Abstract

The forward model in diffuse optical tomography (DOT) describes how light propagates through a turbid medium. It is often approximated by a diffusion equation (DE) that is numerically discretized by the classical finite element method (FEM). We propose a nonlocal diffusion equation (NDE) as a new forward model for DOT, the discretization of which is carried out with an efficient graph-based numerical method (GNM). To quantitatively evaluate the new forward model, we first conduct experiments on a homogeneous slab, where the numerical accuracy of both NDE and DE is compared against the existing analytical solution. We further evaluate NDE by comparing its image reconstruction performance (inverse problem) to that of DE. Our experiments show that NDE is quantitatively comparable to DE and is up to 64% faster due to the efficient graph-based representation that can be implemented identically for geometries in different dimensions. The proposed discretization method can be easily applied to other imaging techniques like diffuse correlation spectroscopy which are normally modeled by the diffusion equation.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]

2019 (1)

2018 (1)

2017 (1)

H. Zhang, D. Zeng, H. Zhang, J. Wang, Z. R. Liang, and J. H. Ma, “Applications of nonlocal means algorithm in low-dose X-ray CT image processing and reconstruction: A review,” Med. Phys. 44(3), 1168–1185 (2017).
[Crossref]

2016 (2)

W. Q. Lu, J. M. Duan, Z. W. Qiu, Z. K. Pan, Q. W. Liu, and L. Bai, “Implementation of high-order variational models made easy for image processing,” Math. Meth. Appl. Sci. 39(14), 4208–4233 (2016).
[Crossref]

M. Bhatt, K. R. Ayyalasomayajula, and P. K. Yalavarthy, “Generalized Beer–Lambert model for near-infrared light propagation in thick biological tissues,” J. Biomed. Opt. 21(7), 076012 (2016).
[Crossref]

2015 (1)

J. M. Duan, Z. K. Pan, B. C. Zhang, W. Q. Liu, and X. C. Tai, “Fast algorithm for color texture image inpainting using the non-local CTV model,” J Glob Optim 62(4), 853–876 (2015).
[Crossref]

2014 (2)

X. Bresson, X. C. Tai, T. F. Chan, and A. Szlam, “Multi-class transductive learning based on l1 relaxations of Cheeger cut and Mumford-Shah-Potts model,” J Math Imaging Vis 49(1), 191–201 (2014).
[Crossref]

W. B. Baker, A. B. Parthasarathy, D. R. Busch, R. C. Mesquita, H. Joel, and A. G. Yodh, “Modified Beer-Lambert law for blood flow,” Biomed. Opt. Express 5(11), 4053–4075 (2014).
[Crossref]

2013 (2)

J. M. Duan, Z. K. Pan, W. Q. Liu, and X. C. Tai, “Color texture image inpainting using the non local CTV model,” JSIP 04(03), 43–51 (2013).
[Crossref]

E. Merkurjev, T. Kostic, and A. L. Bertozzi, “An MBO scheme on graphs for classification and image processing,” SIAM J. Imaging Sci. 6(4), 1903–1930 (2013).
[Crossref]

2012 (2)

A. L. Bertozzi and A. Flenner, “Diffuse interface models on graphs for classification of high dimensional data,” Multiscale Model. Simul. 10(3), 1090–1118 (2012).
[Crossref]

A. T. Eggebrecht, B. R. White, S. L. Ferradal, C. X. Chen, Y. X. Zhan, A. Z. Snyder, H. Dehghani, and J. P. Culver, “A quantitative spatial comparison of high-density diffuse optical tomography and fmri cortical mapping,” NeuroImage 61(4), 1120–1128 (2012).
[Crossref]

2010 (2)

M. Gunzburger and R. B. Lehoucq, “A nonlocal vector calculus with application to nonlocal boundary value problems,” Multiscale Model. Simul. 8(5), 1581–1598 (2010).
[Crossref]

A. D. Klose, “The forward and inverse problem in tissue optics based on the radiative transfer equation: a brief review,” J. Quant. Spectrosc. Radiat. Transfer 111(11), 1852–1853 (2010).
[Crossref]

2009 (5)

2006 (1)

L. Kocsis, P. Herman, and A. Eke, “The modified Beer–Lambert law revisited,” Phys. Med. Biol. 51(5), N91–N98 (2006).
[Crossref]

2004 (1)

D. A. Boas, A. M. Dale, and M. A. Franceschini, “Diffuse optical imaging of brain activation: approaches to optimizing image sensitivity, resolution, and accuracy,” NeuroImage 23, S275–S288 (2004).
[Crossref]

2001 (1)

D. A. Boas, T. Gaudette, G. Strangman, X. F. Cheng, J. J. A.. Marota, and J. B. Mandeville, “The accuracy of near infrared spectroscopy and imaging during focal changes in cerebral hemodynamics,” NeuroImage 13(1), 76–90 (2001).
[Crossref]

1999 (1)

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15(2), R41–R93 (1999).
[Crossref]

1998 (1)

1995 (1)

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22(11), 1779–1792 (1995).
[Crossref]

1992 (2)

T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19(4), 879–888 (1992).
[Crossref]

S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37(7), 1531–1560 (1992).
[Crossref]

1991 (1)

V. Allen and A. L. McKenzie, “The modified diffusion dipole model,” Phys. Med. Biol. 36(12), 1621–1638 (1991).
[Crossref]

1983 (1)

B. C. Wilson and G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10(6), 824–830 (1983).
[Crossref]

1978 (1)

G. Eason, A. R. Veitch, R. M. Nisbet, and F. W. Turnbull, “The theory of the back-scattering of light by blood,” J. Phys. D: Appl. Phys. 11(10), 1463–1479 (1978).
[Crossref]

1971 (1)

W. H. Reed, “New difference schemes for the neutron transport equation,” Nucl. Sci. Eng. 46(2), 309–314 (1971).
[Crossref]

Adam, G.

B. C. Wilson and G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10(6), 824–830 (1983).
[Crossref]

Allen, V.

V. Allen and A. L. McKenzie, “The modified diffusion dipole model,” Phys. Med. Biol. 36(12), 1621–1638 (1991).
[Crossref]

Arridge, S. R.

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15(2), R41–R93 (1999).
[Crossref]

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22(11), 1779–1792 (1995).
[Crossref]

S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37(7), 1531–1560 (1992).
[Crossref]

Ayyalasomayajula, K. R.

M. Bhatt, K. R. Ayyalasomayajula, and P. K. Yalavarthy, “Generalized Beer–Lambert model for near-infrared light propagation in thick biological tissues,” J. Biomed. Opt. 21(7), 076012 (2016).
[Crossref]

Bai, L.

W. Q. Lu, J. M. Duan, Z. W. Qiu, Z. K. Pan, Q. W. Liu, and L. Bai, “Implementation of high-order variational models made easy for image processing,” Math. Meth. Appl. Sci. 39(14), 4208–4233 (2016).
[Crossref]

Baker, W. B.

Bays, R.

Bertozzi, A. L.

E. Merkurjev, T. Kostic, and A. L. Bertozzi, “An MBO scheme on graphs for classification and image processing,” SIAM J. Imaging Sci. 6(4), 1903–1930 (2013).
[Crossref]

A. L. Bertozzi and A. Flenner, “Diffuse interface models on graphs for classification of high dimensional data,” Multiscale Model. Simul. 10(3), 1090–1118 (2012).
[Crossref]

Bhatt, M.

M. Bhatt, K. R. Ayyalasomayajula, and P. K. Yalavarthy, “Generalized Beer–Lambert model for near-infrared light propagation in thick biological tissues,” J. Biomed. Opt. 21(7), 076012 (2016).
[Crossref]

Boas, D. A.

D. A. Boas, A. M. Dale, and M. A. Franceschini, “Diffuse optical imaging of brain activation: approaches to optimizing image sensitivity, resolution, and accuracy,” NeuroImage 23, S275–S288 (2004).
[Crossref]

D. A. Boas, T. Gaudette, G. Strangman, X. F. Cheng, J. J. A.. Marota, and J. B. Mandeville, “The accuracy of near infrared spectroscopy and imaging during focal changes in cerebral hemodynamics,” NeuroImage 13(1), 76–90 (2001).
[Crossref]

D. A. Boas, “Diffuse photon probes of structural and dynamical properties of turbid media: theory and biomedical applications,” Doctoral dissertation, Graduate School of Arts and Sciences, University of Pennsylvania (1996).

Bresson, X.

X. Bresson, X. C. Tai, T. F. Chan, and A. Szlam, “Multi-class transductive learning based on l1 relaxations of Cheeger cut and Mumford-Shah-Potts model,” J Math Imaging Vis 49(1), 191–201 (2014).
[Crossref]

Buades, A.

A. Buades, B. Coll, and J. M. Morel, “A non-local algorithm for image denoising,” 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR’05)2, 60–65 (2005).

Busch, D. R.

Carpenter, C. M.

H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using NIRFAST: Algorithm for numerical model and image reconstruction,” Commun. Numer. Meth. Engng. 25(6), 711–732 (2009).
[Crossref]

Chan, T. F.

X. Bresson, X. C. Tai, T. F. Chan, and A. Szlam, “Multi-class transductive learning based on l1 relaxations of Cheeger cut and Mumford-Shah-Potts model,” J Math Imaging Vis 49(1), 191–201 (2014).
[Crossref]

Chen, C. X.

A. T. Eggebrecht, B. R. White, S. L. Ferradal, C. X. Chen, Y. X. Zhan, A. Z. Snyder, H. Dehghani, and J. P. Culver, “A quantitative spatial comparison of high-density diffuse optical tomography and fmri cortical mapping,” NeuroImage 61(4), 1120–1128 (2012).
[Crossref]

Cheng, X. F.

D. A. Boas, T. Gaudette, G. Strangman, X. F. Cheng, J. J. A.. Marota, and J. B. Mandeville, “The accuracy of near infrared spectroscopy and imaging during focal changes in cerebral hemodynamics,” NeuroImage 13(1), 76–90 (2001).
[Crossref]

Coll, B.

A. Buades, B. Coll, and J. M. Morel, “A non-local algorithm for image denoising,” 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR’05)2, 60–65 (2005).

Cope, M.

S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37(7), 1531–1560 (1992).
[Crossref]

Culver, J. P.

A. T. Eggebrecht, B. R. White, S. L. Ferradal, C. X. Chen, Y. X. Zhan, A. Z. Snyder, H. Dehghani, and J. P. Culver, “A quantitative spatial comparison of high-density diffuse optical tomography and fmri cortical mapping,” NeuroImage 61(4), 1120–1128 (2012).
[Crossref]

H. Dehghani, B. R. White, B. W. Zeff, A. Tizzard, and J. P. Culver, “Depth sensitivity and image reconstruction analysis of dense imaging arrays for mapping brain function with diffuse optical tomography,” Appl. Opt. 48(10), D137–D143 (2009).
[Crossref]

Dale, A. M.

D. A. Boas, A. M. Dale, and M. A. Franceschini, “Diffuse optical imaging of brain activation: approaches to optimizing image sensitivity, resolution, and accuracy,” NeuroImage 23, S275–S288 (2004).
[Crossref]

Davis, S. C.

H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using NIRFAST: Algorithm for numerical model and image reconstruction,” Commun. Numer. Meth. Engng. 25(6), 711–732 (2009).
[Crossref]

Dehghani, H.

A. T. Eggebrecht, B. R. White, S. L. Ferradal, C. X. Chen, Y. X. Zhan, A. Z. Snyder, H. Dehghani, and J. P. Culver, “A quantitative spatial comparison of high-density diffuse optical tomography and fmri cortical mapping,” NeuroImage 61(4), 1120–1128 (2012).
[Crossref]

H. Dehghani, B. R. White, B. W. Zeff, A. Tizzard, and J. P. Culver, “Depth sensitivity and image reconstruction analysis of dense imaging arrays for mapping brain function with diffuse optical tomography,” Appl. Opt. 48(10), D137–D143 (2009).
[Crossref]

H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using NIRFAST: Algorithm for numerical model and image reconstruction,” Commun. Numer. Meth. Engng. 25(6), 711–732 (2009).
[Crossref]

Delpy, D. T.

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22(11), 1779–1792 (1995).
[Crossref]

S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37(7), 1531–1560 (1992).
[Crossref]

Dögnitz, N.

Duan, J. M.

W. Q. Lu, J. M. Duan, D. Orive-Miguel, L. Herve, and I. B. Styles, “Graph-and finite element-based total variation models for the inverse problem in diffuse optical tomography,” Biomed. Opt. Express 10(6), 2684–2707 (2019).
[Crossref]

W. Q. Lu, J. M. Duan, Z. W. Qiu, Z. K. Pan, Q. W. Liu, and L. Bai, “Implementation of high-order variational models made easy for image processing,” Math. Meth. Appl. Sci. 39(14), 4208–4233 (2016).
[Crossref]

J. M. Duan, Z. K. Pan, B. C. Zhang, W. Q. Liu, and X. C. Tai, “Fast algorithm for color texture image inpainting using the non-local CTV model,” J Glob Optim 62(4), 853–876 (2015).
[Crossref]

J. M. Duan, Z. K. Pan, W. Q. Liu, and X. C. Tai, “Color texture image inpainting using the non local CTV model,” JSIP 04(03), 43–51 (2013).
[Crossref]

Duderstadt, J. J.

J. J. Duderstadt and W. R. Martin, Transport Theory (John Wiley & Sons Chichester, 1979).

Eames, M. E.

H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using NIRFAST: Algorithm for numerical model and image reconstruction,” Commun. Numer. Meth. Engng. 25(6), 711–732 (2009).
[Crossref]

Eason, G.

G. Eason, A. R. Veitch, R. M. Nisbet, and F. W. Turnbull, “The theory of the back-scattering of light by blood,” J. Phys. D: Appl. Phys. 11(10), 1463–1479 (1978).
[Crossref]

Eggebrecht, A. T.

A. T. Eggebrecht, B. R. White, S. L. Ferradal, C. X. Chen, Y. X. Zhan, A. Z. Snyder, H. Dehghani, and J. P. Culver, “A quantitative spatial comparison of high-density diffuse optical tomography and fmri cortical mapping,” NeuroImage 61(4), 1120–1128 (2012).
[Crossref]

Eke, A.

L. Kocsis, P. Herman, and A. Eke, “The modified Beer–Lambert law revisited,” Phys. Med. Biol. 51(5), N91–N98 (2006).
[Crossref]

Farrell, T. J.

T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19(4), 879–888 (1992).
[Crossref]

Ferradal, S. L.

A. T. Eggebrecht, B. R. White, S. L. Ferradal, C. X. Chen, Y. X. Zhan, A. Z. Snyder, H. Dehghani, and J. P. Culver, “A quantitative spatial comparison of high-density diffuse optical tomography and fmri cortical mapping,” NeuroImage 61(4), 1120–1128 (2012).
[Crossref]

Flenner, A.

A. L. Bertozzi and A. Flenner, “Diffuse interface models on graphs for classification of high dimensional data,” Multiscale Model. Simul. 10(3), 1090–1118 (2012).
[Crossref]

Franceschini, M. A.

D. A. Boas, A. M. Dale, and M. A. Franceschini, “Diffuse optical imaging of brain activation: approaches to optimizing image sensitivity, resolution, and accuracy,” NeuroImage 23, S275–S288 (2004).
[Crossref]

Gaudette, T.

D. A. Boas, T. Gaudette, G. Strangman, X. F. Cheng, J. J. A.. Marota, and J. B. Mandeville, “The accuracy of near infrared spectroscopy and imaging during focal changes in cerebral hemodynamics,” NeuroImage 13(1), 76–90 (2001).
[Crossref]

Gilboa, G.

G. Gilboa and S. Osher, “Nonlocal operators with applications to image processing,” Multiscale Model. Simul. 7(3), 1005–1028 (2009).
[Crossref]

González-Rodríguez, P.

Grant, P. E.

Gunzburger, M.

M. Gunzburger and R. B. Lehoucq, “A nonlocal vector calculus with application to nonlocal boundary value problems,” Multiscale Model. Simul. 8(5), 1581–1598 (2010).
[Crossref]

Heiskala, J.

Herman, P.

L. Kocsis, P. Herman, and A. Eke, “The modified Beer–Lambert law revisited,” Phys. Med. Biol. 51(5), N91–N98 (2006).
[Crossref]

Herve, L.

Hiraoka, M.

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22(11), 1779–1792 (1995).
[Crossref]

Ichai, C.

C. Ichai, H. Quintard, and J. C. Orban, Metabolic Disorders and Critically Ill Patients: From Pathophysiology to Treatment (Springer, 2017).

Joel, H.

Kienle, A.

Kim, A. D.

Klose, A. D.

A. D. Klose, “The forward and inverse problem in tissue optics based on the radiative transfer equation: a brief review,” J. Quant. Spectrosc. Radiat. Transfer 111(11), 1852–1853 (2010).
[Crossref]

Kocsis, L.

L. Kocsis, P. Herman, and A. Eke, “The modified Beer–Lambert law revisited,” Phys. Med. Biol. 51(5), N91–N98 (2006).
[Crossref]

Kostic, T.

E. Merkurjev, T. Kostic, and A. L. Bertozzi, “An MBO scheme on graphs for classification and image processing,” SIAM J. Imaging Sci. 6(4), 1903–1930 (2013).
[Crossref]

Lehoucq, R. B.

M. Gunzburger and R. B. Lehoucq, “A nonlocal vector calculus with application to nonlocal boundary value problems,” Multiscale Model. Simul. 8(5), 1581–1598 (2010).
[Crossref]

Liang, Z. R.

H. Zhang, D. Zeng, H. Zhang, J. Wang, Z. R. Liang, and J. H. Ma, “Applications of nonlocal means algorithm in low-dose X-ray CT image processing and reconstruction: A review,” Med. Phys. 44(3), 1168–1185 (2017).
[Crossref]

Lighter, D.

Liu, Q. W.

W. Q. Lu, J. M. Duan, Z. W. Qiu, Z. K. Pan, Q. W. Liu, and L. Bai, “Implementation of high-order variational models made easy for image processing,” Math. Meth. Appl. Sci. 39(14), 4208–4233 (2016).
[Crossref]

Liu, W. Q.

J. M. Duan, Z. K. Pan, B. C. Zhang, W. Q. Liu, and X. C. Tai, “Fast algorithm for color texture image inpainting using the non-local CTV model,” J Glob Optim 62(4), 853–876 (2015).
[Crossref]

J. M. Duan, Z. K. Pan, W. Q. Liu, and X. C. Tai, “Color texture image inpainting using the non local CTV model,” JSIP 04(03), 43–51 (2013).
[Crossref]

Lu, W. Q.

Lynch, D. R.

D. R. Lynch, Numerical Partial Differential Equations for Environmental Scientists and Engineers: A First Practical Course (Springer Science & Business Media, 2004).

Ma, J. H.

H. Zhang, D. Zeng, H. Zhang, J. Wang, Z. R. Liang, and J. H. Ma, “Applications of nonlocal means algorithm in low-dose X-ray CT image processing and reconstruction: A review,” Med. Phys. 44(3), 1168–1185 (2017).
[Crossref]

Mandeville, J. B.

D. A. Boas, T. Gaudette, G. Strangman, X. F. Cheng, J. J. A.. Marota, and J. B. Mandeville, “The accuracy of near infrared spectroscopy and imaging during focal changes in cerebral hemodynamics,” NeuroImage 13(1), 76–90 (2001).
[Crossref]

Marota, J. J. A..

D. A. Boas, T. Gaudette, G. Strangman, X. F. Cheng, J. J. A.. Marota, and J. B. Mandeville, “The accuracy of near infrared spectroscopy and imaging during focal changes in cerebral hemodynamics,” NeuroImage 13(1), 76–90 (2001).
[Crossref]

Martin, W. R.

J. J. Duderstadt and W. R. Martin, Transport Theory (John Wiley & Sons Chichester, 1979).

McKenzie, A. L.

V. Allen and A. L. McKenzie, “The modified diffusion dipole model,” Phys. Med. Biol. 36(12), 1621–1638 (1991).
[Crossref]

Merkurjev, E.

E. Merkurjev, T. Kostic, and A. L. Bertozzi, “An MBO scheme on graphs for classification and image processing,” SIAM J. Imaging Sci. 6(4), 1903–1930 (2013).
[Crossref]

Mesquita, R. C.

Metsäranta, M.

Morel, J. M.

A. Buades, B. Coll, and J. M. Morel, “A non-local algorithm for image denoising,” 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR’05)2, 60–65 (2005).

Nisbet, R. M.

G. Eason, A. R. Veitch, R. M. Nisbet, and F. W. Turnbull, “The theory of the back-scattering of light by blood,” J. Phys. D: Appl. Phys. 11(10), 1463–1479 (1978).
[Crossref]

Nissilä, I.

Orban, J. C.

C. Ichai, H. Quintard, and J. C. Orban, Metabolic Disorders and Critically Ill Patients: From Pathophysiology to Treatment (Springer, 2017).

Orive-Miguel, D.

Osher, S.

G. Gilboa and S. Osher, “Nonlocal operators with applications to image processing,” Multiscale Model. Simul. 7(3), 1005–1028 (2009).
[Crossref]

Pan, Z. K.

W. Q. Lu, J. M. Duan, Z. W. Qiu, Z. K. Pan, Q. W. Liu, and L. Bai, “Implementation of high-order variational models made easy for image processing,” Math. Meth. Appl. Sci. 39(14), 4208–4233 (2016).
[Crossref]

J. M. Duan, Z. K. Pan, B. C. Zhang, W. Q. Liu, and X. C. Tai, “Fast algorithm for color texture image inpainting using the non-local CTV model,” J Glob Optim 62(4), 853–876 (2015).
[Crossref]

J. M. Duan, Z. K. Pan, W. Q. Liu, and X. C. Tai, “Color texture image inpainting using the non local CTV model,” JSIP 04(03), 43–51 (2013).
[Crossref]

Parthasarathy, A. B.

Patterson, M. S.

A. Kienle, M. S. Patterson, N. Dögnitz, R. Bays, G. Wagnières, and H. van Den Bergh, “Noninvasive determination of the optical properties of two-layered turbid media,” Appl. Opt. 37(4), 779–791 (1998).
[Crossref]

T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19(4), 879–888 (1992).
[Crossref]

Paulsen, K. D.

H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using NIRFAST: Algorithm for numerical model and image reconstruction,” Commun. Numer. Meth. Engng. 25(6), 711–732 (2009).
[Crossref]

Pogue, B. W.

H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using NIRFAST: Algorithm for numerical model and image reconstruction,” Commun. Numer. Meth. Engng. 25(6), 711–732 (2009).
[Crossref]

Pollari, M.

Qiu, Z. W.

W. Q. Lu, J. M. Duan, Z. W. Qiu, Z. K. Pan, Q. W. Liu, and L. Bai, “Implementation of high-order variational models made easy for image processing,” Math. Meth. Appl. Sci. 39(14), 4208–4233 (2016).
[Crossref]

Quintard, H.

C. Ichai, H. Quintard, and J. C. Orban, Metabolic Disorders and Critically Ill Patients: From Pathophysiology to Treatment (Springer, 2017).

Reed, W. H.

W. H. Reed, “New difference schemes for the neutron transport equation,” Nucl. Sci. Eng. 46(2), 309–314 (1971).
[Crossref]

Schweiger, M.

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22(11), 1779–1792 (1995).
[Crossref]

Snyder, A. Z.

A. T. Eggebrecht, B. R. White, S. L. Ferradal, C. X. Chen, Y. X. Zhan, A. Z. Snyder, H. Dehghani, and J. P. Culver, “A quantitative spatial comparison of high-density diffuse optical tomography and fmri cortical mapping,” NeuroImage 61(4), 1120–1128 (2012).
[Crossref]

Srinivasan, S.

H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using NIRFAST: Algorithm for numerical model and image reconstruction,” Commun. Numer. Meth. Engng. 25(6), 711–732 (2009).
[Crossref]

Strangman, G.

D. A. Boas, T. Gaudette, G. Strangman, X. F. Cheng, J. J. A.. Marota, and J. B. Mandeville, “The accuracy of near infrared spectroscopy and imaging during focal changes in cerebral hemodynamics,” NeuroImage 13(1), 76–90 (2001).
[Crossref]

Styles, I. B.

Szlam, A.

X. Bresson, X. C. Tai, T. F. Chan, and A. Szlam, “Multi-class transductive learning based on l1 relaxations of Cheeger cut and Mumford-Shah-Potts model,” J Math Imaging Vis 49(1), 191–201 (2014).
[Crossref]

Tai, X. C.

J. M. Duan, Z. K. Pan, B. C. Zhang, W. Q. Liu, and X. C. Tai, “Fast algorithm for color texture image inpainting using the non-local CTV model,” J Glob Optim 62(4), 853–876 (2015).
[Crossref]

X. Bresson, X. C. Tai, T. F. Chan, and A. Szlam, “Multi-class transductive learning based on l1 relaxations of Cheeger cut and Mumford-Shah-Potts model,” J Math Imaging Vis 49(1), 191–201 (2014).
[Crossref]

J. M. Duan, Z. K. Pan, W. Q. Liu, and X. C. Tai, “Color texture image inpainting using the non local CTV model,” JSIP 04(03), 43–51 (2013).
[Crossref]

Tizzard, A.

Turnbull, F. W.

G. Eason, A. R. Veitch, R. M. Nisbet, and F. W. Turnbull, “The theory of the back-scattering of light by blood,” J. Phys. D: Appl. Phys. 11(10), 1463–1479 (1978).
[Crossref]

van Den Bergh, H.

Veitch, A. R.

G. Eason, A. R. Veitch, R. M. Nisbet, and F. W. Turnbull, “The theory of the back-scattering of light by blood,” J. Phys. D: Appl. Phys. 11(10), 1463–1479 (1978).
[Crossref]

Wagnières, G.

Wang, J.

H. Zhang, D. Zeng, H. Zhang, J. Wang, Z. R. Liang, and J. H. Ma, “Applications of nonlocal means algorithm in low-dose X-ray CT image processing and reconstruction: A review,” Med. Phys. 44(3), 1168–1185 (2017).
[Crossref]

White, B. R.

A. T. Eggebrecht, B. R. White, S. L. Ferradal, C. X. Chen, Y. X. Zhan, A. Z. Snyder, H. Dehghani, and J. P. Culver, “A quantitative spatial comparison of high-density diffuse optical tomography and fmri cortical mapping,” NeuroImage 61(4), 1120–1128 (2012).
[Crossref]

H. Dehghani, B. R. White, B. W. Zeff, A. Tizzard, and J. P. Culver, “Depth sensitivity and image reconstruction analysis of dense imaging arrays for mapping brain function with diffuse optical tomography,” Appl. Opt. 48(10), D137–D143 (2009).
[Crossref]

Wilson, B.

T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19(4), 879–888 (1992).
[Crossref]

Wilson, B. C.

B. C. Wilson and G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10(6), 824–830 (1983).
[Crossref]

Yalavarthy, P. K.

M. Bhatt, K. R. Ayyalasomayajula, and P. K. Yalavarthy, “Generalized Beer–Lambert model for near-infrared light propagation in thick biological tissues,” J. Biomed. Opt. 21(7), 076012 (2016).
[Crossref]

H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using NIRFAST: Algorithm for numerical model and image reconstruction,” Commun. Numer. Meth. Engng. 25(6), 711–732 (2009).
[Crossref]

Yodh, A. G.

Zeff, B. W.

Zeng, D.

H. Zhang, D. Zeng, H. Zhang, J. Wang, Z. R. Liang, and J. H. Ma, “Applications of nonlocal means algorithm in low-dose X-ray CT image processing and reconstruction: A review,” Med. Phys. 44(3), 1168–1185 (2017).
[Crossref]

Zhan, Y. X.

A. T. Eggebrecht, B. R. White, S. L. Ferradal, C. X. Chen, Y. X. Zhan, A. Z. Snyder, H. Dehghani, and J. P. Culver, “A quantitative spatial comparison of high-density diffuse optical tomography and fmri cortical mapping,” NeuroImage 61(4), 1120–1128 (2012).
[Crossref]

Zhang, B. C.

J. M. Duan, Z. K. Pan, B. C. Zhang, W. Q. Liu, and X. C. Tai, “Fast algorithm for color texture image inpainting using the non-local CTV model,” J Glob Optim 62(4), 853–876 (2015).
[Crossref]

Zhang, H.

H. Zhang, D. Zeng, H. Zhang, J. Wang, Z. R. Liang, and J. H. Ma, “Applications of nonlocal means algorithm in low-dose X-ray CT image processing and reconstruction: A review,” Med. Phys. 44(3), 1168–1185 (2017).
[Crossref]

H. Zhang, D. Zeng, H. Zhang, J. Wang, Z. R. Liang, and J. H. Ma, “Applications of nonlocal means algorithm in low-dose X-ray CT image processing and reconstruction: A review,” Med. Phys. 44(3), 1168–1185 (2017).
[Crossref]

Appl. Opt. (2)

Biomed. Opt. Express (3)

Commun. Numer. Meth. Engng. (1)

H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using NIRFAST: Algorithm for numerical model and image reconstruction,” Commun. Numer. Meth. Engng. 25(6), 711–732 (2009).
[Crossref]

Inverse Probl. (1)

S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15(2), R41–R93 (1999).
[Crossref]

J Glob Optim (1)

J. M. Duan, Z. K. Pan, B. C. Zhang, W. Q. Liu, and X. C. Tai, “Fast algorithm for color texture image inpainting using the non-local CTV model,” J Glob Optim 62(4), 853–876 (2015).
[Crossref]

J Math Imaging Vis (1)

X. Bresson, X. C. Tai, T. F. Chan, and A. Szlam, “Multi-class transductive learning based on l1 relaxations of Cheeger cut and Mumford-Shah-Potts model,” J Math Imaging Vis 49(1), 191–201 (2014).
[Crossref]

J. Biomed. Opt. (1)

M. Bhatt, K. R. Ayyalasomayajula, and P. K. Yalavarthy, “Generalized Beer–Lambert model for near-infrared light propagation in thick biological tissues,” J. Biomed. Opt. 21(7), 076012 (2016).
[Crossref]

J. Phys. D: Appl. Phys. (1)

G. Eason, A. R. Veitch, R. M. Nisbet, and F. W. Turnbull, “The theory of the back-scattering of light by blood,” J. Phys. D: Appl. Phys. 11(10), 1463–1479 (1978).
[Crossref]

J. Quant. Spectrosc. Radiat. Transfer (1)

A. D. Klose, “The forward and inverse problem in tissue optics based on the radiative transfer equation: a brief review,” J. Quant. Spectrosc. Radiat. Transfer 111(11), 1852–1853 (2010).
[Crossref]

JSIP (1)

J. M. Duan, Z. K. Pan, W. Q. Liu, and X. C. Tai, “Color texture image inpainting using the non local CTV model,” JSIP 04(03), 43–51 (2013).
[Crossref]

Math. Meth. Appl. Sci. (1)

W. Q. Lu, J. M. Duan, Z. W. Qiu, Z. K. Pan, Q. W. Liu, and L. Bai, “Implementation of high-order variational models made easy for image processing,” Math. Meth. Appl. Sci. 39(14), 4208–4233 (2016).
[Crossref]

Med. Phys. (4)

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. 22(11), 1779–1792 (1995).
[Crossref]

H. Zhang, D. Zeng, H. Zhang, J. Wang, Z. R. Liang, and J. H. Ma, “Applications of nonlocal means algorithm in low-dose X-ray CT image processing and reconstruction: A review,” Med. Phys. 44(3), 1168–1185 (2017).
[Crossref]

B. C. Wilson and G. Adam, “A Monte Carlo model for the absorption and flux distributions of light in tissue,” Med. Phys. 10(6), 824–830 (1983).
[Crossref]

T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19(4), 879–888 (1992).
[Crossref]

Multiscale Model. Simul. (3)

A. L. Bertozzi and A. Flenner, “Diffuse interface models on graphs for classification of high dimensional data,” Multiscale Model. Simul. 10(3), 1090–1118 (2012).
[Crossref]

G. Gilboa and S. Osher, “Nonlocal operators with applications to image processing,” Multiscale Model. Simul. 7(3), 1005–1028 (2009).
[Crossref]

M. Gunzburger and R. B. Lehoucq, “A nonlocal vector calculus with application to nonlocal boundary value problems,” Multiscale Model. Simul. 8(5), 1581–1598 (2010).
[Crossref]

NeuroImage (3)

A. T. Eggebrecht, B. R. White, S. L. Ferradal, C. X. Chen, Y. X. Zhan, A. Z. Snyder, H. Dehghani, and J. P. Culver, “A quantitative spatial comparison of high-density diffuse optical tomography and fmri cortical mapping,” NeuroImage 61(4), 1120–1128 (2012).
[Crossref]

D. A. Boas, A. M. Dale, and M. A. Franceschini, “Diffuse optical imaging of brain activation: approaches to optimizing image sensitivity, resolution, and accuracy,” NeuroImage 23, S275–S288 (2004).
[Crossref]

D. A. Boas, T. Gaudette, G. Strangman, X. F. Cheng, J. J. A.. Marota, and J. B. Mandeville, “The accuracy of near infrared spectroscopy and imaging during focal changes in cerebral hemodynamics,” NeuroImage 13(1), 76–90 (2001).
[Crossref]

Nucl. Sci. Eng. (1)

W. H. Reed, “New difference schemes for the neutron transport equation,” Nucl. Sci. Eng. 46(2), 309–314 (1971).
[Crossref]

Opt. Express (2)

Phys. Med. Biol. (3)

L. Kocsis, P. Herman, and A. Eke, “The modified Beer–Lambert law revisited,” Phys. Med. Biol. 51(5), N91–N98 (2006).
[Crossref]

S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37(7), 1531–1560 (1992).
[Crossref]

V. Allen and A. L. McKenzie, “The modified diffusion dipole model,” Phys. Med. Biol. 36(12), 1621–1638 (1991).
[Crossref]

SIAM J. Imaging Sci. (1)

E. Merkurjev, T. Kostic, and A. L. Bertozzi, “An MBO scheme on graphs for classification and image processing,” SIAM J. Imaging Sci. 6(4), 1903–1930 (2013).
[Crossref]

Other (5)

C. Ichai, H. Quintard, and J. C. Orban, Metabolic Disorders and Critically Ill Patients: From Pathophysiology to Treatment (Springer, 2017).

D. R. Lynch, Numerical Partial Differential Equations for Environmental Scientists and Engineers: A First Practical Course (Springer Science & Business Media, 2004).

A. Buades, B. Coll, and J. M. Morel, “A non-local algorithm for image denoising,” 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR’05)2, 60–65 (2005).

D. A. Boas, “Diffuse photon probes of structural and dynamical properties of turbid media: theory and biomedical applications,” Doctoral dissertation, Graduate School of Arts and Sciences, University of Pennsylvania (1996).

J. J. Duderstadt and W. R. Martin, Transport Theory (John Wiley & Sons Chichester, 1979).

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Figures (10)

Fig. 1.
Fig. 1. Rectangular-slab mesh with one source (red dot) and six detectors (green dots). The distance between the source and the six detectors varies from 15 mm to 40 mm, in 5 mm increments.
Fig. 2.
Fig. 2. The flux measurements on the boundary versus the source-detector distance. (a): NBF; (b): Percentage of error based on NBF.
Fig. 3.
Fig. 3. (a)–(c): NFR at each vertex in the ROI calculated using the analytical solution, forward models based on FEM and the one based on GNM, respectively; (d)–(f): logarithm of the NFRs, corresponding to (a)–(c).
Fig. 4.
Fig. 4. Descending tendency of the NFR from the source to the medium along the z axis.
Fig. 5.
Fig. 5. Scheme of the heterogeneous rectangular-slab model.
Fig. 6.
Fig. 6. The flux measurements on the boundary versus the source-detector distance. (a): NBF; (b): Percentage of error based on NBF.
Fig. 7.
Fig. 7. CPU time (s) consumed at one source-detector channel using different forward models. ’A’ represents the FEM approach while ’B’ represents the GNM approach. Right figure is the zoomed-in plot of the area in the green dash line of the left figure.
Fig. 8.
Fig. 8. (a): A typical circle mesh with sixteen co-located sources and detectors; (b): True distribution of $\mu _a$; (c): Images reconstruction of $\mu _a$ using the forward model based on FEM and GNM (from left to right column) on $0\%$ (top part) and $1\%$ (bottom part) noisy data.
Fig. 9.
Fig. 9. 1D cross sections of images recovered in Fig. 8 along the horizontal line across the centre of the target. Left to right column: $0\%$ and $1\%$ added Gaussian noise.
Fig. 10.
Fig. 10. (a): Three-dimensional head mesh and distribution of the rectangular imaging array with 36 sources (red dots) and 37 detectors (green dots); (b): Ground truth; (c) Reconstruction with the forward model based on FEM and GNM, respectively.

Tables (2)

Tables Icon

Table 1. Evaluation metrics for the recovered results using FEM and GNM on data with 0% and 1% added noise.

Tables Icon

Table 2. Evaluation metrics for μ a on the recovered results shown in Figure 10.

Equations (15)

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

( κ ( x ) Φ ( x ) ) + μ a ( x ) Φ ( x ) = q 0 ( x ) for x Ω .
2 A n ^ ( κ ( x ) Φ ( x ) ) + Φ ( x ) = 0 for x Γ ,
d i v w ( κ ( x ) w Φ ( x ) ) + μ a ( x ) Φ ( x ) = q 0 ( x ) for x Ω .
2 A N w ( κ ( x ) w Φ ( x ) ) + Φ ( x ) = 0 for x Γ .
j Φ i ( Φ j Φ i ) w i j : V × V R .
w Φ i , j ( Φ j Φ i ) w i j : V × V R .
d i v w ν i j = 1 N ( ν i j ν j i ) w i j : V Ω R ,
N w ν i j = 1 N ( ν i j ν j i ) w i j : V Γ R .
Δ w Φ i 1 2 d i v w ( w Φ i ) = j = 1 N ( Φ j Φ i ) w i j : V R .
j N i ( κ i + κ j ) ( Φ i Φ j ) w i j + μ a i Φ i = q 0 i for i Ω 2 A j N i ( κ i + κ j ) ( Φ i Φ j ) w i j + Φ i = 0 for i Γ
M Φ = Q .
M i , j = { j N i ( κ i + κ j ) w i j + μ a i if i = j Ω j N i ( κ i + κ j ) w i j + 1 2 A if i = j Γ ( κ i + κ j ) w i j if i j and j N i 0 otherwise .
I ( ρ ) = 1 4 π [ 1 μ a + μ s ( μ e f f + 1 r 1 ) e μ e f f r 1 r 1 2 + 3 + 4 A 3 ( μ a + μ s ) ( μ e f f + 1 r 2 ) e μ e f f r 2 r 2 2 ] ,
Φ ( r , z ) = P μ e f f 2 4 π μ a [ ( exp { μ e f f [ ( z z 0 ) 2 + r 2 ] 1 / 2 } μ e f f [ ( z z 0 ) 2 + r 2 ] 1 / 2 ) ( exp { μ e f f [ ( z + z 0 ) 2 + r 2 ] 1 / 2 } μ e f f [ ( z + z 0 ) 2 + r 2 ] 1 / 2 ) ] ,
μ a = arg min μ a { Φ M F ( μ a ) 2 2 + λ R ( μ a ) } ,

Metrics