Abstract

We develop iterative diffraction tomography algorithms, which are similar to the distorted Born algorithms, for inverting scattered intensity data. Within the Born approximation, the unknown scattered field is expressed as a multiplicative perturbation to the incident field. With this, the forward equation becomes stable, which helps us compute nearly oscillation-free solutions that have immediate bearing on the accuracy of the Jacobian computed for use in a deterministic Gauss–Newton (GN) reconstruction. However, since the data are inherently noisy and the sensitivity of measurement to refractive index away from the detectors is poor, we report a derivative-free evolutionary stochastic scheme, providing strictly additive updates in order to bridge the measurement-prediction misfit, to arrive at the refractive index distribution from intensity transport data. The superiority of the stochastic algorithm over the GN scheme for similar settings is demonstrated by the reconstruction of the refractive index profile from simulated and experimentally acquired intensity data.

© 2014 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. C. Kak and M. Slaney, “Tomographic imaging with diffracting sources,” in Principles of Computerized Tomographic Imaging (IEEE, 1988), pp. 203–273.
  2. I. H. Lira and C. M. Vest, “Refraction correction in holographic interferometry and tomography of transparent objects,” Appl. Opt. 26, 3919–3928 (1987).
    [Crossref]
  3. A. H. Andersen, “Tomography transform and inverse in geometrical optics,” J. Opt. Soc. Am. A 4, 1385–1395 (1987).
    [Crossref]
  4. A. C. Kak, “Tomographic imaging with diffracting and non-diffracting sources,” in Array Signal Processing, S. Haykin, ed. (Prentice-Hall, 1985).
  5. A. J. Devaney and G. Beylkin, “Diffraction tomography using arbitrary transmitter and receiver surfaces,” Ultrason. Imag. 6, 181–193 (1984).
  6. M. Slaney, A. C. Kak, and L. E. Larsen, “Limitations of imaging with first-order diffraction tomography,” IEEE Trans. Microw. Theory Technol. 32, 860–874 (1984).
    [Crossref]
  7. P. Guo and A. J. Devaney, “Comparison of reconstruction algorithms for optical diffraction tomography,” J. Opt. Soc. Am. A 22, 2338–2347 (2005).
    [Crossref]
  8. G. A. Tsihrintzis and A. J. Devaney, “Higher order (nonlinear) diffraction tomography: inversion of the Rytov series,” IEEE Trans. Inf. Theory 46, 1748–1761 (2000).
    [Crossref]
  9. W. C. Chew and Y. M. Wang, “Reconstruction of two-dimensional permittivity distribution using the distorted Born iterative method,” IEEE Trans. Med. Imag. 9, 218–225 (1990).
    [Crossref]
  10. A. H. Hielscher, A. D. Klose, and K. M. Hansen, “Gradient-based iterative image reconstruction scheme for time-resolved optical tomography,” IEEE Trans. Med. Imag. 18, 262–271 (1999).
    [Crossref]
  11. K. Belkebir and A. G. Tijhuis, “Modified gradient method and modified Born method for solving a two-dimensional inverse scattering problem,” Inverse Probl. 17, 1671–1688 (2001).
    [Crossref]
  12. R. F. Harrington, Field Computation by Moment Method (Macmillan, 1968).
  13. F. Natterer and F. Wübbeling, “A propagation-backpropagation method for ultrasound tomography,” Inverse Probl. 11, 1225–1232 (1995).
    [Crossref]
  14. N. Pimprikar, J. Teresa, D. Roy, R. M. Vasu, and K. Rajan, “An approximately H1-optimal Petrov-Galerkin mesh-free method: application to computation of scattered light for optical tomography,” CMES—Computer Modeling in Engineering and Sciences 92, 33–61 (2013).
  15. P. Guo and A. J. Devaney, “Digital microscopy using phase-shifting digital holography with two reference waves,” Opt. Lett. 29, 857–859 (2004).
    [Crossref]
  16. N. Jayashree, G. K. Datta, and R. M. Vasu, “Optical tomographic microscope for quantitative imaging of phase objects,” Appl. Opt. 39, 277–283 (2000).
    [Crossref]
  17. M. H. Maleki and A. J. Devaney, “Phase retrieval and intensity-only reconstruction algorithm for optical diffraction tomography,” J. Opt. Soc. Am. A 10, 1086–1092 (1993).
    [Crossref]
  18. G. Gbur and E. Wolf, “Hybrid diffraction tomography without phase information,” J. Opt. Soc. Am. A 19, 2194–2202 (2002).
    [Crossref]
  19. M. A. Anastasio, D. Shi, Y. Huang, and G. Gbur, “Image reconstruction in spherical-wave intensity diffraction tomography,” J. Opt. Soc. Am. A 22, 2651–2661 (2005).
    [Crossref]
  20. H. M. Varma, R. M. Vasu, and A. K. Nandakumaran, “Direct reconstruction of complex refractive index distribution from boundary measurement of intensity and normal derivative of intensity,” J. Opt. Soc. Am. A 24, 3089–3099 (2007).
    [Crossref]
  21. A. M. Stuart, “Inverse problems: a Bayesian perspective,” Acta Numerica 19, 451–559 (2010).
    [Crossref]
  22. M. Venugopal, D. Roy, and R. M. Vasu, “A new evolutionary Bayesian approach incorporating additive path correction for nonlinear inverse problems,” arxiv.org/pdf/1305.2702.
  23. D. L. Marks, “A family of approximations scanning the Born and Rytov scattering series,” Opt. Express 14, 8837–8848 (2006).
    [Crossref]
  24. F. C. Klebaner, Introduction to Stochastic Calculus with Applications, 2nd ed. (Imperial College, 2005).
  25. T. Raveendran, S. Sarkar, D. Roy, and R. M. Vasu, “A novel filtering framework through Girsanov correction for the identification of nonlinear dynamical systems,” Inverse Probl. 29, 065002 (2013).
    [Crossref]
  26. S. Sarkar, S. R. Chowdhury, M. Venugopal, R. M. Vasu, and D. Roy, “A Kushner-Stratonovich Monte Carlo filter applied to nonlinear dynamical system identification,” Physica D 270, 46–59 (2014).
    [Crossref]
  27. D. M. Livings, S. L. Dance, and N. K. Nichols, “Unbiased ensemble square root filters,” Physica D 237, 1021–1028 (2008).
    [Crossref]
  28. B. Banerjee, D. Roy, and R. M. Vasu, “A pseudo-dynamic sub-optimal filter for elastography under static loading and measurements,” Phys. Med. Biol. 54, 285–305 (2009).
    [Crossref]
  29. D. Roy, “A family of lower- and higher-order transversal linearization techniques in non-linear stochastic engineering dynamics,” Int. J. Numer. Methods Eng. 61, 764–790 (2004).
    [Crossref]
  30. L. S. Ramachandra and D. Roy, “A new method for nonlinear two-point boundary value problems in solid mechanics,” J. Appl. Mech. 68, 776–786 (2001).
    [Crossref]
  31. D. Roy and L. S. Ramachandra, “A semi-analytical locally transversal linearization method for non-linear dynamical systems,” Int. J. Numer. Methods Eng. 51, 203–224 (2001).
    [Crossref]

2014 (1)

S. Sarkar, S. R. Chowdhury, M. Venugopal, R. M. Vasu, and D. Roy, “A Kushner-Stratonovich Monte Carlo filter applied to nonlinear dynamical system identification,” Physica D 270, 46–59 (2014).
[Crossref]

2013 (2)

T. Raveendran, S. Sarkar, D. Roy, and R. M. Vasu, “A novel filtering framework through Girsanov correction for the identification of nonlinear dynamical systems,” Inverse Probl. 29, 065002 (2013).
[Crossref]

N. Pimprikar, J. Teresa, D. Roy, R. M. Vasu, and K. Rajan, “An approximately H1-optimal Petrov-Galerkin mesh-free method: application to computation of scattered light for optical tomography,” CMES—Computer Modeling in Engineering and Sciences 92, 33–61 (2013).

2010 (1)

A. M. Stuart, “Inverse problems: a Bayesian perspective,” Acta Numerica 19, 451–559 (2010).
[Crossref]

2009 (1)

B. Banerjee, D. Roy, and R. M. Vasu, “A pseudo-dynamic sub-optimal filter for elastography under static loading and measurements,” Phys. Med. Biol. 54, 285–305 (2009).
[Crossref]

2008 (1)

D. M. Livings, S. L. Dance, and N. K. Nichols, “Unbiased ensemble square root filters,” Physica D 237, 1021–1028 (2008).
[Crossref]

2007 (1)

2006 (1)

2005 (2)

2004 (2)

P. Guo and A. J. Devaney, “Digital microscopy using phase-shifting digital holography with two reference waves,” Opt. Lett. 29, 857–859 (2004).
[Crossref]

D. Roy, “A family of lower- and higher-order transversal linearization techniques in non-linear stochastic engineering dynamics,” Int. J. Numer. Methods Eng. 61, 764–790 (2004).
[Crossref]

2002 (1)

2001 (3)

K. Belkebir and A. G. Tijhuis, “Modified gradient method and modified Born method for solving a two-dimensional inverse scattering problem,” Inverse Probl. 17, 1671–1688 (2001).
[Crossref]

L. S. Ramachandra and D. Roy, “A new method for nonlinear two-point boundary value problems in solid mechanics,” J. Appl. Mech. 68, 776–786 (2001).
[Crossref]

D. Roy and L. S. Ramachandra, “A semi-analytical locally transversal linearization method for non-linear dynamical systems,” Int. J. Numer. Methods Eng. 51, 203–224 (2001).
[Crossref]

2000 (2)

G. A. Tsihrintzis and A. J. Devaney, “Higher order (nonlinear) diffraction tomography: inversion of the Rytov series,” IEEE Trans. Inf. Theory 46, 1748–1761 (2000).
[Crossref]

N. Jayashree, G. K. Datta, and R. M. Vasu, “Optical tomographic microscope for quantitative imaging of phase objects,” Appl. Opt. 39, 277–283 (2000).
[Crossref]

1999 (1)

A. H. Hielscher, A. D. Klose, and K. M. Hansen, “Gradient-based iterative image reconstruction scheme for time-resolved optical tomography,” IEEE Trans. Med. Imag. 18, 262–271 (1999).
[Crossref]

1995 (1)

F. Natterer and F. Wübbeling, “A propagation-backpropagation method for ultrasound tomography,” Inverse Probl. 11, 1225–1232 (1995).
[Crossref]

1993 (1)

1990 (1)

W. C. Chew and Y. M. Wang, “Reconstruction of two-dimensional permittivity distribution using the distorted Born iterative method,” IEEE Trans. Med. Imag. 9, 218–225 (1990).
[Crossref]

1987 (2)

1984 (2)

A. J. Devaney and G. Beylkin, “Diffraction tomography using arbitrary transmitter and receiver surfaces,” Ultrason. Imag. 6, 181–193 (1984).

M. Slaney, A. C. Kak, and L. E. Larsen, “Limitations of imaging with first-order diffraction tomography,” IEEE Trans. Microw. Theory Technol. 32, 860–874 (1984).
[Crossref]

Anastasio, M. A.

Andersen, A. H.

Banerjee, B.

B. Banerjee, D. Roy, and R. M. Vasu, “A pseudo-dynamic sub-optimal filter for elastography under static loading and measurements,” Phys. Med. Biol. 54, 285–305 (2009).
[Crossref]

Belkebir, K.

K. Belkebir and A. G. Tijhuis, “Modified gradient method and modified Born method for solving a two-dimensional inverse scattering problem,” Inverse Probl. 17, 1671–1688 (2001).
[Crossref]

Beylkin, G.

A. J. Devaney and G. Beylkin, “Diffraction tomography using arbitrary transmitter and receiver surfaces,” Ultrason. Imag. 6, 181–193 (1984).

Chew, W. C.

W. C. Chew and Y. M. Wang, “Reconstruction of two-dimensional permittivity distribution using the distorted Born iterative method,” IEEE Trans. Med. Imag. 9, 218–225 (1990).
[Crossref]

Chowdhury, S. R.

S. Sarkar, S. R. Chowdhury, M. Venugopal, R. M. Vasu, and D. Roy, “A Kushner-Stratonovich Monte Carlo filter applied to nonlinear dynamical system identification,” Physica D 270, 46–59 (2014).
[Crossref]

Dance, S. L.

D. M. Livings, S. L. Dance, and N. K. Nichols, “Unbiased ensemble square root filters,” Physica D 237, 1021–1028 (2008).
[Crossref]

Datta, G. K.

Devaney, A. J.

Gbur, G.

Guo, P.

Hansen, K. M.

A. H. Hielscher, A. D. Klose, and K. M. Hansen, “Gradient-based iterative image reconstruction scheme for time-resolved optical tomography,” IEEE Trans. Med. Imag. 18, 262–271 (1999).
[Crossref]

Harrington, R. F.

R. F. Harrington, Field Computation by Moment Method (Macmillan, 1968).

Hielscher, A. H.

A. H. Hielscher, A. D. Klose, and K. M. Hansen, “Gradient-based iterative image reconstruction scheme for time-resolved optical tomography,” IEEE Trans. Med. Imag. 18, 262–271 (1999).
[Crossref]

Huang, Y.

Jayashree, N.

Kak, A. C.

M. Slaney, A. C. Kak, and L. E. Larsen, “Limitations of imaging with first-order diffraction tomography,” IEEE Trans. Microw. Theory Technol. 32, 860–874 (1984).
[Crossref]

A. C. Kak, “Tomographic imaging with diffracting and non-diffracting sources,” in Array Signal Processing, S. Haykin, ed. (Prentice-Hall, 1985).

A. C. Kak and M. Slaney, “Tomographic imaging with diffracting sources,” in Principles of Computerized Tomographic Imaging (IEEE, 1988), pp. 203–273.

Klebaner, F. C.

F. C. Klebaner, Introduction to Stochastic Calculus with Applications, 2nd ed. (Imperial College, 2005).

Klose, A. D.

A. H. Hielscher, A. D. Klose, and K. M. Hansen, “Gradient-based iterative image reconstruction scheme for time-resolved optical tomography,” IEEE Trans. Med. Imag. 18, 262–271 (1999).
[Crossref]

Larsen, L. E.

M. Slaney, A. C. Kak, and L. E. Larsen, “Limitations of imaging with first-order diffraction tomography,” IEEE Trans. Microw. Theory Technol. 32, 860–874 (1984).
[Crossref]

Lira, I. H.

Livings, D. M.

D. M. Livings, S. L. Dance, and N. K. Nichols, “Unbiased ensemble square root filters,” Physica D 237, 1021–1028 (2008).
[Crossref]

Maleki, M. H.

Marks, D. L.

Nandakumaran, A. K.

Natterer, F.

F. Natterer and F. Wübbeling, “A propagation-backpropagation method for ultrasound tomography,” Inverse Probl. 11, 1225–1232 (1995).
[Crossref]

Nichols, N. K.

D. M. Livings, S. L. Dance, and N. K. Nichols, “Unbiased ensemble square root filters,” Physica D 237, 1021–1028 (2008).
[Crossref]

Pimprikar, N.

N. Pimprikar, J. Teresa, D. Roy, R. M. Vasu, and K. Rajan, “An approximately H1-optimal Petrov-Galerkin mesh-free method: application to computation of scattered light for optical tomography,” CMES—Computer Modeling in Engineering and Sciences 92, 33–61 (2013).

Rajan, K.

N. Pimprikar, J. Teresa, D. Roy, R. M. Vasu, and K. Rajan, “An approximately H1-optimal Petrov-Galerkin mesh-free method: application to computation of scattered light for optical tomography,” CMES—Computer Modeling in Engineering and Sciences 92, 33–61 (2013).

Ramachandra, L. S.

L. S. Ramachandra and D. Roy, “A new method for nonlinear two-point boundary value problems in solid mechanics,” J. Appl. Mech. 68, 776–786 (2001).
[Crossref]

D. Roy and L. S. Ramachandra, “A semi-analytical locally transversal linearization method for non-linear dynamical systems,” Int. J. Numer. Methods Eng. 51, 203–224 (2001).
[Crossref]

Raveendran, T.

T. Raveendran, S. Sarkar, D. Roy, and R. M. Vasu, “A novel filtering framework through Girsanov correction for the identification of nonlinear dynamical systems,” Inverse Probl. 29, 065002 (2013).
[Crossref]

Roy, D.

S. Sarkar, S. R. Chowdhury, M. Venugopal, R. M. Vasu, and D. Roy, “A Kushner-Stratonovich Monte Carlo filter applied to nonlinear dynamical system identification,” Physica D 270, 46–59 (2014).
[Crossref]

T. Raveendran, S. Sarkar, D. Roy, and R. M. Vasu, “A novel filtering framework through Girsanov correction for the identification of nonlinear dynamical systems,” Inverse Probl. 29, 065002 (2013).
[Crossref]

N. Pimprikar, J. Teresa, D. Roy, R. M. Vasu, and K. Rajan, “An approximately H1-optimal Petrov-Galerkin mesh-free method: application to computation of scattered light for optical tomography,” CMES—Computer Modeling in Engineering and Sciences 92, 33–61 (2013).

B. Banerjee, D. Roy, and R. M. Vasu, “A pseudo-dynamic sub-optimal filter for elastography under static loading and measurements,” Phys. Med. Biol. 54, 285–305 (2009).
[Crossref]

D. Roy, “A family of lower- and higher-order transversal linearization techniques in non-linear stochastic engineering dynamics,” Int. J. Numer. Methods Eng. 61, 764–790 (2004).
[Crossref]

L. S. Ramachandra and D. Roy, “A new method for nonlinear two-point boundary value problems in solid mechanics,” J. Appl. Mech. 68, 776–786 (2001).
[Crossref]

D. Roy and L. S. Ramachandra, “A semi-analytical locally transversal linearization method for non-linear dynamical systems,” Int. J. Numer. Methods Eng. 51, 203–224 (2001).
[Crossref]

M. Venugopal, D. Roy, and R. M. Vasu, “A new evolutionary Bayesian approach incorporating additive path correction for nonlinear inverse problems,” arxiv.org/pdf/1305.2702.

Sarkar, S.

S. Sarkar, S. R. Chowdhury, M. Venugopal, R. M. Vasu, and D. Roy, “A Kushner-Stratonovich Monte Carlo filter applied to nonlinear dynamical system identification,” Physica D 270, 46–59 (2014).
[Crossref]

T. Raveendran, S. Sarkar, D. Roy, and R. M. Vasu, “A novel filtering framework through Girsanov correction for the identification of nonlinear dynamical systems,” Inverse Probl. 29, 065002 (2013).
[Crossref]

Shi, D.

Slaney, M.

M. Slaney, A. C. Kak, and L. E. Larsen, “Limitations of imaging with first-order diffraction tomography,” IEEE Trans. Microw. Theory Technol. 32, 860–874 (1984).
[Crossref]

A. C. Kak and M. Slaney, “Tomographic imaging with diffracting sources,” in Principles of Computerized Tomographic Imaging (IEEE, 1988), pp. 203–273.

Stuart, A. M.

A. M. Stuart, “Inverse problems: a Bayesian perspective,” Acta Numerica 19, 451–559 (2010).
[Crossref]

Teresa, J.

N. Pimprikar, J. Teresa, D. Roy, R. M. Vasu, and K. Rajan, “An approximately H1-optimal Petrov-Galerkin mesh-free method: application to computation of scattered light for optical tomography,” CMES—Computer Modeling in Engineering and Sciences 92, 33–61 (2013).

Tijhuis, A. G.

K. Belkebir and A. G. Tijhuis, “Modified gradient method and modified Born method for solving a two-dimensional inverse scattering problem,” Inverse Probl. 17, 1671–1688 (2001).
[Crossref]

Tsihrintzis, G. A.

G. A. Tsihrintzis and A. J. Devaney, “Higher order (nonlinear) diffraction tomography: inversion of the Rytov series,” IEEE Trans. Inf. Theory 46, 1748–1761 (2000).
[Crossref]

Varma, H. M.

Vasu, R. M.

S. Sarkar, S. R. Chowdhury, M. Venugopal, R. M. Vasu, and D. Roy, “A Kushner-Stratonovich Monte Carlo filter applied to nonlinear dynamical system identification,” Physica D 270, 46–59 (2014).
[Crossref]

T. Raveendran, S. Sarkar, D. Roy, and R. M. Vasu, “A novel filtering framework through Girsanov correction for the identification of nonlinear dynamical systems,” Inverse Probl. 29, 065002 (2013).
[Crossref]

N. Pimprikar, J. Teresa, D. Roy, R. M. Vasu, and K. Rajan, “An approximately H1-optimal Petrov-Galerkin mesh-free method: application to computation of scattered light for optical tomography,” CMES—Computer Modeling in Engineering and Sciences 92, 33–61 (2013).

B. Banerjee, D. Roy, and R. M. Vasu, “A pseudo-dynamic sub-optimal filter for elastography under static loading and measurements,” Phys. Med. Biol. 54, 285–305 (2009).
[Crossref]

H. M. Varma, R. M. Vasu, and A. K. Nandakumaran, “Direct reconstruction of complex refractive index distribution from boundary measurement of intensity and normal derivative of intensity,” J. Opt. Soc. Am. A 24, 3089–3099 (2007).
[Crossref]

N. Jayashree, G. K. Datta, and R. M. Vasu, “Optical tomographic microscope for quantitative imaging of phase objects,” Appl. Opt. 39, 277–283 (2000).
[Crossref]

M. Venugopal, D. Roy, and R. M. Vasu, “A new evolutionary Bayesian approach incorporating additive path correction for nonlinear inverse problems,” arxiv.org/pdf/1305.2702.

Venugopal, M.

S. Sarkar, S. R. Chowdhury, M. Venugopal, R. M. Vasu, and D. Roy, “A Kushner-Stratonovich Monte Carlo filter applied to nonlinear dynamical system identification,” Physica D 270, 46–59 (2014).
[Crossref]

M. Venugopal, D. Roy, and R. M. Vasu, “A new evolutionary Bayesian approach incorporating additive path correction for nonlinear inverse problems,” arxiv.org/pdf/1305.2702.

Vest, C. M.

Wang, Y. M.

W. C. Chew and Y. M. Wang, “Reconstruction of two-dimensional permittivity distribution using the distorted Born iterative method,” IEEE Trans. Med. Imag. 9, 218–225 (1990).
[Crossref]

Wolf, E.

Wübbeling, F.

F. Natterer and F. Wübbeling, “A propagation-backpropagation method for ultrasound tomography,” Inverse Probl. 11, 1225–1232 (1995).
[Crossref]

Acta Numerica (1)

A. M. Stuart, “Inverse problems: a Bayesian perspective,” Acta Numerica 19, 451–559 (2010).
[Crossref]

Appl. Opt. (2)

CMES—Computer Modeling in Engineering and Sciences (1)

N. Pimprikar, J. Teresa, D. Roy, R. M. Vasu, and K. Rajan, “An approximately H1-optimal Petrov-Galerkin mesh-free method: application to computation of scattered light for optical tomography,” CMES—Computer Modeling in Engineering and Sciences 92, 33–61 (2013).

IEEE Trans. Inf. Theory (1)

G. A. Tsihrintzis and A. J. Devaney, “Higher order (nonlinear) diffraction tomography: inversion of the Rytov series,” IEEE Trans. Inf. Theory 46, 1748–1761 (2000).
[Crossref]

IEEE Trans. Med. Imag. (2)

W. C. Chew and Y. M. Wang, “Reconstruction of two-dimensional permittivity distribution using the distorted Born iterative method,” IEEE Trans. Med. Imag. 9, 218–225 (1990).
[Crossref]

A. H. Hielscher, A. D. Klose, and K. M. Hansen, “Gradient-based iterative image reconstruction scheme for time-resolved optical tomography,” IEEE Trans. Med. Imag. 18, 262–271 (1999).
[Crossref]

IEEE Trans. Microw. Theory Technol. (1)

M. Slaney, A. C. Kak, and L. E. Larsen, “Limitations of imaging with first-order diffraction tomography,” IEEE Trans. Microw. Theory Technol. 32, 860–874 (1984).
[Crossref]

Int. J. Numer. Methods Eng. (2)

D. Roy, “A family of lower- and higher-order transversal linearization techniques in non-linear stochastic engineering dynamics,” Int. J. Numer. Methods Eng. 61, 764–790 (2004).
[Crossref]

D. Roy and L. S. Ramachandra, “A semi-analytical locally transversal linearization method for non-linear dynamical systems,” Int. J. Numer. Methods Eng. 51, 203–224 (2001).
[Crossref]

Inverse Probl. (3)

T. Raveendran, S. Sarkar, D. Roy, and R. M. Vasu, “A novel filtering framework through Girsanov correction for the identification of nonlinear dynamical systems,” Inverse Probl. 29, 065002 (2013).
[Crossref]

F. Natterer and F. Wübbeling, “A propagation-backpropagation method for ultrasound tomography,” Inverse Probl. 11, 1225–1232 (1995).
[Crossref]

K. Belkebir and A. G. Tijhuis, “Modified gradient method and modified Born method for solving a two-dimensional inverse scattering problem,” Inverse Probl. 17, 1671–1688 (2001).
[Crossref]

J. Appl. Mech. (1)

L. S. Ramachandra and D. Roy, “A new method for nonlinear two-point boundary value problems in solid mechanics,” J. Appl. Mech. 68, 776–786 (2001).
[Crossref]

J. Opt. Soc. Am. A (6)

Opt. Express (1)

Opt. Lett. (1)

Phys. Med. Biol. (1)

B. Banerjee, D. Roy, and R. M. Vasu, “A pseudo-dynamic sub-optimal filter for elastography under static loading and measurements,” Phys. Med. Biol. 54, 285–305 (2009).
[Crossref]

Physica D (2)

S. Sarkar, S. R. Chowdhury, M. Venugopal, R. M. Vasu, and D. Roy, “A Kushner-Stratonovich Monte Carlo filter applied to nonlinear dynamical system identification,” Physica D 270, 46–59 (2014).
[Crossref]

D. M. Livings, S. L. Dance, and N. K. Nichols, “Unbiased ensemble square root filters,” Physica D 237, 1021–1028 (2008).
[Crossref]

Ultrason. Imag. (1)

A. J. Devaney and G. Beylkin, “Diffraction tomography using arbitrary transmitter and receiver surfaces,” Ultrason. Imag. 6, 181–193 (1984).

Other (5)

A. C. Kak and M. Slaney, “Tomographic imaging with diffracting sources,” in Principles of Computerized Tomographic Imaging (IEEE, 1988), pp. 203–273.

A. C. Kak, “Tomographic imaging with diffracting and non-diffracting sources,” in Array Signal Processing, S. Haykin, ed. (Prentice-Hall, 1985).

R. F. Harrington, Field Computation by Moment Method (Macmillan, 1968).

F. C. Klebaner, Introduction to Stochastic Calculus with Applications, 2nd ed. (Imperial College, 2005).

M. Venugopal, D. Roy, and R. M. Vasu, “A new evolutionary Bayesian approach incorporating additive path correction for nonlinear inverse problems,” arxiv.org/pdf/1305.2702.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1.
Fig. 1.

Domain Ω of interest is the region enclosed by the square virtual boundary. The inhomogeneity in refractive index lies inside the dotted circle.

Fig. 2.
Fig. 2.

Domain considered for simulation and experiment.

Fig. 3.
Fig. 3.

Flow chart for the GN algorithm. The outer loop sets up the perturbation equation and the inner iteration solves it through a procedure like the conjugate gradient search.

Fig. 4.
Fig. 4.

Flowchart explaining the recursion used by the evolutionary stochastic search scheme (ESM).

Fig. 5.
Fig. 5.

Schematic diagram of the experimental setup.

Fig. 6.
Fig. 6.

Comparison of the experimental and the simulated normal derivative of intensity data.

Fig. 7.
Fig. 7.

(a) Cross-section (through the center) of the reconstructed refractive index distributions with simulated data with 1% noise. (b) Cross-section (through the center) of the reconstructed refractive index distributions with simulated data with 4% noise.

Fig. 8.
Fig. 8.

Comparison of the computed data (normal derivative of intensity) obtained from the reconstructed objects, the cross sections of which are shown in Fig. 7(b).

Fig. 9.
Fig. 9.

(a) Reference refractive index profile of fiber. (b) Reconstructed refractive index profile using the FBP algorithm. The data for FBP was the total field obtained from the measured intensity through phase retrieval. (c) Reconstructed refractive index profile using the GN method. (d) Reconstructed refractive index profile using ESM.

Fig. 10.
Fig. 10.

Cross-section (through the center) of the reconstructed refractive index distributions of Fig. 9.

Equations (29)

Equations on this page are rendered with MathJax. Learn more.

(2+k(r)2)u(r)=0.
n(r)=n0+δn(r)
k2(r)=k02(1+δn(r)n0)2.
(2+k02)u(r)=k02[(n(r)n0)21]u(r).
(2+k02)us(r)=o(r)u(r).
us(r)=Ωg(rr)o(r)u0(r)dr.
[2+2ikθ^·k02f(r)]V(r)=k02f(r).
[V(m)+ik0{(cosθcosθ˜(m))x^+(sinθsinθ˜(m))y^}V(m)]·n^=0,
Δfi+1=Δfi+(JTJ+λI)1JTΔM.
(2+2ik0θ^·k02f(r))δV(r)=k02(1+V(r))d(r).
f[I(Ω)n^]=limd0n^[(1+V*(Ω))δV(Ω)]+n^[(1+V(Ω))δV*(Ω)]d.
(2+2ik0yk02f(r))V(r)=k02f(r)
V(m)x[ik0cosθ˜(m)]V(m)=0onx+x
V(m)y+ik0[1sinθ˜(m)]V(m)=0ony+y.
(2+2ik0yk02f(r))δV(r)=k02(1+V(r))d(r)
(2ψ+2ik0ψyk02f(r)ψ)=0inΩ,
ψy+ik0(1+sinθ˜)ψ=qony+y
ψx+(ik0cosθ˜)ψ=0onx+x.
y+[δVyq]d(Ω)=Ωψ*[k02(1+V)d(r)]dΩ.
δV(m0)y=ΩGψ*(r,m0)[k02(1+V(r))d(r)]dΩ.
Ji,j=Gψ*(j,i)[k02(1+V(j)]+Gψ**(j,i)[k02(1+V*(j)].
dfτ=dξτ
Mτ=H(fτ)+ητ.
fk+1=f^k+Δξk
Mk+1=H(fk+1)+ηk+1,
f^k+1(j)=fk+1(j)+Gk+1(Mk+1H(fk+1(j))),j=1,,nE.
Fk+1=1nE1[fk+1(1)1nEj=1nEfk+1(j),,fk+1(nE)1nEj=1nEfk+1(j)]
Mk+1=1nE1[H(fk+1(1))1nEj=1nEH(fk+1(j)),,H(fk+1(nE))1nEj=1nEH(fk+1(j))].
f^¯k+1,nE=1nEj=1nEf^k+1(j).

Metrics