Abstract

We present a Born iterative method for reconstructing optical properties of turbid media by means of frequency-domain data. The approach is based on iterative solutions of a linear perturbation equation, which is derived from the integral form of the Helmholtz wave equation for photon-density waves. In each iteration the total field and the associated weight matrix are recalculated based on the previous reconstructed image. We then obtain a new estimate by solving the updated perturbation equation. The forward solution, also based on a Helmholtz equation, is obtained by a multigrid finite difference method. The inversion is carried out through a Tikhonov regularized optimization process by the conjugate gradient descent method. Using this method, we first reconstruct the distribution of the complex wave numbers in a test medium, from which the absorption and the scattering distributions are then derived. Simulation results with two-dimensional test media have shown that this method can yield quantitatively (in terms of coefficient values) as well as qualitatively (in terms of object location and shape) accurate reconstructions of absorption and scattering distributions in cases in which the first-order Born approximation cannot work well. Both full-angle and limited-angle measurement schemes have been simulated to examine the effect of the location of detectors and sources. The robustness of the algorithm to noise has also been evaluated.

© 1997 Optical Society of America

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1996

1995

B. W. Pogue and M. S. Patterson, “Forward and inverse calculations for 3-D frequency-domain diffuse optical tomography,” in Optical Tomography: Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance and R. R. Alfano, eds., Proc. SPIE 2389, 328–339 (1995).
[CrossRef]

Y. Yao, Y. Wang, Y. Pei, W. Zhu, and 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. Barbour, M. Carlin, and M. Fiddy, eds., Proc. SPIE 2570, 96–107 (1995).
[CrossRef]

Y. Yao, Y. Wang, Y. Pei, W. Zhu, J. Hu, and 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. Barbour, M. Carvlin, and M. Fiddy, eds., Proc. SPIE 2570, 254–266 (1995).
[CrossRef]

M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Simultaneous scattering and absorption images of heterogeneous media using diffusive waves with the Rytov approximation,”B. Chance and R. R. Alfano, eds. in Optical Tomography: Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, Proc. SPIE 2389, 320–327 (1995).
[CrossRef]

Optical Tomography: Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance and R. R. Alfano, eds., Proc. SPIE 2389 (1995).

1994

D. A. Boas, M. A. O’Leary, B. Chance, and A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[CrossRef]

1993

J. Fishkin and E. Gratton, “Propagation of photon-density waves in strongly scattering media containing an absorbing semi-infinite plane bounded by a straight edge,” J. Opt. Soc. Am. A 10, 127–140 (1993).
[CrossRef] [PubMed]

B. J. Tromberg, L. O. Svaasand, T. T. Tsay, and R. C. Haskell, “Properties of photon density waves in multiple-scattering media,” Appl. Opt. 32, 607–616 (1993).
[CrossRef] [PubMed]

S. A. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delay, “Performance of an iterative reconstruction algorithm for near infrared absorption and scatter imaging,” in Photon Migration, and Imaging in Random Medium and Tissues, Proc. SPIE 1888, 360–371 (1993).
[CrossRef]

S. A. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delay, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
[CrossRef] [PubMed]

1992

M. Moghaddadam and W. C. Chew, “Nonlinear two-dimensional velocity profile inversion using time-domain data,” IEEE Trans. Geosci. Remote Sens. 30, 146–156 (1992).
[CrossRef]

1991

N. Joachimowicz, C. Pichot, and J. P. Hugonin, “Inverse scattering: an iterative numerical method for electromagnetic imaging,” IEEE Trans. Antennas Propag. 39, 1742–1752 (1991).
[CrossRef]

S. R. Arridge, P. van der Zee, M. Cope, and D. T. Delpy, “Reconstruction methods for infra-red absorption imaging,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance and A. Katzir, eds., Proc. SPIE 1431, 204–215 (1991).
[CrossRef]

E. M. Sevick and B. Chance, “Photon migration in a model of the head measured using time- and frequency-domain techniques: potentials of spectroscopy and imaging,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance and A. Katzir, eds., Proc. SPIE 1431, 84–96 (1991).
[CrossRef]

R. L. Barbour, H. L. Graber, R. Aronson, and J. Lubowsky, “Imaging of subsurface regions of random media by remote sensing,” in Time-resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Proc. SPIE 1431, 192–203 (1991).
[CrossRef]

1990

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

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

1989

J. C. Adams, “MUDPACK: multigrid portable FORTRAN software for the efficient solution of linear elliptic partial differential equation,” Appl. Math. Comput. 34, 113–146 (1989).
[CrossRef]

1981

R. E. Alcouffe, A. Brandt, J. E. Dendy, and J. W. Painter, “The multi-grid method for the diffusion equation with strongly discontinuous coefficients,” SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. 2, 430–454 (1981).
[CrossRef]

Adams, J. C.

J. C. Adams, “MUDPACK: multigrid portable FORTRAN software for the efficient solution of linear elliptic partial differential equation,” Appl. Math. Comput. 34, 113–146 (1989).
[CrossRef]

Alcouffe, R. E.

R. E. Alcouffe, A. Brandt, J. E. Dendy, and J. W. Painter, “The multi-grid method for the diffusion equation with strongly discontinuous coefficients,” SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. 2, 430–454 (1981).
[CrossRef]

Aronson, R.

R. L. Barbour, H. L. Graber, R. Aronson, and J. Lubowsky, “Imaging of subsurface regions of random media by remote sensing,” in Time-resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Proc. SPIE 1431, 192–203 (1991).
[CrossRef]

Arridge, S. A.

S. A. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delay, “Performance of an iterative reconstruction algorithm for near infrared absorption and scatter imaging,” in Photon Migration, and Imaging in Random Medium and Tissues, Proc. SPIE 1888, 360–371 (1993).
[CrossRef]

S. A. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delay, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
[CrossRef] [PubMed]

Arridge, S. R.

S. R. Arridge, P. van der Zee, M. Cope, and D. T. Delpy, “Reconstruction methods for infra-red absorption imaging,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance and A. Katzir, eds., Proc. SPIE 1431, 204–215 (1991).
[CrossRef]

Barbour, R. L.

Y. Q. Yao, R. L. Barbour, Y. Wang, H. L. Graber, and J. W. Chang, “Sensitivity studies for imaging a spherical object embedded in a spherically symmetric, two-layer turbid medium with photon-density waves,” Appl. Opt. 35, 735–751 (1996).
[CrossRef] [PubMed]

Y. Yao, Y. Wang, Y. Pei, W. Zhu, J. Hu, and 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. Barbour, M. Carvlin, and M. Fiddy, eds., Proc. SPIE 2570, 254–266 (1995).
[CrossRef]

Y. Yao, Y. Wang, Y. Pei, W. Zhu, and 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. Barbour, M. Carlin, and M. Fiddy, eds., Proc. SPIE 2570, 96–107 (1995).
[CrossRef]

R. L. Barbour, H. L. Graber, R. Aronson, and J. Lubowsky, “Imaging of subsurface regions of random media by remote sensing,” in Time-resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Proc. SPIE 1431, 192–203 (1991).
[CrossRef]

Boas, D. A.

M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Simultaneous scattering and absorption images of heterogeneous media using diffusive waves with the Rytov approximation,”B. Chance and R. R. Alfano, eds. in Optical Tomography: Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, Proc. SPIE 2389, 320–327 (1995).
[CrossRef]

D. A. Boas, M. A. O’Leary, B. Chance, and A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[CrossRef]

Brandt, A.

R. E. Alcouffe, A. Brandt, J. E. Dendy, and J. W. Painter, “The multi-grid method for the diffusion equation with strongly discontinuous coefficients,” SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. 2, 430–454 (1981).
[CrossRef]

Chance, B.

M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Simultaneous scattering and absorption images of heterogeneous media using diffusive waves with the Rytov approximation,”B. Chance and R. R. Alfano, eds. in Optical Tomography: Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, Proc. SPIE 2389, 320–327 (1995).
[CrossRef]

D. A. Boas, M. A. O’Leary, B. Chance, and A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[CrossRef]

E. M. Sevick and B. Chance, “Photon migration in a model of the head measured using time- and frequency-domain techniques: potentials of spectroscopy and imaging,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance and A. Katzir, eds., Proc. SPIE 1431, 84–96 (1991).
[CrossRef]

Chang, J. W.

Chew, W. C.

M. Moghaddadam and W. C. Chew, “Nonlinear two-dimensional velocity profile inversion using time-domain data,” IEEE Trans. Geosci. Remote Sens. 30, 146–156 (1992).
[CrossRef]

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

Cope, M.

S. R. Arridge, P. van der Zee, M. Cope, and D. T. Delpy, “Reconstruction methods for infra-red absorption imaging,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance and A. Katzir, eds., Proc. SPIE 1431, 204–215 (1991).
[CrossRef]

Delay, D. T.

S. A. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delay, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
[CrossRef] [PubMed]

S. A. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delay, “Performance of an iterative reconstruction algorithm for near infrared absorption and scatter imaging,” in Photon Migration, and Imaging in Random Medium and Tissues, Proc. SPIE 1888, 360–371 (1993).
[CrossRef]

Delpy, D. T.

S. R. Arridge, P. van der Zee, M. Cope, and D. T. Delpy, “Reconstruction methods for infra-red absorption imaging,” in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance and A. Katzir, eds., Proc. SPIE 1431, 204–215 (1991).
[CrossRef]

Dendy, J. E.

R. E. Alcouffe, A. Brandt, J. E. Dendy, and J. W. Painter, “The multi-grid method for the diffusion equation with strongly discontinuous coefficients,” SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. 2, 430–454 (1981).
[CrossRef]

Fishkin, J.

Graber, H. L.

Y. Q. Yao, R. L. Barbour, Y. Wang, H. L. Graber, and J. W. Chang, “Sensitivity studies for imaging a spherical object embedded in a spherically symmetric, two-layer turbid medium with photon-density waves,” Appl. Opt. 35, 735–751 (1996).
[CrossRef] [PubMed]

R. L. Barbour, H. L. Graber, R. Aronson, and J. Lubowsky, “Imaging of subsurface regions of random media by remote sensing,” in Time-resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Proc. SPIE 1431, 192–203 (1991).
[CrossRef]

Gratton, E.

Grunbaum, F. A.

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

Haskell, R. C.

Hiraoka, M.

S. A. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delay, “Performance of an iterative reconstruction algorithm for near infrared absorption and scatter imaging,” in Photon Migration, and Imaging in Random Medium and Tissues, Proc. SPIE 1888, 360–371 (1993).
[CrossRef]

S. A. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delay, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
[CrossRef] [PubMed]

Hu, J.

Y. Yao, Y. Wang, Y. Pei, W. Zhu, J. Hu, and 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. Barbour, M. Carvlin, and M. Fiddy, eds., Proc. SPIE 2570, 254–266 (1995).
[CrossRef]

Hugonin, J. P.

N. Joachimowicz, C. Pichot, and J. P. Hugonin, “Inverse scattering: an iterative numerical method for electromagnetic imaging,” IEEE Trans. Antennas Propag. 39, 1742–1752 (1991).
[CrossRef]

Joachimowicz, N.

N. Joachimowicz, C. Pichot, and J. P. Hugonin, “Inverse scattering: an iterative numerical method for electromagnetic imaging,” IEEE Trans. Antennas Propag. 39, 1742–1752 (1991).
[CrossRef]

Kohn, P.

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

Lubowsky, J.

R. L. Barbour, H. L. Graber, R. Aronson, and J. Lubowsky, “Imaging of subsurface regions of random media by remote sensing,” in Time-resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Proc. SPIE 1431, 192–203 (1991).
[CrossRef]

Moghaddadam, M.

M. Moghaddadam and W. C. Chew, “Nonlinear two-dimensional velocity profile inversion using time-domain data,” IEEE Trans. Geosci. Remote Sens. 30, 146–156 (1992).
[CrossRef]

O’Leary, M. A.

M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Simultaneous scattering and absorption images of heterogeneous media using diffusive waves with the Rytov approximation,”B. Chance and R. R. Alfano, eds. in Optical Tomography: Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, Proc. SPIE 2389, 320–327 (1995).
[CrossRef]

D. A. Boas, M. A. O’Leary, B. Chance, and A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[CrossRef]

Painter, J. W.

R. E. Alcouffe, A. Brandt, J. E. Dendy, and J. W. Painter, “The multi-grid method for the diffusion equation with strongly discontinuous coefficients,” SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. 2, 430–454 (1981).
[CrossRef]

Patterson, M. S.

B. W. Pogue and M. S. Patterson, “Forward and inverse calculations for 3-D frequency-domain diffuse optical tomography,” in Optical Tomography: Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance and R. R. Alfano, eds., Proc. SPIE 2389, 328–339 (1995).
[CrossRef]

Pei, Y.

Y. Yao, Y. Wang, Y. Pei, W. Zhu, J. Hu, and 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. Barbour, M. Carvlin, and M. Fiddy, eds., Proc. SPIE 2570, 254–266 (1995).
[CrossRef]

Y. Yao, Y. Wang, Y. Pei, W. Zhu, and 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. Barbour, M. Carlin, and M. Fiddy, eds., Proc. SPIE 2570, 96–107 (1995).
[CrossRef]

Pichot, C.

N. Joachimowicz, C. Pichot, and J. P. Hugonin, “Inverse scattering: an iterative numerical method for electromagnetic imaging,” IEEE Trans. Antennas Propag. 39, 1742–1752 (1991).
[CrossRef]

Pogue, B. W.

B. W. Pogue and M. S. Patterson, “Forward and inverse calculations for 3-D frequency-domain diffuse optical tomography,” in Optical Tomography: Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance and R. R. Alfano, eds., Proc. SPIE 2389, 328–339 (1995).
[CrossRef]

Schweiger, M.

S. A. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delay, “Performance of an iterative reconstruction algorithm for near infrared absorption and scatter imaging,” in Photon Migration, and Imaging in Random Medium and Tissues, Proc. SPIE 1888, 360–371 (1993).
[CrossRef]

S. A. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delay, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
[CrossRef] [PubMed]

Sevick, E. M.

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Figures (22)

Fig. 1
Fig. 1

Full multigrid algorithm for a problem with M=4 grids.

Fig. 2
Fig. 2

(a) Amplitude and (b) phase shift of the incident wave obtained with different grid sizes in a 10cm×10cm homogeneous medium with μab=0.05cm-1 and μsb=10.0cm-1. (c) Amplitude and (d) phase shift of the scattered field obtained with different grid spacings, caused by a 0.7-cm-radius object, with μa=0.2cm-1 and μs=30.0cm-1. The modulation frequency is 200 MHz.

Fig. 3
Fig. 3

Flow chart of the BIM (RLS, regularized least-squares method; CGD, conjugate gradient descent).

Fig. 4
Fig. 4

(a) Data-acquisition geometry for full-angle profile inversion with 10 sources and 20 detectors in a uniform ring geometry. (b) Data-acquisition geometry for limited-angle profile inversion with 10 sources and 20 detectors distributed uniformly along a semicircle ranging from -65° to 65°. In both cases the radius of the source–detector ring is 4 cm. The object radius is 0.7 cm. The object is located either at the center or at an off-center location. O(r) is the object function.

Fig. 5
Fig. 5

Reconstruction of the absorption coefficient for test case I: (a) image of the true perturbation; (b)–(f) reconstructed images after 1 (i.e., the Born approximation), 2, 3, 5, and 15 iterations, respectively.

Fig. 6
Fig. 6

Reconstruction of the scattering coefficient for test case I: (a) image of the true perturbation; (b)–(f) reconstructed images after 1 (i.e., the Born approximation), 2, 3, 5, and 15 iterations, respectively.

Fig. 7
Fig. 7

Cross-sectional cuts along the y axis passing through the object center for (a) the absorption reconstruction shown in Fig. 5 and (b) the scattering reconstruction shown in Fig. 6. (c) Relative RMSE’s of the reconstructed absorption distribution (circles) and the scattering distribution (squares) as functions of iteration steps.

Fig. 8
Fig. 8

Reconstruction of the absorption coefficient for test case II: (a) image of the true perturbation; (b)–(f) reconstructed images after 1, 2, 5, 8, and 15 iterations, respectively.

Fig. 9
Fig. 9

Reconstruction of the scattering coefficient for test case II: (a) image of the true perturbation; (b)–(f) reconstructed images after 1, 2, 5, 10, and 15 iterations, respectively.

Fig. 10
Fig. 10

Cross-sectional cuts for reconstruction results shown in Figs. 8 and 9: (a) and (c) are along the y axis and the x axis, respectively, through the object center for the absorption reconstruction shown in Fig. 8; (b) and (d) are for the scattering reconstruction shown in Fig. 9. (e) Relative RMSE’s of the reconstructed absorption distribution (circles) shown in Fig. 8 and the scattering distribution (squares) in Fig. 9 as functions of iteration steps.

Fig. 11
Fig. 11

Reconstruction of the absorption distribution for test case III: (a) image of the true perturbation; (b)–(f) reconstructed images after 1, 2, 3, 5, and 15 iterations, respectively.

Fig. 12
Fig. 12

Reconstruction of the scattering distributions for test case III: (a) image of the true perturbation; (b)–(f) reconstructed images after 1, 2, 5, 10, and 15 iterations, respectively.

Fig. 13
Fig. 13

Cross-sectional cuts along the x axis passing through the centers of the two objects for (a) the absorption reconstruction shown in Fig. 11 and (b) the scattering reconstruction shown in Fig. 12. (c) Relative RMSE’s of the reconstructed absorption distribution (circles) in Fig. 11 and the scattering distribution (squares) shown in Fig. 12 as functions of iteration steps.

Fig. 14
Fig. 14

Reconstruction of the absorption distributions from noisy data with a SNR of 25 dB. (a)–(d) Reconstructed images after 1, 2, 5, and 15 iterations, respectively. The target medium is shown in Fig. 8(a).

Fig. 15
Fig. 15

Reconstruction of the scattering distributions from noisy data with a SNR of 25 dB. (a)–(d) Reconstructed images after 1, 2, 5, and 15 iterations, respectively. The target medium is shown in Fig. 9(a).

Fig. 16
Fig. 16

Cross-sectional cuts along the x axis passing through the centers of the object for (a) the absorption reconstruction shown in Fig. 14 and (b) the scattering reconstruction shown in Fig. 15.

Fig. 17
Fig. 17

Reconstruction of the absorption distributions from noisy data with a SNR of 25 dB. (a)–(d) Reconstructed images after 1, 2, 5, and 15 iterations, respectively. The target medium is shown in Fig. 11(a).

Fig. 18
Fig. 18

Reconstruction of the scattering distributions from noisy data with a SNR of 25 dB. (a)–(d) Reconstructed images after 1, 2, 5, and 15 iterations, respectively. The target medium is shown in Fig. 12(a).

Fig. 19
Fig. 19

Cross-sectional cuts along the x axis passing through the centers of the two objects for (a) the absorption reconstruction shown in Fig. 17 and (b) the scattering reconstruction shown in Fig. 18.

Fig. 20
Fig. 20

Limited-angle reconstruction of the absorption coefficient for test case IV: (a) image of the true perturbation; (b)–(f) reconstructed images after 1, 2, 5, 8, and 15 iterations, respectively.

Fig. 21
Fig. 21

Limited-angle reconstruction of the scattering coefficient for test case IV: (a) image of the true perturbation; (b)–(f) reconstructed images after 1, 2, 5, 10, and 15 iterations, respectively.

Fig. 22
Fig. 22

Cross-sectional cuts along the y axis passing through the object center for (a) the absorption reconstruction shown in Fig. 20 and (b) the scattering reconstruction shown in Fig. 21.

Tables (1)

Tables Icon

Table 1 Medium Properties and Measurement Schemes in Different Simulation Studies

Equations (27)

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

·[D(r)u(r)]+[-μa(r)+i(ω/v)]u(r)=-Q(r)
rΩ,
2u(r)+k2(r)u(r)=-S(r)-u(r)·D(r)D(r),
k(r)=[-μa(r)/D(r)+iω/vD(r)]1/2
2u(r)+k2(r)u(r)=-S(r).
ui+1,j,k+ui-1,j,kΔx2+ui,j+1,k+ui,j-1,kΔy2+ui,j,k+1+ui,j,k-1Δz2-2Δx2+2Δy2+2Δz2-ki,j,k2ui, j, k=-Si, j, k,
Lu=f,
Lkuk=fkk=1,2,, M.
Lk-1=(Ik-1k)TLkIk-1k,
fk-1=(Ik-1k)Trk,
rk=fk-Lkuk
ukGk(u˜k, fk),
O(r)=k2(r)-kb2=-3(μab+δμa)(δμa+δμs)-δμaDb+i3ωc(δμa+δμs),
2u(r)+kb2u(r)=S(r)-O(r)u(r).
ΩG(r, r)O(r)u(r)d2r=u(r)-ub(r),
2G(r, r)+kb2G(r, r)=-δ(r-r).
us(r)=u(r)-ub(r)
W(m×n)O(n×1)=us(m×1)
Wij=G(rdi, rj)u(rj, rsi)δvi=1,,m,
j=1,,n,
Re[O(r)]=-3(μab+δμa)(δμa+δμs)-[(δμa)/Db],
Im[O(r)]=[(3ω)/c](δμa+δμs).
δμa(r)=-μabIm[O(r)]ω/c+Re[O(r)]Im[O(r)]ω/c+3(μab+μsb),
δμs(r)=Re[O(r)]+3(3μab+μsb)+Im[O(r)]ω/cIm[O(r)]3ω/cIm[O(r)]ω/c+3(μab+μsb).
RMSE(i)=Σj[αj(i)-αj]2Σj(αj)21/2,
RMSE(i+1)-RMSE(i)RMSE(i)<10-6,
SNR=10 logPuPN,

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