J. J. L. Carson, “Three-dimensional photoacoustic imaging by sparse-array detection and iterative image reconstruction,” J. Biomed. Opt. 13, 054052 (2008).
[CrossRef] [PubMed]

Cox, B. T.

Ephrat, P.

P. Ephrat, L. Keenliside, A. Seabrook, F. S. Prato, and J. J. L. Carson, “Three-dimensional photoacoustic imaging by sparse-array detection and iterative image reconstruction,” J. Biomed. Opt. 13, 054052 (2008).
[CrossRef] [PubMed]

Ermilov, S.

K. Wang, S. Ermilov, R. Su, H. Brecht, A. Oraevsky, and M. Anastasio, “An imaging model incorporating ultrasonic transducer properties for three-dimensional optoacoustic tomography,” IEEE Trans. Med. Imag. 30, 203–214 (2010).
[CrossRef]

Fang, R.

R. A. Kruger, P. Liu, R. Fang, and C. Appledorn, “Photoacoustic ultrasound (PAUS) reconstruction tomography,” Med. Phys. 22, 1605–1609 (1995).
[CrossRef] [PubMed]

Feng, D.

Y. Xu, D. Feng, and L. V. Wang, “Exact frequency-domain reconstruction for thermoacoustic tomography: I. Planar geometry,” IEEE Trans. Med. Imag. 21, 823–828 (2002).
[CrossRef]

Finch, D.

D. Finch, M. Haltmeier, and Rakesh, “Inversion of spherical means and the wave equation in even dimensions,” SIAM J. Appl. Math. 68, 392–412 (2007).
[CrossRef]

D. Finch, S. Patch, and Rakesh, “Determining a function from its mean values over a family of spheres,” SIAM J. Math. Anal. 35, 1213–1240 (2004).
[CrossRef]

Frenz, M.

K. P. Köstli, M. Frenz, H. Bebie, and H. P. Weber, “Temporal backward projection of optoacoustic pressure transients using Fourier transform methods,” Phys. Med. Biol. 46, 1863–1872(2001).
[CrossRef] [PubMed]

Haltmeier, M.

M. Haltmeier, O. Scherzer, and G. Zangerl, “A reconstruction algorithm for photoacoustic imaging based on the nonuniform FFT,” IEEE Trans. Med. Imag. 28, 1727–1735 (2009).
[CrossRef]

D. Finch, M. Haltmeier, and Rakesh, “Inversion of spherical means and the wave equation in even dimensions,” SIAM J. Appl. Math. 68, 392–412 (2007).
[CrossRef]

M. Haltmeier, O. Scherzer, P. Burgholzer, and G. Paltauf, “Thermoacoustic computed tomography with large planar receivers,” Inverse Probl. 20, 1663–1673 (2004).
[CrossRef]

Hristova, Y.

Y. Hristova, P. Kuchment, and L. Nguyen, “Reconstruction and time reversal in thermoacoustic tomography in acoustically homogeneous and inhomogeneous media,” Inverse Probl. 24, 055006 (2008).
[CrossRef]

Ingard, K.

P. Morse and K. Ingard, Theoretical Acoustics (Princeton University Press, 1986).

Jin, X.

X. Jin and L. V. Wang, “Thermoacoustic tomography with correction for acoustic speed variations,” Phys. Med. Biol. 51, 6437–6448 (2006).
[CrossRef] [PubMed]

Karabutov, A. A.

A. A. Oraevsky and A. A. Karabutov, “Optoacoustic tomography,” in Biomedical Photonics Handbook, T.Vo-Dinh, ed. (CRC Press, 2003).

Keenliside, L.

P. Ephrat, L. Keenliside, A. Seabrook, F. S. Prato, and J. J. L. Carson, “Three-dimensional photoacoustic imaging by sparse-array detection and iterative image reconstruction,” J. Biomed. Opt. 13, 054052 (2008).
[CrossRef] [PubMed]

Keng, G.

M. A. Anastasio, J. Zhang, X. Pan, Y. Zou, G. Keng, and L. V. Wang, “Half-time image reconstruction in thermoacoustic tomography,” IEEE Trans. Med. Imag. 24, 199–210 (2005).
[CrossRef]

Köstli, K. P.

B. T. Cox, S. R. Arridge, K. P. Köstli, and P. C. Beard, “Two-dimensional quantitative photoacoustic image reconstruction of absorption distributions in scattering media by use of a simple iterative method,” Appl. Opt. 45, 1866–1875 (2006).
[CrossRef] [PubMed]

K. P. Köstli, M. Frenz, H. Bebie, and H. P. Weber, “Temporal backward projection of optoacoustic pressure transients using Fourier transform methods,” Phys. Med. Biol. 46, 1863–1872(2001).
[CrossRef] [PubMed]

Kruger, G.

R. Kruger, D. Reinecke, and G. Kruger, “Thermoacoustic computed tomography—technical considerations,” Med. Phys. 26, 1832–1837 (1999).
[CrossRef] [PubMed]

Kruger, R.

R. Kruger, D. Reinecke, and G. Kruger, “Thermoacoustic computed tomography—technical considerations,” Med. Phys. 26, 1832–1837 (1999).
[CrossRef] [PubMed]

Kruger, R. A.

R. A. Kruger, P. Liu, R. Fang, and C. Appledorn, “Photoacoustic ultrasound (PAUS) reconstruction tomography,” Med. Phys. 22, 1605–1609 (1995).
[CrossRef] [PubMed]

Kuchment, P.

Y. Hristova, P. Kuchment, and L. Nguyen, “Reconstruction and time reversal in thermoacoustic tomography in acoustically homogeneous and inhomogeneous media,” Inverse Probl. 24, 055006 (2008).
[CrossRef]

M. Agranovsky and P. Kuchment, “Uniqueness of reconstruction and an inversion procedure for thermoacoustic and photoacoustic tomography with variable sound speed,” Inverse Probl. 23, 2089–2102 (2007).
[CrossRef]

Kunyansky, L. A.

L. A. Kunyansky, “Explicit inversion formulae for the spherical mean radon transform,” Inverse Probl. 23, 373–383 (2007).
[CrossRef]

Kupinski, M.

M. A. Anastasio, M. Kupinski, and X. Pan, “Noise properties of reconstructed images in ultrasound diffraction tomography,” IEEE Trans. Nucl. Sci. 45, 2216–2223 (1998).
[CrossRef]

Liu, P.

R. A. Kruger, P. Liu, R. Fang, and C. Appledorn, “Photoacoustic ultrasound (PAUS) reconstruction tomography,” Med. Phys. 22, 1605–1609 (1995).
[CrossRef] [PubMed]

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Manohar, S.

R. Willemink, S. Manohar, Y. Purwar, C. Slump, F. van der Heijden, and T. van Leeuwen, “Imaging of acoustic attenuation and speed of sound maps using photoacoustic measurements,” Proc. SPIE 6920, 692013 (2008).
[CrossRef]

Modgil, D.

D. Modgil, M. A. Anastasio, and P. J. L. Rivière, “Image reconstruction in photoacoustic tomography with variable speed of sound using a higher-order geometrical acoustics approximation,” J. Biomed. Opt. 15, 021308 (2010).
[CrossRef] [PubMed]

M. A. Anastasio, J. Zhang, D. Modgil, and P. J. L. Riviere, “Application of inverse source concepts to photoacoustic tomography,” Inverse Probl. 23, S21–S35 (2007).
[CrossRef]

Morse, P.

P. Morse and K. Ingard, Theoretical Acoustics (Princeton University Press, 1986).

Nguyen, L.

Y. Hristova, P. Kuchment, and L. Nguyen, “Reconstruction and time reversal in thermoacoustic tomography in acoustically homogeneous and inhomogeneous media,” Inverse Probl. 24, 055006 (2008).
[CrossRef]

Ntziachristos, V.

V. Ntziachristos and D. Razansky, “Molecular imaging by means of multispectral optoacoustic tomography (MSOT),” Chem. Rev. 110, 2783–2794 (2010).
[CrossRef] [PubMed]

Oraevsky, A.

K. Wang, S. Ermilov, R. Su, H. Brecht, A. Oraevsky, and M. Anastasio, “An imaging model incorporating ultrasonic transducer properties for three-dimensional optoacoustic tomography,” IEEE Trans. Med. Imag. 30, 203–214 (2010).
[CrossRef]

Oraevsky, A. A.

A. A. Oraevsky and A. A. Karabutov, “Optoacoustic tomography,” in Biomedical Photonics Handbook, T.Vo-Dinh, ed. (CRC Press, 2003).

Paltauf, G.

M. Haltmeier, O. Scherzer, P. Burgholzer, and G. Paltauf, “Thermoacoustic computed tomography with large planar receivers,” Inverse Probl. 20, 1663–1673 (2004).
[CrossRef]

Pan, X.

M. A. Anastasio, J. Zhang, X. Pan, Y. Zou, G. Keng, and L. V. Wang, “Half-time image reconstruction in thermoacoustic tomography,” IEEE Trans. Med. Imag. 24, 199–210 (2005).
[CrossRef]

M. A. Anastasio, M. Kupinski, and X. Pan, “Noise properties of reconstructed images in ultrasound diffraction tomography,” IEEE Trans. Nucl. Sci. 45, 2216–2223 (1998).
[CrossRef]

Patch, S.

D. Finch, S. Patch, and Rakesh, “Determining a function from its mean values over a family of spheres,” SIAM J. Math. Anal. 35, 1213–1240 (2004).
[CrossRef]

Prato, F. S.

P. Ephrat, L. Keenliside, A. Seabrook, F. S. Prato, and J. J. L. Carson, “Three-dimensional photoacoustic imaging by sparse-array detection and iterative image reconstruction,” J. Biomed. Opt. 13, 054052 (2008).
[CrossRef] [PubMed]

Purwar, Y.

R. Willemink, S. Manohar, Y. Purwar, C. Slump, F. van der Heijden, and T. van Leeuwen, “Imaging of acoustic attenuation and speed of sound maps using photoacoustic measurements,” Proc. SPIE 6920, 692013 (2008).
[CrossRef]

Rakesh,

D. Finch, M. Haltmeier, and Rakesh, “Inversion of spherical means and the wave equation in even dimensions,” SIAM J. Appl. Math. 68, 392–412 (2007).
[CrossRef]

D. Finch, S. Patch, and Rakesh, “Determining a function from its mean values over a family of spheres,” SIAM J. Math. Anal. 35, 1213–1240 (2004).
[CrossRef]

Razansky, D.

V. Ntziachristos and D. Razansky, “Molecular imaging by means of multispectral optoacoustic tomography (MSOT),” Chem. Rev. 110, 2783–2794 (2010).
[CrossRef] [PubMed]

Reinecke, D.

R. Kruger, D. Reinecke, and G. Kruger, “Thermoacoustic computed tomography—technical considerations,” Med. Phys. 26, 1832–1837 (1999).
[CrossRef] [PubMed]

Riviere, P. J. L.

M. A. Anastasio, J. Zhang, D. Modgil, and P. J. L. Riviere, “Application of inverse source concepts to photoacoustic tomography,” Inverse Probl. 23, S21–S35 (2007).
[CrossRef]

P. J. L. Riviere, J. Zhang, and M. A. Anastasio, “Image reconstruction in optoacoustic tomography for dispersive acoustic media,” Opt. Lett. 31, 781–783 (2006).
[CrossRef] [PubMed]

Rivière, P. J. L.

D. Modgil, M. A. Anastasio, and P. J. L. Rivière, “Image reconstruction in photoacoustic tomography with variable speed of sound using a higher-order geometrical acoustics approximation,” J. Biomed. Opt. 15, 021308 (2010).
[CrossRef] [PubMed]

Scherzer, O.

M. Haltmeier, O. Scherzer, and G. Zangerl, “A reconstruction algorithm for photoacoustic imaging based on the nonuniform FFT,” IEEE Trans. Med. Imag. 28, 1727–1735 (2009).
[CrossRef]

M. Haltmeier, O. Scherzer, P. Burgholzer, and G. Paltauf, “Thermoacoustic computed tomography with large planar receivers,” Inverse Probl. 20, 1663–1673 (2004).
[CrossRef]

Seabrook, A.

P. Ephrat, L. Keenliside, A. Seabrook, F. S. Prato, and J. J. L. Carson, “Three-dimensional photoacoustic imaging by sparse-array detection and iterative image reconstruction,” J. Biomed. Opt. 13, 054052 (2008).
[CrossRef] [PubMed]

Slump, C.

R. Willemink, S. Manohar, Y. Purwar, C. Slump, F. van der Heijden, and T. van Leeuwen, “Imaging of acoustic attenuation and speed of sound maps using photoacoustic measurements,” Proc. SPIE 6920, 692013 (2008).
[CrossRef]

Stefanov, P.

P. Stefanov and G. Uhlmann, “Thermoacoustic tomography with variable sound speed,” Inverse Probl. 25, 075011 (2009).
[CrossRef]

Su, R.

K. Wang, S. Ermilov, R. Su, H. Brecht, A. Oraevsky, and M. Anastasio, “An imaging model incorporating ultrasonic transducer properties for three-dimensional optoacoustic tomography,” IEEE Trans. Med. Imag. 30, 203–214 (2010).
[CrossRef]

Tai, C.

C. Tai, Dyadic Green Functions in Electromagnetic Theory (IEEE, 1994).

Uhlmann, G.

P. Stefanov and G. Uhlmann, “Thermoacoustic tomography with variable sound speed,” Inverse Probl. 25, 075011 (2009).
[CrossRef]

van der Heijden, F.

R. Willemink, S. Manohar, Y. Purwar, C. Slump, F. van der Heijden, and T. van Leeuwen, “Imaging of acoustic attenuation and speed of sound maps using photoacoustic measurements,” Proc. SPIE 6920, 692013 (2008).
[CrossRef]

van Leeuwen, T.

R. Willemink, S. Manohar, Y. Purwar, C. Slump, F. van der Heijden, and T. van Leeuwen, “Imaging of acoustic attenuation and speed of sound maps using photoacoustic measurements,” Proc. SPIE 6920, 692013 (2008).
[CrossRef]

Wang, K.

K. Wang, S. Ermilov, R. Su, H. Brecht, A. Oraevsky, and M. Anastasio, “An imaging model incorporating ultrasonic transducer properties for three-dimensional optoacoustic tomography,” IEEE Trans. Med. Imag. 30, 203–214 (2010).
[CrossRef]

Wang, L. V.

L. V. Wang, “Prospects of photoacoustic tomography,” Med. Phys. 35, 5758 (2008).
[CrossRef]

X. Jin and L. V. Wang, “Thermoacoustic tomography with correction for acoustic speed variations,” Phys. Med. Biol. 51, 6437–6448 (2006).
[CrossRef] [PubMed]

M. Xu and L. V. Wang, “Biomedical photoacoustics,” Rev. Sci. Instrum. 77, 041101 (2006).
[CrossRef]

M. A. Anastasio, J. Zhang, X. Pan, Y. Zou, G. Keng, and L. V. Wang, “Half-time image reconstruction in thermoacoustic tomography,” IEEE Trans. Med. Imag. 24, 199–210 (2005).
[CrossRef]

M. Xu and L. V. Wang, “Universal back-projection algorithm for photoacoustic computed tomography,” Phys. Rev. E 71, 016706 (2005).
[CrossRef]

M. Xu and L. V. Wang, “Analytic explanation of spatial resolution related to bandwidth and detector aperture size in thermoacoustic or photoacoustic reconstruction,” Phys. Rev. E 67, 056605 (2003).
[CrossRef]

Y. Xu and L. V. Wang, “Effects of acoustic heterogeneity in breast thermoacoustic tomography,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 50, 1134–1146 (2003).
[CrossRef] [PubMed]

Y. Xu, D. Feng, and L. V. Wang, “Exact frequency-domain reconstruction for thermoacoustic tomography: I. Planar geometry,” IEEE Trans. Med. Imag. 21, 823–828 (2002).
[CrossRef]

Weber, H. P.

K. P. Köstli, M. Frenz, H. Bebie, and H. P. Weber, “Temporal backward projection of optoacoustic pressure transients using Fourier transform methods,” Phys. Med. Biol. 46, 1863–1872(2001).
[CrossRef] [PubMed]

Willemink, R.

R. Willemink, S. Manohar, Y. Purwar, C. Slump, F. van der Heijden, and T. van Leeuwen, “Imaging of acoustic attenuation and speed of sound maps using photoacoustic measurements,” Proc. SPIE 6920, 692013 (2008).
[CrossRef]

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Xu, M.

M. Xu and L. V. Wang, “Biomedical photoacoustics,” Rev. Sci. Instrum. 77, 041101 (2006).
[CrossRef]

M. Xu and L. V. Wang, “Universal back-projection algorithm for photoacoustic computed tomography,” Phys. Rev. E 71, 016706 (2005).
[CrossRef]

M. Xu and L. V. Wang, “Analytic explanation of spatial resolution related to bandwidth and detector aperture size in thermoacoustic or photoacoustic reconstruction,” Phys. Rev. E 67, 056605 (2003).
[CrossRef]

Xu, Y.

Y. Xu and L. V. Wang, “Effects of acoustic heterogeneity in breast thermoacoustic tomography,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 50, 1134–1146 (2003).
[CrossRef] [PubMed]

Y. Xu, D. Feng, and L. V. Wang, “Exact frequency-domain reconstruction for thermoacoustic tomography: I. Planar geometry,” IEEE Trans. Med. Imag. 21, 823–828 (2002).
[CrossRef]

Zangerl, G.

M. Haltmeier, O. Scherzer, and G. Zangerl, “A reconstruction algorithm for photoacoustic imaging based on the nonuniform FFT,” IEEE Trans. Med. Imag. 28, 1727–1735 (2009).
[CrossRef]

Zhang, J.

M. A. Anastasio, J. Zhang, D. Modgil, and P. J. L. Riviere, “Application of inverse source concepts to photoacoustic tomography,” Inverse Probl. 23, S21–S35 (2007).
[CrossRef]

P. J. L. Riviere, J. Zhang, and M. A. Anastasio, “Image reconstruction in optoacoustic tomography for dispersive acoustic media,” Opt. Lett. 31, 781–783 (2006).
[CrossRef] [PubMed]

M. A. Anastasio, J. Zhang, X. Pan, Y. Zou, G. Keng, and L. V. Wang, “Half-time image reconstruction in thermoacoustic tomography,” IEEE Trans. Med. Imag. 24, 199–210 (2005).
[CrossRef]

Zou, Y.

M. A. Anastasio, J. Zhang, X. Pan, Y. Zou, G. Keng, and L. V. Wang, “Half-time image reconstruction in thermoacoustic tomography,” IEEE Trans. Med. Imag. 24, 199–210 (2005).
[CrossRef]

Appl. Opt. (1)

Chem. Rev. (1)

V. Ntziachristos and D. Razansky, “Molecular imaging by means of multispectral optoacoustic tomography (MSOT),” Chem. Rev. 110, 2783–2794 (2010).
[CrossRef] [PubMed]

IEEE Trans. Med. Imag. (4)

M. Haltmeier, O. Scherzer, and G. Zangerl, “A reconstruction algorithm for photoacoustic imaging based on the nonuniform FFT,” IEEE Trans. Med. Imag. 28, 1727–1735 (2009).
[CrossRef]

K. Wang, S. Ermilov, R. Su, H. Brecht, A. Oraevsky, and M. Anastasio, “An imaging model incorporating ultrasonic transducer properties for three-dimensional optoacoustic tomography,” IEEE Trans. Med. Imag. 30, 203–214 (2010).
[CrossRef]

Y. Xu, D. Feng, and L. V. Wang, “Exact frequency-domain reconstruction for thermoacoustic tomography: I. Planar geometry,” IEEE Trans. Med. Imag. 21, 823–828 (2002).
[CrossRef]

M. A. Anastasio, J. Zhang, X. Pan, Y. Zou, G. Keng, and L. V. Wang, “Half-time image reconstruction in thermoacoustic tomography,” IEEE Trans. Med. Imag. 24, 199–210 (2005).
[CrossRef]

IEEE Trans. Nucl. Sci. (1)

M. A. Anastasio, M. Kupinski, and X. Pan, “Noise properties of reconstructed images in ultrasound diffraction tomography,” IEEE Trans. Nucl. Sci. 45, 2216–2223 (1998).
[CrossRef]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control (1)

Y. Xu and L. V. Wang, “Effects of acoustic heterogeneity in breast thermoacoustic tomography,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 50, 1134–1146 (2003).
[CrossRef] [PubMed]

Inverse Probl. (6)

M. Agranovsky and P. Kuchment, “Uniqueness of reconstruction and an inversion procedure for thermoacoustic and photoacoustic tomography with variable sound speed,” Inverse Probl. 23, 2089–2102 (2007).
[CrossRef]

Y. Hristova, P. Kuchment, and L. Nguyen, “Reconstruction and time reversal in thermoacoustic tomography in acoustically homogeneous and inhomogeneous media,” Inverse Probl. 24, 055006 (2008).
[CrossRef]

P. Stefanov and G. Uhlmann, “Thermoacoustic tomography with variable sound speed,” Inverse Probl. 25, 075011 (2009).
[CrossRef]

M. A. Anastasio, J. Zhang, D. Modgil, and P. J. L. Riviere, “Application of inverse source concepts to photoacoustic tomography,” Inverse Probl. 23, S21–S35 (2007).
[CrossRef]

L. A. Kunyansky, “Explicit inversion formulae for the spherical mean radon transform,” Inverse Probl. 23, 373–383 (2007).
[CrossRef]

M. Haltmeier, O. Scherzer, P. Burgholzer, and G. Paltauf, “Thermoacoustic computed tomography with large planar receivers,” Inverse Probl. 20, 1663–1673 (2004).
[CrossRef]

J. Biomed. Opt. (2)

P. Ephrat, L. Keenliside, A. Seabrook, F. S. Prato, and J. J. L. Carson, “Three-dimensional photoacoustic imaging by sparse-array detection and iterative image reconstruction,” J. Biomed. Opt. 13, 054052 (2008).
[CrossRef] [PubMed]

D. Modgil, M. A. Anastasio, and P. J. L. Rivière, “Image reconstruction in photoacoustic tomography with variable speed of sound using a higher-order geometrical acoustics approximation,” J. Biomed. Opt. 15, 021308 (2010).
[CrossRef] [PubMed]

Med. Phys. (3)

R. A. Kruger, P. Liu, R. Fang, and C. Appledorn, “Photoacoustic ultrasound (PAUS) reconstruction tomography,” Med. Phys. 22, 1605–1609 (1995).
[CrossRef] [PubMed]

R. Kruger, D. Reinecke, and G. Kruger, “Thermoacoustic computed tomography—technical considerations,” Med. Phys. 26, 1832–1837 (1999).
[CrossRef] [PubMed]

L. V. Wang, “Prospects of photoacoustic tomography,” Med. Phys. 35, 5758 (2008).
[CrossRef]

Opt. Lett. (1)

Phys. Med. Biol. (2)

X. Jin and L. V. Wang, “Thermoacoustic tomography with correction for acoustic speed variations,” Phys. Med. Biol. 51, 6437–6448 (2006).
[CrossRef] [PubMed]

K. P. Köstli, M. Frenz, H. Bebie, and H. P. Weber, “Temporal backward projection of optoacoustic pressure transients using Fourier transform methods,” Phys. Med. Biol. 46, 1863–1872(2001).
[CrossRef] [PubMed]

Phys. Rev. E (2)

M. Xu and L. V. Wang, “Universal back-projection algorithm for photoacoustic computed tomography,” Phys. Rev. E 71, 016706 (2005).
[CrossRef]

M. Xu and L. V. Wang, “Analytic explanation of spatial resolution related to bandwidth and detector aperture size in thermoacoustic or photoacoustic reconstruction,” Phys. Rev. E 67, 056605 (2003).
[CrossRef]

Proc. SPIE (1)

R. Willemink, S. Manohar, Y. Purwar, C. Slump, F. van der Heijden, and T. van Leeuwen, “Imaging of acoustic attenuation and speed of sound maps using photoacoustic measurements,” Proc. SPIE 6920, 692013 (2008).
[CrossRef]

Rev. Sci. Instrum. (1)

M. Xu and L. V. Wang, “Biomedical photoacoustics,” Rev. Sci. Instrum. 77, 041101 (2006).
[CrossRef]

SIAM J. Appl. Math. (1)

D. Finch, M. Haltmeier, and Rakesh, “Inversion of spherical means and the wave equation in even dimensions,” SIAM J. Appl. Math. 68, 392–412 (2007).
[CrossRef]

SIAM J. Math. Anal. (1)

D. Finch, S. Patch, and Rakesh, “Determining a function from its mean values over a family of spheres,” SIAM J. Math. Anal. 35, 1213–1240 (2004).
[CrossRef]

Other (5)

C. Tai, Dyadic Green Functions in Electromagnetic Theory (IEEE, 1994).

A. A. Oraevsky and A. A. Karabutov, “Optoacoustic tomography,” in Biomedical Photonics Handbook, T.Vo-Dinh, ed. (CRC Press, 2003).

L.Wang, ed., Photoacoustic Imaging and Spectroscopy (CRC Press, 2009).
[CrossRef]

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

P. Morse and K. Ingard, Theoretical Acoustics (Princeton University Press, 1986).

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Figures (6)

Fig. 1
Fig. 1

Example of a layered medium. Each layer, bounded between z m and z m 1 and having length d m , may have different speed of sound, c m and particle density ρ m . The detection plane is assumed to be at z 0 = 0 , and the object is contained in layer n.

Fig. 2
Fig. 2

(top left) Image of the numerical phantom in the plane z = 1.37 cm reconstructed by use of Eq. (27) from noiseless data. (top right) Profile through the reconstructed image (black) and original phantom (blue dashed) along the line x = 0.75 cm and z = 1.37 cm . (bottom left) Corresponding profiles along the line y = 1.33 cm and z = 1.51 cm . (bottom right) Corresponding profiles along the line y = 1.42 cm and z = 1.42 cm .

Fig. 3
Fig. 3

(top left) Image of the numerical phantom in the plane z = 1.37 cm reconstructed by use of Eq. (27) from noiseless data using the assumption of a homogeneous medium with speed of sound consistent with the first layer of the actual layered structure. (top right) Profile through the reconstructed image (black) and original phantom (blue dashed) along the line x = 0.75 cm and z = 1.37 cm . (bottom left) Corresponding profiles along the line y = 1.33 cm and z = 1.51 cm . (bottom right) Corresponding profiles along the line y = 1.42 cm and z = 1.42 cm .

Fig. 4
Fig. 4

(top left) image of the numerical phantom in the plane z = 1.37 cm reconstructed by use of Eq. (27) from noiseless data when the speed of sound in the first layer of the structure is incorrectly estimated. (top right) profile through the reconstructed image (black) and original phantom (blue dashed) along the line x = 0.75 cm and z = 1.37 cm . (bottom left) Corresponding profiles along the line y = 1.33 cm and z = 1.51 cm . (bottom right) Corresponding profiles along the line y = 1.42 cm and z = 1.42 cm .

Fig. 5
Fig. 5

(top left) Image of the numerical phantom in the plane z = 1.37 cm reconstructed by use of Eq. (27) from noiseless data when the speed of sound in the second and third layer of the structure are incorrectly estimated. (top right) Profile through the reconstructed image (black) and original phantom (blue dashed) along the line x = 0.75 cm and z = 1.37 cm . (bottom left) Corresponding profiles along the line y = 1.33 cm and z = 1.51 cm . (bottom right) Corresponding profiles along the line y = 1.42 cm and z = 1.42 cm .

Fig. 6
Fig. 6

(top left) Image of the numerical phantom in the plane z = 1.37 cm reconstructed by use of Eq. (27) when 1% noise is added to the acoustic signal. (top right) Profile of the variance of the reconstructed object for the case when the object is embedded in a layered medium (blue dashed) and a profile of the variance of the reconstructed object when the object is embedded in a homogeneous medium (black) along the line x = 0.75 cm and z = 1.37 cm . (bottom left) Corresponding profiles along the line y = 1.33 cm and z = 1.51 cm . (bottom right) Corresponding profiles along the line y = 1.42 cm and z = 1.42 cm .

Equations (33)

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p ˜ ( r , ω ) = d t p ( r , t ) e i ω t ,
[ 2 + k 2 ( r ) ] p ˜ ( r , ω ) = i ω β C P A ( r ) H ( ω ) ,
p ˜ ( r , ω ) = i ω β H ( ω ) C P V d 3 r G ( r , r , ω ) A ( r ) ,
p ¯ ( k x , k y , ω ) = d x d y p ˜ ( x , y , z = 0 , ω ) e i ( k x x + k y y ) .
A ( k x , k y , k z ) = d x d y d z A ( x , y , z ) e i ( k x x + k y y + k z z ) .
A ( k x , k y , ω 2 / c 2 k x 2 k y 2 ) = 2 C p p ¯ ( k x , k y , ω ) ω β H ( ω ) ω 2 / c 2 k x 2 k y 2 ,
ρ ( r ) = ρ m for     z m z < z m 1 ,
c ( r ) = c m for     z m z < z m 1 ,
( 2 + k m 2 ) p ˜ ( r , ω ) = 0 for     z m z < z m 1 ,
p ˜ ( r , ω ) | z m = p ˜ ( r , ω ) | z m +     m ,
1 ρ m + 1 p ˜ z | z m = 1 ρ m p ˜ z | z m +     m .
( 2 + k 2 ) G 0 ( r , r ; ω ) = δ ( r r ) ,
G 0 ( r , r ; ω ) = exp ( i k | r r | ) 4 π | r r | = d 2 k ( 2 π ) 2 i 2 k z e i k · ( r r ) e i k z | z z | .
( 2 + k P 2 ) G P ( r , r ; ω ) = δ ( r r ) for     z P z < z P 1 ,
( 2 + k m 2 ) G P ( r , r ; ω ) = 0 for     z m z < z m 1 , m P .
G P ( r , r ; ω ) = d 2 k ( 2 π ) 2 { e i k · r [ a m ( k ) e i k z ( m ) z + b m ( k ) e i k z ( m ) z ] + i 2 k z ( m ) e i k · ( r r ) e i k z ( m ) | z z | δ m P } for     z m z < z m 1 ,
a m + 1 + b m + 1 a m exp ( i k z ( m ) d m ) b m exp ( i k z ( m ) d m ) = S 1 δ P , m + 1 + S 2 δ P m ,
k z ( m + 1 ) ρ m + 1 [ a m + 1 b m + 1 ] k z ( m ) ρ m [ a m exp ( i k z ( m ) d m ) b m exp ( i k z ( m ) d m ) ] = k z ( m + 1 ) ρ m + 1 S 1 δ P , m + 1 k z ( m ) ρ m S 2 δ P m ,
S 1 i 2 k z ( m + 1 ) exp ( i k · r ) exp [ i k z ( m + 1 ) ( z m z ) ] ,
S 2 i 2 k z ( m ) exp ( i k · r ) exp [ i k z ( m ) ( z z m ) ] ,
[ s ] 2 P 3 = i 2 k z ( P ) exp ( i k · r ) exp [ i k z ( P ) ( z P 1 z ) ] ,
[ s ] 2 P 2 = i 2 ρ P exp ( i k · r ) exp [ i k z ( P ) ( z P 1 z ) ] ,
[ s ] 2 P 1 = i 2 k z ( P ) exp ( i k · r ) exp [ i k z ( P ) ( z z P ) ] ,
[ s ] 2 P = i 2 ρ P exp ( i k · r ) exp [ i k z ( P ) ( z z P ) ] ,
g = M ̲ 1 · s .
[ g ] n ( k ; r ) = T n ( k ) e i k · r e i k z ( P ) z + R n ( k ) e i k · r e i k z ( P ) z , n = 1 , 2 , 3 , , 2 N 2 ,
G P meas ( r , r ; ω ) = d 2 k ( 2 π ) 2 e i k · r a 1 ( k ; r ) ,
p ( r , ω ) = i ω β H ( ω ) C P V d 3 r G P meas ( r , r ; ω ) A ( r ) .
p ˜ ( k , ω ) = i ω β H ( ω ) C P [ T 1 ( k ) A ( k ; k z ( P ) ) + R 1 ( k ) A ( k ; k z ( P ) ) ] ,
A ( k ; k z ( P ) ) = C P i ω β H ( ω ) [ T 1 * ( k ) p ˜ ( k , ω ) R 1 ( k ) p ˜ * ( k , ω ) ] | T 1 | 2 | R 1 | 2 .
A ( k ; k z ( m ) ) = 2 C P k z ( m ) i ω β H ( ω ) 1 + e 2 i k z ( f ) d f r m f ( k , ω ) r f s ( k , ω ) e i k z ( s ) d e i k z ( f ) d f t m f ( k , ω ) t f s ( k , ω ) p ˜ ( k , ω ) ,
r i j ( k , ω ) = ρ i k z ( j ) ( k , ω ) ρ j k z ( i ) ( k , ω ) ρ i k z ( j ) ( k , ω ) + ρ j k z ( i ) ( k , ω ) ,
t i j ( k , ω ) = 2 ρ i k z ( j ) ( k , ω ) ρ i k z ( j ) ( k , ω ) + ρ j k z ( i ) ( k , ω ) .

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