Abstract

We conducted a systematic investigation of the reflectance diffuse optical tomography using continuous wave (CW) measurements and nonlinear reconstruction algorithms. We illustrated and suggested how to fine-tune the nonlinear reconstruction methods in order to optimize target localization with depth-adaptive regularizations, reduce boundary noises in the reconstructed images using a logarithm based objective function, improve reconstruction quantification using transport models, and resolve crosstalk problems between absorption and scattering contrasts with the CW reflectance measurements. The upgraded nonlinear reconstruction algorithms were evaluated with a series of numerical and experimental tests, which show the potentials of the proposed approaches for imaging both absorption and scattering contrasts in the deep targets with enhanced image quality.

© 2014 Optical Society of America

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References

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  1. B. Yodh and B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48(3), 34–40 (1995).
    [Crossref]
  2. H. Jiang, N. V. Iftimia, Y. Xu, J. A. Eggert, L. L. Fajardo, and K. L. Klove, “Near-infrared optical imaging of the breast with model-based reconstruction,” Acad. Radiol. 9(2), 186–194 (2002).
    [Crossref] [PubMed]
  3. Z. Yuan, Q. Zhang, E. S. Sobel, and H. Jiang, “Image-guided optical spectroscopy in diagnosis of osteoarthritis: A clinical study,” Biomed. Opt. Express 1(1), 74–86 (2010).
    [Crossref] [PubMed]
  4. Z. Yuan, “Combing independent component analysis and Granger causality to capture brain network dynamics with fNIRS measurements,” Biomed. Opt. Express 4(11), 2629–2643 (2013).
    [Crossref] [PubMed]
  5. R. C. Mesquita, M. A. Franceschini, and D. A. Boas, “Resting state functional connectivity of the whole head with near-infrared spectroscopy,” Biomed. Opt. Express 1(1), 324–336 (2010).
    [Crossref] [PubMed]
  6. S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “Interpreting hemoglobin and water concentration, oxygen saturation, and scattering measured in vivo by near-infrared breast tomography,” Proc. Natl. Acad. Sci. U.S.A. 100(21), 12349–12354 (2003).
    [Crossref] [PubMed]
  7. Z. Yuan and H. Jiang, “Image reconstruction scheme that combines modified Newton method and efficient initial guess estimation for optical tomography of finger joints,” Appl. Opt. 46(14), 2757–2768 (2007).
    [Crossref] [PubMed]
  8. G. Xu and D. Piao, “A geometric-differential-sensitivity based algorithm improves object depth-localization for diffuse optical tomography in a circular-array outward-imaging geometry,” Med. Phys. 40(1), 0131011 (2013).
    [Crossref]
  9. R. Endoh, M. Fuji, and K. Nakayama, “Depth-adaptive regularized reconstruction for reflection diffuse optical tomography,” Opt. Rev. 15(1), 51–56 (2008).
    [Crossref]
  10. A. Puszka, L. Hervé, A. Planat-Chrétien, A. Koenig, J. Derouard, and J.-M. Dinten, “Time-domain reflectance diffuse optical tomography with Mellin-Laplace transform for experimental detection and depth localization of a single absorbing inclusion,” Biomed. Opt. Express 4(4), 569–583 (2013).
    [Crossref] [PubMed]
  11. Z. Yuan, X. H. Hu, and H. Jiang, “A higher order diffusion model for three-dimensional photon migration and image reconstruction in optical tomography,” Phys. Med. Biol. 54(1), 67–88 (2009).
    [Crossref] [PubMed]
  12. Z. Yuan, Q. Zhang, E. Sobel, and H. Jiang, “Comparison of diffusion approximation and higher order diffusion equation for optical tomography of osteoarthritis,” J. Biomed. Opt. 14(5), 0540131 (2009).
    [Crossref]
  13. V. C. Kavuri, Z.-J. Lin, F. Tian, and H. Liu, “Sparsity enhanced spatial resolution and depth localization in diffuse optical tomography,” Biomed. Opt. Express 3(5), 943–957 (2012).
    [Crossref] [PubMed]
  14. Z. Yuan, X. Li, and L. Xi, “Listening to light scattering in turbid media: quantitative optical scattering imaging using photoacoustic measurement with one-wavelength illumination,” J. Opt. 16(6), 0653011 (2014).
    [Crossref]
  15. P. Meaney, Q. Fang, T. Rubaek, E. Demidenko, and K. D. Paulsen, “Log transformation benefits parameter estimation in microwave tomographic imaging,” Med. Phys. 34(6), 2013–2023 (2007).

2014 (1)

Z. Yuan, X. Li, and L. Xi, “Listening to light scattering in turbid media: quantitative optical scattering imaging using photoacoustic measurement with one-wavelength illumination,” J. Opt. 16(6), 0653011 (2014).
[Crossref]

2013 (3)

2012 (1)

2010 (2)

2009 (2)

Z. Yuan, X. H. Hu, and H. Jiang, “A higher order diffusion model for three-dimensional photon migration and image reconstruction in optical tomography,” Phys. Med. Biol. 54(1), 67–88 (2009).
[Crossref] [PubMed]

Z. Yuan, Q. Zhang, E. Sobel, and H. Jiang, “Comparison of diffusion approximation and higher order diffusion equation for optical tomography of osteoarthritis,” J. Biomed. Opt. 14(5), 0540131 (2009).
[Crossref]

2008 (1)

R. Endoh, M. Fuji, and K. Nakayama, “Depth-adaptive regularized reconstruction for reflection diffuse optical tomography,” Opt. Rev. 15(1), 51–56 (2008).
[Crossref]

2007 (2)

P. Meaney, Q. Fang, T. Rubaek, E. Demidenko, and K. D. Paulsen, “Log transformation benefits parameter estimation in microwave tomographic imaging,” Med. Phys. 34(6), 2013–2023 (2007).

Z. Yuan and H. Jiang, “Image reconstruction scheme that combines modified Newton method and efficient initial guess estimation for optical tomography of finger joints,” Appl. Opt. 46(14), 2757–2768 (2007).
[Crossref] [PubMed]

2003 (1)

S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “Interpreting hemoglobin and water concentration, oxygen saturation, and scattering measured in vivo by near-infrared breast tomography,” Proc. Natl. Acad. Sci. U.S.A. 100(21), 12349–12354 (2003).
[Crossref] [PubMed]

2002 (1)

H. Jiang, N. V. Iftimia, Y. Xu, J. A. Eggert, L. L. Fajardo, and K. L. Klove, “Near-infrared optical imaging of the breast with model-based reconstruction,” Acad. Radiol. 9(2), 186–194 (2002).
[Crossref] [PubMed]

1995 (1)

B. Yodh and B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48(3), 34–40 (1995).
[Crossref]

Boas, D. A.

Chance, B.

B. Yodh and B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48(3), 34–40 (1995).
[Crossref]

Dehghani, H.

S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “Interpreting hemoglobin and water concentration, oxygen saturation, and scattering measured in vivo by near-infrared breast tomography,” Proc. Natl. Acad. Sci. U.S.A. 100(21), 12349–12354 (2003).
[Crossref] [PubMed]

Demidenko, E.

P. Meaney, Q. Fang, T. Rubaek, E. Demidenko, and K. D. Paulsen, “Log transformation benefits parameter estimation in microwave tomographic imaging,” Med. Phys. 34(6), 2013–2023 (2007).

Derouard, J.

Dinten, J.-M.

Eggert, J. A.

H. Jiang, N. V. Iftimia, Y. Xu, J. A. Eggert, L. L. Fajardo, and K. L. Klove, “Near-infrared optical imaging of the breast with model-based reconstruction,” Acad. Radiol. 9(2), 186–194 (2002).
[Crossref] [PubMed]

Endoh, R.

R. Endoh, M. Fuji, and K. Nakayama, “Depth-adaptive regularized reconstruction for reflection diffuse optical tomography,” Opt. Rev. 15(1), 51–56 (2008).
[Crossref]

Fajardo, L. L.

H. Jiang, N. V. Iftimia, Y. Xu, J. A. Eggert, L. L. Fajardo, and K. L. Klove, “Near-infrared optical imaging of the breast with model-based reconstruction,” Acad. Radiol. 9(2), 186–194 (2002).
[Crossref] [PubMed]

Fang, Q.

P. Meaney, Q. Fang, T. Rubaek, E. Demidenko, and K. D. Paulsen, “Log transformation benefits parameter estimation in microwave tomographic imaging,” Med. Phys. 34(6), 2013–2023 (2007).

Franceschini, M. A.

Fuji, M.

R. Endoh, M. Fuji, and K. Nakayama, “Depth-adaptive regularized reconstruction for reflection diffuse optical tomography,” Opt. Rev. 15(1), 51–56 (2008).
[Crossref]

Gibson, J. J.

S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “Interpreting hemoglobin and water concentration, oxygen saturation, and scattering measured in vivo by near-infrared breast tomography,” Proc. Natl. Acad. Sci. U.S.A. 100(21), 12349–12354 (2003).
[Crossref] [PubMed]

Hervé, L.

Hu, X. H.

Z. Yuan, X. H. Hu, and H. Jiang, “A higher order diffusion model for three-dimensional photon migration and image reconstruction in optical tomography,” Phys. Med. Biol. 54(1), 67–88 (2009).
[Crossref] [PubMed]

Iftimia, N. V.

H. Jiang, N. V. Iftimia, Y. Xu, J. A. Eggert, L. L. Fajardo, and K. L. Klove, “Near-infrared optical imaging of the breast with model-based reconstruction,” Acad. Radiol. 9(2), 186–194 (2002).
[Crossref] [PubMed]

Jiang, H.

Z. Yuan, Q. Zhang, E. S. Sobel, and H. Jiang, “Image-guided optical spectroscopy in diagnosis of osteoarthritis: A clinical study,” Biomed. Opt. Express 1(1), 74–86 (2010).
[Crossref] [PubMed]

Z. Yuan, Q. Zhang, E. Sobel, and H. Jiang, “Comparison of diffusion approximation and higher order diffusion equation for optical tomography of osteoarthritis,” J. Biomed. Opt. 14(5), 0540131 (2009).
[Crossref]

Z. Yuan, X. H. Hu, and H. Jiang, “A higher order diffusion model for three-dimensional photon migration and image reconstruction in optical tomography,” Phys. Med. Biol. 54(1), 67–88 (2009).
[Crossref] [PubMed]

Z. Yuan and H. Jiang, “Image reconstruction scheme that combines modified Newton method and efficient initial guess estimation for optical tomography of finger joints,” Appl. Opt. 46(14), 2757–2768 (2007).
[Crossref] [PubMed]

H. Jiang, N. V. Iftimia, Y. Xu, J. A. Eggert, L. L. Fajardo, and K. L. Klove, “Near-infrared optical imaging of the breast with model-based reconstruction,” Acad. Radiol. 9(2), 186–194 (2002).
[Crossref] [PubMed]

Jiang, S.

S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “Interpreting hemoglobin and water concentration, oxygen saturation, and scattering measured in vivo by near-infrared breast tomography,” Proc. Natl. Acad. Sci. U.S.A. 100(21), 12349–12354 (2003).
[Crossref] [PubMed]

Kavuri, V. C.

Klove, K. L.

H. Jiang, N. V. Iftimia, Y. Xu, J. A. Eggert, L. L. Fajardo, and K. L. Klove, “Near-infrared optical imaging of the breast with model-based reconstruction,” Acad. Radiol. 9(2), 186–194 (2002).
[Crossref] [PubMed]

Koenig, A.

Kogel, C.

S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “Interpreting hemoglobin and water concentration, oxygen saturation, and scattering measured in vivo by near-infrared breast tomography,” Proc. Natl. Acad. Sci. U.S.A. 100(21), 12349–12354 (2003).
[Crossref] [PubMed]

Li, X.

Z. Yuan, X. Li, and L. Xi, “Listening to light scattering in turbid media: quantitative optical scattering imaging using photoacoustic measurement with one-wavelength illumination,” J. Opt. 16(6), 0653011 (2014).
[Crossref]

Lin, Z.-J.

Liu, H.

Meaney, P.

P. Meaney, Q. Fang, T. Rubaek, E. Demidenko, and K. D. Paulsen, “Log transformation benefits parameter estimation in microwave tomographic imaging,” Med. Phys. 34(6), 2013–2023 (2007).

Mesquita, R. C.

Nakayama, K.

R. Endoh, M. Fuji, and K. Nakayama, “Depth-adaptive regularized reconstruction for reflection diffuse optical tomography,” Opt. Rev. 15(1), 51–56 (2008).
[Crossref]

Paulsen, K. D.

P. Meaney, Q. Fang, T. Rubaek, E. Demidenko, and K. D. Paulsen, “Log transformation benefits parameter estimation in microwave tomographic imaging,” Med. Phys. 34(6), 2013–2023 (2007).

S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “Interpreting hemoglobin and water concentration, oxygen saturation, and scattering measured in vivo by near-infrared breast tomography,” Proc. Natl. Acad. Sci. U.S.A. 100(21), 12349–12354 (2003).
[Crossref] [PubMed]

Piao, D.

G. Xu and D. Piao, “A geometric-differential-sensitivity based algorithm improves object depth-localization for diffuse optical tomography in a circular-array outward-imaging geometry,” Med. Phys. 40(1), 0131011 (2013).
[Crossref]

Planat-Chrétien, A.

Pogue, B. W.

S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “Interpreting hemoglobin and water concentration, oxygen saturation, and scattering measured in vivo by near-infrared breast tomography,” Proc. Natl. Acad. Sci. U.S.A. 100(21), 12349–12354 (2003).
[Crossref] [PubMed]

Poplack, S. P.

S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “Interpreting hemoglobin and water concentration, oxygen saturation, and scattering measured in vivo by near-infrared breast tomography,” Proc. Natl. Acad. Sci. U.S.A. 100(21), 12349–12354 (2003).
[Crossref] [PubMed]

Puszka, A.

Rubaek, T.

P. Meaney, Q. Fang, T. Rubaek, E. Demidenko, and K. D. Paulsen, “Log transformation benefits parameter estimation in microwave tomographic imaging,” Med. Phys. 34(6), 2013–2023 (2007).

Sobel, E.

Z. Yuan, Q. Zhang, E. Sobel, and H. Jiang, “Comparison of diffusion approximation and higher order diffusion equation for optical tomography of osteoarthritis,” J. Biomed. Opt. 14(5), 0540131 (2009).
[Crossref]

Sobel, E. S.

Soho, S.

S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “Interpreting hemoglobin and water concentration, oxygen saturation, and scattering measured in vivo by near-infrared breast tomography,” Proc. Natl. Acad. Sci. U.S.A. 100(21), 12349–12354 (2003).
[Crossref] [PubMed]

Srinivasan, S.

S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “Interpreting hemoglobin and water concentration, oxygen saturation, and scattering measured in vivo by near-infrared breast tomography,” Proc. Natl. Acad. Sci. U.S.A. 100(21), 12349–12354 (2003).
[Crossref] [PubMed]

Tian, F.

Tosteson, T. D.

S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “Interpreting hemoglobin and water concentration, oxygen saturation, and scattering measured in vivo by near-infrared breast tomography,” Proc. Natl. Acad. Sci. U.S.A. 100(21), 12349–12354 (2003).
[Crossref] [PubMed]

Xi, L.

Z. Yuan, X. Li, and L. Xi, “Listening to light scattering in turbid media: quantitative optical scattering imaging using photoacoustic measurement with one-wavelength illumination,” J. Opt. 16(6), 0653011 (2014).
[Crossref]

Xu, G.

G. Xu and D. Piao, “A geometric-differential-sensitivity based algorithm improves object depth-localization for diffuse optical tomography in a circular-array outward-imaging geometry,” Med. Phys. 40(1), 0131011 (2013).
[Crossref]

Xu, Y.

H. Jiang, N. V. Iftimia, Y. Xu, J. A. Eggert, L. L. Fajardo, and K. L. Klove, “Near-infrared optical imaging of the breast with model-based reconstruction,” Acad. Radiol. 9(2), 186–194 (2002).
[Crossref] [PubMed]

Yodh, B.

B. Yodh and B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48(3), 34–40 (1995).
[Crossref]

Yuan, Z.

Z. Yuan, X. Li, and L. Xi, “Listening to light scattering in turbid media: quantitative optical scattering imaging using photoacoustic measurement with one-wavelength illumination,” J. Opt. 16(6), 0653011 (2014).
[Crossref]

Z. Yuan, “Combing independent component analysis and Granger causality to capture brain network dynamics with fNIRS measurements,” Biomed. Opt. Express 4(11), 2629–2643 (2013).
[Crossref] [PubMed]

Z. Yuan, Q. Zhang, E. S. Sobel, and H. Jiang, “Image-guided optical spectroscopy in diagnosis of osteoarthritis: A clinical study,” Biomed. Opt. Express 1(1), 74–86 (2010).
[Crossref] [PubMed]

Z. Yuan, Q. Zhang, E. Sobel, and H. Jiang, “Comparison of diffusion approximation and higher order diffusion equation for optical tomography of osteoarthritis,” J. Biomed. Opt. 14(5), 0540131 (2009).
[Crossref]

Z. Yuan, X. H. Hu, and H. Jiang, “A higher order diffusion model for three-dimensional photon migration and image reconstruction in optical tomography,” Phys. Med. Biol. 54(1), 67–88 (2009).
[Crossref] [PubMed]

Z. Yuan and H. Jiang, “Image reconstruction scheme that combines modified Newton method and efficient initial guess estimation for optical tomography of finger joints,” Appl. Opt. 46(14), 2757–2768 (2007).
[Crossref] [PubMed]

Zhang, Q.

Z. Yuan, Q. Zhang, E. S. Sobel, and H. Jiang, “Image-guided optical spectroscopy in diagnosis of osteoarthritis: A clinical study,” Biomed. Opt. Express 1(1), 74–86 (2010).
[Crossref] [PubMed]

Z. Yuan, Q. Zhang, E. Sobel, and H. Jiang, “Comparison of diffusion approximation and higher order diffusion equation for optical tomography of osteoarthritis,” J. Biomed. Opt. 14(5), 0540131 (2009).
[Crossref]

Acad. Radiol. (1)

H. Jiang, N. V. Iftimia, Y. Xu, J. A. Eggert, L. L. Fajardo, and K. L. Klove, “Near-infrared optical imaging of the breast with model-based reconstruction,” Acad. Radiol. 9(2), 186–194 (2002).
[Crossref] [PubMed]

Appl. Opt. (1)

Biomed. Opt. Express (5)

J. Biomed. Opt. (1)

Z. Yuan, Q. Zhang, E. Sobel, and H. Jiang, “Comparison of diffusion approximation and higher order diffusion equation for optical tomography of osteoarthritis,” J. Biomed. Opt. 14(5), 0540131 (2009).
[Crossref]

J. Opt. (1)

Z. Yuan, X. Li, and L. Xi, “Listening to light scattering in turbid media: quantitative optical scattering imaging using photoacoustic measurement with one-wavelength illumination,” J. Opt. 16(6), 0653011 (2014).
[Crossref]

Med. Phys. (2)

P. Meaney, Q. Fang, T. Rubaek, E. Demidenko, and K. D. Paulsen, “Log transformation benefits parameter estimation in microwave tomographic imaging,” Med. Phys. 34(6), 2013–2023 (2007).

G. Xu and D. Piao, “A geometric-differential-sensitivity based algorithm improves object depth-localization for diffuse optical tomography in a circular-array outward-imaging geometry,” Med. Phys. 40(1), 0131011 (2013).
[Crossref]

Opt. Rev. (1)

R. Endoh, M. Fuji, and K. Nakayama, “Depth-adaptive regularized reconstruction for reflection diffuse optical tomography,” Opt. Rev. 15(1), 51–56 (2008).
[Crossref]

Phys. Med. Biol. (1)

Z. Yuan, X. H. Hu, and H. Jiang, “A higher order diffusion model for three-dimensional photon migration and image reconstruction in optical tomography,” Phys. Med. Biol. 54(1), 67–88 (2009).
[Crossref] [PubMed]

Phys. Today (1)

B. Yodh and B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48(3), 34–40 (1995).
[Crossref]

Proc. Natl. Acad. Sci. U.S.A. (1)

S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “Interpreting hemoglobin and water concentration, oxygen saturation, and scattering measured in vivo by near-infrared breast tomography,” Proc. Natl. Acad. Sci. U.S.A. 100(21), 12349–12354 (2003).
[Crossref] [PubMed]

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Figures (8)

Fig. 1
Fig. 1

Profile (a) and half profile (b) of the simulation geometry. (c) A schematic demonstration of the distribution of laser sources on the top surface of the background medium.

Fig. 2
Fig. 2

Reconstructed absorption images at selected planes (Z = 0-4 mm from left to right) using our old algorithm. The target and background optical properties for this test are the same as in the cases S1 and S2. The dotted circles represent the exact positions and sizes of the targets.

Fig. 3
Fig. 3

Reconstructed absorption images at selected planes (Z = 0-4 mm from left to right) for test S1 with the logarithm based objective function as in Eq. (4). The dotted circles represent the exact positions and sizes of the targets.

Fig. 4
Fig. 4

Reconstructed absorption images at selected planes (Z = 0-4 mm from left to right) for test S2 using the forward modeling based on Eqs. (15)-(20). The dotted circles represent the exact positions and sizes of the targets.

Fig. 5
Fig. 5

Reconstructed absorption (top row) and scattering images (bottom row) at selected planes (Z = 0-4 mm from left to right) for test S3 using absorption priors to resolve the crosstalk issue between absorption and scattering contrasts. The dotted circles represent the exact positions and sizes of the targets.

Fig. 6
Fig. 6

Reconstructed absorption images at selected planes (Z = 3-7 mm from left to right) for test S4 with depth adaptive regularization (bottom row) and with old algorithm (top row). The dotted circles represent the exact positions and sizes of the targets.

Fig. 7
Fig. 7

A schematic demonstration of the CCD-based CW experimental system.

Fig. 8
Fig. 8

Reconstructed absorption images at selected planes (Z = 2-6 mm from top to bottom row) for experimental test using improved algorithm with logarithm objective function, transport forward modeling and depth adaptive regularization (left column) and previous algorithm (right column). The images on the 1st-5th rows are for the slices at Z = 2-6 mm, respectively. The dotted circles represent the exact positions and sizes of the target located in the central slice.

Tables (1)

Tables Icon

Table 1 Recovered quantitative optical properties, sizes and locations of the targets for the four numerical simulations (S1-S4) and one phantom (P1) tests using improved (New) and existing (Old) algorithms. Note. GT: Ground truth; N/A: Not available. LT: Left target with Y = 6 in Fig. 1(c); RT: Right target with Y = 15 in Fig. 1(c). S1-S4: simulation tests 1-4; P1:Phantom test.

Equations (22)

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

Min:F= i=1 M ( Φ i m Φ i c ) 2
D(r)Φ(r) μ a (r)Φ(r)=S(r) , DΦn=αΦ
( T +λI)Δχ= T ( Φ o Φ c )
Min:F= i=1 M ( ln Φ i m ln Φ i c ) 2
F ζ = F ζ ( ζ 0 )+Δζ d dζ [ F ζ ( ζ 0 )]+...
ζ= ζ 0 { d dζ [ F ζ ( ζ 0 )]} 1 F ζ ( ζ 0 )
ζ i = ζ i1 { d dζ [ F ζ ( ζ i1 )]} 1 F ζ ( ζ i1 )
F ζ =2 ( Ln Φ c ζ ) T (Ln Φ c Ln Φ m )
2 F ζ 2 =2 ( Ln Φ c ζ ) T Ln Φ c ζ +2 ( 2 Ln Φ c ζ 2 ) T (Ln Φ c Ln Φ m )
ζ i = ζ i1 + {2 ( Ln Φ c ζ ) T Ln Φ c ζ } 1 2 ( Ln Φ c ζ ) T (Ln Φ m Ln Φ c )
LnΦ ζ = Φ/ζ Φ
( J T J+λI)Δχ= J T (ln Φ o ln Φ c )
ΩΦ(r,Ω)+ σ t (r)Φ( r,Ω )= σ s ( Ω ' ,Ω,r)Φ( r, Ω ' )d Ω ' +s(r,Ω)
Φ( r,Ω )= l=0 N m=0 l (2l+1) p l m ( cosθ ) [ ψ lm ( r )cos( mϕ )+ γ lm (r)sin(mϕ) ]
D( 2 ψ 1 2 x + 2 ψ 1 2 y + 2 ψ 1 2 z ) μ a ψ 1 +D(2 2 ψ 2 2 z 2 ψ 2 2 x 2 ψ 2 2 y )+6D( 2 ψ 3 2 x )+6D( 2 ψ 3 2 y ) +6D 2 ψ 5 zx +6D 2 ψ 6 zy +6D 2 ψ 4 xy +6D 2 ψ 4 yx =S(r)
25 7 D( 2 ψ 2 2 x + 2 ψ 2 2 y )+2D( 2 ψ 1 2 z 1 2 2 ψ 1 2 x 1 2 2 ψ 1 2 y )+ + 60 7 D( 2 ψ 3 2 y 2 ψ 3 2 x 2 ψ 4 xy 2 ψ 4 yx )+ 55 7 D 2 ψ 2 2 z + 30 7 D( 2 ψ 5 zx + 2 ψ 6 zy )5 μ ' t ψ 2 =0
D( 2 ψ 1 2 x 2 ψ 1 2 y ) 10 7 D( 2 ψ 2 2 x 2 ψ 2 2 y )+ 30 7 D( 2 ψ 5 xz 2 ψ 6 yz )+ 90 7 D( 2 ψ 3 2 x + 2 ψ 3 2 y + 1 3 2 ψ 3 2 z )10 μ ' t ψ 3 =0
1 2 D 2 ψ 1 xy + 1 2 D 2 ψ 1 yx 5 7 D 2 ψ 2 xy 5 7 D 2 ψ 2 yx + 45 7 D( 2 ψ 4 2 x + 2 ψ 4 2 y + 1 3 2 ψ 4 2 z )+ 15 7 D( 2 ψ 5 yz + 2 ψ 6 xz )5 μ ' t ψ 4 =0
2D 2 ψ 1 xz + 10 7 D 2 ψ 2 xz + 45 7 D( 2 ψ 5 2 z + 2 ψ 5 2 x + 1 3 2 ψ 5 2 y )+ 60 7 D( 2 ψ 3 xz + 2 ψ 4 yz + 1 2 2 ψ 6 zy )5 μ ' t ψ 5 =0
2D 2 ψ 1 yz + 10 7 D 2 ψ 2 yz + 45 7 D( 2 ψ 6 2 z + 1 3 2 ψ 6 2 x + 2 ψ 6 2 y )+ 60 7 D( 2 ψ 4 xz 2 ψ 3 yz + 1 2 2 ψ 5 xy )5 μ ' t ψ 6 =0
D ψ 1 n=α ψ 1 , ψ 2 = ψ 3 = ψ 4 = ψ 5 = ψ 6 =0
( T + D λ I)Δχ= T (ln Φ o ln Φ c )

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