Abstract

We present an iterative total least-squares algorithm for computing images of the interior structure of highly scattering media by using the conjugate gradient method. For imaging the dense scattering media in optical tomography, a perturbation approach has been described previously [Y. Wang et al., Proc. SPIE 1641, 58 (1992); R. L. Barbour et al., in Medical Optical Tomography: Functional Imaging and Monitoring (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 87–120], which solves a perturbation equation of the form WΔx=ΔI. In order to solve this equation, least-squares or regularized least-squares solvers have been used in the past to determine best fits to the measurement data ΔI while assuming that the operator matrix W is accurate. In practice, errors also occur in the operator matrix. Here we propose an iterative total least-squares (ITLS) method that minimizes the errors in both weights and detector readings. Theoretically, the total least-squares (TLS) solution is given by the singular vector of the matrix [WI] associated with the smallest singular value. The proposed ITLS method obtains this solution by using a conjugate gradient method that is particularly suitable for very large matrices. Simulation results have shown that the TLS method can yield a significantly more accurate result than the least-squares method.

© 1997 Optical Society of America

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  1. R. L. Barbour, H. L. Graber, J. Chang, S. Barbour, P. C. Koo, R. Aronson, “MR guided optical tomography: prospects and computation for a new imaging method,” in IEEE Computational Science Engineering Magazine, Winter1995, pp. 63–77.
  2. Y. Wang, J. Chang, R. Aronson, R. Barbour, H. Graber, J. Lubowsky, “Imaging scattering media by diffusion tomography: an iterative perturbation approach,” in Physiological Monitoring and Early Detection Diagnostic Methods, T. S. Mang, ed., Proc. SPIE1641, 58–71 (1992).
    [CrossRef]
  3. R. L. Barbour, H. L. Graber, Y. Wang, J. Chang, R. Aronson, “A perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. H. Mueller, B. Chance, R. R. Alfano, S. R. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. R. Masters, S. Svanberg, P. van der Zee, eds., Vol. IS11 of Institute Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 87–120.
  4. J. Chang, Y. Wang, R. Aronson, H. L. Graber, R. L. Barbour, “A layer-stripping approach for recovery of scattering media from time-resolved data,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc. SPIE1767, 384–395 (1992).
    [CrossRef]
  5. W. Zhu, Y. Wang, H. L. Graber, R. L. Barbour, J. Chang, “A regularized progressive expansion algorithm for recovery of scattering media from time-resolved data,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed. (Optical Society of America, Washington, D.C., 1994), pp. 211–216.
  6. W. Zhu, Y. Wang, Y. Deng, Y. Yao, R. Barbour, “Multiresolution regularized least squares image reconstruction based on wavelet in optical tomography,” in Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, R. L. Barbour, M. J. Carvlin, M. A. Fiddy, eds., Proc. SPIE2570, 186–196 (1995).
    [CrossRef]
  7. S. R. Arridge, “The forward and inverse problems in time resolved infra-red imaging,” in Medical Optical Tomography: Functional Imaging and Monitoring, Vol. IS11 of Institute Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 35–64.
  8. Y. Yao, Y. Wang, Y. Pei, W. Zhu, R. L. Barbour, “Simultaneous reconstruction of absorption and scattering distributions in turbid media using a Born iterative method,” in Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, R. L. Barbour, M. J. Carvlin, M. A. Fiddy, eds., Proc. SPIE2570, 96–107 (1995).
    [CrossRef]
  9. H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, M. S. Patterson, “Optical image reconstruction using frequency domain data: simulations and experiments,” J. Opt. Soc. Am. A 13, 253–266 (1996).
    [CrossRef]
  10. H. L. Graber, R. L. Barbour, J. Chang, R. Aronson, “Identification of the functional form of nonlinear effects of localized finite absorption in a diffusing medium,” in Optical Tomography; Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 669–681 (1995).
    [CrossRef]
  11. J. H. Justice, A. A. Vassiliou, “Diffraction tomography for geophysical monitoring of hydrocarbon reservoirs,” Proc. IEEE 78, 711–722 (1990).
    [CrossRef]
  12. V. Z. Mesarović, N. P. Galatsanos, A. Katsaggelos, “Regularized constrained total least squares image restoration,” IEEE Trans. Image Process. 4, 1096–1108 (1995).
    [CrossRef] [PubMed]
  13. P. Li, S. W. Flax, E. S. Ebbini, M. O’Donnell, “Blocked element compensation in phased array imaging,” IEEE Trans. Ultrason. Frequencies Freq. Control 40, 282–292 (1993).
  14. K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), Chap. 8, pp. 194–231.
  15. G. H. Golub, C. F. Van Loan, “An analysis of the total least squares problem,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal.17, 883–893 (1980).
  16. M. T. Silvia, E. C. Tacker, “Regularization of Marchenko’s integral equation by total least squares,” J. Acoust. Soc. Am. 72, 1202–1207 (1982).
    [CrossRef]
  17. G. H. Golub, “Some modified matrix eigenvalue problems,” SIAM Rev. 15, 318–334 (1973).
    [CrossRef]
  18. S. Van Huffel, J. Vandewalle, The Total Least Squares Problem: Computational Aspects and Analysis (SIAM Press, Philadelphia, 1991).
  19. C. E. Davila, “An efficient recursive total least squares algorithm for FIR adaptive filtering,” IEEE Trans. Signal Process. 42, 268–280 (1994).
    [CrossRef]
  20. N. K. Bose, H. C. Kim, H. M. Valenzuela, “Recursive total least squares algorithm for image reconstruction,” Multidimens. Syst. Signal Process. 4, 253–268 (1993).
  21. H. Chen, T. K. Sarkar, S. A. Dianat, J. D. Brule, “Adaptive spectral estimation by the conjugate gradient method,” IEEE Trans. Acoust. Speech Signal Process. ASSP-34, 272–284 (1986).
    [CrossRef]
  22. X. Yang, T. K. Sarkar, E. Arvas, “A survey of conjugate gradient algorithms for solution of extreme eigen-problem of a symmetric matrix,” IEEE Trans. Acoust. Speech Signal Process. 37, 1550–1556 (1989).
    [CrossRef]
  23. I. Shavitt, C. F. Bender, A. Pipano, R. P. Hosteny, “The iterative calculation of several of the lowest or highest eigenvalues and corresponding eigenvector of very large symmetric matrices,” J. Comput. Phys. 11, 90–108 (1973).
    [CrossRef]
  24. R. Fletcher, C. M. Reeves, “Function minimization by conjugate gradients,” Comput. J. 7, 149–154 (1964).
  25. R. L. Barbour, H. L. Graber, R. Aronson, J. Lubowsky, “Imaging of subsurface regions of random media by remote sensing,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. SPIE1431, 192–203 (1991).
    [CrossRef]
  26. Y. Q. Yao, Y. Wang, R. L. Barbour, H. L. Graber, J. W. Chang, “Scattering characteristics of photon density waves from an object in a spherically two-layer medium,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 291–303 (1995).
    [CrossRef]
  27. Y. Q. Yao, Y. Wang, Y. L. Pei, W. W. Zhu, J. H. Hu, R. L. Barbour, “Frequency domain optical tomography in human tissue,” in Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, R. L. Barbour, M. J. Carvlin, M. A. Fiddy, eds., Proc. SPIE2570, 254–266 (1995).
    [CrossRef]
  28. W. Zhu, Y. Wang, Y. Yao, R. L. Barbour, “Wavelet based multigrid reconstruction algorithm for optical tomography,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. Fujimoto, eds. (Optical Society of America, Washington, D.C., 1996), pp. 278–281.
  29. T. J. Abatzoglou, J. M. Mendel, G. A. Harada, “The constrained total least squares technique and its applications to harmonic superresolution,” IEEE Trans. Signal Process. 39, 1070–1087 (1991).
    [CrossRef]

1996

1995

V. Z. Mesarović, N. P. Galatsanos, A. Katsaggelos, “Regularized constrained total least squares image restoration,” IEEE Trans. Image Process. 4, 1096–1108 (1995).
[CrossRef] [PubMed]

1994

C. E. Davila, “An efficient recursive total least squares algorithm for FIR adaptive filtering,” IEEE Trans. Signal Process. 42, 268–280 (1994).
[CrossRef]

1993

N. K. Bose, H. C. Kim, H. M. Valenzuela, “Recursive total least squares algorithm for image reconstruction,” Multidimens. Syst. Signal Process. 4, 253–268 (1993).

P. Li, S. W. Flax, E. S. Ebbini, M. O’Donnell, “Blocked element compensation in phased array imaging,” IEEE Trans. Ultrason. Frequencies Freq. Control 40, 282–292 (1993).

1991

T. J. Abatzoglou, J. M. Mendel, G. A. Harada, “The constrained total least squares technique and its applications to harmonic superresolution,” IEEE Trans. Signal Process. 39, 1070–1087 (1991).
[CrossRef]

1990

J. H. Justice, A. A. Vassiliou, “Diffraction tomography for geophysical monitoring of hydrocarbon reservoirs,” Proc. IEEE 78, 711–722 (1990).
[CrossRef]

1989

X. Yang, T. K. Sarkar, E. Arvas, “A survey of conjugate gradient algorithms for solution of extreme eigen-problem of a symmetric matrix,” IEEE Trans. Acoust. Speech Signal Process. 37, 1550–1556 (1989).
[CrossRef]

1986

H. Chen, T. K. Sarkar, S. A. Dianat, J. D. Brule, “Adaptive spectral estimation by the conjugate gradient method,” IEEE Trans. Acoust. Speech Signal Process. ASSP-34, 272–284 (1986).
[CrossRef]

1982

M. T. Silvia, E. C. Tacker, “Regularization of Marchenko’s integral equation by total least squares,” J. Acoust. Soc. Am. 72, 1202–1207 (1982).
[CrossRef]

1973

G. H. Golub, “Some modified matrix eigenvalue problems,” SIAM Rev. 15, 318–334 (1973).
[CrossRef]

I. Shavitt, C. F. Bender, A. Pipano, R. P. Hosteny, “The iterative calculation of several of the lowest or highest eigenvalues and corresponding eigenvector of very large symmetric matrices,” J. Comput. Phys. 11, 90–108 (1973).
[CrossRef]

1964

R. Fletcher, C. M. Reeves, “Function minimization by conjugate gradients,” Comput. J. 7, 149–154 (1964).

Abatzoglou, T. J.

T. J. Abatzoglou, J. M. Mendel, G. A. Harada, “The constrained total least squares technique and its applications to harmonic superresolution,” IEEE Trans. Signal Process. 39, 1070–1087 (1991).
[CrossRef]

Aronson, R.

R. L. Barbour, H. L. Graber, J. Chang, S. Barbour, P. C. Koo, R. Aronson, “MR guided optical tomography: prospects and computation for a new imaging method,” in IEEE Computational Science Engineering Magazine, Winter1995, pp. 63–77.

H. L. Graber, R. L. Barbour, J. Chang, R. Aronson, “Identification of the functional form of nonlinear effects of localized finite absorption in a diffusing medium,” in Optical Tomography; Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 669–681 (1995).
[CrossRef]

R. L. Barbour, H. L. Graber, R. Aronson, J. Lubowsky, “Imaging of subsurface regions of random media by remote sensing,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. SPIE1431, 192–203 (1991).
[CrossRef]

R. L. Barbour, H. L. Graber, Y. Wang, J. Chang, R. Aronson, “A perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. H. Mueller, B. Chance, R. R. Alfano, S. R. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. R. Masters, S. Svanberg, P. van der Zee, eds., Vol. IS11 of Institute Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 87–120.

J. Chang, Y. Wang, R. Aronson, H. L. Graber, R. L. Barbour, “A layer-stripping approach for recovery of scattering media from time-resolved data,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc. SPIE1767, 384–395 (1992).
[CrossRef]

Y. Wang, J. Chang, R. Aronson, R. Barbour, H. Graber, J. Lubowsky, “Imaging scattering media by diffusion tomography: an iterative perturbation approach,” in Physiological Monitoring and Early Detection Diagnostic Methods, T. S. Mang, ed., Proc. SPIE1641, 58–71 (1992).
[CrossRef]

Arridge, S. R.

S. R. Arridge, “The forward and inverse problems in time resolved infra-red imaging,” in Medical Optical Tomography: Functional Imaging and Monitoring, Vol. IS11 of Institute Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 35–64.

Arvas, E.

X. Yang, T. K. Sarkar, E. Arvas, “A survey of conjugate gradient algorithms for solution of extreme eigen-problem of a symmetric matrix,” IEEE Trans. Acoust. Speech Signal Process. 37, 1550–1556 (1989).
[CrossRef]

Barbour, R.

Y. Wang, J. Chang, R. Aronson, R. Barbour, H. Graber, J. Lubowsky, “Imaging scattering media by diffusion tomography: an iterative perturbation approach,” in Physiological Monitoring and Early Detection Diagnostic Methods, T. S. Mang, ed., Proc. SPIE1641, 58–71 (1992).
[CrossRef]

W. Zhu, Y. Wang, Y. Deng, Y. Yao, R. Barbour, “Multiresolution regularized least squares image reconstruction based on wavelet in optical tomography,” in Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, R. L. Barbour, M. J. Carvlin, M. A. Fiddy, eds., Proc. SPIE2570, 186–196 (1995).
[CrossRef]

Barbour, R. L.

W. Zhu, Y. Wang, H. L. Graber, R. L. Barbour, J. Chang, “A regularized progressive expansion algorithm for recovery of scattering media from time-resolved data,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed. (Optical Society of America, Washington, D.C., 1994), pp. 211–216.

Y. Q. Yao, Y. Wang, Y. L. Pei, W. W. Zhu, J. H. Hu, R. L. Barbour, “Frequency domain optical tomography in human tissue,” in Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, R. L. Barbour, M. J. Carvlin, M. A. Fiddy, eds., Proc. SPIE2570, 254–266 (1995).
[CrossRef]

Y. Q. Yao, Y. Wang, R. L. Barbour, H. L. Graber, J. W. Chang, “Scattering characteristics of photon density waves from an object in a spherically two-layer medium,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 291–303 (1995).
[CrossRef]

W. Zhu, Y. Wang, Y. Yao, R. L. Barbour, “Wavelet based multigrid reconstruction algorithm for optical tomography,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. Fujimoto, eds. (Optical Society of America, Washington, D.C., 1996), pp. 278–281.

R. L. Barbour, H. L. Graber, Y. Wang, J. Chang, R. Aronson, “A perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. H. Mueller, B. Chance, R. R. Alfano, S. R. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. R. Masters, S. Svanberg, P. van der Zee, eds., Vol. IS11 of Institute Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 87–120.

J. Chang, Y. Wang, R. Aronson, H. L. Graber, R. L. Barbour, “A layer-stripping approach for recovery of scattering media from time-resolved data,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc. SPIE1767, 384–395 (1992).
[CrossRef]

R. L. Barbour, H. L. Graber, R. Aronson, J. Lubowsky, “Imaging of subsurface regions of random media by remote sensing,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. SPIE1431, 192–203 (1991).
[CrossRef]

Y. Yao, Y. Wang, Y. Pei, W. Zhu, R. L. Barbour, “Simultaneous reconstruction of absorption and scattering distributions in turbid media using a Born iterative method,” in Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, R. L. Barbour, M. J. Carvlin, M. A. Fiddy, eds., Proc. SPIE2570, 96–107 (1995).
[CrossRef]

H. L. Graber, R. L. Barbour, J. Chang, R. Aronson, “Identification of the functional form of nonlinear effects of localized finite absorption in a diffusing medium,” in Optical Tomography; Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 669–681 (1995).
[CrossRef]

R. L. Barbour, H. L. Graber, J. Chang, S. Barbour, P. C. Koo, R. Aronson, “MR guided optical tomography: prospects and computation for a new imaging method,” in IEEE Computational Science Engineering Magazine, Winter1995, pp. 63–77.

Barbour, S.

R. L. Barbour, H. L. Graber, J. Chang, S. Barbour, P. C. Koo, R. Aronson, “MR guided optical tomography: prospects and computation for a new imaging method,” in IEEE Computational Science Engineering Magazine, Winter1995, pp. 63–77.

Bender, C. F.

I. Shavitt, C. F. Bender, A. Pipano, R. P. Hosteny, “The iterative calculation of several of the lowest or highest eigenvalues and corresponding eigenvector of very large symmetric matrices,” J. Comput. Phys. 11, 90–108 (1973).
[CrossRef]

Bose, N. K.

N. K. Bose, H. C. Kim, H. M. Valenzuela, “Recursive total least squares algorithm for image reconstruction,” Multidimens. Syst. Signal Process. 4, 253–268 (1993).

Brule, J. D.

H. Chen, T. K. Sarkar, S. A. Dianat, J. D. Brule, “Adaptive spectral estimation by the conjugate gradient method,” IEEE Trans. Acoust. Speech Signal Process. ASSP-34, 272–284 (1986).
[CrossRef]

Case, K. M.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), Chap. 8, pp. 194–231.

Chang, J.

H. L. Graber, R. L. Barbour, J. Chang, R. Aronson, “Identification of the functional form of nonlinear effects of localized finite absorption in a diffusing medium,” in Optical Tomography; Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 669–681 (1995).
[CrossRef]

R. L. Barbour, H. L. Graber, J. Chang, S. Barbour, P. C. Koo, R. Aronson, “MR guided optical tomography: prospects and computation for a new imaging method,” in IEEE Computational Science Engineering Magazine, Winter1995, pp. 63–77.

R. L. Barbour, H. L. Graber, Y. Wang, J. Chang, R. Aronson, “A perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. H. Mueller, B. Chance, R. R. Alfano, S. R. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. R. Masters, S. Svanberg, P. van der Zee, eds., Vol. IS11 of Institute Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 87–120.

J. Chang, Y. Wang, R. Aronson, H. L. Graber, R. L. Barbour, “A layer-stripping approach for recovery of scattering media from time-resolved data,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc. SPIE1767, 384–395 (1992).
[CrossRef]

W. Zhu, Y. Wang, H. L. Graber, R. L. Barbour, J. Chang, “A regularized progressive expansion algorithm for recovery of scattering media from time-resolved data,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed. (Optical Society of America, Washington, D.C., 1994), pp. 211–216.

Y. Wang, J. Chang, R. Aronson, R. Barbour, H. Graber, J. Lubowsky, “Imaging scattering media by diffusion tomography: an iterative perturbation approach,” in Physiological Monitoring and Early Detection Diagnostic Methods, T. S. Mang, ed., Proc. SPIE1641, 58–71 (1992).
[CrossRef]

Chang, J. W.

Y. Q. Yao, Y. Wang, R. L. Barbour, H. L. Graber, J. W. Chang, “Scattering characteristics of photon density waves from an object in a spherically two-layer medium,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 291–303 (1995).
[CrossRef]

Chen, H.

H. Chen, T. K. Sarkar, S. A. Dianat, J. D. Brule, “Adaptive spectral estimation by the conjugate gradient method,” IEEE Trans. Acoust. Speech Signal Process. ASSP-34, 272–284 (1986).
[CrossRef]

Davila, C. E.

C. E. Davila, “An efficient recursive total least squares algorithm for FIR adaptive filtering,” IEEE Trans. Signal Process. 42, 268–280 (1994).
[CrossRef]

Deng, Y.

W. Zhu, Y. Wang, Y. Deng, Y. Yao, R. Barbour, “Multiresolution regularized least squares image reconstruction based on wavelet in optical tomography,” in Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, R. L. Barbour, M. J. Carvlin, M. A. Fiddy, eds., Proc. SPIE2570, 186–196 (1995).
[CrossRef]

Dianat, S. A.

H. Chen, T. K. Sarkar, S. A. Dianat, J. D. Brule, “Adaptive spectral estimation by the conjugate gradient method,” IEEE Trans. Acoust. Speech Signal Process. ASSP-34, 272–284 (1986).
[CrossRef]

Ebbini, E. S.

P. Li, S. W. Flax, E. S. Ebbini, M. O’Donnell, “Blocked element compensation in phased array imaging,” IEEE Trans. Ultrason. Frequencies Freq. Control 40, 282–292 (1993).

Flax, S. W.

P. Li, S. W. Flax, E. S. Ebbini, M. O’Donnell, “Blocked element compensation in phased array imaging,” IEEE Trans. Ultrason. Frequencies Freq. Control 40, 282–292 (1993).

Fletcher, R.

R. Fletcher, C. M. Reeves, “Function minimization by conjugate gradients,” Comput. J. 7, 149–154 (1964).

Galatsanos, N. P.

V. Z. Mesarović, N. P. Galatsanos, A. Katsaggelos, “Regularized constrained total least squares image restoration,” IEEE Trans. Image Process. 4, 1096–1108 (1995).
[CrossRef] [PubMed]

Golub, G. H.

G. H. Golub, “Some modified matrix eigenvalue problems,” SIAM Rev. 15, 318–334 (1973).
[CrossRef]

G. H. Golub, C. F. Van Loan, “An analysis of the total least squares problem,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal.17, 883–893 (1980).

Graber, H.

Y. Wang, J. Chang, R. Aronson, R. Barbour, H. Graber, J. Lubowsky, “Imaging scattering media by diffusion tomography: an iterative perturbation approach,” in Physiological Monitoring and Early Detection Diagnostic Methods, T. S. Mang, ed., Proc. SPIE1641, 58–71 (1992).
[CrossRef]

Graber, H. L.

W. Zhu, Y. Wang, H. L. Graber, R. L. Barbour, J. Chang, “A regularized progressive expansion algorithm for recovery of scattering media from time-resolved data,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed. (Optical Society of America, Washington, D.C., 1994), pp. 211–216.

Y. Q. Yao, Y. Wang, R. L. Barbour, H. L. Graber, J. W. Chang, “Scattering characteristics of photon density waves from an object in a spherically two-layer medium,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 291–303 (1995).
[CrossRef]

H. L. Graber, R. L. Barbour, J. Chang, R. Aronson, “Identification of the functional form of nonlinear effects of localized finite absorption in a diffusing medium,” in Optical Tomography; Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 669–681 (1995).
[CrossRef]

R. L. Barbour, H. L. Graber, J. Chang, S. Barbour, P. C. Koo, R. Aronson, “MR guided optical tomography: prospects and computation for a new imaging method,” in IEEE Computational Science Engineering Magazine, Winter1995, pp. 63–77.

J. Chang, Y. Wang, R. Aronson, H. L. Graber, R. L. Barbour, “A layer-stripping approach for recovery of scattering media from time-resolved data,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc. SPIE1767, 384–395 (1992).
[CrossRef]

R. L. Barbour, H. L. Graber, Y. Wang, J. Chang, R. Aronson, “A perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. H. Mueller, B. Chance, R. R. Alfano, S. R. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. R. Masters, S. Svanberg, P. van der Zee, eds., Vol. IS11 of Institute Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 87–120.

R. L. Barbour, H. L. Graber, R. Aronson, J. Lubowsky, “Imaging of subsurface regions of random media by remote sensing,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. SPIE1431, 192–203 (1991).
[CrossRef]

Harada, G. A.

T. J. Abatzoglou, J. M. Mendel, G. A. Harada, “The constrained total least squares technique and its applications to harmonic superresolution,” IEEE Trans. Signal Process. 39, 1070–1087 (1991).
[CrossRef]

Hosteny, R. P.

I. Shavitt, C. F. Bender, A. Pipano, R. P. Hosteny, “The iterative calculation of several of the lowest or highest eigenvalues and corresponding eigenvector of very large symmetric matrices,” J. Comput. Phys. 11, 90–108 (1973).
[CrossRef]

Hu, J. H.

Y. Q. Yao, Y. Wang, Y. L. Pei, W. W. Zhu, J. H. Hu, R. L. Barbour, “Frequency domain optical tomography in human tissue,” in Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, R. L. Barbour, M. J. Carvlin, M. A. Fiddy, eds., Proc. SPIE2570, 254–266 (1995).
[CrossRef]

Jiang, H.

Justice, J. H.

J. H. Justice, A. A. Vassiliou, “Diffraction tomography for geophysical monitoring of hydrocarbon reservoirs,” Proc. IEEE 78, 711–722 (1990).
[CrossRef]

Katsaggelos, A.

V. Z. Mesarović, N. P. Galatsanos, A. Katsaggelos, “Regularized constrained total least squares image restoration,” IEEE Trans. Image Process. 4, 1096–1108 (1995).
[CrossRef] [PubMed]

Kim, H. C.

N. K. Bose, H. C. Kim, H. M. Valenzuela, “Recursive total least squares algorithm for image reconstruction,” Multidimens. Syst. Signal Process. 4, 253–268 (1993).

Koo, P. C.

R. L. Barbour, H. L. Graber, J. Chang, S. Barbour, P. C. Koo, R. Aronson, “MR guided optical tomography: prospects and computation for a new imaging method,” in IEEE Computational Science Engineering Magazine, Winter1995, pp. 63–77.

Li, P.

P. Li, S. W. Flax, E. S. Ebbini, M. O’Donnell, “Blocked element compensation in phased array imaging,” IEEE Trans. Ultrason. Frequencies Freq. Control 40, 282–292 (1993).

Lubowsky, J.

Y. Wang, J. Chang, R. Aronson, R. Barbour, H. Graber, J. Lubowsky, “Imaging scattering media by diffusion tomography: an iterative perturbation approach,” in Physiological Monitoring and Early Detection Diagnostic Methods, T. S. Mang, ed., Proc. SPIE1641, 58–71 (1992).
[CrossRef]

R. L. Barbour, H. L. Graber, R. Aronson, J. Lubowsky, “Imaging of subsurface regions of random media by remote sensing,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. SPIE1431, 192–203 (1991).
[CrossRef]

Mendel, J. M.

T. J. Abatzoglou, J. M. Mendel, G. A. Harada, “The constrained total least squares technique and its applications to harmonic superresolution,” IEEE Trans. Signal Process. 39, 1070–1087 (1991).
[CrossRef]

Mesarovic, V. Z.

V. Z. Mesarović, N. P. Galatsanos, A. Katsaggelos, “Regularized constrained total least squares image restoration,” IEEE Trans. Image Process. 4, 1096–1108 (1995).
[CrossRef] [PubMed]

O’Donnell, M.

P. Li, S. W. Flax, E. S. Ebbini, M. O’Donnell, “Blocked element compensation in phased array imaging,” IEEE Trans. Ultrason. Frequencies Freq. Control 40, 282–292 (1993).

Osterberg, U. L.

Patterson, M. S.

Paulsen, K. D.

Pei, Y.

Y. Yao, Y. Wang, Y. Pei, W. Zhu, R. L. Barbour, “Simultaneous reconstruction of absorption and scattering distributions in turbid media using a Born iterative method,” in Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, R. L. Barbour, M. J. Carvlin, M. A. Fiddy, eds., Proc. SPIE2570, 96–107 (1995).
[CrossRef]

Pei, Y. L.

Y. Q. Yao, Y. Wang, Y. L. Pei, W. W. Zhu, J. H. Hu, R. L. Barbour, “Frequency domain optical tomography in human tissue,” in Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, R. L. Barbour, M. J. Carvlin, M. A. Fiddy, eds., Proc. SPIE2570, 254–266 (1995).
[CrossRef]

Pipano, A.

I. Shavitt, C. F. Bender, A. Pipano, R. P. Hosteny, “The iterative calculation of several of the lowest or highest eigenvalues and corresponding eigenvector of very large symmetric matrices,” J. Comput. Phys. 11, 90–108 (1973).
[CrossRef]

Pogue, B. W.

Reeves, C. M.

R. Fletcher, C. M. Reeves, “Function minimization by conjugate gradients,” Comput. J. 7, 149–154 (1964).

Sarkar, T. K.

X. Yang, T. K. Sarkar, E. Arvas, “A survey of conjugate gradient algorithms for solution of extreme eigen-problem of a symmetric matrix,” IEEE Trans. Acoust. Speech Signal Process. 37, 1550–1556 (1989).
[CrossRef]

H. Chen, T. K. Sarkar, S. A. Dianat, J. D. Brule, “Adaptive spectral estimation by the conjugate gradient method,” IEEE Trans. Acoust. Speech Signal Process. ASSP-34, 272–284 (1986).
[CrossRef]

Shavitt, I.

I. Shavitt, C. F. Bender, A. Pipano, R. P. Hosteny, “The iterative calculation of several of the lowest or highest eigenvalues and corresponding eigenvector of very large symmetric matrices,” J. Comput. Phys. 11, 90–108 (1973).
[CrossRef]

Silvia, M. T.

M. T. Silvia, E. C. Tacker, “Regularization of Marchenko’s integral equation by total least squares,” J. Acoust. Soc. Am. 72, 1202–1207 (1982).
[CrossRef]

Tacker, E. C.

M. T. Silvia, E. C. Tacker, “Regularization of Marchenko’s integral equation by total least squares,” J. Acoust. Soc. Am. 72, 1202–1207 (1982).
[CrossRef]

Valenzuela, H. M.

N. K. Bose, H. C. Kim, H. M. Valenzuela, “Recursive total least squares algorithm for image reconstruction,” Multidimens. Syst. Signal Process. 4, 253–268 (1993).

Van Huffel, S.

S. Van Huffel, J. Vandewalle, The Total Least Squares Problem: Computational Aspects and Analysis (SIAM Press, Philadelphia, 1991).

Van Loan, C. F.

G. H. Golub, C. F. Van Loan, “An analysis of the total least squares problem,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal.17, 883–893 (1980).

Vandewalle, J.

S. Van Huffel, J. Vandewalle, The Total Least Squares Problem: Computational Aspects and Analysis (SIAM Press, Philadelphia, 1991).

Vassiliou, A. A.

J. H. Justice, A. A. Vassiliou, “Diffraction tomography for geophysical monitoring of hydrocarbon reservoirs,” Proc. IEEE 78, 711–722 (1990).
[CrossRef]

Wang, Y.

W. Zhu, Y. Wang, Y. Yao, R. L. Barbour, “Wavelet based multigrid reconstruction algorithm for optical tomography,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. Fujimoto, eds. (Optical Society of America, Washington, D.C., 1996), pp. 278–281.

Y. Q. Yao, Y. Wang, R. L. Barbour, H. L. Graber, J. W. Chang, “Scattering characteristics of photon density waves from an object in a spherically two-layer medium,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 291–303 (1995).
[CrossRef]

W. Zhu, Y. Wang, Y. Deng, Y. Yao, R. Barbour, “Multiresolution regularized least squares image reconstruction based on wavelet in optical tomography,” in Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, R. L. Barbour, M. J. Carvlin, M. A. Fiddy, eds., Proc. SPIE2570, 186–196 (1995).
[CrossRef]

W. Zhu, Y. Wang, H. L. Graber, R. L. Barbour, J. Chang, “A regularized progressive expansion algorithm for recovery of scattering media from time-resolved data,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed. (Optical Society of America, Washington, D.C., 1994), pp. 211–216.

Y. Q. Yao, Y. Wang, Y. L. Pei, W. W. Zhu, J. H. Hu, R. L. Barbour, “Frequency domain optical tomography in human tissue,” in Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, R. L. Barbour, M. J. Carvlin, M. A. Fiddy, eds., Proc. SPIE2570, 254–266 (1995).
[CrossRef]

Y. Wang, J. Chang, R. Aronson, R. Barbour, H. Graber, J. Lubowsky, “Imaging scattering media by diffusion tomography: an iterative perturbation approach,” in Physiological Monitoring and Early Detection Diagnostic Methods, T. S. Mang, ed., Proc. SPIE1641, 58–71 (1992).
[CrossRef]

R. L. Barbour, H. L. Graber, Y. Wang, J. Chang, R. Aronson, “A perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. H. Mueller, B. Chance, R. R. Alfano, S. R. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. R. Masters, S. Svanberg, P. van der Zee, eds., Vol. IS11 of Institute Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 87–120.

J. Chang, Y. Wang, R. Aronson, H. L. Graber, R. L. Barbour, “A layer-stripping approach for recovery of scattering media from time-resolved data,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc. SPIE1767, 384–395 (1992).
[CrossRef]

Y. Yao, Y. Wang, Y. Pei, W. Zhu, R. L. Barbour, “Simultaneous reconstruction of absorption and scattering distributions in turbid media using a Born iterative method,” in Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, R. L. Barbour, M. J. Carvlin, M. A. Fiddy, eds., Proc. SPIE2570, 96–107 (1995).
[CrossRef]

Yang, X.

X. Yang, T. K. Sarkar, E. Arvas, “A survey of conjugate gradient algorithms for solution of extreme eigen-problem of a symmetric matrix,” IEEE Trans. Acoust. Speech Signal Process. 37, 1550–1556 (1989).
[CrossRef]

Yao, Y.

Y. Yao, Y. Wang, Y. Pei, W. Zhu, R. L. Barbour, “Simultaneous reconstruction of absorption and scattering distributions in turbid media using a Born iterative method,” in Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, R. L. Barbour, M. J. Carvlin, M. A. Fiddy, eds., Proc. SPIE2570, 96–107 (1995).
[CrossRef]

W. Zhu, Y. Wang, Y. Deng, Y. Yao, R. Barbour, “Multiresolution regularized least squares image reconstruction based on wavelet in optical tomography,” in Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, R. L. Barbour, M. J. Carvlin, M. A. Fiddy, eds., Proc. SPIE2570, 186–196 (1995).
[CrossRef]

W. Zhu, Y. Wang, Y. Yao, R. L. Barbour, “Wavelet based multigrid reconstruction algorithm for optical tomography,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. Fujimoto, eds. (Optical Society of America, Washington, D.C., 1996), pp. 278–281.

Yao, Y. Q.

Y. Q. Yao, Y. Wang, R. L. Barbour, H. L. Graber, J. W. Chang, “Scattering characteristics of photon density waves from an object in a spherically two-layer medium,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 291–303 (1995).
[CrossRef]

Y. Q. Yao, Y. Wang, Y. L. Pei, W. W. Zhu, J. H. Hu, R. L. Barbour, “Frequency domain optical tomography in human tissue,” in Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, R. L. Barbour, M. J. Carvlin, M. A. Fiddy, eds., Proc. SPIE2570, 254–266 (1995).
[CrossRef]

Zhu, W.

W. Zhu, Y. Wang, H. L. Graber, R. L. Barbour, J. Chang, “A regularized progressive expansion algorithm for recovery of scattering media from time-resolved data,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed. (Optical Society of America, Washington, D.C., 1994), pp. 211–216.

W. Zhu, Y. Wang, Y. Deng, Y. Yao, R. Barbour, “Multiresolution regularized least squares image reconstruction based on wavelet in optical tomography,” in Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, R. L. Barbour, M. J. Carvlin, M. A. Fiddy, eds., Proc. SPIE2570, 186–196 (1995).
[CrossRef]

W. Zhu, Y. Wang, Y. Yao, R. L. Barbour, “Wavelet based multigrid reconstruction algorithm for optical tomography,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. Fujimoto, eds. (Optical Society of America, Washington, D.C., 1996), pp. 278–281.

Y. Yao, Y. Wang, Y. Pei, W. Zhu, R. L. Barbour, “Simultaneous reconstruction of absorption and scattering distributions in turbid media using a Born iterative method,” in Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, R. L. Barbour, M. J. Carvlin, M. A. Fiddy, eds., Proc. SPIE2570, 96–107 (1995).
[CrossRef]

Zhu, W. W.

Y. Q. Yao, Y. Wang, Y. L. Pei, W. W. Zhu, J. H. Hu, R. L. Barbour, “Frequency domain optical tomography in human tissue,” in Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, R. L. Barbour, M. J. Carvlin, M. A. Fiddy, eds., Proc. SPIE2570, 254–266 (1995).
[CrossRef]

Zweifel, P. F.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), Chap. 8, pp. 194–231.

Comput. J.

R. Fletcher, C. M. Reeves, “Function minimization by conjugate gradients,” Comput. J. 7, 149–154 (1964).

IEEE Trans. Acoust. Speech Signal Process.

H. Chen, T. K. Sarkar, S. A. Dianat, J. D. Brule, “Adaptive spectral estimation by the conjugate gradient method,” IEEE Trans. Acoust. Speech Signal Process. ASSP-34, 272–284 (1986).
[CrossRef]

X. Yang, T. K. Sarkar, E. Arvas, “A survey of conjugate gradient algorithms for solution of extreme eigen-problem of a symmetric matrix,” IEEE Trans. Acoust. Speech Signal Process. 37, 1550–1556 (1989).
[CrossRef]

IEEE Trans. Image Process.

V. Z. Mesarović, N. P. Galatsanos, A. Katsaggelos, “Regularized constrained total least squares image restoration,” IEEE Trans. Image Process. 4, 1096–1108 (1995).
[CrossRef] [PubMed]

IEEE Trans. Signal Process.

C. E. Davila, “An efficient recursive total least squares algorithm for FIR adaptive filtering,” IEEE Trans. Signal Process. 42, 268–280 (1994).
[CrossRef]

T. J. Abatzoglou, J. M. Mendel, G. A. Harada, “The constrained total least squares technique and its applications to harmonic superresolution,” IEEE Trans. Signal Process. 39, 1070–1087 (1991).
[CrossRef]

IEEE Trans. Ultrason. Frequencies Freq. Control

P. Li, S. W. Flax, E. S. Ebbini, M. O’Donnell, “Blocked element compensation in phased array imaging,” IEEE Trans. Ultrason. Frequencies Freq. Control 40, 282–292 (1993).

J. Acoust. Soc. Am.

M. T. Silvia, E. C. Tacker, “Regularization of Marchenko’s integral equation by total least squares,” J. Acoust. Soc. Am. 72, 1202–1207 (1982).
[CrossRef]

J. Comput. Phys.

I. Shavitt, C. F. Bender, A. Pipano, R. P. Hosteny, “The iterative calculation of several of the lowest or highest eigenvalues and corresponding eigenvector of very large symmetric matrices,” J. Comput. Phys. 11, 90–108 (1973).
[CrossRef]

J. Opt. Soc. Am. A

Multidimens. Syst. Signal Process.

N. K. Bose, H. C. Kim, H. M. Valenzuela, “Recursive total least squares algorithm for image reconstruction,” Multidimens. Syst. Signal Process. 4, 253–268 (1993).

Proc. IEEE

J. H. Justice, A. A. Vassiliou, “Diffraction tomography for geophysical monitoring of hydrocarbon reservoirs,” Proc. IEEE 78, 711–722 (1990).
[CrossRef]

SIAM Rev.

G. H. Golub, “Some modified matrix eigenvalue problems,” SIAM Rev. 15, 318–334 (1973).
[CrossRef]

Other

S. Van Huffel, J. Vandewalle, The Total Least Squares Problem: Computational Aspects and Analysis (SIAM Press, Philadelphia, 1991).

H. L. Graber, R. L. Barbour, J. Chang, R. Aronson, “Identification of the functional form of nonlinear effects of localized finite absorption in a diffusing medium,” in Optical Tomography; Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 669–681 (1995).
[CrossRef]

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967), Chap. 8, pp. 194–231.

G. H. Golub, C. F. Van Loan, “An analysis of the total least squares problem,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal.17, 883–893 (1980).

R. L. Barbour, H. L. Graber, R. Aronson, J. Lubowsky, “Imaging of subsurface regions of random media by remote sensing,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, A. Katzir, eds., Proc. SPIE1431, 192–203 (1991).
[CrossRef]

Y. Q. Yao, Y. Wang, R. L. Barbour, H. L. Graber, J. W. Chang, “Scattering characteristics of photon density waves from an object in a spherically two-layer medium,” in Optical Tomography, Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. SPIE2389, 291–303 (1995).
[CrossRef]

Y. Q. Yao, Y. Wang, Y. L. Pei, W. W. Zhu, J. H. Hu, R. L. Barbour, “Frequency domain optical tomography in human tissue,” in Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, R. L. Barbour, M. J. Carvlin, M. A. Fiddy, eds., Proc. SPIE2570, 254–266 (1995).
[CrossRef]

W. Zhu, Y. Wang, Y. Yao, R. L. Barbour, “Wavelet based multigrid reconstruction algorithm for optical tomography,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, J. Fujimoto, eds. (Optical Society of America, Washington, D.C., 1996), pp. 278–281.

R. L. Barbour, H. L. Graber, J. Chang, S. Barbour, P. C. Koo, R. Aronson, “MR guided optical tomography: prospects and computation for a new imaging method,” in IEEE Computational Science Engineering Magazine, Winter1995, pp. 63–77.

Y. Wang, J. Chang, R. Aronson, R. Barbour, H. Graber, J. Lubowsky, “Imaging scattering media by diffusion tomography: an iterative perturbation approach,” in Physiological Monitoring and Early Detection Diagnostic Methods, T. S. Mang, ed., Proc. SPIE1641, 58–71 (1992).
[CrossRef]

R. L. Barbour, H. L. Graber, Y. Wang, J. Chang, R. Aronson, “A perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. H. Mueller, B. Chance, R. R. Alfano, S. R. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. R. Masters, S. Svanberg, P. van der Zee, eds., Vol. IS11 of Institute Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 87–120.

J. Chang, Y. Wang, R. Aronson, H. L. Graber, R. L. Barbour, “A layer-stripping approach for recovery of scattering media from time-resolved data,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc. SPIE1767, 384–395 (1992).
[CrossRef]

W. Zhu, Y. Wang, H. L. Graber, R. L. Barbour, J. Chang, “A regularized progressive expansion algorithm for recovery of scattering media from time-resolved data,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed. (Optical Society of America, Washington, D.C., 1994), pp. 211–216.

W. Zhu, Y. Wang, Y. Deng, Y. Yao, R. Barbour, “Multiresolution regularized least squares image reconstruction based on wavelet in optical tomography,” in Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, R. L. Barbour, M. J. Carvlin, M. A. Fiddy, eds., Proc. SPIE2570, 186–196 (1995).
[CrossRef]

S. R. Arridge, “The forward and inverse problems in time resolved infra-red imaging,” in Medical Optical Tomography: Functional Imaging and Monitoring, Vol. IS11 of Institute Series (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 35–64.

Y. Yao, Y. Wang, Y. Pei, W. Zhu, R. L. Barbour, “Simultaneous reconstruction of absorption and scattering distributions in turbid media using a Born iterative method,” in Experimental and Numerical Methods for Solving Ill-Posed Inverse Problems: Medical and Nonmedical Applications, R. L. Barbour, M. J. Carvlin, M. A. Fiddy, eds., Proc. SPIE2570, 96–107 (1995).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Position and orientation of detectors about the source. The source is placed in the center. The open circles and the crosses indicate positions of detectors that are inclined 10° and 80° from the normal, respectively. The small solid circles indicate positions where measurements are made in both orientations. The grid size is 3 mfp × 3 mfp.

Fig. 2
Fig. 2

Cross section of the original three-layer medium. The darkest gray level represents Δxmax, the maximum value of Δx. Here Δxmax=5.0×10-4.

Fig. 3
Fig. 3

Reconstruction results with use of the three-layer weight function and calculated data. Images in (a), (b), and (c) are the LS reconstruction results under noise levels 0.1%, 0.2%, and 1% in detector readings and 0.001%, 0.002%, and 0.01% in weights, respectively. The maximum reconstruction values are Δxmax=6.8×10-4, 6.7×10-4, and 1.9×10-3. Images in (d), (e), and (f) are the ITLS reconstruction results under the corresponding noise levels, with Δxmax=5.6×10-4, 5.6×10-4, and 5.8×10-4, respectively.

Fig. 4
Fig. 4

Reconstruction for the three-layer medium with use of the half-space weights and calculated data: (a) LS reconstruction result by the CGD method, with Δxmax=8.7×10-3; (b) ITLS reconstruction result, with Δxmax=3.9×10-4.

Fig. 5
Fig. 5

Reconstruction for the three-layer medium with use of the three-layer weights and simulated data: (a) LS reconstruction result, with Δxmax=8.4×10-3; (b) ITLS reconstruction result, with Δxmax=8.4×10-3.

Fig. 6
Fig. 6

Reconstruction for the three-layer medium with use of the half-space weights and simulated data: (a) LS reconstruction result, with Δxmax=6.5×10-1; (b) ITLS reconstruction result, with Δxmax=1.7×10-3.

Fig. 7
Fig. 7

Source–detector configurations for the cylindrical rod computation.

Fig. 8
Fig. 8

Histogram of the noise levels in the weights. The horizontal axis represents the noise level defined by the ratio of the noise standard deviation to the actual weight value, which is plotted on a log10 scale. The vertical axis represents the fraction of weights having a particular noise level.

Fig. 9
Fig. 9

Contour plot of noise levels in weights for different source–detector pairs: (a) contour plot for source–detector pair (1, 2); (b) contour plot for source–detector pair (1, 6).

Fig. 10
Fig. 10

Reconstruction for the cylindrical rod: (a), (b), and (c) are the original and reconstruction results by LS and ITLS, respectively, for the centered case; (d), (e), and (f) are the original medium and reconstruction results by LS and TLS, respectively, for the off-center case. The added noise level is 3% for the detector reading and 0.01% for the weights.

Tables (1)

Tables Icon

Table 1 MSE and RMSE of the Reconstructed Images by the LS and TLS Approaches

Equations (43)

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

WΔx=ΔI,
-Ω·u(r, Ω)-μt(r, Ω)u(r, Ω)+ μs(r, ΩΩ)u(r, Ω)dΩ+s(r, Ω)=0,
Lu=s,
(L+ΔL)(u+Δu)=s,
LΔu=s-Lu-ΔLu-ΔLΔu=-ΔLu-ΔLΔu.
LΔu=-ΔLu.
s=-ΔLu.
Δu(rd, Ωd, rs, Ωs)=V Δμa(r)wa(rd, Ωd, rs, Ωs; r)d3r,
wa(rd, Ωd, rs, Ωs; r)=4π G(r, Ω; rs, Ωs)G(r, -Ω; rd, -Ωd)dΩ
·[D(r)u(r)]-μa(r)u(r)=-S(r),
Δu(rd, rs)=V Δμa(r)wa(rd, rs; r)d3r,
wa(rd, rs; r)=G(rd-r)G(rs-r),
·[D(r)G(r-rs)]-μa(r)G(r-rs)=-δ(r-rs),
WΔx=ΔI,
minimizeEΔI2,
subjecttoΔI+EΔI  Range(W).
WΔx=ΔI+EΔI.
WTWΔx=WTΔI.
minimizeEW|EΔIF,
subjecttoΔI+EΔIRange(W+EW).
(W+Ew)Δx=ΔI+EΔI
minimizeEF,
subjectto(A+E)q=0,
A=[W|ΔI],E=[Ew|EΔI],
q=Δx-1.
minimizeF(q)=qTATAqqTq,
q(k+1)=q(k)+α(k)p(k),
D[α(k)]2+Bα(k)+C=0.
α(k)=(-B+B2-4CD)/(2D),
D=Pb(k)Pc(k)-Pa(k)Pd(k),
B=Pb(k)-λ(k)Pd(k),
C=Pa(k)-λ(k)Pc(k),
λ(k)=Aq(k), Aq(k),
Pa(k)=Aq(k), Ap(k),
Pb(k)=Ap(k), Ap(k),
Pc(k)=p(k), q(k),
Pd(k)=p(k), p(k).
p(k+1)=r(k+1)+β(k)p(k),
r(k+1)=λ(k)q(k+1)-ATAq(k+1).
β(k)=r(k+1), r(k+1)/r(k), r(k)
ATAp(k), p(k+1)=0.
MSE=1nΔx-Δxˆ2,
RMSE=1nΔx-Δxˆ2Δx2,

Metrics