Abstract

We have extended our investigation on the use of a linear algorithm for enhancing the accuracy of diffuse optical tomography (DOT) images, to include spatial maps of the diffusion coefficient. The results show that the corrected images are markedly improved in terms of estimated size, spatial resolution, two-object resolving power, and quantitative accuracy. These image-enhancing effects are significant at expected levels of diffusion-coefficient contrast in tissue and noise levels typical of experimental DOT data. Overall, the types and magnitudes of image-enhancing effects obtained here are qualitatively similar to those seen in previous studies on μa perturbations. The implications for practical implementations of DOT time-series imaging are discussed.

© 2007 Optical Society of America

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  1. Y. Pei, H. L. Graber, and R. L. Barbour, "Influence of systematic errors in reference states on image quality and on stability of derived information for DC optical imaging," Appl. Opt. 40, 5755-5769 (2001).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  6. Y. Xu, Y. Pei, H. L. Graber, and R. L. Barbour, "Image quality improvement via spatial deconvolution in optical tomography: Time-series imaging," J. Biomed. Opt. 10, 051701 (2005).
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    [CrossRef] [PubMed]
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    [CrossRef]
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2007 (1)

Y. Xu, H. L. Graber, and R. L. Barbour, "Image correction algorithm for functional three-dimensional diffuse optical tomography brain imaging," Appl. Opt. 46, 1693-1704 (2007).
[CrossRef] [PubMed]

2005 (4)

2004 (2)

R. L. Barbour, H. L. Graber, Y. Xu, Y. Pei, and R. Aronson, "Strategies for imaging diffusing media," Transp. Theory Stat. Phys. 33, 361-371 (2004).
[CrossRef]

M. Huang and Q. Zhu, "Dual-mesh optical tomography reconstruction method with a depth correction that uses a priori ultrasound information," Appl. Opt. 43, 1654-1662 (2004).
[CrossRef] [PubMed]

2002 (2)

M. J. Cassidy and W. D. Penny, "Bayesian nonstationary autoregressive models for biomedical signal analysis," IEEE Trans. Biomed. Eng. 49, 1142-1152 (2002).
[CrossRef] [PubMed]

H. L. Graber, Y. Pei, and R. L. Barbour, "Imaging of spatiotemporal coincident states by DC optical tomography," IEEE Trans. Med. Imaging 21, 852-866 (2002).
[CrossRef] [PubMed]

2001 (2)

1999 (1)

1998 (1)

P. Xu, "Truncated SVD methods for discrete linear ill-posed problems," Geophys. J. Int. 135, 505-514 (1998).
[CrossRef]

1997 (1)

S. R. Arridge and J. C. Hebden, "Optical imaging in medicine II: Modelling and reconstruction," Phys. Med. Biol. 42, 841-853 (1997).
[CrossRef] [PubMed]

1994 (1)

1962 (1)

B. Chance, P. Cohen, F. Jöbsis, and B. Schoener, "Intracellular oxidation-reduction statesin vivo," Science 137, 499-508 (1962).
[CrossRef] [PubMed]

Ansari, R.

R. L. Barbour, H. L. Graber, Y. Xu, Y. Pei, R. Ansari, M. B. Levin, and M. Farber, "Diffuse Optical Tissue Simulator (DOTS): An experimental calibrating system for functional DOT imaging," in Proceedings of Fifth Inter-Institute Workshop on Optical Diagnostic Imaging from Bench to Bedside (National Institutes of Health, 2006), Poster No. 76.
[PubMed]

Aronson, R.

R. L. Barbour, H. L. Graber, Y. Xu, Y. Pei, and R. Aronson, "Strategies for imaging diffusing media," Transp. Theory Stat. Phys. 33, 361-371 (2004).
[CrossRef]

Arridge, S. R.

A. P. Gibson, J. C. Hebden, and S. R. Arridge, "Recent advances in diffuse optical imaging," Phys. Med. Biol. 50, R1-R43 (2005).
[CrossRef] [PubMed]

S. R. Arridge and J. C. Hebden, "Optical imaging in medicine II: Modelling and reconstruction," Phys. Med. Biol. 42, 841-853 (1997).
[CrossRef] [PubMed]

Arsenin, V. Y.

A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (V. H. Winston, 1977).

Barbour, R. L.

Y. Xu, H. L. Graber, and R. L. Barbour, "Image correction algorithm for functional three-dimensional diffuse optical tomography brain imaging," Appl. Opt. 46, 1693-1704 (2007).
[CrossRef] [PubMed]

Y. Xu, Y. Pei, H. L. Graber, and R. L. Barbour, "Image quality improvement via spatial deconvolution in optical tomography: Time-series imaging," J. Biomed. Opt. 10, 051701 (2005).
[CrossRef] [PubMed]

H. L. Graber, Y. Xu, Y. Pei, and R. L. Barbour, "Spatial deconvolution technique to improve the accuracy of reconstructed three-dimensional diffuse optical tomographic images," Appl. Opt. 44, 941-953 (2005).
[CrossRef] [PubMed]

Y. Xu, H. L. Graber, Y. Pei, and R. L. Barbour, "Improved accuracy of reconstructed diffuse optical tomographic images by means of spatial deconvolution: two-dimensional quantitative characterization," Appl. Opt. 44, 2115-2139 (2005).
[CrossRef] [PubMed]

R. L. Barbour, H. L. Graber, Y. Xu, Y. Pei, and R. Aronson, "Strategies for imaging diffusing media," Transp. Theory Stat. Phys. 33, 361-371 (2004).
[CrossRef]

H. L. Graber, Y. Pei, and R. L. Barbour, "Imaging of spatiotemporal coincident states by DC optical tomography," IEEE Trans. Med. Imaging 21, 852-866 (2002).
[CrossRef] [PubMed]

Y. Pei, H. L. Graber, and R. L. Barbour, "Normalized-constraint algorithm for minimizing inter-parameter cross talk in DC optical tomography," Opt. Express 9, 97-109 (2001).
[CrossRef] [PubMed]

Y. Pei, H. L. Graber, and R. L. Barbour, "Influence of systematic errors in reference states on image quality and on stability of derived information for DC optical imaging," Appl. Opt. 40, 5755-5769 (2001).
[CrossRef]

R. L. Barbour, H. L. Graber, Y. Xu, Y. Pei, R. Ansari, M. B. Levin, and M. Farber, "Diffuse Optical Tissue Simulator (DOTS): An experimental calibrating system for functional DOT imaging," in Proceedings of Fifth Inter-Institute Workshop on Optical Diagnostic Imaging from Bench to Bedside (National Institutes of Health, 2006), Poster No. 76.
[PubMed]

Cassidy, M. J.

M. J. Cassidy and W. D. Penny, "Bayesian nonstationary autoregressive models for biomedical signal analysis," IEEE Trans. Biomed. Eng. 49, 1142-1152 (2002).
[CrossRef] [PubMed]

Chance, B.

B. Chance, P. Cohen, F. Jöbsis, and B. Schoener, "Intracellular oxidation-reduction statesin vivo," Science 137, 499-508 (1962).
[CrossRef] [PubMed]

Cohen, P.

B. Chance, P. Cohen, F. Jöbsis, and B. Schoener, "Intracellular oxidation-reduction statesin vivo," Science 137, 499-508 (1962).
[CrossRef] [PubMed]

Colton, D.

D. Colton, H. W. Engl, A. K. Louis, J. R. McLaughlin, and W. Rundell, eds., Surveys on Solution Methods for Inverse Problems (Springer, 2000).

Engl, H. W.

D. Colton, H. W. Engl, A. K. Louis, J. R. McLaughlin, and W. Rundell, eds., Surveys on Solution Methods for Inverse Problems (Springer, 2000).

Fantini, S.

Farber, M.

R. L. Barbour, H. L. Graber, Y. Xu, Y. Pei, R. Ansari, M. B. Levin, and M. Farber, "Diffuse Optical Tissue Simulator (DOTS): An experimental calibrating system for functional DOT imaging," in Proceedings of Fifth Inter-Institute Workshop on Optical Diagnostic Imaging from Bench to Bedside (National Institutes of Health, 2006), Poster No. 76.
[PubMed]

Franceschini, M. A.

Gibson, A. P.

A. P. Gibson, J. C. Hebden, and S. R. Arridge, "Recent advances in diffuse optical imaging," Phys. Med. Biol. 50, R1-R43 (2005).
[CrossRef] [PubMed]

Graber, H. L.

Y. Xu, H. L. Graber, and R. L. Barbour, "Image correction algorithm for functional three-dimensional diffuse optical tomography brain imaging," Appl. Opt. 46, 1693-1704 (2007).
[CrossRef] [PubMed]

Y. Xu, Y. Pei, H. L. Graber, and R. L. Barbour, "Image quality improvement via spatial deconvolution in optical tomography: Time-series imaging," J. Biomed. Opt. 10, 051701 (2005).
[CrossRef] [PubMed]

H. L. Graber, Y. Xu, Y. Pei, and R. L. Barbour, "Spatial deconvolution technique to improve the accuracy of reconstructed three-dimensional diffuse optical tomographic images," Appl. Opt. 44, 941-953 (2005).
[CrossRef] [PubMed]

Y. Xu, H. L. Graber, Y. Pei, and R. L. Barbour, "Improved accuracy of reconstructed diffuse optical tomographic images by means of spatial deconvolution: two-dimensional quantitative characterization," Appl. Opt. 44, 2115-2139 (2005).
[CrossRef] [PubMed]

R. L. Barbour, H. L. Graber, Y. Xu, Y. Pei, and R. Aronson, "Strategies for imaging diffusing media," Transp. Theory Stat. Phys. 33, 361-371 (2004).
[CrossRef]

H. L. Graber, Y. Pei, and R. L. Barbour, "Imaging of spatiotemporal coincident states by DC optical tomography," IEEE Trans. Med. Imaging 21, 852-866 (2002).
[CrossRef] [PubMed]

Y. Pei, H. L. Graber, and R. L. Barbour, "Normalized-constraint algorithm for minimizing inter-parameter cross talk in DC optical tomography," Opt. Express 9, 97-109 (2001).
[CrossRef] [PubMed]

Y. Pei, H. L. Graber, and R. L. Barbour, "Influence of systematic errors in reference states on image quality and on stability of derived information for DC optical imaging," Appl. Opt. 40, 5755-5769 (2001).
[CrossRef]

R. L. Barbour, H. L. Graber, Y. Xu, Y. Pei, R. Ansari, M. B. Levin, and M. Farber, "Diffuse Optical Tissue Simulator (DOTS): An experimental calibrating system for functional DOT imaging," in Proceedings of Fifth Inter-Institute Workshop on Optical Diagnostic Imaging from Bench to Bedside (National Institutes of Health, 2006), Poster No. 76.
[PubMed]

Gratton, E.

Graves, M. J.

D. W. McRobbie, E. A. Moore, M. J. Graves, and M. R. Prince, MRI from Picture to Proton, 2nd ed. (Cambridge U. Press, 2006.)

Gu, X.

X. Gu, J. Masciotti, and A. H. Hielscher, "Parametric reconstruction method in optical tomography," in Proceedings of 28th IEEE EMBS Annual International Conference (IEEE, 2006), Paper FrB13.4.

Hebden, J. C.

A. P. Gibson, J. C. Hebden, and S. R. Arridge, "Recent advances in diffuse optical imaging," Phys. Med. Biol. 50, R1-R43 (2005).
[CrossRef] [PubMed]

S. R. Arridge and J. C. Hebden, "Optical imaging in medicine II: Modelling and reconstruction," Phys. Med. Biol. 42, 841-853 (1997).
[CrossRef] [PubMed]

Hielscher, A. H.

X. Gu, J. Masciotti, and A. H. Hielscher, "Parametric reconstruction method in optical tomography," in Proceedings of 28th IEEE EMBS Annual International Conference (IEEE, 2006), Paper FrB13.4.

Huang, M.

Jöbsis, F.

B. Chance, P. Cohen, F. Jöbsis, and B. Schoener, "Intracellular oxidation-reduction statesin vivo," Science 137, 499-508 (1962).
[CrossRef] [PubMed]

Levin, M. B.

R. L. Barbour, H. L. Graber, Y. Xu, Y. Pei, R. Ansari, M. B. Levin, and M. Farber, "Diffuse Optical Tissue Simulator (DOTS): An experimental calibrating system for functional DOT imaging," in Proceedings of Fifth Inter-Institute Workshop on Optical Diagnostic Imaging from Bench to Bedside (National Institutes of Health, 2006), Poster No. 76.
[PubMed]

Louis, A. K.

D. Colton, H. W. Engl, A. K. Louis, J. R. McLaughlin, and W. Rundell, eds., Surveys on Solution Methods for Inverse Problems (Springer, 2000).

Maier, J. S.

Masciotti, J.

X. Gu, J. Masciotti, and A. H. Hielscher, "Parametric reconstruction method in optical tomography," in Proceedings of 28th IEEE EMBS Annual International Conference (IEEE, 2006), Paper FrB13.4.

McBride, T. O.

McLaughlin, J. R.

D. Colton, H. W. Engl, A. K. Louis, J. R. McLaughlin, and W. Rundell, eds., Surveys on Solution Methods for Inverse Problems (Springer, 2000).

McRobbie, D. W.

D. W. McRobbie, E. A. Moore, M. J. Graves, and M. R. Prince, MRI from Picture to Proton, 2nd ed. (Cambridge U. Press, 2006.)

Moore, E. A.

D. W. McRobbie, E. A. Moore, M. J. Graves, and M. R. Prince, MRI from Picture to Proton, 2nd ed. (Cambridge U. Press, 2006.)

Osterberg, U. L.

Paulsen, K. D.

Pei, Y.

Y. Xu, H. L. Graber, Y. Pei, and R. L. Barbour, "Improved accuracy of reconstructed diffuse optical tomographic images by means of spatial deconvolution: two-dimensional quantitative characterization," Appl. Opt. 44, 2115-2139 (2005).
[CrossRef] [PubMed]

H. L. Graber, Y. Xu, Y. Pei, and R. L. Barbour, "Spatial deconvolution technique to improve the accuracy of reconstructed three-dimensional diffuse optical tomographic images," Appl. Opt. 44, 941-953 (2005).
[CrossRef] [PubMed]

Y. Xu, Y. Pei, H. L. Graber, and R. L. Barbour, "Image quality improvement via spatial deconvolution in optical tomography: Time-series imaging," J. Biomed. Opt. 10, 051701 (2005).
[CrossRef] [PubMed]

R. L. Barbour, H. L. Graber, Y. Xu, Y. Pei, and R. Aronson, "Strategies for imaging diffusing media," Transp. Theory Stat. Phys. 33, 361-371 (2004).
[CrossRef]

H. L. Graber, Y. Pei, and R. L. Barbour, "Imaging of spatiotemporal coincident states by DC optical tomography," IEEE Trans. Med. Imaging 21, 852-866 (2002).
[CrossRef] [PubMed]

Y. Pei, H. L. Graber, and R. L. Barbour, "Influence of systematic errors in reference states on image quality and on stability of derived information for DC optical imaging," Appl. Opt. 40, 5755-5769 (2001).
[CrossRef]

Y. Pei, H. L. Graber, and R. L. Barbour, "Normalized-constraint algorithm for minimizing inter-parameter cross talk in DC optical tomography," Opt. Express 9, 97-109 (2001).
[CrossRef] [PubMed]

R. L. Barbour, H. L. Graber, Y. Xu, Y. Pei, R. Ansari, M. B. Levin, and M. Farber, "Diffuse Optical Tissue Simulator (DOTS): An experimental calibrating system for functional DOT imaging," in Proceedings of Fifth Inter-Institute Workshop on Optical Diagnostic Imaging from Bench to Bedside (National Institutes of Health, 2006), Poster No. 76.
[PubMed]

Penny, W. D.

M. J. Cassidy and W. D. Penny, "Bayesian nonstationary autoregressive models for biomedical signal analysis," IEEE Trans. Biomed. Eng. 49, 1142-1152 (2002).
[CrossRef] [PubMed]

Pogue, B. W.

Prince, M. R.

D. W. McRobbie, E. A. Moore, M. J. Graves, and M. R. Prince, MRI from Picture to Proton, 2nd ed. (Cambridge U. Press, 2006.)

Rundell, W.

D. Colton, H. W. Engl, A. K. Louis, J. R. McLaughlin, and W. Rundell, eds., Surveys on Solution Methods for Inverse Problems (Springer, 2000).

Schoener, B.

B. Chance, P. Cohen, F. Jöbsis, and B. Schoener, "Intracellular oxidation-reduction statesin vivo," Science 137, 499-508 (1962).
[CrossRef] [PubMed]

Tikhonov, A. N.

A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (V. H. Winston, 1977).

Walker, S. A.

Xu, P.

P. Xu, "Truncated SVD methods for discrete linear ill-posed problems," Geophys. J. Int. 135, 505-514 (1998).
[CrossRef]

Xu, Y.

Y. Xu, H. L. Graber, and R. L. Barbour, "Image correction algorithm for functional three-dimensional diffuse optical tomography brain imaging," Appl. Opt. 46, 1693-1704 (2007).
[CrossRef] [PubMed]

Y. Xu, Y. Pei, H. L. Graber, and R. L. Barbour, "Image quality improvement via spatial deconvolution in optical tomography: Time-series imaging," J. Biomed. Opt. 10, 051701 (2005).
[CrossRef] [PubMed]

H. L. Graber, Y. Xu, Y. Pei, and R. L. Barbour, "Spatial deconvolution technique to improve the accuracy of reconstructed three-dimensional diffuse optical tomographic images," Appl. Opt. 44, 941-953 (2005).
[CrossRef] [PubMed]

Y. Xu, H. L. Graber, Y. Pei, and R. L. Barbour, "Improved accuracy of reconstructed diffuse optical tomographic images by means of spatial deconvolution: two-dimensional quantitative characterization," Appl. Opt. 44, 2115-2139 (2005).
[CrossRef] [PubMed]

R. L. Barbour, H. L. Graber, Y. Xu, Y. Pei, and R. Aronson, "Strategies for imaging diffusing media," Transp. Theory Stat. Phys. 33, 361-371 (2004).
[CrossRef]

R. L. Barbour, H. L. Graber, Y. Xu, Y. Pei, R. Ansari, M. B. Levin, and M. Farber, "Diffuse Optical Tissue Simulator (DOTS): An experimental calibrating system for functional DOT imaging," in Proceedings of Fifth Inter-Institute Workshop on Optical Diagnostic Imaging from Bench to Bedside (National Institutes of Health, 2006), Poster No. 76.
[PubMed]

Zhu, Q.

Appl. Opt. (5)

Geophys. J. Int. (1)

P. Xu, "Truncated SVD methods for discrete linear ill-posed problems," Geophys. J. Int. 135, 505-514 (1998).
[CrossRef]

IEEE Trans. Biomed. Eng. (1)

M. J. Cassidy and W. D. Penny, "Bayesian nonstationary autoregressive models for biomedical signal analysis," IEEE Trans. Biomed. Eng. 49, 1142-1152 (2002).
[CrossRef] [PubMed]

IEEE Trans. Med. Imaging (1)

H. L. Graber, Y. Pei, and R. L. Barbour, "Imaging of spatiotemporal coincident states by DC optical tomography," IEEE Trans. Med. Imaging 21, 852-866 (2002).
[CrossRef] [PubMed]

J. Biomed. Opt. (1)

Y. Xu, Y. Pei, H. L. Graber, and R. L. Barbour, "Image quality improvement via spatial deconvolution in optical tomography: Time-series imaging," J. Biomed. Opt. 10, 051701 (2005).
[CrossRef] [PubMed]

Opt. Express (2)

Opt. Lett. (1)

Phys. Med. Biol. (2)

A. P. Gibson, J. C. Hebden, and S. R. Arridge, "Recent advances in diffuse optical imaging," Phys. Med. Biol. 50, R1-R43 (2005).
[CrossRef] [PubMed]

S. R. Arridge and J. C. Hebden, "Optical imaging in medicine II: Modelling and reconstruction," Phys. Med. Biol. 42, 841-853 (1997).
[CrossRef] [PubMed]

Science (1)

B. Chance, P. Cohen, F. Jöbsis, and B. Schoener, "Intracellular oxidation-reduction statesin vivo," Science 137, 499-508 (1962).
[CrossRef] [PubMed]

Transp. Theory Stat. Phys. (1)

R. L. Barbour, H. L. Graber, Y. Xu, Y. Pei, and R. Aronson, "Strategies for imaging diffusing media," Transp. Theory Stat. Phys. 33, 361-371 (2004).
[CrossRef]

Other (5)

D. W. McRobbie, E. A. Moore, M. J. Graves, and M. R. Prince, MRI from Picture to Proton, 2nd ed. (Cambridge U. Press, 2006.)

X. Gu, J. Masciotti, and A. H. Hielscher, "Parametric reconstruction method in optical tomography," in Proceedings of 28th IEEE EMBS Annual International Conference (IEEE, 2006), Paper FrB13.4.

D. Colton, H. W. Engl, A. K. Louis, J. R. McLaughlin, and W. Rundell, eds., Surveys on Solution Methods for Inverse Problems (Springer, 2000).

A. N. Tikhonov and V. Y. Arsenin, Solutions of Ill-Posed Problems (V. H. Winston, 1977).

R. L. Barbour, H. L. Graber, Y. Xu, Y. Pei, R. Ansari, M. B. Levin, and M. Farber, "Diffuse Optical Tissue Simulator (DOTS): An experimental calibrating system for functional DOT imaging," in Proceedings of Fifth Inter-Institute Workshop on Optical Diagnostic Imaging from Bench to Bedside (National Institutes of Health, 2006), Poster No. 76.
[PubMed]

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Figures (14)

Fig. 1
Fig. 1

Model specifications for DOT forward problem. (a) FEM mesh (981 nodes) used for all inverse-problem computations, with the locations of the isotropic point sources and detectors on the boundary also indicated. (b) Inclusion locations and size ranges modeled in single-inclusion studies. The center was placed at either ( x c , y c ) = ( 1 , 0 ) (solid circles) or ( x c , y c ) = ( 3 , 0 ) (dot-dashed circles). Smallest (0.4 cm diam.) and largest (1.8 or 2.0 cm diam.) inclusion sizes are shown. (c) Inclusion locations modeled in two-inclusion studies. Solid circles and arrows indicate the range of positions used in the fixed y c , increasing x c study. Dashed circles and arrows indicate the range of positions used in the fixed x c , increasing y c study.

Fig. 2
Fig. 2

Definitions of several parameters used in calculating indices of image accuracy. δ D = D bkgr D incl , δ D 0 = maximal value of δ D , Δ ( δ D ) = depth of the notch between the inclusions. δ X 0 = true interinclusion separation distance, δ X = image interinclusion separation distance.

Fig. 3
Fig. 3

Low-contrast-medium images, recovered from noise-free data. The 0.6 cm inclusions have a point of contact at ( x , y ) = ( 0 , 0 ) . (a) Uncorrected image when Δ D > 0 in both inclusions. (b) Corrected image, Δ D > 0 in both inclusions. (c) Uncorrected image, Δ D > 0 in one inclusion, and < 0 in the other. (d) Corrected image, Δ D > 0 in one inclusion, and < 0 in the other.

Fig. 4
Fig. 4

Low-contrast-medium images, recovered from noise-free data. The distance between the centers of the 0.6 cm inclusions 2 cm. Δ D > 0 in both inclusions. (a) Uncorrected image. (b) Corrected image.

Fig. 5
Fig. 5

Plots of spatial correlation value versus scattering-coefficient contrast (i.e., ratio of μ s in the inclusion and background), for uncorrected (dashed line) and corrected (solid curve) images. Target medium contains a single, 1.2 cm diameter inclusion centered at ( x , y ) = ( 1 , 0 ) . Data noise level is ε k = 1 % .

Fig. 6
Fig. 6

Representative single-inclusion target media [(a), (d)], uncorrected images [(b), (e)], and corrected [(c), (f)] images. The inclusion is centered at ( x c , y c ) = ( 1 , 0 ) , with diameter of the inclusion is 0.4 cm in (a)–(c), and 2.0 cm in (d)–(f).

Fig. 7
Fig. 7

(a) Normalized detector readings η R ( t ) / R r , for a source located near the 3 o'clock position in Fig. 1(a), for all 32 detectors at t = 100 . The dot-dashed curve is computed for the target medium shown in Fig. 3(a). Dashed and solid curves are computed for the images in Figs. 3(b) and 3(c), respectively. (b) Root-mean-squared difference between the η curves for the target medium and image (i.e., RMSD i = [ j = 1 32 ( η i j target η i j image ) 2 / 32 ] 1 / 2 , where i and j are the source and detector indices, respectively). Dashed and solid curves are the results for the uncorrected and corrected images, respectively.

Fig. 8
Fig. 8

(a) One-dimensional sections, along the y = 0 diameter, through one-inclusion target medium, uncorrected image, and corrected image, for 0.4 cm diameter inclusion [Figs. 3(a)–3(c)]. (b) Analogous 1D sections for a medium with a 1.2 cm diameter inclusion. (c) One-dimensional sections for a medium with a 2.0 cm diameter inclusion [Figs. 3(d)–3(f)]. (d) Recovered versus true inclusion diameter (i.e., FWHM), for the uncorrected and corrected images of all single-inclusion target media with ( x c , y c ) = ( 1 , 0 ) . (e) FWHM of the uncorrected and corrected images, for all ( x c , y c ) = ( 3 , 0 ) target media. d = uncorrected images , + d = corrected images.

Fig. 9
Fig. 9

Spatial correlation [(a), (b)] and temporal RMSD [(c), (d)] accuracy indices, for all single-inclusion target media. The results shown are mean values over 100 time frames. (a), (c) ( x c , y c ) = ( 1 , 0 ) . (b), (d) ( x c , y c ) = ( 3 , 0 ) . Thin lines in (b) are mean ± SD ; in the other panels, SD is smaller than the line thickness. d = uncorrected images , + d = corrected images.

Fig. 10
Fig. 10

(a) One-dimensional sections along the line y = 1 , through a two-inclusion target medium and corresponding images, where ( x c , y c ) = ( ± 0.5 , 1 ) . (b) Analogous 1D sections along the line y = 2 , for the ( x c , y c ) = ( ± 0.5 , 2 ) case. (c) Analogous 1D sections along the line y = 3 , for the ( x c , y c ) = ( ± 0.5 , 3 ) case. Dot-dashed, dashed, and solid curves are sections through target media, uncorrected images, and corrected images, respectively.

Fig. 11
Fig. 11

(a) Resolving-power index P d , as a function of the inclusion-center coordinate y c , for images of the two-inclusion target media considered in Fig. 10 (i.e., | x c | has a fixed value of 0.5, y c is variable). (b) Location-bias index P s versus y c , for the same set of media. (c) Spatial correlation between target medium and image as a function of y c . (d) Temporal RMSD versus y c . Thin lines in (c) are mean ± SD ; in the other panels SD is smaller than the line thickness. Vertical dot-dashed lines in (c) and (d) indicate the distance from the center, above which the inclusions can be resolved. d =  uncorrected images, + d = corrected images.

Fig. 12
Fig. 12

(a) Resolving-power index P d , as a function of the inclusion-center coordinate x c , for recovered images in the second two-inclusion study (i.e., x c is variable, y c has a fixed value of 0). (b) Location-bias index P s versus x c for the same set of media. (c) Spatial correlation versus x c . (d) Temporal RMSD versus x c . Thin lines in (c) are mean ± SD ; in the other panels, SD is smaller than the line thickness. Vertical dot-dashed lines in (c) and (d) indicate the separation distance, above which the inclusions can be resolved. d = uncorrected images , + d =  corrected images.

Fig. 13
Fig. 13

Recovered images of a two-inclusion target medium, with ( x c , y c ) = ( ± 0.9 , 0 ) . (a)–(f) Detector data are noise-free; (g)–(l) detector data noise level is 1%. (a) and (g) uncorrected, unfiltered images; (b) and (h) corrected, unfiltered images; (d) and (j) uncorrected, low-pass filtered images; (e) and (k) corrected, low-pass filtered images. (c), (f), (i), (l) One-dimensional sections through the plotted 2D images along the y = 0 diameter.

Fig. 14
Fig. 14

Spatial mean value of the time-varying Δ D , within the inclusion, for the Fig. 1(b) one-inclusion case. Dot-dashed curves show the true Δ D ( r ) versus t of the target medium. Dashed curves are the corresponding Δ D ( r ) versus t plots for uncorrected images. Solid curves are the corresponding plots for the corrected images. (a) and (b) Detector data noise level is 1%. (c) and (d) Detector data noise level is 3%. (a) and (c) Unfiltered image data; (b) and (d) low-pass filtered image data.

Tables (2)

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Table 1 Indices of Qualitative and Quantitative Spatial Accuracy

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Table 2 Temporal Correlation Coefficients

Equations (4)

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P d = Δ ( δ D ) δ D 0 ,
P s = δ X δ X 0 ,
[ W μ a | W D ] W [ Δ μ a u Δ D u ] = [ Δ R ] [ Δ μ a u Δ D u ] = [ X μ a X D ] X [ Δ R ] ,
[ Δ μ a c Δ D c ] = [ | F μ a μ a F D μ a | F μ a D F D D ] F [ X μ a X D ] [ Δ R ] [ Y μ a Y D ] Y [ Δ R ] ,

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