Abstract

We explore a pseudodynamic form of the quadratic parameter update equation for diffuse optical tomographic reconstruction from noisy data. A few explicit and implicit strategies for obtaining the parameter updates via a semianalytical integration of the pseudodynamic equations are proposed. Despite the ill-posedness of the inverse problem associated with diffuse optical tomography, adoption of the quadratic update scheme combined with the pseudotime integration appears not only to yield higher convergence, but also a muted sensitivity to the regularization parameters, which include the pseudotime step size for integration. These observations are validated through reconstructions with both numerically generated and experimentally acquired data.

© 2011 Optical Society of America

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  9. A. D. Close and A. H. Heilscher, “Optical tomography using time independent equation of radiative transfer—Part 2: inverse model,” J. Quant. Spectrosc. Radiat. Transfer 72715–732(2002).
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    [CrossRef]
  13. K. van den Doel and U. Ascher, “On level set regularization for highly ill-posed distributed parameter estimation problems,” J. Comput. Phys. 216, 707–723 (2006).
    [CrossRef]
  14. C. Vogel, Computational Methods for Inverse Problems (SIAM, 2002).
    [CrossRef]
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  16. M. Schweiger, S. R. Arridge, and I. Nissila, “Gauss–Newton method for image reconstruction in diffuse optical tomography,” Phys. Med. Biol. 50, 2365–2386 (2005).
    [CrossRef] [PubMed]
  17. S. K. Biswas, K. Rajan, and R. M. Vasu, “Interior photon absorption based adaptive regularization improves diffuse optical tomography,” Proc. SPIE 7546, 754611 (2010).
    [CrossRef]
  18. A. D. Klose and A. H. Heilscher, “Quasi-Newton methods in optical tomographic image reconstruction,” Inverse Probl. 19387–409 (2003).
    [CrossRef]
  19. T. J. Ypma, “Historical development of the Newton-Raphson method,” SIAM Rev. 37, 531–551 (1995).
    [CrossRef]
  20. F. Hettlich and W. Rundell, “A second degree method for nonlinear inverse problem,” SIAM J. Numer. Anal. 37, 587–620(2000).
    [CrossRef]
  21. B. Kanmani and R. M. Vasu, “Diffuse optical tomography through solving a system of quadratic equations: theory and simulations,” Phys. Med. Biol. 51, 981–998 (2006).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  23. B. Banerjee, D. Roy, and R. M. Vasu, “A pseudo-dynamical systems approach to a class of inverse problems in engineering,” Proc. R. Soc. London Ser. A 465, 1561–1579 (2009).
    [CrossRef]
  24. 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]
  25. G. H. Gulub, P. C. Hansen, and D. O’Leary, “Tikhonov regularization and total least squares,” SIAM. J. Matrix Anal. Appl. 21, 185–194 (1999).
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  26. D. Roy, “A numeric-analytic technique non-linear deterministic and stochastic dynamical systems,” Proc. R. Soc. London Ser. A 457, 539–566 (2001).
    [CrossRef]
  27. D. Roy, “Phase space linearization for non-linear oscillator: deterministic and stochastic systems,” J. Sound Vib. 231, 307–341(2000).
    [CrossRef]
  28. S. K. Biswas, K. Rajan, and R. M. Vasu, “Accelerated gradient based diffuse optical tomographic image reconstruction,” Med. Phys. 38, 539–547(2011).
    [CrossRef] [PubMed]
  29. B. Kanmani and R. M. Vasu, “Noise-tolerance analysis for detection and reconstruction of absorbing inhomogeneities with diffuse optical tomography using single- and phase-correlated dual-source schemes,” Phys. Med. Biol. 52, 1409–1429 (2007).
    [CrossRef] [PubMed]
  30. K. D. Paulsen and H. Jiang, “Spatially-varying optical property reconstruction using a finite element diffusion equation approximation,” Med. Phys. 22, 691–701 (1995).
    [CrossRef] [PubMed]
  31. S. K. Biswas, K. Rajan, and R. M. Vasu, “Diffuse optical tomographic imager using a single light source,” J. Appl. Phys. 105, 024702 (2009).
    [CrossRef]
  32. M. Autiero, R. Liuzzi, P. Riccio, and G. Roberti, “Determination of the concentration scaling law of the scattering coefficient of water solutions of Intralipid at 832 nm by comparision between collimated detection and Monte Carlo simulations,” Lasers Surg. Med. 36, 414–422 (2005).
    [CrossRef] [PubMed]

2011

S. K. Biswas, K. Rajan, and R. M. Vasu, “Accelerated gradient based diffuse optical tomographic image reconstruction,” Med. Phys. 38, 539–547(2011).
[CrossRef] [PubMed]

2010

S. K. Biswas, K. Rajan, and R. M. Vasu, “Interior photon absorption based adaptive regularization improves diffuse optical tomography,” Proc. SPIE 7546, 754611 (2010).
[CrossRef]

2009

B. Banerjee, D. Roy, and R. M. Vasu, “A pseudo-dynamical systems approach to a class of inverse problems in engineering,” Proc. R. Soc. London Ser. A 465, 1561–1579 (2009).
[CrossRef]

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]

S. K. Biswas, K. Rajan, and R. M. Vasu, “Diffuse optical tomographic imager using a single light source,” J. Appl. Phys. 105, 024702 (2009).
[CrossRef]

2007

B. Kanmani and R. M. Vasu, “Noise-tolerance analysis for detection and reconstruction of absorbing inhomogeneities with diffuse optical tomography using single- and phase-correlated dual-source schemes,” Phys. Med. Biol. 52, 1409–1429 (2007).
[CrossRef] [PubMed]

2006

B. Kanmani and R. M. Vasu, “Diffuse optical tomography through solving a system of quadratic equations: theory and simulations,” Phys. Med. Biol. 51, 981–998 (2006).
[CrossRef] [PubMed]

U. Ascher, E. Haber, and H. Huang, “On effective methods for implicit piecewise smooth surface recovery,” SIAM J. Comput. 28, 339–358 (2006).
[CrossRef]

K. van den Doel and U. Ascher, “On level set regularization for highly ill-posed distributed parameter estimation problems,” J. Comput. Phys. 216, 707–723 (2006).
[CrossRef]

2005

M. Schweiger, S. R. Arridge, and I. Nissila, “Gauss–Newton method for image reconstruction in diffuse optical tomography,” Phys. Med. Biol. 50, 2365–2386 (2005).
[CrossRef] [PubMed]

A. P. Gibson, J. Hebden, and Arridge, “Recent advantages in diffuse optical tomography,” Phys. Med. Biol. 50, R1–R43(2005).
[CrossRef] [PubMed]

M. Autiero, R. Liuzzi, P. Riccio, and G. Roberti, “Determination of the concentration scaling law of the scattering coefficient of water solutions of Intralipid at 832 nm by comparision between collimated detection and Monte Carlo simulations,” Lasers Surg. Med. 36, 414–422 (2005).
[CrossRef] [PubMed]

2004

L. Muzi, A. P. Lyons, and E. Pouliquen, “Use of X-ray computed tomography for the estimation of parameters relevant to the modeling of acoustic scattering from the seafloor,” Nucl. Instrum. Methods Phys. Res. B 213, 491–497 (2004).
[CrossRef]

2003

A. D. Klose and A. H. Heilscher, “Quasi-Newton methods in optical tomographic image reconstruction,” Inverse Probl. 19387–409 (2003).
[CrossRef]

2002

A. D. Close and A. H. Heilscher, “Optical tomography using time independent equation of radiative transfer—Part 2: inverse model,” J. Quant. Spectrosc. Radiat. Transfer 72715–732(2002).
[CrossRef]

2001

D. Roy, “A numeric-analytic technique non-linear deterministic and stochastic dynamical systems,” Proc. R. Soc. London Ser. A 457, 539–566 (2001).
[CrossRef]

2000

D. Roy, “Phase space linearization for non-linear oscillator: deterministic and stochastic systems,” J. Sound Vib. 231, 307–341(2000).
[CrossRef]

F. Hettlich and W. Rundell, “A second degree method for nonlinear inverse problem,” SIAM J. Numer. Anal. 37, 587–620(2000).
[CrossRef]

1999

G. H. Gulub, P. C. Hansen, and D. O’Leary, “Tikhonov regularization and total least squares,” SIAM. J. Matrix Anal. Appl. 21, 185–194 (1999).
[CrossRef]

A. H. Heilscher, A. D. Close, and K. M. Hansen, “Gradient based iterative image reconstruction scheme for time resolved optical tomography,” IEEE Trans. Med. Imaging 18, 262–271(1999).
[CrossRef]

S. R. Arridge and M. Schweiger, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
[CrossRef]

M. Cheny, D. Issacson, and J. C. Newell, “Electrical impedance tomography,” SIAM Rev. 41, 85–101 (1999).
[CrossRef]

1998

1995

K. D. Paulsen and H. Jiang, “Spatially-varying optical property reconstruction using a finite element diffusion equation approximation,” Med. Phys. 22, 691–701 (1995).
[CrossRef] [PubMed]

T. J. Ypma, “Historical development of the Newton-Raphson method,” SIAM Rev. 37, 531–551 (1995).
[CrossRef]

1989

1988

A. Whitten and J. E. Molyneux, “Geophysical imaging with arbitrary source illumination,” IEEE Trans. Geosci. Remote Sens. 26, 409–419 (1988).
[CrossRef]

1963

D. W. Marquardt, “An algorithm for the least-square estimation of non-linear parameters,” SIAM J. Appl. Math. 11, 431–441(1963).
[CrossRef]

1944

K. Levenberg, “A method for the solution of certain non-linear problems in least-squares,” Q. J. Appl. Math. 2, 164–168(1944).

Arridge,

A. P. Gibson, J. Hebden, and Arridge, “Recent advantages in diffuse optical tomography,” Phys. Med. Biol. 50, R1–R43(2005).
[CrossRef] [PubMed]

Arridge, S. R.

M. Schweiger, S. R. Arridge, and I. Nissila, “Gauss–Newton method for image reconstruction in diffuse optical tomography,” Phys. Med. Biol. 50, 2365–2386 (2005).
[CrossRef] [PubMed]

S. R. Arridge and M. Schweiger, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
[CrossRef]

S. R. Arridge and M. Schweiger, “A gradient based optimization scheme for optical tomography,” Opt. Express 2, 213–226(1998).
[CrossRef] [PubMed]

Ascher, U.

U. Ascher, E. Haber, and H. Huang, “On effective methods for implicit piecewise smooth surface recovery,” SIAM J. Comput. 28, 339–358 (2006).
[CrossRef]

K. van den Doel and U. Ascher, “On level set regularization for highly ill-posed distributed parameter estimation problems,” J. Comput. Phys. 216, 707–723 (2006).
[CrossRef]

Autiero, M.

M. Autiero, R. Liuzzi, P. Riccio, and G. Roberti, “Determination of the concentration scaling law of the scattering coefficient of water solutions of Intralipid at 832 nm by comparision between collimated detection and Monte Carlo simulations,” Lasers Surg. Med. 36, 414–422 (2005).
[CrossRef] [PubMed]

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]

B. Banerjee, D. Roy, and R. M. Vasu, “A pseudo-dynamical systems approach to a class of inverse problems in engineering,” Proc. R. Soc. London Ser. A 465, 1561–1579 (2009).
[CrossRef]

Biswas, S. K.

S. K. Biswas, K. Rajan, and R. M. Vasu, “Accelerated gradient based diffuse optical tomographic image reconstruction,” Med. Phys. 38, 539–547(2011).
[CrossRef] [PubMed]

S. K. Biswas, K. Rajan, and R. M. Vasu, “Interior photon absorption based adaptive regularization improves diffuse optical tomography,” Proc. SPIE 7546, 754611 (2010).
[CrossRef]

S. K. Biswas, K. Rajan, and R. M. Vasu, “Diffuse optical tomographic imager using a single light source,” J. Appl. Phys. 105, 024702 (2009).
[CrossRef]

Chance, B.

Cheny, M.

M. Cheny, D. Issacson, and J. C. Newell, “Electrical impedance tomography,” SIAM Rev. 41, 85–101 (1999).
[CrossRef]

Close, A. D.

A. D. Close and A. H. Heilscher, “Optical tomography using time independent equation of radiative transfer—Part 2: inverse model,” J. Quant. Spectrosc. Radiat. Transfer 72715–732(2002).
[CrossRef]

A. H. Heilscher, A. D. Close, and K. M. Hansen, “Gradient based iterative image reconstruction scheme for time resolved optical tomography,” IEEE Trans. Med. Imaging 18, 262–271(1999).
[CrossRef]

Dennis, J. E.

J. E. Dennis, Jr., and R. B. Schnabel, “Quasi Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Prentice-Hall, 1983).

Gibson, A. P.

A. P. Gibson, J. Hebden, and Arridge, “Recent advantages in diffuse optical tomography,” Phys. Med. Biol. 50, R1–R43(2005).
[CrossRef] [PubMed]

Gulub, G. H.

G. H. Gulub, P. C. Hansen, and D. O’Leary, “Tikhonov regularization and total least squares,” SIAM. J. Matrix Anal. Appl. 21, 185–194 (1999).
[CrossRef]

Haber, E.

U. Ascher, E. Haber, and H. Huang, “On effective methods for implicit piecewise smooth surface recovery,” SIAM J. Comput. 28, 339–358 (2006).
[CrossRef]

Hansen, K. M.

A. H. Heilscher, A. D. Close, and K. M. Hansen, “Gradient based iterative image reconstruction scheme for time resolved optical tomography,” IEEE Trans. Med. Imaging 18, 262–271(1999).
[CrossRef]

Hansen, P. C.

G. H. Gulub, P. C. Hansen, and D. O’Leary, “Tikhonov regularization and total least squares,” SIAM. J. Matrix Anal. Appl. 21, 185–194 (1999).
[CrossRef]

Hebden, J.

A. P. Gibson, J. Hebden, and Arridge, “Recent advantages in diffuse optical tomography,” Phys. Med. Biol. 50, R1–R43(2005).
[CrossRef] [PubMed]

Heilscher, A. H.

A. D. Klose and A. H. Heilscher, “Quasi-Newton methods in optical tomographic image reconstruction,” Inverse Probl. 19387–409 (2003).
[CrossRef]

A. D. Close and A. H. Heilscher, “Optical tomography using time independent equation of radiative transfer—Part 2: inverse model,” J. Quant. Spectrosc. Radiat. Transfer 72715–732(2002).
[CrossRef]

A. H. Heilscher, A. D. Close, and K. M. Hansen, “Gradient based iterative image reconstruction scheme for time resolved optical tomography,” IEEE Trans. Med. Imaging 18, 262–271(1999).
[CrossRef]

Hettlich, F.

F. Hettlich and W. Rundell, “A second degree method for nonlinear inverse problem,” SIAM J. Numer. Anal. 37, 587–620(2000).
[CrossRef]

Huang, H.

U. Ascher, E. Haber, and H. Huang, “On effective methods for implicit piecewise smooth surface recovery,” SIAM J. Comput. 28, 339–358 (2006).
[CrossRef]

Ishimaru, A.

Issacson, D.

M. Cheny, D. Issacson, and J. C. Newell, “Electrical impedance tomography,” SIAM Rev. 41, 85–101 (1999).
[CrossRef]

Jiang, H.

K. D. Paulsen and H. Jiang, “Spatially-varying optical property reconstruction using a finite element diffusion equation approximation,” Med. Phys. 22, 691–701 (1995).
[CrossRef] [PubMed]

Kanmani, B.

B. Kanmani and R. M. Vasu, “Noise-tolerance analysis for detection and reconstruction of absorbing inhomogeneities with diffuse optical tomography using single- and phase-correlated dual-source schemes,” Phys. Med. Biol. 52, 1409–1429 (2007).
[CrossRef] [PubMed]

B. Kanmani and R. M. Vasu, “Diffuse optical tomography through solving a system of quadratic equations: theory and simulations,” Phys. Med. Biol. 51, 981–998 (2006).
[CrossRef] [PubMed]

Klose, A. D.

A. D. Klose and A. H. Heilscher, “Quasi-Newton methods in optical tomographic image reconstruction,” Inverse Probl. 19387–409 (2003).
[CrossRef]

Levenberg, K.

K. Levenberg, “A method for the solution of certain non-linear problems in least-squares,” Q. J. Appl. Math. 2, 164–168(1944).

Liuzzi, R.

M. Autiero, R. Liuzzi, P. Riccio, and G. Roberti, “Determination of the concentration scaling law of the scattering coefficient of water solutions of Intralipid at 832 nm by comparision between collimated detection and Monte Carlo simulations,” Lasers Surg. Med. 36, 414–422 (2005).
[CrossRef] [PubMed]

Lyons, A. P.

L. Muzi, A. P. Lyons, and E. Pouliquen, “Use of X-ray computed tomography for the estimation of parameters relevant to the modeling of acoustic scattering from the seafloor,” Nucl. Instrum. Methods Phys. Res. B 213, 491–497 (2004).
[CrossRef]

Marquardt, D. W.

D. W. Marquardt, “An algorithm for the least-square estimation of non-linear parameters,” SIAM J. Appl. Math. 11, 431–441(1963).
[CrossRef]

Molyneux, J. E.

A. Whitten and J. E. Molyneux, “Geophysical imaging with arbitrary source illumination,” IEEE Trans. Geosci. Remote Sens. 26, 409–419 (1988).
[CrossRef]

Muzi, L.

L. Muzi, A. P. Lyons, and E. Pouliquen, “Use of X-ray computed tomography for the estimation of parameters relevant to the modeling of acoustic scattering from the seafloor,” Nucl. Instrum. Methods Phys. Res. B 213, 491–497 (2004).
[CrossRef]

Newell, J. C.

M. Cheny, D. Issacson, and J. C. Newell, “Electrical impedance tomography,” SIAM Rev. 41, 85–101 (1999).
[CrossRef]

Nissila, I.

M. Schweiger, S. R. Arridge, and I. Nissila, “Gauss–Newton method for image reconstruction in diffuse optical tomography,” Phys. Med. Biol. 50, 2365–2386 (2005).
[CrossRef] [PubMed]

O’Leary, D.

G. H. Gulub, P. C. Hansen, and D. O’Leary, “Tikhonov regularization and total least squares,” SIAM. J. Matrix Anal. Appl. 21, 185–194 (1999).
[CrossRef]

Patterson, M. S.

Paulsen, K. D.

K. D. Paulsen and H. Jiang, “Spatially-varying optical property reconstruction using a finite element diffusion equation approximation,” Med. Phys. 22, 691–701 (1995).
[CrossRef] [PubMed]

Pouliquen, E.

L. Muzi, A. P. Lyons, and E. Pouliquen, “Use of X-ray computed tomography for the estimation of parameters relevant to the modeling of acoustic scattering from the seafloor,” Nucl. Instrum. Methods Phys. Res. B 213, 491–497 (2004).
[CrossRef]

Rajan, K.

S. K. Biswas, K. Rajan, and R. M. Vasu, “Accelerated gradient based diffuse optical tomographic image reconstruction,” Med. Phys. 38, 539–547(2011).
[CrossRef] [PubMed]

S. K. Biswas, K. Rajan, and R. M. Vasu, “Interior photon absorption based adaptive regularization improves diffuse optical tomography,” Proc. SPIE 7546, 754611 (2010).
[CrossRef]

S. K. Biswas, K. Rajan, and R. M. Vasu, “Diffuse optical tomographic imager using a single light source,” J. Appl. Phys. 105, 024702 (2009).
[CrossRef]

Riccio, P.

M. Autiero, R. Liuzzi, P. Riccio, and G. Roberti, “Determination of the concentration scaling law of the scattering coefficient of water solutions of Intralipid at 832 nm by comparision between collimated detection and Monte Carlo simulations,” Lasers Surg. Med. 36, 414–422 (2005).
[CrossRef] [PubMed]

Roberti, G.

M. Autiero, R. Liuzzi, P. Riccio, and G. Roberti, “Determination of the concentration scaling law of the scattering coefficient of water solutions of Intralipid at 832 nm by comparision between collimated detection and Monte Carlo simulations,” Lasers Surg. Med. 36, 414–422 (2005).
[CrossRef] [PubMed]

Roy, D.

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]

B. Banerjee, D. Roy, and R. M. Vasu, “A pseudo-dynamical systems approach to a class of inverse problems in engineering,” Proc. R. Soc. London Ser. A 465, 1561–1579 (2009).
[CrossRef]

D. Roy, “A numeric-analytic technique non-linear deterministic and stochastic dynamical systems,” Proc. R. Soc. London Ser. A 457, 539–566 (2001).
[CrossRef]

D. Roy, “Phase space linearization for non-linear oscillator: deterministic and stochastic systems,” J. Sound Vib. 231, 307–341(2000).
[CrossRef]

Rundell, W.

F. Hettlich and W. Rundell, “A second degree method for nonlinear inverse problem,” SIAM J. Numer. Anal. 37, 587–620(2000).
[CrossRef]

Schnabel, R. B.

J. E. Dennis, Jr., and R. B. Schnabel, “Quasi Numerical Methods for Unconstrained Optimization and Nonlinear Equations (Prentice-Hall, 1983).

Schweiger, M.

M. Schweiger, S. R. Arridge, and I. Nissila, “Gauss–Newton method for image reconstruction in diffuse optical tomography,” Phys. Med. Biol. 50, 2365–2386 (2005).
[CrossRef] [PubMed]

S. R. Arridge and M. Schweiger, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
[CrossRef]

S. R. Arridge and M. Schweiger, “A gradient based optimization scheme for optical tomography,” Opt. Express 2, 213–226(1998).
[CrossRef] [PubMed]

van den Doel, K.

K. van den Doel and U. Ascher, “On level set regularization for highly ill-posed distributed parameter estimation problems,” J. Comput. Phys. 216, 707–723 (2006).
[CrossRef]

Vasu, R. M.

S. K. Biswas, K. Rajan, and R. M. Vasu, “Accelerated gradient based diffuse optical tomographic image reconstruction,” Med. Phys. 38, 539–547(2011).
[CrossRef] [PubMed]

S. K. Biswas, K. Rajan, and R. M. Vasu, “Interior photon absorption based adaptive regularization improves diffuse optical tomography,” Proc. SPIE 7546, 754611 (2010).
[CrossRef]

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]

B. Banerjee, D. Roy, and R. M. Vasu, “A pseudo-dynamical systems approach to a class of inverse problems in engineering,” Proc. R. Soc. London Ser. A 465, 1561–1579 (2009).
[CrossRef]

S. K. Biswas, K. Rajan, and R. M. Vasu, “Diffuse optical tomographic imager using a single light source,” J. Appl. Phys. 105, 024702 (2009).
[CrossRef]

B. Kanmani and R. M. Vasu, “Noise-tolerance analysis for detection and reconstruction of absorbing inhomogeneities with diffuse optical tomography using single- and phase-correlated dual-source schemes,” Phys. Med. Biol. 52, 1409–1429 (2007).
[CrossRef] [PubMed]

B. Kanmani and R. M. Vasu, “Diffuse optical tomography through solving a system of quadratic equations: theory and simulations,” Phys. Med. Biol. 51, 981–998 (2006).
[CrossRef] [PubMed]

Vogel, C.

C. Vogel, Computational Methods for Inverse Problems (SIAM, 2002).
[CrossRef]

Whitten, A.

A. Whitten and J. E. Molyneux, “Geophysical imaging with arbitrary source illumination,” IEEE Trans. Geosci. Remote Sens. 26, 409–419 (1988).
[CrossRef]

Wilson, B. C.

Ypma, T. J.

T. J. Ypma, “Historical development of the Newton-Raphson method,” SIAM Rev. 37, 531–551 (1995).
[CrossRef]

Appl. Opt.

IEEE Trans. Geosci. Remote Sens.

A. Whitten and J. E. Molyneux, “Geophysical imaging with arbitrary source illumination,” IEEE Trans. Geosci. Remote Sens. 26, 409–419 (1988).
[CrossRef]

IEEE Trans. Med. Imaging

A. H. Heilscher, A. D. Close, and K. M. Hansen, “Gradient based iterative image reconstruction scheme for time resolved optical tomography,” IEEE Trans. Med. Imaging 18, 262–271(1999).
[CrossRef]

Inverse Probl.

S. R. Arridge and M. Schweiger, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).
[CrossRef]

A. D. Klose and A. H. Heilscher, “Quasi-Newton methods in optical tomographic image reconstruction,” Inverse Probl. 19387–409 (2003).
[CrossRef]

J. Appl. Phys.

S. K. Biswas, K. Rajan, and R. M. Vasu, “Diffuse optical tomographic imager using a single light source,” J. Appl. Phys. 105, 024702 (2009).
[CrossRef]

J. Comput. Phys.

K. van den Doel and U. Ascher, “On level set regularization for highly ill-posed distributed parameter estimation problems,” J. Comput. Phys. 216, 707–723 (2006).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transfer

A. D. Close and A. H. Heilscher, “Optical tomography using time independent equation of radiative transfer—Part 2: inverse model,” J. Quant. Spectrosc. Radiat. Transfer 72715–732(2002).
[CrossRef]

J. Sound Vib.

D. Roy, “Phase space linearization for non-linear oscillator: deterministic and stochastic systems,” J. Sound Vib. 231, 307–341(2000).
[CrossRef]

Lasers Surg. Med.

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Figures (8)

Fig. 1
Fig. 1

(a) Data gathering geometry. (b) Target image. (c) Reconstructed image using GN method. (d) Reconstructed image using pseudodynamic time marching algorithm on the linear perturbation equation. (e) Same as in (d), but obtained through the quadratic perturbation equation. (f) Cross-sectional plots through the centers of the reconstructed inhomogeneities.

Fig. 2
Fig. 2

(a) Pseudotime evolution of the absorption coefficient. (b) Plots showing the variation of the MSE at convergence, with λ for the two methods. It is evident that the pseudotime scheme is unaffected by changes in λ. (c) Variation of MSE with recursion/iteration for the GN and pseudodynamic methods.

Fig. 3
Fig. 3

Comparison of reconstructed absorption coefficients using the GN method for various noise levels in data. The included inhomogeneity is of diameter 10 mm . The noise level varies from 2% to 10% in intervals of 2% along the rows. (f) Cross-sectional plots through the centers of the reconstructed inhomogenieties.

Fig. 4
Fig. 4

Comparison of reconstructed absorption coefficients using pseudotime integration of the linear perturbation equation, for various noise levels in data. The included inhomogeneity is of diameter 10 mm . The noise level varies from 2% to 10% in intervals of 2% along the rows. (f) Cross-sectional plots through the centers of the reconstructed inhomogenieties.

Fig. 5
Fig. 5

Comparison of reconstructed absorption coefficients using pseudotime integration of the quadratic perturbation equation, for various noise levels in data. The included inhomogeneity is of diameter 10 mm . The noise level varies from 2% to 10% in intervals of 2% along the rows. (f) Cross-sectional plots through the centers of the reconstructed inhomogenieties.

Fig. 6
Fig. 6

Comparison of performance of the various reconstruction strategies; cross-sectional plots through the reconstructed inhomogeneities from (a) 6% noisy data and (b) 8% noisy data. It is seen that the quadratic algorithm using pseudotime integration recovers the contrast the best.

Fig. 7
Fig. 7

Results of μ a recovery shown as cross-sectional plots through the center of the inhomogeneities, when the background μ s is varied (a) using the GN method (b) using the linear pseudodynamic method, and (c) using the nonlinear pseudodynamic method. Note that the reduction in contrast is the least when the nonlinear pseudodynamic algorithm is used.

Fig. 8
Fig. 8

Experimental setup and the reconstruction results using experimental data. (a) Schematic diagram of the DOT imaging system and (b) its photograph. Reconstructed images using (c) the GN algorithm, (d) the linear pseudotime marching algorithm, and (e) the nonlinear pseudotime marching algorithm. (f) Cross-sectional plots through the reconstructed inhomogeneities.

Equations (30)

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F ( μ a ( r ) , κ ( r ) ) = M ,
· ( κ ( r ) Φ ( r ) ) + μ a ( r ) Φ ( r ) = Q 0 δ ( r r 0 ) ,
2 A κ ( m ) Φ ( m ) n + Φ ( m ) = 0.
Δ M = M e M c = F ( μ b ) Δ μ .
Δ M = F ( μ b ) Δ μ ,
[ F ( μ b ) T F ( μ b ) + λ I ] Δ μ = F ( μ b ) T Δ M .
e := [ F ( μ b ) T F ( μ b ) + λ I ] Δ μ F ( μ b ) T Δ M 2 .
Δ M = F ( μ b ) Δ μ + Δ μ T F ( μ b ) Δ μ .
e := [ F ( μ b ) T F ( μ b ) ] Δ μ + [ F ( μ b ) T ( Δ μ T F ( μ b ) ] Δ μ + λ I Δ μ F ( μ b ) T Δ M 2 .
χ = 1 2 M e F ( μ a , μ s ) 2 = i = 1 d r i 2 ,
[ F T F i = 1 d F i Δ M i ] Δ μ F T Δ M = 0.
F T F Δ μ + F Δ M Δ μ = F T Δ M ,
F T F Δ μ + F T Δ μ T F Δ μ = F T Δ M .
μ ˙ + S ( μ i , λ ) [ μ ( t ) μ i ] + V = 0 ,
G ( μ i , λ ) := F T ( μ i ) F ( μ i ) + F T Δ μ T F + λ I .
μ i + 1 = exp ( S ( μ i , λ ) μ i + t i t i + 1 exp ( S ( μ i , λ ) ( t i + 1 t ) f ( t ) d t ,
μ ˙ + G ( μ , λ ) ( μ μ i ) + V ( μ i ) = 0 ,
μ ˙ + G ( μ i , λ ) ( μ μ i ) + V ( μ i ) = 0 ,
Δ μ i + 1 = [ F T ( μ i ) F ( μ i ) + λ ˜ I ] 1 F T ( μ i ) Δ M ; μ ˜ i + 1 = μ i + Δ μ ˜ i + 1 .
μ ˙ + G ( μ ˜ i + 1 , λ ) ( μ μ i ) + V ( μ ˜ i + 1 ) = 0.
μ ˙ + G ( μ i + 1 , λ ) ( μ μ i ) + V ( μ i + 1 ) = 0.
μ i + 1 exp ( G ( μ i + 1 , λ ) μ i + t i t i + 1 exp ( G ( μ i + 1 , λ ) ( t i + 1 t ) f ( μ i + 1 , t ) d t = 0 ,
F ( μ 2 ) F ( μ 1 ) F ( μ 1 ) ( μ 2 μ 1 ) C μ 2 μ 1 F ( μ 2 ) F ( μ 1 ) ,
F C ,
F C ,
r 0 := μ ^ μ 0 ρ C + C C 2 λ ˜ ,
λ > ρ 1 ρ ( C + C C 2 λ ˜ ( sup μ B ( μ ^ , r 0 ) M e F ( μ ) + δ ) ) 2 ,
M e ( δ ) F ( μ k ) τ δ M e ( δ ) F ( μ j ) for all     j [ 0 , k 1 ] ,
τ > 1 + C r 0 ρ r 0 ( C + C C 2 λ ˜ ) > 1.
M e ( δ ) = M c + N 100 ( 1 2 R ) M c ,

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