Abstract

We present a method to obtain quantitatively accurate images of small obstacles or inhomogeneities situated near the surface of a strongly scattering medium. The method uses time-resolved measurements of backscattered light to form the images. Using the asymptotic solution of the radiative transfer equation for this problem, we determine that the key information content in measurements is modeled by a diffusion approximation that is valid for small source-detector distances, and shallow penetration depths. We simplify this model further by linearizing the effect of the inhomogeneities about the known background optical properties using the Born approximation. The resulting model is used in a two-stage imaging algorithm. First, the spatial location of the inhomogeneities are determined using a modification of the multiple signal classification (MUSIC) method. Using those results, we then determine the quantitative values of the inhomogeneities through a least-squares approximation. We find that this two-stage method is most effective for reconstructing a sequence of one-dimensional images along the penetration depth corresponding to null source-detector separations rather than simultaneously using measurements over several source-detector distances. This method is limited to penetration depths and distances between boundary measurements on the order of the scattering mean-free path.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  3. M. Moscoso, J. B. Keller, and G. Papanicolaou, “Depolarization and blurring of optical images by biological tissue,” J. Opt. Soc. America A 18, 948–960 (2001).
    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  9. D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
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    [Crossref]
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    [Crossref]
  13. R. Griesmaier and C. Schmiedecke, “A multifrequency MUSIC algorithm for locating small inhomogeneities in inverse scattering,” Inverse Probl. 33, 035015 (2017).
    [Crossref]
  14. W. Liao and A. Fannjiang, “A MUSIC for single-snapshot spectral estimation: stability and super-resolution,” Appl. Comput. Harmon. Anal. 40, 33–67 (2016).
    [Crossref]
  15. B. V. P. Dileep, T. Das, and P. K. Dutta, “Modified CS-MUSIC for diffuse optical tomography using joint sparsity,” Optik 158, 1478–1490 (2018).
    [Crossref]
  16. S. B. Rohde and A. D. Kim, “Backscattering of continuous and pulsed beams,” Multiscale Model. Simul. 15, 1356–1375 (2017).
    [Crossref]
  17. L. V. Wang and H.-I. Wu, Biomedical Optics: Principles and Imaging (John Wiley & Sons, 2012).
  18. A. Ishimaru, Wave Propagation and Scattering in Random Media (Wiley-IEEE, 1997).
  19. L. Spinelli, F. Martelli, S. Del Bianco, A. Pifferi, A. Torricelli, R. Cubeddu, and G. Zaccanti, “Absorption and scattering perturbations in homogeneous and layered diffusive media probed by time-resolved reflectance at null source-detector separation,” Phys. Rev. E 74, 021919 (2006).
    [Crossref]
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    [Crossref] [PubMed]
  21. M. Moscoso, A. Novikov, G. Papanicolaou, and C. Tsogka, “Robust multifrequency imaging with MUSIC,” submitted, 2018.
  22. R. Prony, “Essai experimental et analytique sur les lois de la dilatabilite de fluides elastiques et sur celles del la force expansive de la vapeur de l’alkool a differentes temperatures,” J. L’Ecole Polytech.1, 24–76 (1795).

2018 (1)

B. V. P. Dileep, T. Das, and P. K. Dutta, “Modified CS-MUSIC for diffuse optical tomography using joint sparsity,” Optik 158, 1478–1490 (2018).
[Crossref]

2017 (2)

S. B. Rohde and A. D. Kim, “Backscattering of continuous and pulsed beams,” Multiscale Model. Simul. 15, 1356–1375 (2017).
[Crossref]

R. Griesmaier and C. Schmiedecke, “A multifrequency MUSIC algorithm for locating small inhomogeneities in inverse scattering,” Inverse Probl. 33, 035015 (2017).
[Crossref]

2016 (1)

W. Liao and A. Fannjiang, “A MUSIC for single-snapshot spectral estimation: stability and super-resolution,” Appl. Comput. Harmon. Anal. 40, 33–67 (2016).
[Crossref]

2015 (1)

D. Contini, A. D. Mora, L. Spinelli, A. Farina, A. Torricelli, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, G. Boso, F. Zappa, and A. Pifferi, “Effect of time-gated detection in diffuse optical imaging at short source-detector separation,” J. Phys. D: Appl. Phys. 48, 045401 (2015).
[Crossref]

2013 (1)

2008 (1)

A. Pifferi, A. Torricelli, L. Spinelli, D. Contini, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, A. D. Mora, F. Zappa, and S. Cova, “Time-resolved diffuse reflectance using small source-detector separation and fast single-photon gating,” Phys. Rev. Lett. 100, 138101 (2008).
[Crossref] [PubMed]

2006 (1)

L. Spinelli, F. Martelli, S. Del Bianco, A. Pifferi, A. Torricelli, R. Cubeddu, and G. Zaccanti, “Absorption and scattering perturbations in homogeneous and layered diffusive media probed by time-resolved reflectance at null source-detector separation,” Phys. Rev. E 74, 021919 (2006).
[Crossref]

2005 (1)

A. Devaney, E. Marengo, and F. Gruber, “Time-reversal-based imaging and inverse scattering of multiply scattering point targets,” J. Acoust. Soc. Am. 118, 3129–3138 (2005).
[Crossref]

2001 (1)

M. Moscoso, J. B. Keller, and G. Papanicolaou, “Depolarization and blurring of optical images by biological tissue,” J. Opt. Soc. America A 18, 948–960 (2001).
[Crossref]

2000 (1)

1997 (1)

1993 (1)

1992 (2)

1991 (1)

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

1986 (1)

R. O. Schmidt, “Multiple emitter location and signal parameter estimation,” IEEE Trans. Antennas Propag. 34, 276–280 (1986).
[Crossref]

Alfano, R. R.

Bonner, R. F.

Boso, G.

D. Contini, A. D. Mora, L. Spinelli, A. Farina, A. Torricelli, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, G. Boso, F. Zappa, and A. Pifferi, “Effect of time-gated detection in diffuse optical imaging at short source-detector separation,” J. Phys. D: Appl. Phys. 48, 045401 (2015).
[Crossref]

Chang, W.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Contini, D.

D. Contini, A. D. Mora, L. Spinelli, A. Farina, A. Torricelli, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, G. Boso, F. Zappa, and A. Pifferi, “Effect of time-gated detection in diffuse optical imaging at short source-detector separation,” J. Phys. D: Appl. Phys. 48, 045401 (2015).
[Crossref]

A. Pifferi, A. Torricelli, L. Spinelli, D. Contini, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, A. D. Mora, F. Zappa, and S. Cova, “Time-resolved diffuse reflectance using small source-detector separation and fast single-photon gating,” Phys. Rev. Lett. 100, 138101 (2008).
[Crossref] [PubMed]

Cova, S.

A. Pifferi, A. Torricelli, L. Spinelli, D. Contini, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, A. D. Mora, F. Zappa, and S. Cova, “Time-resolved diffuse reflectance using small source-detector separation and fast single-photon gating,” Phys. Rev. Lett. 100, 138101 (2008).
[Crossref] [PubMed]

Cubeddu, R.

D. Contini, A. D. Mora, L. Spinelli, A. Farina, A. Torricelli, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, G. Boso, F. Zappa, and A. Pifferi, “Effect of time-gated detection in diffuse optical imaging at short source-detector separation,” J. Phys. D: Appl. Phys. 48, 045401 (2015).
[Crossref]

A. Pifferi, A. Torricelli, L. Spinelli, D. Contini, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, A. D. Mora, F. Zappa, and S. Cova, “Time-resolved diffuse reflectance using small source-detector separation and fast single-photon gating,” Phys. Rev. Lett. 100, 138101 (2008).
[Crossref] [PubMed]

L. Spinelli, F. Martelli, S. Del Bianco, A. Pifferi, A. Torricelli, R. Cubeddu, and G. Zaccanti, “Absorption and scattering perturbations in homogeneous and layered diffusive media probed by time-resolved reflectance at null source-detector separation,” Phys. Rev. E 74, 021919 (2006).
[Crossref]

Das, B. B.

Das, T.

B. V. P. Dileep, T. Das, and P. K. Dutta, “Modified CS-MUSIC for diffuse optical tomography using joint sparsity,” Optik 158, 1478–1490 (2018).
[Crossref]

Del Bianco, S.

L. Spinelli, F. Martelli, S. Del Bianco, A. Pifferi, A. Torricelli, R. Cubeddu, and G. Zaccanti, “Absorption and scattering perturbations in homogeneous and layered diffusive media probed by time-resolved reflectance at null source-detector separation,” Phys. Rev. E 74, 021919 (2006).
[Crossref]

Demos, S. G.

Devaney, A.

A. Devaney, E. Marengo, and F. Gruber, “Time-reversal-based imaging and inverse scattering of multiply scattering point targets,” J. Acoust. Soc. Am. 118, 3129–3138 (2005).
[Crossref]

Dileep, B. V. P.

B. V. P. Dileep, T. Das, and P. K. Dutta, “Modified CS-MUSIC for diffuse optical tomography using joint sparsity,” Optik 158, 1478–1490 (2018).
[Crossref]

Dutta, P. K.

B. V. P. Dileep, T. Das, and P. K. Dutta, “Modified CS-MUSIC for diffuse optical tomography using joint sparsity,” Optik 158, 1478–1490 (2018).
[Crossref]

Fannjiang, A.

W. Liao and A. Fannjiang, “A MUSIC for single-snapshot spectral estimation: stability and super-resolution,” Appl. Comput. Harmon. Anal. 40, 33–67 (2016).
[Crossref]

Farina, A.

D. Contini, A. D. Mora, L. Spinelli, A. Farina, A. Torricelli, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, G. Boso, F. Zappa, and A. Pifferi, “Effect of time-gated detection in diffuse optical imaging at short source-detector separation,” J. Phys. D: Appl. Phys. 48, 045401 (2015).
[Crossref]

Fujimoto, J. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Gandjbakche, A. H.

González-Rodríguez, P.

Griesmaier, R.

R. Griesmaier and C. Schmiedecke, “A multifrequency MUSIC algorithm for locating small inhomogeneities in inverse scattering,” Inverse Probl. 33, 035015 (2017).
[Crossref]

Gruber, F.

A. Devaney, E. Marengo, and F. Gruber, “Time-reversal-based imaging and inverse scattering of multiply scattering point targets,” J. Acoust. Soc. Am. 118, 3129–3138 (2005).
[Crossref]

Hayes, M. H.

M. H. Hayes, Statistical Digital Signal Processing and Modeling (John Wiley & Sons, 1996).

Huang, D.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Wiley-IEEE, 1997).

Keller, J. B.

M. Moscoso, J. B. Keller, and G. Papanicolaou, “Depolarization and blurring of optical images by biological tissue,” J. Opt. Soc. America A 18, 948–960 (2001).
[Crossref]

Kim, A. D.

Liao, W.

W. Liao and A. Fannjiang, “A MUSIC for single-snapshot spectral estimation: stability and super-resolution,” Appl. Comput. Harmon. Anal. 40, 33–67 (2016).
[Crossref]

Lin, C. P.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Marengo, E.

A. Devaney, E. Marengo, and F. Gruber, “Time-reversal-based imaging and inverse scattering of multiply scattering point targets,” J. Acoust. Soc. Am. 118, 3129–3138 (2005).
[Crossref]

Martelli, F.

D. Contini, A. D. Mora, L. Spinelli, A. Farina, A. Torricelli, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, G. Boso, F. Zappa, and A. Pifferi, “Effect of time-gated detection in diffuse optical imaging at short source-detector separation,” J. Phys. D: Appl. Phys. 48, 045401 (2015).
[Crossref]

A. Pifferi, A. Torricelli, L. Spinelli, D. Contini, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, A. D. Mora, F. Zappa, and S. Cova, “Time-resolved diffuse reflectance using small source-detector separation and fast single-photon gating,” Phys. Rev. Lett. 100, 138101 (2008).
[Crossref] [PubMed]

L. Spinelli, F. Martelli, S. Del Bianco, A. Pifferi, A. Torricelli, R. Cubeddu, and G. Zaccanti, “Absorption and scattering perturbations in homogeneous and layered diffusive media probed by time-resolved reflectance at null source-detector separation,” Phys. Rev. E 74, 021919 (2006).
[Crossref]

Mora, A. D.

D. Contini, A. D. Mora, L. Spinelli, A. Farina, A. Torricelli, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, G. Boso, F. Zappa, and A. Pifferi, “Effect of time-gated detection in diffuse optical imaging at short source-detector separation,” J. Phys. D: Appl. Phys. 48, 045401 (2015).
[Crossref]

A. Pifferi, A. Torricelli, L. Spinelli, D. Contini, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, A. D. Mora, F. Zappa, and S. Cova, “Time-resolved diffuse reflectance using small source-detector separation and fast single-photon gating,” Phys. Rev. Lett. 100, 138101 (2008).
[Crossref] [PubMed]

Moscoso, M.

P. González-Rodríguez, A. D. Kim, and M. Moscoso, “Robust depth selectivity in mesoscopic scattering regimes using angle-resolved measurements,” Opt. Lett. 38, 787–789 (2013).
[Crossref] [PubMed]

M. Moscoso, J. B. Keller, and G. Papanicolaou, “Depolarization and blurring of optical images by biological tissue,” J. Opt. Soc. America A 18, 948–960 (2001).
[Crossref]

M. Moscoso, A. Novikov, G. Papanicolaou, and C. Tsogka, “Robust multifrequency imaging with MUSIC,” submitted, 2018.

Novikov, A.

M. Moscoso, A. Novikov, G. Papanicolaou, and C. Tsogka, “Robust multifrequency imaging with MUSIC,” submitted, 2018.

Papanicolaou, G.

M. Moscoso, J. B. Keller, and G. Papanicolaou, “Depolarization and blurring of optical images by biological tissue,” J. Opt. Soc. America A 18, 948–960 (2001).
[Crossref]

M. Moscoso, A. Novikov, G. Papanicolaou, and C. Tsogka, “Robust multifrequency imaging with MUSIC,” submitted, 2018.

Pifferi, A.

D. Contini, A. D. Mora, L. Spinelli, A. Farina, A. Torricelli, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, G. Boso, F. Zappa, and A. Pifferi, “Effect of time-gated detection in diffuse optical imaging at short source-detector separation,” J. Phys. D: Appl. Phys. 48, 045401 (2015).
[Crossref]

A. Pifferi, A. Torricelli, L. Spinelli, D. Contini, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, A. D. Mora, F. Zappa, and S. Cova, “Time-resolved diffuse reflectance using small source-detector separation and fast single-photon gating,” Phys. Rev. Lett. 100, 138101 (2008).
[Crossref] [PubMed]

L. Spinelli, F. Martelli, S. Del Bianco, A. Pifferi, A. Torricelli, R. Cubeddu, and G. Zaccanti, “Absorption and scattering perturbations in homogeneous and layered diffusive media probed by time-resolved reflectance at null source-detector separation,” Phys. Rev. E 74, 021919 (2006).
[Crossref]

Prony, R.

R. Prony, “Essai experimental et analytique sur les lois de la dilatabilite de fluides elastiques et sur celles del la force expansive de la vapeur de l’alkool a differentes temperatures,” J. L’Ecole Polytech.1, 24–76 (1795).

Radousky, H. B.

Rohde, S. B.

S. B. Rohde and A. D. Kim, “Backscattering of continuous and pulsed beams,” Multiscale Model. Simul. 15, 1356–1375 (2017).
[Crossref]

Schmidt, R. O.

R. O. Schmidt, “Multiple emitter location and signal parameter estimation,” IEEE Trans. Antennas Propag. 34, 276–280 (1986).
[Crossref]

Schmiedecke, C.

R. Griesmaier and C. Schmiedecke, “A multifrequency MUSIC algorithm for locating small inhomogeneities in inverse scattering,” Inverse Probl. 33, 035015 (2017).
[Crossref]

Schmitt, J. M.

Schuman, J. S.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Spinelli, L.

D. Contini, A. D. Mora, L. Spinelli, A. Farina, A. Torricelli, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, G. Boso, F. Zappa, and A. Pifferi, “Effect of time-gated detection in diffuse optical imaging at short source-detector separation,” J. Phys. D: Appl. Phys. 48, 045401 (2015).
[Crossref]

A. Pifferi, A. Torricelli, L. Spinelli, D. Contini, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, A. D. Mora, F. Zappa, and S. Cova, “Time-resolved diffuse reflectance using small source-detector separation and fast single-photon gating,” Phys. Rev. Lett. 100, 138101 (2008).
[Crossref] [PubMed]

L. Spinelli, F. Martelli, S. Del Bianco, A. Pifferi, A. Torricelli, R. Cubeddu, and G. Zaccanti, “Absorption and scattering perturbations in homogeneous and layered diffusive media probed by time-resolved reflectance at null source-detector separation,” Phys. Rev. E 74, 021919 (2006).
[Crossref]

Stinson, W. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Swanson, E. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, and J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Torricelli, A.

D. Contini, A. D. Mora, L. Spinelli, A. Farina, A. Torricelli, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, G. Boso, F. Zappa, and A. Pifferi, “Effect of time-gated detection in diffuse optical imaging at short source-detector separation,” J. Phys. D: Appl. Phys. 48, 045401 (2015).
[Crossref]

A. Pifferi, A. Torricelli, L. Spinelli, D. Contini, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, A. D. Mora, F. Zappa, and S. Cova, “Time-resolved diffuse reflectance using small source-detector separation and fast single-photon gating,” Phys. Rev. Lett. 100, 138101 (2008).
[Crossref] [PubMed]

L. Spinelli, F. Martelli, S. Del Bianco, A. Pifferi, A. Torricelli, R. Cubeddu, and G. Zaccanti, “Absorption and scattering perturbations in homogeneous and layered diffusive media probed by time-resolved reflectance at null source-detector separation,” Phys. Rev. E 74, 021919 (2006).
[Crossref]

Tosi, A.

D. Contini, A. D. Mora, L. Spinelli, A. Farina, A. Torricelli, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, G. Boso, F. Zappa, and A. Pifferi, “Effect of time-gated detection in diffuse optical imaging at short source-detector separation,” J. Phys. D: Appl. Phys. 48, 045401 (2015).
[Crossref]

A. Pifferi, A. Torricelli, L. Spinelli, D. Contini, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, A. D. Mora, F. Zappa, and S. Cova, “Time-resolved diffuse reflectance using small source-detector separation and fast single-photon gating,” Phys. Rev. Lett. 100, 138101 (2008).
[Crossref] [PubMed]

Tsogka, C.

M. Moscoso, A. Novikov, G. Papanicolaou, and C. Tsogka, “Robust multifrequency imaging with MUSIC,” submitted, 2018.

Wang, L. V.

L. V. Wang and H.-I. Wu, Biomedical Optics: Principles and Imaging (John Wiley & Sons, 2012).

Wu, H.-I.

L. V. Wang and H.-I. Wu, Biomedical Optics: Principles and Imaging (John Wiley & Sons, 2012).

Yoo, K. M.

Zaccanti, G.

D. Contini, A. D. Mora, L. Spinelli, A. Farina, A. Torricelli, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, G. Boso, F. Zappa, and A. Pifferi, “Effect of time-gated detection in diffuse optical imaging at short source-detector separation,” J. Phys. D: Appl. Phys. 48, 045401 (2015).
[Crossref]

A. Pifferi, A. Torricelli, L. Spinelli, D. Contini, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, A. D. Mora, F. Zappa, and S. Cova, “Time-resolved diffuse reflectance using small source-detector separation and fast single-photon gating,” Phys. Rev. Lett. 100, 138101 (2008).
[Crossref] [PubMed]

L. Spinelli, F. Martelli, S. Del Bianco, A. Pifferi, A. Torricelli, R. Cubeddu, and G. Zaccanti, “Absorption and scattering perturbations in homogeneous and layered diffusive media probed by time-resolved reflectance at null source-detector separation,” Phys. Rev. E 74, 021919 (2006).
[Crossref]

Zappa, F.

D. Contini, A. D. Mora, L. Spinelli, A. Farina, A. Torricelli, R. Cubeddu, F. Martelli, G. Zaccanti, A. Tosi, G. Boso, F. Zappa, and A. Pifferi, “Effect of time-gated detection in diffuse optical imaging at short source-detector separation,” J. Phys. D: Appl. Phys. 48, 045401 (2015).
[Crossref]

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Figures (4)

Fig. 1
Fig. 1 Top row: Exact distributions of perturbations δD to be reconstructed. Bottom row: minimum 2-norm reconstructions using all source-detector pairs with a 2 GHz system bandwidth.
Fig. 2
Fig. 2 Minimum 2-norm reconstructions using collocated sources and detectors for the same cases as shown in Fig. 1 with a 2 GHz system bandwidth.
Fig. 3
Fig. 3 Top row: Results from the first stage of the imaging algorithm showing MUSIC reconstructions of the support of δD for the same cases shown in Fig. 1. Bottom row: Results from the second stage of the imaging algorithm in which δD quantities are recovored over the support results from the first stage by a least-squares approximation. Here, we have used a 2 GHz system bandwidth.
Fig. 4
Fig. 4 Reconstructions of δD using a broader bandwidth of 2 THz with sampling rate 0.13 THz. The left plot shows the minimum 2-norm solution using all source-detector pairs. The middle plot shows the minimum 2-norm solution using null source-detector separation data. The right plot shows the MUSIC + least-squares reconstruction.

Equations (35)

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1 c I t + s ^ I + μ a I + μ s I = μ s 4 π f ( s ^ s ^ ) I ( s ^ , r , t ) d s ^
I | t = 0 = 0 ,
I | z = 0 = T 0 δ ( s ^ z ^ ) h ( x , y , t ) + R [ I ]
r ( x , y , t ) = NA T out [ I ] ( s ^ , x , y , 0 , t ) s ^ z ^ d s ^ .
1 c u t + μ a u ( D u ) = 0 ,
r ~ r 0 h + r 1 n u | z = 0 ,
u | t = 0 = 0 ,
u | z = 0 = c 0 h ( x , y , t ) .
r ˜ = n u | z = 0 ,
U ( r , ω ) = 1 2 π u ( r , t ) e i ω t d t ,
( D U ) ( μ a + i ω / c ) U = 0
U | z = 0 = c 0 H ( x , y , ω ) ,
2 U 0 k 2 U 0 = 0
2 ( U U 0 ) k 2 ( U U 0 ) = D 0 1 ( δ D U ) ,
( U U 0 ) | z = 0 = 0 .
2 G k 2 G = D 0 1 δ ( r r )
G | z = 0 = 0 .
G ( r , r ) = e k r 4 π D 0 r e k r * 4 π D 0 r * ,
U U 0 = z > 0 G ( r , r ) [ ( δ D U ) ] d r .
U U 0 + z > 0 G ( r , r ) [ ( δ D U 0 ) ] d r .
R ˜ z U 0 | z = 0 z > 0 [ G z ( r , r ) U 0 ] δ D d r ,
2 G ˜ k 2 G ˜ = 0
G ˜ | z = 0 = δ ( x x ) δ ( y y ) .
R ˜ z U 0 | z = 0 + D 0 z > 0 [ G z ( r d , r ) G z ( r , r s ) ] δ D d r .
𝒜 s x = b s
𝒜 s ( i + ( l 1 ) N d , n ) = G z ( r i , r n ; ω l ) G z ( r n , r s ; ω l )
b l = z > 0 G z ( r d , r ; ω l ) G z ( r , r s ; ω l ) δ D ( r ) d r ,
b l n = 1 N G z z ( z 0 ; z n ; ω l ) G z z ( z n , z 0 ; ω l ) δ D n Δ z ,
G ( z , z ; ω l ) = e k l | z z | 4 π D 0 | z z | e k l | z + z | 4 π D 0 | z + z |
𝒜 x = b ,
𝒜 ( l , n ) = G z z 2 ( z 0 , z n ; ω l )
= ( b 1 b 2 b S b 2 b 3 b S + 1 b S b S + 1 b 2 S )
= U Σ V * = j = 1 S σ j u j v j * .
n MUSIC = a ˜ n 2 j = M + 1 N | a ˜ n * u j | 2 , n = 1 , , N ,
a ˜ n = [ G z z 2 ( z 0 , z n , ω 1 ) , G z z 2 ( z 0 , z n , ω 2 ) , , G z z 2 ( z 0 , z n , ω S ) ] t .

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