Abstract

In a previous paper [Zhu et al., J. Opt. Soc. Am. A 14, 799 (1997)] an iterative algorithm for obtaining the total least-squares (TLS) solution of a linear system based on the Rayleigh quotient formulation was presented. Here we derive what to our knowledge are the first statistical properties of this solution. It is shown that the Rayleigh-quotient-form TLS (RQF-TLS) estimator is equivalent to the maximum-likelihood estimator when noise terms in both data and operator elements are independent and identically distributed Gaussian. A perturbation analysis of the RQF-TLS solution is derived, and from it the mean square error of the RQF-TLS solution is obtained in closed form, which is valid at small noise levels. We then present a wavelet-based multiresolution scheme for obtaining the TLS solution. This method was employed with a multigrid algorithm to solve the linear perturbation equation encountered in optical tomography. Results from numerical simulations show that this method requires substantially less computation than the previously reported one-grid TLS algorithm. The method also allows one to identify regions of interest quickly from a coarse-level reconstruction and restrict the reconstruction in the following fine resolutions to those regions. Finally, the method is less sensitive to noise than the one-grid TLS and multigrid least-squares algorithms.

© 1998 Optical Society of America

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  1. 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. Mueller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. van der Zee, eds., Vol. IS11 of Institute Series (Society of Photo-optical Instrumentation Engineers, Bellingham, Washington, 1993), pp. 87–120.
  2. Y. Wang, J. Chang, R. Aronson, R. L. Barbour, H. L. 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. Y. Q. Yao, Y. Wang, Y. L. Pei, 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 Application, R. L. Barbour, M. J. Carvlin, M. A. Fiddy eds., Proc. SPIE2570, 254–266 (1995).
    [CrossRef]
  4. J. R. Singer, F. A. Grünbaum, P. Kohn, J. P. Zubelli, “Image reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).
    [CrossRef] [PubMed]
  5. S. R. Arridge, P. van der Zee, M. Cope, D. Delpy, “New results for the development of infrared-red absorption imaging,” in Biomedical Image Processing, A. C. Bovik, W. E. Higgins, eds., Proc. SPIE1245, 92–103 (1990).
    [CrossRef]
  6. S. R. Arridge, “The forward and inverse problems in time resolved infra-red imaging,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Mueller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. van der Zee, eds., Vol. IS11 of Institute Series (Society of Photo-optical Instrumentation Engineers, Bellingham, Washington, 1993), pp. 35–64.
  7. W. Zhu, Y. Wang, Y. Deng, Y. Yao, R. Barbour, “A wavelet-based multiresolution regularized least squares reconstruction approach for optical tomography,” IEEE Trans. Med. Imag. 16, 210–217 (1997).
    [CrossRef]
  8. W. Zhu, Y. Wang, Y. Yao, J. Chang, H. L. Graber, R. Barbour, “Iterative total least-squares image reconstruction algorithm for optical tomography by the conjugate gradient method,” J. Opt. Soc. Am. A 14, 799–807 (1997).
    [CrossRef]
  9. 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]
  10. J. Chang, W. Zhu, Y. Wang, H. L. Graber, R. L. Barbour, “Regularized progressive expansion algorithm for recovery of scattering media from time-resolved data,” J. Opt. Soc. Am. A 14, 306–312 (1997).
    [CrossRef]
  11. G. H. Golub, C. F. Van Loan, “An analysis of the total least squares problem,” SIAM (Soc. Ind. Appl. Math.) J. Math. Anal. 17, 883–893 (1980).
  12. V. Z. Measarvić, N. P. Galatsanos, A. Katsaggelos, “Regularized constrained total least squares image restoration,” IEEE Trans. Image Process. 4, 1096–1108 (1995).
    [CrossRef]
  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. G. Wang, J. Zhang, G. W. Pan, “Solution of inverse problems in image processing by wavelet expansion,” IEEE Trans. Image Process. 4, 579–593 (1995).
    [CrossRef] [PubMed]
  15. L. Blanc-Feraud, P. Charbonnier, P. Lobel, M. Barlaud, “A fast tomographic reconstruction algorithm in the 2D wavelet transform domain,” in Proceedings of the 1994 IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1994), pp. 305–308.
  16. A. H. Delaney, Y. Bresler, “Multiresolution tomographic reconstruction using wavelets,” IEEE Trans. Image Process. 4, 799–813 (1995).
    [CrossRef] [PubMed]
  17. M. R. Banham, Nikolas P. Galatsanos, H. L. Gonzalez, A. K. Katsaggelos, “Multichannel restoration of single channel images using a wavelet-based subband decomposition,” IEEE Trans. Image Process. 3, 821–833 (1994).
    [CrossRef] [PubMed]
  18. 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]
  19. G. H. Golub, “Some modified matrix eigenvalue problems,” SIAM (Soc. Ind. Appl. Math.) Rev. 15, 318–334 (1973).
  20. 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. 34, 272–284 (1986).
    [CrossRef]
  21. 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]
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  23. 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]
  24. V. Z. Measarvić, N. P. Galatsanos, “map and regularized constrained total least squares image restoration,” in Proceedings of the 1994 IEEE International Conference on Image Processing (Institute of Electrical and Electronics Engineers, Piscatway, N.J., 1994), Vol. 3, pp. 177–181.
  25. S. G. Mallat, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern. Anal. Mach. Intell. 11, 674–693 (1989).
    [CrossRef]
  26. W. Hackbusch, Multigrid Methods and Applications (Springer-Verlag, Berlin, 1985).
  27. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).
  28. 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]
  29. W. L. Briggs, A Multigrid Tutorial (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1987).

1997

1995

G. Wang, J. Zhang, G. W. Pan, “Solution of inverse problems in image processing by wavelet expansion,” IEEE Trans. Image Process. 4, 579–593 (1995).
[CrossRef] [PubMed]

A. H. Delaney, Y. Bresler, “Multiresolution tomographic reconstruction using wavelets,” IEEE Trans. Image Process. 4, 799–813 (1995).
[CrossRef] [PubMed]

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

1994

M. R. Banham, Nikolas P. Galatsanos, H. L. Gonzalez, A. K. Katsaggelos, “Multichannel restoration of single channel images using a wavelet-based subband decomposition,” IEEE Trans. Image Process. 3, 821–833 (1994).
[CrossRef] [PubMed]

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. R. Singer, F. A. Grünbaum, P. Kohn, J. P. Zubelli, “Image reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).
[CrossRef] [PubMed]

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]

S. G. Mallat, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern. Anal. Mach. Intell. 11, 674–693 (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. 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]

1980

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

1973

G. H. Golub, “Some modified matrix eigenvalue problems,” SIAM (Soc. Ind. Appl. Math.) Rev. 15, 318–334 (1973).

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, 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. Mueller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. van der Zee, eds., Vol. IS11 of Institute Series (Society of Photo-optical Instrumentation Engineers, Bellingham, Washington, 1993), pp. 87–120.

Y. Wang, J. Chang, R. Aronson, R. L. Barbour, H. L. 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]

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]

Arridge, S. R.

S. R. Arridge, P. van der Zee, M. Cope, D. Delpy, “New results for the development of infrared-red absorption imaging,” in Biomedical Image Processing, A. C. Bovik, W. E. Higgins, eds., Proc. SPIE1245, 92–103 (1990).
[CrossRef]

S. R. Arridge, “The forward and inverse problems in time resolved infra-red imaging,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Mueller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. van der Zee, eds., Vol. IS11 of Institute Series (Society of Photo-optical Instrumentation Engineers, Bellingham, Washington, 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]

Banham, M. R.

M. R. Banham, Nikolas P. Galatsanos, H. L. Gonzalez, A. K. Katsaggelos, “Multichannel restoration of single channel images using a wavelet-based subband decomposition,” IEEE Trans. Image Process. 3, 821–833 (1994).
[CrossRef] [PubMed]

Barbour, R.

W. Zhu, Y. Wang, Y. Deng, Y. Yao, R. Barbour, “A wavelet-based multiresolution regularized least squares reconstruction approach for optical tomography,” IEEE Trans. Med. Imag. 16, 210–217 (1997).
[CrossRef]

W. Zhu, Y. Wang, Y. Yao, J. Chang, H. L. Graber, R. Barbour, “Iterative total least-squares image reconstruction algorithm for optical tomography by the conjugate gradient method,” J. Opt. Soc. Am. A 14, 799–807 (1997).
[CrossRef]

Barbour, R. L.

J. Chang, W. Zhu, Y. Wang, H. L. Graber, R. L. Barbour, “Regularized progressive expansion algorithm for recovery of scattering media from time-resolved data,” J. Opt. Soc. Am. A 14, 306–312 (1997).
[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. Mueller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. van der Zee, eds., Vol. IS11 of Institute Series (Society of Photo-optical Instrumentation Engineers, Bellingham, Washington, 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. 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. 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 Application, R. L. Barbour, M. J. Carvlin, M. A. Fiddy eds., Proc. SPIE2570, 254–266 (1995).
[CrossRef]

Y. Wang, J. Chang, R. Aronson, R. L. Barbour, H. L. 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]

Barlaud, M.

L. Blanc-Feraud, P. Charbonnier, P. Lobel, M. Barlaud, “A fast tomographic reconstruction algorithm in the 2D wavelet transform domain,” in Proceedings of the 1994 IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1994), pp. 305–308.

Blanc-Feraud, L.

L. Blanc-Feraud, P. Charbonnier, P. Lobel, M. Barlaud, “A fast tomographic reconstruction algorithm in the 2D wavelet transform domain,” in Proceedings of the 1994 IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1994), pp. 305–308.

Bresler, Y.

A. H. Delaney, Y. Bresler, “Multiresolution tomographic reconstruction using wavelets,” IEEE Trans. Image Process. 4, 799–813 (1995).
[CrossRef] [PubMed]

Briggs, W. L.

W. L. Briggs, A Multigrid Tutorial (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1987).

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. 34, 272–284 (1986).
[CrossRef]

Chang, J.

J. Chang, W. Zhu, Y. Wang, H. L. Graber, R. L. Barbour, “Regularized progressive expansion algorithm for recovery of scattering media from time-resolved data,” J. Opt. Soc. Am. A 14, 306–312 (1997).
[CrossRef]

W. Zhu, Y. Wang, Y. Yao, J. Chang, H. L. Graber, R. Barbour, “Iterative total least-squares image reconstruction algorithm for optical tomography by the conjugate gradient method,” J. Opt. Soc. Am. A 14, 799–807 (1997).
[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. Mueller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. van der Zee, eds., Vol. IS11 of Institute Series (Society of Photo-optical Instrumentation Engineers, Bellingham, Washington, 1993), pp. 87–120.

Y. Wang, J. Chang, R. Aronson, R. L. Barbour, H. L. 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]

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]

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]

Charbonnier, P.

L. Blanc-Feraud, P. Charbonnier, P. Lobel, M. Barlaud, “A fast tomographic reconstruction algorithm in the 2D wavelet transform domain,” in Proceedings of the 1994 IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1994), pp. 305–308.

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. 34, 272–284 (1986).
[CrossRef]

Cope, M.

S. R. Arridge, P. van der Zee, M. Cope, D. Delpy, “New results for the development of infrared-red absorption imaging,” in Biomedical Image Processing, A. C. Bovik, W. E. Higgins, eds., Proc. SPIE1245, 92–103 (1990).
[CrossRef]

Delaney, A. H.

A. H. Delaney, Y. Bresler, “Multiresolution tomographic reconstruction using wavelets,” IEEE Trans. Image Process. 4, 799–813 (1995).
[CrossRef] [PubMed]

Delpy, D.

S. R. Arridge, P. van der Zee, M. Cope, D. Delpy, “New results for the development of infrared-red absorption imaging,” in Biomedical Image Processing, A. C. Bovik, W. E. Higgins, eds., Proc. SPIE1245, 92–103 (1990).
[CrossRef]

Deng, Y.

W. Zhu, Y. Wang, Y. Deng, Y. Yao, R. Barbour, “A wavelet-based multiresolution regularized least squares reconstruction approach for optical tomography,” IEEE Trans. Med. Imag. 16, 210–217 (1997).
[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. 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).

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

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).

Galatsanos, N. P.

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

V. Z. Measarvić, N. P. Galatsanos, “map and regularized constrained total least squares image restoration,” in Proceedings of the 1994 IEEE International Conference on Image Processing (Institute of Electrical and Electronics Engineers, Piscatway, N.J., 1994), Vol. 3, pp. 177–181.

Galatsanos, Nikolas P.

M. R. Banham, Nikolas P. Galatsanos, H. L. Gonzalez, A. K. Katsaggelos, “Multichannel restoration of single channel images using a wavelet-based subband decomposition,” IEEE Trans. Image Process. 3, 821–833 (1994).
[CrossRef] [PubMed]

Golub, G. H.

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

G. H. Golub, “Some modified matrix eigenvalue problems,” SIAM (Soc. Ind. Appl. Math.) Rev. 15, 318–334 (1973).

Gonzalez, H. L.

M. R. Banham, Nikolas P. Galatsanos, H. L. Gonzalez, A. K. Katsaggelos, “Multichannel restoration of single channel images using a wavelet-based subband decomposition,” IEEE Trans. Image Process. 3, 821–833 (1994).
[CrossRef] [PubMed]

Graber, H. L.

W. Zhu, Y. Wang, Y. Yao, J. Chang, H. L. Graber, R. Barbour, “Iterative total least-squares image reconstruction algorithm for optical tomography by the conjugate gradient method,” J. Opt. Soc. Am. A 14, 799–807 (1997).
[CrossRef]

J. Chang, W. Zhu, Y. Wang, H. L. Graber, R. L. Barbour, “Regularized progressive expansion algorithm for recovery of scattering media from time-resolved data,” J. Opt. Soc. Am. A 14, 306–312 (1997).
[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. Mueller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. van der Zee, eds., Vol. IS11 of Institute Series (Society of Photo-optical Instrumentation Engineers, Bellingham, Washington, 1993), pp. 87–120.

Y. Wang, J. Chang, R. Aronson, R. L. Barbour, H. L. 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]

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. 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]

Grünbaum, F. A.

J. R. Singer, F. A. Grünbaum, P. Kohn, J. P. Zubelli, “Image reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).
[CrossRef] [PubMed]

Hackbusch, W.

W. Hackbusch, Multigrid Methods and Applications (Springer-Verlag, Berlin, 1985).

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]

Hu, J. H.

Y. Q. Yao, Y. Wang, Y. L. Pei, 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 Application, R. L. Barbour, M. J. Carvlin, M. A. Fiddy eds., Proc. SPIE2570, 254–266 (1995).
[CrossRef]

Katsaggelos, A.

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

Katsaggelos, A. K.

M. R. Banham, Nikolas P. Galatsanos, H. L. Gonzalez, A. K. Katsaggelos, “Multichannel restoration of single channel images using a wavelet-based subband decomposition,” IEEE Trans. Image Process. 3, 821–833 (1994).
[CrossRef] [PubMed]

Kohn, P.

J. R. Singer, F. A. Grünbaum, P. Kohn, J. P. Zubelli, “Image reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).
[CrossRef] [PubMed]

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).

Lobel, P.

L. Blanc-Feraud, P. Charbonnier, P. Lobel, M. Barlaud, “A fast tomographic reconstruction algorithm in the 2D wavelet transform domain,” in Proceedings of the 1994 IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1994), pp. 305–308.

Lubowsky, J.

Y. Wang, J. Chang, R. Aronson, R. L. Barbour, H. L. 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]

Mallat, S. G.

S. G. Mallat, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern. Anal. Mach. Intell. 11, 674–693 (1989).
[CrossRef]

Measarvic, V. Z.

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

V. Z. Measarvić, N. P. Galatsanos, “map and regularized constrained total least squares image restoration,” in Proceedings of the 1994 IEEE International Conference on Image Processing (Institute of Electrical and Electronics Engineers, Piscatway, N.J., 1994), Vol. 3, pp. 177–181.

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]

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).

Pan, G. W.

G. Wang, J. Zhang, G. W. Pan, “Solution of inverse problems in image processing by wavelet expansion,” IEEE Trans. Image Process. 4, 579–593 (1995).
[CrossRef] [PubMed]

Pei, Y. L.

Y. Q. Yao, Y. Wang, Y. L. Pei, 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 Application, R. L. Barbour, M. J. Carvlin, M. A. Fiddy eds., Proc. SPIE2570, 254–266 (1995).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

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. 34, 272–284 (1986).
[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]

Singer, J. R.

J. R. Singer, F. A. Grünbaum, P. Kohn, J. P. Zubelli, “Image reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).
[CrossRef] [PubMed]

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]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

van der Zee, P.

S. R. Arridge, P. van der Zee, M. Cope, D. Delpy, “New results for the development of infrared-red absorption imaging,” in Biomedical Image Processing, A. C. Bovik, W. E. Higgins, eds., Proc. SPIE1245, 92–103 (1990).
[CrossRef]

Van Huffel, S.

S. Van Huffel, J. Vandewalle, The Total Least Squares Problem: Computational Aspects and Analysis (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 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. Math. Anal. 17, 883–893 (1980).

Vandewalle, J.

S. Van Huffel, J. Vandewalle, The Total Least Squares Problem: Computational Aspects and Analysis (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1991).

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

Wang, G.

G. Wang, J. Zhang, G. W. Pan, “Solution of inverse problems in image processing by wavelet expansion,” IEEE Trans. Image Process. 4, 579–593 (1995).
[CrossRef] [PubMed]

Wang, Y.

J. Chang, W. Zhu, Y. Wang, H. L. Graber, R. L. Barbour, “Regularized progressive expansion algorithm for recovery of scattering media from time-resolved data,” J. Opt. Soc. Am. A 14, 306–312 (1997).
[CrossRef]

W. Zhu, Y. Wang, Y. Yao, J. Chang, H. L. Graber, R. Barbour, “Iterative total least-squares image reconstruction algorithm for optical tomography by the conjugate gradient method,” J. Opt. Soc. Am. A 14, 799–807 (1997).
[CrossRef]

W. Zhu, Y. Wang, Y. Deng, Y. Yao, R. Barbour, “A wavelet-based multiresolution regularized least squares reconstruction approach for optical tomography,” IEEE Trans. Med. Imag. 16, 210–217 (1997).
[CrossRef]

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. 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. 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 Application, R. L. Barbour, M. J. Carvlin, M. A. Fiddy eds., Proc. SPIE2570, 254–266 (1995).
[CrossRef]

Y. Wang, J. Chang, R. Aronson, R. L. Barbour, H. L. 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. Mueller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. van der Zee, eds., Vol. IS11 of Institute Series (Society of Photo-optical Instrumentation Engineers, Bellingham, Washington, 1993), pp. 87–120.

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.

W. Zhu, Y. Wang, Y. Yao, J. Chang, H. L. Graber, R. Barbour, “Iterative total least-squares image reconstruction algorithm for optical tomography by the conjugate gradient method,” J. Opt. Soc. Am. A 14, 799–807 (1997).
[CrossRef]

W. Zhu, Y. Wang, Y. Deng, Y. Yao, R. Barbour, “A wavelet-based multiresolution regularized least squares reconstruction approach for optical tomography,” IEEE Trans. Med. Imag. 16, 210–217 (1997).
[CrossRef]

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. 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 Application, R. L. Barbour, M. J. Carvlin, M. A. Fiddy eds., Proc. SPIE2570, 254–266 (1995).
[CrossRef]

Zhang, J.

G. Wang, J. Zhang, G. W. Pan, “Solution of inverse problems in image processing by wavelet expansion,” IEEE Trans. Image Process. 4, 579–593 (1995).
[CrossRef] [PubMed]

Zhu, W.

J. Chang, W. Zhu, Y. Wang, H. L. Graber, R. L. Barbour, “Regularized progressive expansion algorithm for recovery of scattering media from time-resolved data,” J. Opt. Soc. Am. A 14, 306–312 (1997).
[CrossRef]

W. Zhu, Y. Wang, Y. Yao, J. Chang, H. L. Graber, R. Barbour, “Iterative total least-squares image reconstruction algorithm for optical tomography by the conjugate gradient method,” J. Opt. Soc. Am. A 14, 799–807 (1997).
[CrossRef]

W. Zhu, Y. Wang, Y. Deng, Y. Yao, R. Barbour, “A wavelet-based multiresolution regularized least squares reconstruction approach for optical tomography,” IEEE Trans. Med. Imag. 16, 210–217 (1997).
[CrossRef]

Y. Q. Yao, Y. Wang, Y. L. Pei, 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 Application, R. L. Barbour, M. J. Carvlin, M. A. Fiddy eds., Proc. SPIE2570, 254–266 (1995).
[CrossRef]

Zubelli, J. P.

J. R. Singer, F. A. Grünbaum, P. Kohn, J. P. Zubelli, “Image reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).
[CrossRef] [PubMed]

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

G. Wang, J. Zhang, G. W. Pan, “Solution of inverse problems in image processing by wavelet expansion,” IEEE Trans. Image Process. 4, 579–593 (1995).
[CrossRef] [PubMed]

A. H. Delaney, Y. Bresler, “Multiresolution tomographic reconstruction using wavelets,” IEEE Trans. Image Process. 4, 799–813 (1995).
[CrossRef] [PubMed]

M. R. Banham, Nikolas P. Galatsanos, H. L. Gonzalez, A. K. Katsaggelos, “Multichannel restoration of single channel images using a wavelet-based subband decomposition,” IEEE Trans. Image Process. 3, 821–833 (1994).
[CrossRef] [PubMed]

IEEE Trans. Med. Imag.

W. Zhu, Y. Wang, Y. Deng, Y. Yao, R. Barbour, “A wavelet-based multiresolution regularized least squares reconstruction approach for optical tomography,” IEEE Trans. Med. Imag. 16, 210–217 (1997).
[CrossRef]

IEEE Trans. Pattern. Anal. Mach. Intell.

S. G. Mallat, “A theory for multiresolution signal decomposition: the wavelet representation,” IEEE Trans. Pattern. Anal. Mach. Intell. 11, 674–693 (1989).
[CrossRef]

IEEE Trans. Signal Process.

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. Opt. Soc. Am. A

Science

J. R. Singer, F. A. Grünbaum, P. Kohn, J. P. Zubelli, “Image reconstruction of the interior of bodies that diffuse radiation,” Science 248, 990–993 (1990).
[CrossRef] [PubMed]

SIAM (Soc. Ind. Appl. Math.) J. Math. Anal.

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

SIAM (Soc. Ind. Appl. Math.) Rev.

G. H. Golub, “Some modified matrix eigenvalue problems,” SIAM (Soc. Ind. Appl. Math.) Rev. 15, 318–334 (1973).

Other

L. Blanc-Feraud, P. Charbonnier, P. Lobel, M. Barlaud, “A fast tomographic reconstruction algorithm in the 2D wavelet transform domain,” in Proceedings of the 1994 IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, Piscataway, N.J., 1994), pp. 305–308.

W. Hackbusch, Multigrid Methods and Applications (Springer-Verlag, Berlin, 1985).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C: The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

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. L. Briggs, A Multigrid Tutorial (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1987).

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. Mueller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. van der Zee, eds., Vol. IS11 of Institute Series (Society of Photo-optical Instrumentation Engineers, Bellingham, Washington, 1993), pp. 87–120.

Y. Wang, J. Chang, R. Aronson, R. L. Barbour, H. L. 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]

Y. Q. Yao, Y. Wang, Y. L. Pei, 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 Application, R. L. Barbour, M. J. Carvlin, M. A. Fiddy eds., Proc. SPIE2570, 254–266 (1995).
[CrossRef]

S. R. Arridge, P. van der Zee, M. Cope, D. Delpy, “New results for the development of infrared-red absorption imaging,” in Biomedical Image Processing, A. C. Bovik, W. E. Higgins, eds., Proc. SPIE1245, 92–103 (1990).
[CrossRef]

S. R. Arridge, “The forward and inverse problems in time resolved infra-red imaging,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Mueller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. van der Zee, eds., Vol. IS11 of Institute Series (Society of Photo-optical Instrumentation Engineers, Bellingham, Washington, 1993), pp. 35–64.

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]

V. Z. Measarvić, N. P. Galatsanos, “map and regularized constrained total least squares image restoration,” in Proceedings of the 1994 IEEE International Conference on Image Processing (Institute of Electrical and Electronics Engineers, Piscatway, N.J., 1994), Vol. 3, pp. 177–181.

S. Van Huffel, J. Vandewalle, The Total Least Squares Problem: Computational Aspects and Analysis (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1991).

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Figures (8)

Fig. 1
Fig. 1

Wavelet decomposition of 2D image from level j+1 to j.

Fig. 2
Fig. 2

Modified V-cycle multigrid algorithm.

Fig. 3
Fig. 3

The source–detector configurations of the cylindrical rod.

Fig. 4
Fig. 4

Reconstruction results for a medium with a centered rod. The absorption coefficient distribution is a constant in the rod. The SNR in the data is 30 dB. The weights are corrupted by the same noise as that in the data. (a) Original image, (b) reconstructed image with one-grid TLS with 85 iterations, (c) reconstruction result with one grid with 885 iterations, (d) reconstruction image with two-grid TLS algorithm with 1200 iterations at the coarse grid only, (e) reconstruction image with two-grid algorithm with additional 800 iterations at the fine grid, (f) reconstructed image with 3200 iterations in a localized region at the fine grid with (d) as the initial solution. The total computation time for (b) and (d) are roughly the same, and the total computation time for (c), (e), and (f) are also similar. The time for (d) is approximately 1/10 of (c). Note that although (f) looks the same as (e), it is quantitatively more accurate.

Fig. 5
Fig. 5

Reconstruction results of a medium with an off-center sinelike absorber. The SNR in the data is 30 dB. The weights are corrupted by the same noise as that in the data. (a) Original image, (b) reconstruction result with multigrid LS with 1200 iterations at the coarse grid and 800 at the fine grid, (c) reconstructed image with one-grid TLS with 85 iterations, (d) reconstructed image with one-grid TLS with 885 iterations, (e) reconstructed image with two-grid TLS algorithm with 1200 iterations in the coarse grid only, (f) reconstructed image with the two-grid TLS algorithm with an additional 600 iterations at the fine grid with (e) as initial solution. The computation time for (c) and (e) are roughly same. The time for (f) is approximately 3/4 of (d) and that for (c) or (e) is approximately 1/10 of (d).

Fig. 6
Fig. 6

Comparison of computation time required by multigrid TLS versus one-grid TLS in terms of work units for the test medium considered in Fig. 5.

Fig. 7
Fig. 7

Reconstruction results of a medium containing a centered sinelike absorber. The SNR in the data is 20 dB. The weights are corrupted by the same noise as that in the data. (a) Original image, (b) reconstruction result with multigrid LS with 1200 iterations at the coarse grid and 800 at the fine grid, (c) reconstructed image with one-grid TLS with 85 iterations, (d) reconstructed image with one-grid TLS with 885 iterations, (e) reconstructed image with two-grid algorithm with 1200 iterations at the coarse grid only, (f) reconstructed image with the two-grid algorithm with an additional 600 iterations at the fine grid with (e) as the initial solution. The computation time for (c) and (e) are roughly same. The time for (f) is approximately 3/4 of (d) and that for (c) or (e) is approximately 1/10 of (d).

Fig. 8
Fig. 8

Wavelet-transform analysis. (a) Wavelet transform of Fig. 7(a), (b) wavelet transform of Fig. 7(f), (c) wavelet transform of Fig. 6(a), (d) wavelet transform of Fig. 6(f). To reveal the high-frequency subsignals, we applied a square-root mapping to the signal magnitude when plotting.

Tables (3)

Tables Icon

Table 1 MSE and CC for Different Methods for a Test Medium Containing a Rod Absorber at the Center

Tables Icon

Table 2 MSE and CC for Different Methods for a Test Medium Containing an Off-Centered Sinelike Absorber

Tables Icon

Table 3 MSE and CC for Different Methods for a Test Medium Containing a Centered Sinelike Absorber

Equations (48)

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

Hx=y,
minimizeΔy2,
subjecttoy+Δyrange(H).
Hx=y+Δy
HTHx=HTy,
xLS=Hy,
minimize[ΔH|Δy]F,
subjecttoy+Δyrange(H+ΔH).
BF2=tr(BTB)=i=1mj=1n(bij)2.
(H+ΔH)x=y+Δy
minimizeΔAF,
subjectto(A+ΔA)q=0,
A=[H|y],ΔA=[ΔH|Δy],
q=x-1.
x=-1vn+1,n+1v˜n+1.
x=-i=p+1n+1vi,n+1v˜ii=p+1n+1vi,n+12,
minimizeF(q)=qTATAqqTq=y-Hx2x2+1,
y=(H+ΔH)x+Δy.
Cy/x=E[(y-ηy/x)(y-ηy/x)T]=E[(ΔHx+Δy)(ΔHx+Δy)T]=E[ΔHxxTΔHT]+σ2I,
E[ΔHxxTΔHT]=σ2x2I.
Cy/x=σ2(x2+1)I.
P(y/x)=[(2π)m|Cy/x|]-1/2exp[-12(y-Hx)T×Cy/x-1(y-Hx)],
L(y/x)=-(y-Hx)TCy/x-1(y-Hx)
-log[(2π)m|Cy/x|].
L(y/x)(y-Hx)TCy/x-1(y-Hx).
1σ2(x2+1)(y-Hx)T(y-Hx).
Aoqo=0,
Ao=[Ho|yo],qo=xo-1.
F(q)q=2(qTq)2[qTqATA-qTATAqI]q=0.
qoTqo(AoTAo+AoTΔA+ΔATAo)qo+qoTqoAoTAoΔq+2qoTΔqAoTAoqo=Aoqo2(qo+Δq).
Δq=-(AoTAo)AoTΔAqo,
MSETLS=E[ΔqTΔq]=trace(R),
R=(AoTAo)AoTE[ΔAqoqoTΔAT]Ao(AoTAo)T.
E[ΔAqoqoTΔAT]=σ2qo2I.
R=σ2qo2(AoTAo).
MSETLS=trace(R)=σ2qo2trace[(AoTAo)].
MSETLS=σ2qo2i=1rλi-1=σ2(1+xo2)i=1rλi-1.
x˜N=Wx=[A-1x(N/2)D-1x(N/2)]T,
F˜=WMFWNT.
f˜=(WMWN)f,
F˜(M×N)=A-1F(M/2×N/2)D-11F(M/2×N/2)D-12F(M/2×N/2)D-13F(M/2×N/2).
F˜(M×N)=A-2F(M/4×N/4)D-21F(M/4×N/4)D-11F(M/2×N/2)D-22F(M/4×N/4)D-23F(M/4×N/4)D-12F(M/2×N/2)D-13F(M/2×N/2).
H˜x˜=y˜,
A˜=[H˜y˜],
q˜=x˜-1.
minimizeF(q˜)=q˜TA˜TA˜q˜q˜Tq˜.
MSE=1nx-xˆ2,
CC=x,xˆxxˆ,

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