Abstract

We explore the development and performance of algorithms for hyperspectral diffuse optical tomography (DOT) for which data from hundreds of wavelengths are collected and used to determine the concentration distribution of chromophores in the medium under investigation. An efficient method is detailed for forming the images using iterative algorithms applied to a linearized Born approximation model assuming the scattering coefficient is spatially constant and known. The L-surface framework is employed to select optimal regularization parameters for the inverse problem. We report image reconstructions using 126 wavelengths with estimation error in simulations as low as 0.05 and mean square error of experimental data of 0.18 and 0.29 for ink and dye concentrations, respectively, an improvement over reconstructions using fewer specifically chosen wavelengths.

© 2011 OSA

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2010 (1)

S. Bélanger, M. Abran, X. Intes, C. Casanova, and F. Lesage, “Real-time diffuse optical tomography based on structured illumination,” J. Biomed. Opt. 15, 016006 (2010).
[CrossRef] [PubMed]

2009 (2)

B. Brendel and T. Nielsen, “Selection of optimal wavelengths for spectral reconstruction in diffuse optical tomography,” J. Biomed. Opt. 14, 1–10 (2009).
[CrossRef]

Q. Fang and D. a. Boas, “Monte Carlo simulation of photon migration in 3D turbid media accelerated by graphics processing units,” Opt. Express 17, 20178–20190 (2009).
[CrossRef] [PubMed]

2008 (3)

J. Wang, S. C. Davis, S. Srinivasan, S. Jiang, B. W. Pogue, and K. D. Paulsen, “Spectral tomography with diffuse near-infrared light: inclusion of broadband frequency domain spectral data,” J. Biomed. Opt. 13, 1–10 (2008).
[CrossRef]

S. D. Konecky, R. Choe, A. Corlu, K. Lee, R. Wiener, S. M. Srinivas, J. R. Saffer, R. Freifelder, J. S. Karp, N. Hajjioui, F. Azer, and A. G. Yodh, “Comparison of diffuse optical tomography of human breast with whole-body and breast-only positron emission tomography,” Med. Phys. 35, 446–455 (2008).
[CrossRef] [PubMed]

M. E. Eames, B. W. Pogue, and H. Dehghani, “Wavelength band optimization in spectral near-infrared optical tomography improves accuracy while reducing data acquisition and computational burden,” J. Biomed. Opt. 13, 1–9 (2008).
[CrossRef]

2007 (4)

N. Liu, A. Sassaroli, and S. Fantini, “Paired-wavelength spectral approach to measuring the relative concentrations of two localized chromophores in turbid media: an experimental study,” J. Biomed. Opt. 12, 05160 (2007).
[CrossRef]

G. Boverman, Q. Fang, S. A. Carp, E. L. Miller, D. H. Brooks, J. Selb, R. H. Moore, D. B. Kopans, and D. A. Boas, “Spatio-temporal imaging of the hemoglobin in the compressed breast with diffuse optical tomography,” Phys. Med. Biol. 52, 3619–3641 (2007).
[CrossRef] [PubMed]

P. K. Yalavarthy, B. W. Pogue, H. Dehghani, C. M. Carpenter, S. Jiang, and K. D. Paulsen, “Structural information within regularization matrices improves near infrared diffuse optical tomography,” Opt. Express 15, 8043–8058 (2007).
[CrossRef] [PubMed]

C. Li, S. R. Grobmyer, L. Chen, Q. Zhang, L. L. Fajardo, and H. Jiang, “Multispectral diffuse optical tomography with absorption and scattering spectral constraints,” Appl. Opt. 46, 8229–8236 (2007).
[CrossRef] [PubMed]

2006 (1)

B. Brooksby, B. W. Pogue, S. Jiang, H. Dehghani, S. Srinivasan, C. Kogel, T. D. Tosteson, J. Weaver, S. P. Poplack, and K. D. Paulsen, “Imaging breast adipose and fibroglandular tissue molecular signatures by using hybrid MRI-guided near-infrared spectral tomography,” Proc. Natl. Acad. Sci. U.S.A. 103, 8828–8833 (2006).
[CrossRef] [PubMed]

2005 (6)

M. Guven, B. Yazici, X. Intes, and B. Chance, “Diffuse optical tomography with a priori anatomical information,” Phys. Med. Biol. 50, 2837–2858 (2005).
[CrossRef] [PubMed]

S. Fantini, E. L. Heffer, V. E. Pera, A. Sassaroli, and N. Liu, “Spatial and spectral information in optical mammography,” Technol. Cancer Res. Treat. 4, 471–482 (2005).

G. Boverman, E. L. Miller, A. Li, Q. Zhang, T. Chaves, D. H. Brooks, and D. A. Boas, “Quantitative spectroscopic diffuse optical tomography of the breast guided by imperfect a priori structural information,” Phys. Med. Biol. 50, 3941–3956 (2005).
[CrossRef] [PubMed]

A. Li, G. Boverman, Y. Zhang, D. Brooks, E. L. Miller, M. E. Kilmer, Q. Zhang, E. M. C. Hillman, and D. A. Boas, “Optimal linear inverse solution with multiple priors in diffuse optical tomography,” Appl. Opt. 44, 1948–1956 (2005).
[CrossRef] [PubMed]

A. Corlu, R. Choe, T. Durduran, K. Lee, M. Schweiger, S. R. Arridge, E. M. Hillman, and A. G. Yodh, “Diffuse optical tomography with spectral constraints and wavelength optimization,” Appl. Opt. 44, 2082–2093 (2005).
[CrossRef] [PubMed]

S. Lam, F. Lesage, and X. Intes, “Time domain fluorescent diffuse optical tomography: analytical expressions,” Opt. Express 13, 2263–2275 (2005).
[CrossRef] [PubMed]

2004 (6)

H. Dehghani, B. W. Pogue, S. P. Poplack, and K. D. Paulsen, “Multiwavelength three-dimensional near-infrared tomography of the breast: initial simulation, phantom, and clinical results,” Appl. Opt. 42, 135–145 (2004).
[CrossRef]

A. Li, Q. Zhang, J. P. Culver, E. L. Miller, and D. A. Boas, “Reconstructing chromosphere concentration images directly by continuous diffuse optical tomography,” Opt. Lett. 29, 256–258 (2004).
[CrossRef] [PubMed]

M. E. Kilmer and E. de Sturler, “Recycling subspace information for diffuse optical tomography,” SIAM J. Sci. Comput. 27, 2140–2166 (2004).
[CrossRef]

B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, S. Srinivasan, X. Song, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “Characterization of hemoglobin, water, and NIR scattering in breast tissue: analysis of intersubject variability and menstrual cycles changes,” J. Biomed. Opt. 9, 541–552 (2004).
[CrossRef] [PubMed]

T. Durduran, R. Choe, J. P. culver, L. Zubkov, M. J. Holboke, J. Giammarco, B. Chance, and A. G. Yodh, “Bulk optical properties of healthy female breast tissue,” Phys. Med. Biol. 47, 2847–2861 (2004).
[CrossRef]

L. Spinelli, A. Torricelli, A. Pifferi, P. Taroni, G. M. Danesini, and R. Cubeddu, “Bulk optical properties and tissue components in the female breast from multiwavelength time-resolved mammography,” J. Biomed. Opt. 9, 1137–1142 (2004).
[CrossRef] [PubMed]

2003 (4)

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, 12349–12354 (2003).
[CrossRef] [PubMed]

P. Taroni, A. Pifferi, A. Torricelli, D. Comelli, and R. Cubeddu, “In vivo absorption and scattering spectroscopy of biological tissues,” Photochem. Photobiol. Sci. 2, 124–129 (2003).
[CrossRef] [PubMed]

M. E. Kilmer, E. L. Miller, A. Barbaro, and D. Boas, “Three-dimensional shape-based imaging of absorption perturbation for diffuse optical tomography,” Appl. Opt. 42, 3129–3144 (2003).
[CrossRef] [PubMed]

A. Corlu, T. Durduran, R. Choe, M. Schweiger, E. M. C. Hillman, S. R. Arridge, and A. G. Yodh, “Uniqueness and wavelength optimization in continuous-wave multispectral diffuse optical tomography,” Opt. Lett. 28, 2339–2341 (2003).
[CrossRef] [PubMed]

2002 (2)

J. P. Culver, R. Choe, M. J. Holboke, L. Zubkov, T. Durduran, A. Slemp, and A. G. Yodh, “Three-dimensional diffuse optical tomography in the parallel plane transmission geometry: Evaluation of a hybrid frequency domain/continuous wave clinical system for breast imaging,” Med. Phys. 30, 235–247 (2002).
[CrossRef]

M. Belge, M. E. Kilmer, and E. L. Miller, “Efficient determination of multiple regularization parameters in a generalized l-curve framework,” Inverse Probl. 18, 1161–1183 (2002).
[CrossRef]

2001 (3)

2000 (2)

R. J. Gaudette, D. H. Brooks, C. A. DiMarzio, M. E. Kilmer, E. L. Miller, T. Gaudette, and D. A. Boas, “A comparison study of linear reconstruction techinques for diffuse optical tomographic imaging of absorption coefficient,” Phys. Med. Biol. 45, 1051–1069 (2000).
[CrossRef] [PubMed]

M. A. Franceschini, V. Toronov, M. E. Filiaci, E. Gratton, and S. Fantini, “On-line optical imaging of the human brain with 160-ms temporal resolution,” Opt. Express 6, 49–57 (2000).
[CrossRef] [PubMed]

1999 (4)

M. Schweiger and S. R. Arridge, “Optical tomographic reconstruction in a complex head model using a priori region boundary information,” Phys. Med. Biol. 44, 2703–2721 (1999).
[CrossRef] [PubMed]

S. Fantini, D. Hueber, M. A. Franceschini, E. Gratton, W. Rosenfeld, P. G. Stubblefield, D. Maulik, and M. R. Stankovic, “Non-invasive optical monitoring of the newborn piglet brain using continuous-wave and frequency-domain spectroscopy,” Phys. Med. Biol. 44, 1543–1563 (1999).
[CrossRef] [PubMed]

B. W. Pogue, T. O. McBride, J. Prewitt, U. L. Osterberg, and K. D. Paulsen, “Spatially variant regularization improves diffuse optical tomography,” Appl. Opt. 38, 2950–2961 (1999).
[CrossRef]

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

1998 (1)

B. Chen, J. J. Stamnes, and K. Stamnes, “Reconstruction algorithm for diffraction tomography of diffuse photon density waves in a random medium,” J. Eur. Opt. Soc. A 7, 1161–1180 (1998).
[CrossRef]

1997 (2)

E. Gratton, S. Fantini, M. A. Franceschini, C. Gratton, and M. Fabiani, “Measurements of scattering and absorption changes in muscle and brain,” Philos. Trans. R. Soc. London, Ser. B 352, 727–735 (1997).
[CrossRef] [PubMed]

D. A. Boas, “A fundamental limitation of linearized algorithms for diffuse optical tomography,” Opt. Express 1, 404–413 (1997).
[CrossRef] [PubMed]

1995 (2)

M. A. OLeary, D. A. Boas, B. Chance, and A. G. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusing-photon tomography,” Opt. Lett. 20, 426–428 (1995).
[CrossRef]

S. Fantini, M. A. Franceschini, J. S. Maier, S. A. Walker, B. Barbieri, and E. Gratton, “Frequency-domain multi-channel optical detector for noninvasive tissue spectroscopy and oximetry,” Opt. Eng. 34, 32–42 (1995).
[CrossRef]

1994 (2)

T. Coleman and Y. Li, “On the convergence of reflective newton methods for large-scale nonlinear minimization subject to bounds,” Math. Prog. 61, 189–224 (1994).
[CrossRef]

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

1982 (1)

C. C. Paige and M. A. Saunders, “Lsqr: An algorithm for sparse linear equations and sparse least squares,” ACM Trans. Math. Softw. 8, 43–71 (1982).
[CrossRef]

Abran, M.

S. Bélanger, M. Abran, X. Intes, C. Casanova, and F. Lesage, “Real-time diffuse optical tomography based on structured illumination,” J. Biomed. Opt. 15, 016006 (2010).
[CrossRef] [PubMed]

Arridge, S. R.

Azer, F.

S. D. Konecky, R. Choe, A. Corlu, K. Lee, R. Wiener, S. M. Srinivas, J. R. Saffer, R. Freifelder, J. S. Karp, N. Hajjioui, F. Azer, and A. G. Yodh, “Comparison of diffuse optical tomography of human breast with whole-body and breast-only positron emission tomography,” Med. Phys. 35, 446–455 (2008).
[CrossRef] [PubMed]

Barbaro, A.

Barbieri, B.

S. Fantini, M. A. Franceschini, J. S. Maier, S. A. Walker, B. Barbieri, and E. Gratton, “Frequency-domain multi-channel optical detector for noninvasive tissue spectroscopy and oximetry,” Opt. Eng. 34, 32–42 (1995).
[CrossRef]

Barbour, R. L.

Bélanger, S.

S. Bélanger, M. Abran, X. Intes, C. Casanova, and F. Lesage, “Real-time diffuse optical tomography based on structured illumination,” J. Biomed. Opt. 15, 016006 (2010).
[CrossRef] [PubMed]

Belge, M.

M. Belge, M. E. Kilmer, and E. L. Miller, “Efficient determination of multiple regularization parameters in a generalized l-curve framework,” Inverse Probl. 18, 1161–1183 (2002).
[CrossRef]

Boas, D.

M. E. Kilmer, E. L. Miller, A. Barbaro, and D. Boas, “Three-dimensional shape-based imaging of absorption perturbation for diffuse optical tomography,” Appl. Opt. 42, 3129–3144 (2003).
[CrossRef] [PubMed]

D. Boas, D. Brooks, E. Miller, C. DiMarzio, M. Kilmer, R. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Trans. Signal Process. 18, 57–75 (2001).
[CrossRef]

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

G. Boverman, E. L. Miller, A. Li, Q. Zhang, T. Chaves, D. H. Brooks, and D. A. Boas, “Quantitative spectroscopic diffuse optical tomography of the breast guided by imperfect a priori structural information,” Phys. Med. Biol. 50, 3941–3956 (2005).
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A. Li, G. Boverman, Y. Zhang, D. Brooks, E. L. Miller, M. E. Kilmer, Q. Zhang, E. M. C. Hillman, and D. A. Boas, “Optimal linear inverse solution with multiple priors in diffuse optical tomography,” Appl. Opt. 44, 1948–1956 (2005).
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D. Boas, D. Brooks, E. Miller, C. DiMarzio, M. Kilmer, R. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Trans. Signal Process. 18, 57–75 (2001).
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G. Boverman, Q. Fang, S. A. Carp, E. L. Miller, D. H. Brooks, J. Selb, R. H. Moore, D. B. Kopans, and D. A. Boas, “Spatio-temporal imaging of the hemoglobin in the compressed breast with diffuse optical tomography,” Phys. Med. Biol. 52, 3619–3641 (2007).
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G. Boverman, E. L. Miller, A. Li, Q. Zhang, T. Chaves, D. H. Brooks, and D. A. Boas, “Quantitative spectroscopic diffuse optical tomography of the breast guided by imperfect a priori structural information,” Phys. Med. Biol. 50, 3941–3956 (2005).
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R. J. Gaudette, D. H. Brooks, C. A. DiMarzio, M. E. Kilmer, E. L. Miller, T. Gaudette, and D. A. Boas, “A comparison study of linear reconstruction techinques for diffuse optical tomographic imaging of absorption coefficient,” Phys. Med. Biol. 45, 1051–1069 (2000).
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T. Durduran, R. Choe, J. P. culver, L. Zubkov, M. J. Holboke, J. Giammarco, B. Chance, and A. G. Yodh, “Bulk optical properties of healthy female breast tissue,” Phys. Med. Biol. 47, 2847–2861 (2004).
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G. Boverman, E. L. Miller, A. Li, Q. Zhang, T. Chaves, D. H. Brooks, and D. A. Boas, “Quantitative spectroscopic diffuse optical tomography of the breast guided by imperfect a priori structural information,” Phys. Med. Biol. 50, 3941–3956 (2005).
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J. P. Culver, R. Choe, M. J. Holboke, L. Zubkov, T. Durduran, A. Slemp, and A. G. Yodh, “Three-dimensional diffuse optical tomography in the parallel plane transmission geometry: Evaluation of a hybrid frequency domain/continuous wave clinical system for breast imaging,” Med. Phys. 30, 235–247 (2002).
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A. Li, Q. Zhang, J. P. Culver, E. L. Miller, and D. A. Boas, “Reconstructing chromosphere concentration images directly by continuous diffuse optical tomography,” Opt. Lett. 29, 256–258 (2004).
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J. P. Culver, R. Choe, M. J. Holboke, L. Zubkov, T. Durduran, A. Slemp, and A. G. Yodh, “Three-dimensional diffuse optical tomography in the parallel plane transmission geometry: Evaluation of a hybrid frequency domain/continuous wave clinical system for breast imaging,” Med. Phys. 30, 235–247 (2002).
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J. P. Culver, V. Ntziachristos, M. J. Holboke, and A. G. Yodh, “Optimization of optode arrangements for diffuse optical tomography: a singular-value analysis,” Opt. Lett. 26, 701–703 (2001).
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R. J. Gaudette, D. H. Brooks, C. A. DiMarzio, M. E. Kilmer, E. L. Miller, T. Gaudette, and D. A. Boas, “A comparison study of linear reconstruction techinques for diffuse optical tomographic imaging of absorption coefficient,” Phys. Med. Biol. 45, 1051–1069 (2000).
[CrossRef] [PubMed]

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A. Corlu, R. Choe, T. Durduran, K. Lee, M. Schweiger, S. R. Arridge, E. M. Hillman, and A. G. Yodh, “Diffuse optical tomography with spectral constraints and wavelength optimization,” Appl. Opt. 44, 2082–2093 (2005).
[CrossRef] [PubMed]

T. Durduran, R. Choe, J. P. culver, L. Zubkov, M. J. Holboke, J. Giammarco, B. Chance, and A. G. Yodh, “Bulk optical properties of healthy female breast tissue,” Phys. Med. Biol. 47, 2847–2861 (2004).
[CrossRef]

A. Corlu, T. Durduran, R. Choe, M. Schweiger, E. M. C. Hillman, S. R. Arridge, and A. G. Yodh, “Uniqueness and wavelength optimization in continuous-wave multispectral diffuse optical tomography,” Opt. Lett. 28, 2339–2341 (2003).
[CrossRef] [PubMed]

J. P. Culver, R. Choe, M. J. Holboke, L. Zubkov, T. Durduran, A. Slemp, and A. G. Yodh, “Three-dimensional diffuse optical tomography in the parallel plane transmission geometry: Evaluation of a hybrid frequency domain/continuous wave clinical system for breast imaging,” Med. Phys. 30, 235–247 (2002).
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G. Boverman, Q. Fang, S. A. Carp, E. L. Miller, D. H. Brooks, J. Selb, R. H. Moore, D. B. Kopans, and D. A. Boas, “Spatio-temporal imaging of the hemoglobin in the compressed breast with diffuse optical tomography,” Phys. Med. Biol. 52, 3619–3641 (2007).
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S. Fantini, D. Hueber, M. A. Franceschini, E. Gratton, W. Rosenfeld, P. G. Stubblefield, D. Maulik, and M. R. Stankovic, “Non-invasive optical monitoring of the newborn piglet brain using continuous-wave and frequency-domain spectroscopy,” Phys. Med. Biol. 44, 1543–1563 (1999).
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E. Gratton, S. Fantini, M. A. Franceschini, C. Gratton, and M. Fabiani, “Measurements of scattering and absorption changes in muscle and brain,” Philos. Trans. R. Soc. London, Ser. B 352, 727–735 (1997).
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S. Fantini, M. A. Franceschini, J. S. Maier, S. A. Walker, B. Barbieri, and E. Gratton, “Frequency-domain multi-channel optical detector for noninvasive tissue spectroscopy and oximetry,” Opt. Eng. 34, 32–42 (1995).
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M. A. Franceschini, V. Toronov, M. E. Filiaci, E. Gratton, and S. Fantini, “On-line optical imaging of the human brain with 160-ms temporal resolution,” Opt. Express 6, 49–57 (2000).
[CrossRef] [PubMed]

S. Fantini, D. Hueber, M. A. Franceschini, E. Gratton, W. Rosenfeld, P. G. Stubblefield, D. Maulik, and M. R. Stankovic, “Non-invasive optical monitoring of the newborn piglet brain using continuous-wave and frequency-domain spectroscopy,” Phys. Med. Biol. 44, 1543–1563 (1999).
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E. Gratton, S. Fantini, M. A. Franceschini, C. Gratton, and M. Fabiani, “Measurements of scattering and absorption changes in muscle and brain,” Philos. Trans. R. Soc. London, Ser. B 352, 727–735 (1997).
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S. Fantini, M. A. Franceschini, J. S. Maier, S. A. Walker, B. Barbieri, and E. Gratton, “Frequency-domain multi-channel optical detector for noninvasive tissue spectroscopy and oximetry,” Opt. Eng. 34, 32–42 (1995).
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S. D. Konecky, R. Choe, A. Corlu, K. Lee, R. Wiener, S. M. Srinivas, J. R. Saffer, R. Freifelder, J. S. Karp, N. Hajjioui, F. Azer, and A. G. Yodh, “Comparison of diffuse optical tomography of human breast with whole-body and breast-only positron emission tomography,” Med. Phys. 35, 446–455 (2008).
[CrossRef] [PubMed]

Gaudette, R.

D. Boas, D. Brooks, E. Miller, C. DiMarzio, M. Kilmer, R. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Trans. Signal Process. 18, 57–75 (2001).
[CrossRef]

Gaudette, R. J.

R. J. Gaudette, D. H. Brooks, C. A. DiMarzio, M. E. Kilmer, E. L. Miller, T. Gaudette, and D. A. Boas, “A comparison study of linear reconstruction techinques for diffuse optical tomographic imaging of absorption coefficient,” Phys. Med. Biol. 45, 1051–1069 (2000).
[CrossRef] [PubMed]

Gaudette, T.

R. J. Gaudette, D. H. Brooks, C. A. DiMarzio, M. E. Kilmer, E. L. Miller, T. Gaudette, and D. A. Boas, “A comparison study of linear reconstruction techinques for diffuse optical tomographic imaging of absorption coefficient,” Phys. Med. Biol. 45, 1051–1069 (2000).
[CrossRef] [PubMed]

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T. Durduran, R. Choe, J. P. culver, L. Zubkov, M. J. Holboke, J. Giammarco, B. Chance, and A. G. Yodh, “Bulk optical properties of healthy female breast tissue,” Phys. Med. Biol. 47, 2847–2861 (2004).
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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, 12349–12354 (2003).
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Gratton, C.

E. Gratton, S. Fantini, M. A. Franceschini, C. Gratton, and M. Fabiani, “Measurements of scattering and absorption changes in muscle and brain,” Philos. Trans. R. Soc. London, Ser. B 352, 727–735 (1997).
[CrossRef] [PubMed]

Gratton, E.

M. A. Franceschini, V. Toronov, M. E. Filiaci, E. Gratton, and S. Fantini, “On-line optical imaging of the human brain with 160-ms temporal resolution,” Opt. Express 6, 49–57 (2000).
[CrossRef] [PubMed]

S. Fantini, D. Hueber, M. A. Franceschini, E. Gratton, W. Rosenfeld, P. G. Stubblefield, D. Maulik, and M. R. Stankovic, “Non-invasive optical monitoring of the newborn piglet brain using continuous-wave and frequency-domain spectroscopy,” Phys. Med. Biol. 44, 1543–1563 (1999).
[CrossRef] [PubMed]

E. Gratton, S. Fantini, M. A. Franceschini, C. Gratton, and M. Fabiani, “Measurements of scattering and absorption changes in muscle and brain,” Philos. Trans. R. Soc. London, Ser. B 352, 727–735 (1997).
[CrossRef] [PubMed]

S. Fantini, M. A. Franceschini, J. S. Maier, S. A. Walker, B. Barbieri, and E. Gratton, “Frequency-domain multi-channel optical detector for noninvasive tissue spectroscopy and oximetry,” Opt. Eng. 34, 32–42 (1995).
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M. Guven, B. Yazici, X. Intes, and B. Chance, “Diffuse optical tomography with a priori anatomical information,” Phys. Med. Biol. 50, 2837–2858 (2005).
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Hajjioui, N.

S. D. Konecky, R. Choe, A. Corlu, K. Lee, R. Wiener, S. M. Srinivas, J. R. Saffer, R. Freifelder, J. S. Karp, N. Hajjioui, F. Azer, and A. G. Yodh, “Comparison of diffuse optical tomography of human breast with whole-body and breast-only positron emission tomography,” Med. Phys. 35, 446–455 (2008).
[CrossRef] [PubMed]

Heffer, E. L.

S. Fantini, E. L. Heffer, V. E. Pera, A. Sassaroli, and N. Liu, “Spatial and spectral information in optical mammography,” Technol. Cancer Res. Treat. 4, 471–482 (2005).

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Hillman, E. M. C.

Holboke, M. J.

T. Durduran, R. Choe, J. P. culver, L. Zubkov, M. J. Holboke, J. Giammarco, B. Chance, and A. G. Yodh, “Bulk optical properties of healthy female breast tissue,” Phys. Med. Biol. 47, 2847–2861 (2004).
[CrossRef]

J. P. Culver, R. Choe, M. J. Holboke, L. Zubkov, T. Durduran, A. Slemp, and A. G. Yodh, “Three-dimensional diffuse optical tomography in the parallel plane transmission geometry: Evaluation of a hybrid frequency domain/continuous wave clinical system for breast imaging,” Med. Phys. 30, 235–247 (2002).
[CrossRef]

J. P. Culver, V. Ntziachristos, M. J. Holboke, and A. G. Yodh, “Optimization of optode arrangements for diffuse optical tomography: a singular-value analysis,” Opt. Lett. 26, 701–703 (2001).
[CrossRef]

Hueber, D.

S. Fantini, D. Hueber, M. A. Franceschini, E. Gratton, W. Rosenfeld, P. G. Stubblefield, D. Maulik, and M. R. Stankovic, “Non-invasive optical monitoring of the newborn piglet brain using continuous-wave and frequency-domain spectroscopy,” Phys. Med. Biol. 44, 1543–1563 (1999).
[CrossRef] [PubMed]

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S. Bélanger, M. Abran, X. Intes, C. Casanova, and F. Lesage, “Real-time diffuse optical tomography based on structured illumination,” J. Biomed. Opt. 15, 016006 (2010).
[CrossRef] [PubMed]

M. Guven, B. Yazici, X. Intes, and B. Chance, “Diffuse optical tomography with a priori anatomical information,” Phys. Med. Biol. 50, 2837–2858 (2005).
[CrossRef] [PubMed]

S. Lam, F. Lesage, and X. Intes, “Time domain fluorescent diffuse optical tomography: analytical expressions,” Opt. Express 13, 2263–2275 (2005).
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P. Taroni, A. Pifferi, A. Torricelli, D. Comelli, and R. Cubeddu, “In vivo absorption and scattering spectroscopy of biological tissues,” Photochem. Photobiol. Sci. 2, 124–129 (2003).
[CrossRef] [PubMed]

Toga, A. W.

A. A. Joshi, A. J. Chaudhari, D. W. Shattuck, J. Dutta, R. M. Leahy, and A. W. Toga, “Posture matching and elastic registration of a mouse atlas to surface topography range data,” Proc IEEE Intl. Symp. Biomedical Imaging , 366–369 (2009).

Toronov, V.

Torricelli, A.

L. Spinelli, A. Torricelli, A. Pifferi, P. Taroni, G. M. Danesini, and R. Cubeddu, “Bulk optical properties and tissue components in the female breast from multiwavelength time-resolved mammography,” J. Biomed. Opt. 9, 1137–1142 (2004).
[CrossRef] [PubMed]

P. Taroni, A. Pifferi, A. Torricelli, D. Comelli, and R. Cubeddu, “In vivo absorption and scattering spectroscopy of biological tissues,” Photochem. Photobiol. Sci. 2, 124–129 (2003).
[CrossRef] [PubMed]

Tosteson, T. D.

B. Brooksby, B. W. Pogue, S. Jiang, H. Dehghani, S. Srinivasan, C. Kogel, T. D. Tosteson, J. Weaver, S. P. Poplack, and K. D. Paulsen, “Imaging breast adipose and fibroglandular tissue molecular signatures by using hybrid MRI-guided near-infrared spectral tomography,” Proc. Natl. Acad. Sci. U.S.A. 103, 8828–8833 (2006).
[CrossRef] [PubMed]

B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, S. Srinivasan, X. Song, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “Characterization of hemoglobin, water, and NIR scattering in breast tissue: analysis of intersubject variability and menstrual cycles changes,” J. Biomed. Opt. 9, 541–552 (2004).
[CrossRef] [PubMed]

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, 12349–12354 (2003).
[CrossRef] [PubMed]

Walker, S. A.

S. Fantini, M. A. Franceschini, J. S. Maier, S. A. Walker, B. Barbieri, and E. Gratton, “Frequency-domain multi-channel optical detector for noninvasive tissue spectroscopy and oximetry,” Opt. Eng. 34, 32–42 (1995).
[CrossRef]

Wang, J.

J. Wang, S. C. Davis, S. Srinivasan, S. Jiang, B. W. Pogue, and K. D. Paulsen, “Spectral tomography with diffuse near-infrared light: inclusion of broadband frequency domain spectral data,” J. Biomed. Opt. 13, 1–10 (2008).
[CrossRef]

Weaver, J.

B. Brooksby, B. W. Pogue, S. Jiang, H. Dehghani, S. Srinivasan, C. Kogel, T. D. Tosteson, J. Weaver, S. P. Poplack, and K. D. Paulsen, “Imaging breast adipose and fibroglandular tissue molecular signatures by using hybrid MRI-guided near-infrared spectral tomography,” Proc. Natl. Acad. Sci. U.S.A. 103, 8828–8833 (2006).
[CrossRef] [PubMed]

Wiener, R.

S. D. Konecky, R. Choe, A. Corlu, K. Lee, R. Wiener, S. M. Srinivas, J. R. Saffer, R. Freifelder, J. S. Karp, N. Hajjioui, F. Azer, and A. G. Yodh, “Comparison of diffuse optical tomography of human breast with whole-body and breast-only positron emission tomography,” Med. Phys. 35, 446–455 (2008).
[CrossRef] [PubMed]

Yalavarthy, P. K.

Yazici, B.

M. Guven, B. Yazici, X. Intes, and B. Chance, “Diffuse optical tomography with a priori anatomical information,” Phys. Med. Biol. 50, 2837–2858 (2005).
[CrossRef] [PubMed]

Yodh, A.

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

Yodh, A. G.

S. D. Konecky, R. Choe, A. Corlu, K. Lee, R. Wiener, S. M. Srinivas, J. R. Saffer, R. Freifelder, J. S. Karp, N. Hajjioui, F. Azer, and A. G. Yodh, “Comparison of diffuse optical tomography of human breast with whole-body and breast-only positron emission tomography,” Med. Phys. 35, 446–455 (2008).
[CrossRef] [PubMed]

A. Corlu, R. Choe, T. Durduran, K. Lee, M. Schweiger, S. R. Arridge, E. M. Hillman, and A. G. Yodh, “Diffuse optical tomography with spectral constraints and wavelength optimization,” Appl. Opt. 44, 2082–2093 (2005).
[CrossRef] [PubMed]

T. Durduran, R. Choe, J. P. culver, L. Zubkov, M. J. Holboke, J. Giammarco, B. Chance, and A. G. Yodh, “Bulk optical properties of healthy female breast tissue,” Phys. Med. Biol. 47, 2847–2861 (2004).
[CrossRef]

A. Corlu, T. Durduran, R. Choe, M. Schweiger, E. M. C. Hillman, S. R. Arridge, and A. G. Yodh, “Uniqueness and wavelength optimization in continuous-wave multispectral diffuse optical tomography,” Opt. Lett. 28, 2339–2341 (2003).
[CrossRef] [PubMed]

J. P. Culver, R. Choe, M. J. Holboke, L. Zubkov, T. Durduran, A. Slemp, and A. G. Yodh, “Three-dimensional diffuse optical tomography in the parallel plane transmission geometry: Evaluation of a hybrid frequency domain/continuous wave clinical system for breast imaging,” Med. Phys. 30, 235–247 (2002).
[CrossRef]

J. P. Culver, V. Ntziachristos, M. J. Holboke, and A. G. Yodh, “Optimization of optode arrangements for diffuse optical tomography: a singular-value analysis,” Opt. Lett. 26, 701–703 (2001).
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M. A. OLeary, D. A. Boas, B. Chance, and A. G. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusing-photon tomography,” Opt. Lett. 20, 426–428 (1995).
[CrossRef]

Zhang, Q.

Zhang, Y.

Zhong, S.

Zubkov, L.

T. Durduran, R. Choe, J. P. culver, L. Zubkov, M. J. Holboke, J. Giammarco, B. Chance, and A. G. Yodh, “Bulk optical properties of healthy female breast tissue,” Phys. Med. Biol. 47, 2847–2861 (2004).
[CrossRef]

J. P. Culver, R. Choe, M. J. Holboke, L. Zubkov, T. Durduran, A. Slemp, and A. G. Yodh, “Three-dimensional diffuse optical tomography in the parallel plane transmission geometry: Evaluation of a hybrid frequency domain/continuous wave clinical system for breast imaging,” Med. Phys. 30, 235–247 (2002).
[CrossRef]

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C. C. Paige and M. A. Saunders, “Lsqr: An algorithm for sparse linear equations and sparse least squares,” ACM Trans. Math. Softw. 8, 43–71 (1982).
[CrossRef]

Appl. Opt. (6)

IEEE Trans. Signal Process. (1)

D. Boas, D. Brooks, E. Miller, C. DiMarzio, M. Kilmer, R. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Trans. Signal Process. 18, 57–75 (2001).
[CrossRef]

Inverse Probl. (2)

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

M. Belge, M. E. Kilmer, and E. L. Miller, “Efficient determination of multiple regularization parameters in a generalized l-curve framework,” Inverse Probl. 18, 1161–1183 (2002).
[CrossRef]

J. Biomed. Opt. (7)

N. Liu, A. Sassaroli, and S. Fantini, “Paired-wavelength spectral approach to measuring the relative concentrations of two localized chromophores in turbid media: an experimental study,” J. Biomed. Opt. 12, 05160 (2007).
[CrossRef]

M. E. Eames, B. W. Pogue, and H. Dehghani, “Wavelength band optimization in spectral near-infrared optical tomography improves accuracy while reducing data acquisition and computational burden,” J. Biomed. Opt. 13, 1–9 (2008).
[CrossRef]

B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, S. Srinivasan, X. Song, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “Characterization of hemoglobin, water, and NIR scattering in breast tissue: analysis of intersubject variability and menstrual cycles changes,” J. Biomed. Opt. 9, 541–552 (2004).
[CrossRef] [PubMed]

J. Wang, S. C. Davis, S. Srinivasan, S. Jiang, B. W. Pogue, and K. D. Paulsen, “Spectral tomography with diffuse near-infrared light: inclusion of broadband frequency domain spectral data,” J. Biomed. Opt. 13, 1–10 (2008).
[CrossRef]

B. Brendel and T. Nielsen, “Selection of optimal wavelengths for spectral reconstruction in diffuse optical tomography,” J. Biomed. Opt. 14, 1–10 (2009).
[CrossRef]

L. Spinelli, A. Torricelli, A. Pifferi, P. Taroni, G. M. Danesini, and R. Cubeddu, “Bulk optical properties and tissue components in the female breast from multiwavelength time-resolved mammography,” J. Biomed. Opt. 9, 1137–1142 (2004).
[CrossRef] [PubMed]

S. Bélanger, M. Abran, X. Intes, C. Casanova, and F. Lesage, “Real-time diffuse optical tomography based on structured illumination,” J. Biomed. Opt. 15, 016006 (2010).
[CrossRef] [PubMed]

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B. Chen, J. J. Stamnes, and K. Stamnes, “Reconstruction algorithm for diffraction tomography of diffuse photon density waves in a random medium,” J. Eur. Opt. Soc. A 7, 1161–1180 (1998).
[CrossRef]

J. Opt. Soc. Am. A (1)

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[CrossRef]

Med. Phys. (2)

S. D. Konecky, R. Choe, A. Corlu, K. Lee, R. Wiener, S. M. Srinivas, J. R. Saffer, R. Freifelder, J. S. Karp, N. Hajjioui, F. Azer, and A. G. Yodh, “Comparison of diffuse optical tomography of human breast with whole-body and breast-only positron emission tomography,” Med. Phys. 35, 446–455 (2008).
[CrossRef] [PubMed]

J. P. Culver, R. Choe, M. J. Holboke, L. Zubkov, T. Durduran, A. Slemp, and A. G. Yodh, “Three-dimensional diffuse optical tomography in the parallel plane transmission geometry: Evaluation of a hybrid frequency domain/continuous wave clinical system for breast imaging,” Med. Phys. 30, 235–247 (2002).
[CrossRef]

Opt. Eng. (1)

S. Fantini, M. A. Franceschini, J. S. Maier, S. A. Walker, B. Barbieri, and E. Gratton, “Frequency-domain multi-channel optical detector for noninvasive tissue spectroscopy and oximetry,” Opt. Eng. 34, 32–42 (1995).
[CrossRef]

Opt. Express (5)

Opt. Lett. (4)

Philos. Trans. R. Soc. London, Ser. B (1)

E. Gratton, S. Fantini, M. A. Franceschini, C. Gratton, and M. Fabiani, “Measurements of scattering and absorption changes in muscle and brain,” Philos. Trans. R. Soc. London, Ser. B 352, 727–735 (1997).
[CrossRef] [PubMed]

Photochem. Photobiol. Sci. (1)

P. Taroni, A. Pifferi, A. Torricelli, D. Comelli, and R. Cubeddu, “In vivo absorption and scattering spectroscopy of biological tissues,” Photochem. Photobiol. Sci. 2, 124–129 (2003).
[CrossRef] [PubMed]

Phys. Med. Biol. (7)

M. Guven, B. Yazici, X. Intes, and B. Chance, “Diffuse optical tomography with a priori anatomical information,” Phys. Med. Biol. 50, 2837–2858 (2005).
[CrossRef] [PubMed]

G. Boverman, E. L. Miller, A. Li, Q. Zhang, T. Chaves, D. H. Brooks, and D. A. Boas, “Quantitative spectroscopic diffuse optical tomography of the breast guided by imperfect a priori structural information,” Phys. Med. Biol. 50, 3941–3956 (2005).
[CrossRef] [PubMed]

T. Durduran, R. Choe, J. P. culver, L. Zubkov, M. J. Holboke, J. Giammarco, B. Chance, and A. G. Yodh, “Bulk optical properties of healthy female breast tissue,” Phys. Med. Biol. 47, 2847–2861 (2004).
[CrossRef]

R. J. Gaudette, D. H. Brooks, C. A. DiMarzio, M. E. Kilmer, E. L. Miller, T. Gaudette, and D. A. Boas, “A comparison study of linear reconstruction techinques for diffuse optical tomographic imaging of absorption coefficient,” Phys. Med. Biol. 45, 1051–1069 (2000).
[CrossRef] [PubMed]

G. Boverman, Q. Fang, S. A. Carp, E. L. Miller, D. H. Brooks, J. Selb, R. H. Moore, D. B. Kopans, and D. A. Boas, “Spatio-temporal imaging of the hemoglobin in the compressed breast with diffuse optical tomography,” Phys. Med. Biol. 52, 3619–3641 (2007).
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M. Schweiger and S. R. Arridge, “Optical tomographic reconstruction in a complex head model using a priori region boundary information,” Phys. Med. Biol. 44, 2703–2721 (1999).
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S. Fantini, D. Hueber, M. A. Franceschini, E. Gratton, W. Rosenfeld, P. G. Stubblefield, D. Maulik, and M. R. Stankovic, “Non-invasive optical monitoring of the newborn piglet brain using continuous-wave and frequency-domain spectroscopy,” Phys. Med. Biol. 44, 1543–1563 (1999).
[CrossRef] [PubMed]

Proc. Natl. Acad. Sci. U.S.A. (3)

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

B. Brooksby, B. W. Pogue, S. Jiang, H. Dehghani, S. Srinivasan, C. Kogel, T. D. Tosteson, J. Weaver, S. P. Poplack, and K. D. Paulsen, “Imaging breast adipose and fibroglandular tissue molecular signatures by using hybrid MRI-guided near-infrared spectral tomography,” Proc. Natl. Acad. Sci. U.S.A. 103, 8828–8833 (2006).
[CrossRef] [PubMed]

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, 12349–12354 (2003).
[CrossRef] [PubMed]

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M. E. Kilmer and E. de Sturler, “Recycling subspace information for diffuse optical tomography,” SIAM J. Sci. Comput. 27, 2140–2166 (2004).
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Other (7)

A. A. Joshi, A. J. Chaudhari, D. W. Shattuck, J. Dutta, R. M. Leahy, and A. W. Toga, “Posture matching and elastic registration of a mouse atlas to surface topography range data,” Proc IEEE Intl. Symp. Biomedical Imaging , 366–369 (2009).

J. D. Vylder and W. Philips, “A computational efficient external energy for active contour segmentation using edge propagation,” Intl. con. on Im. Processing , (2010).

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S. Prahl, “Tabulated molar extinction coefficient for hemoglobin in water,” http://omlc.ogi.edu/spectra/hemoglobin/summary.html (Oregon Medical Laser Center, 2007).

F. Larusson, “Hyperspectral imaging for diffuse optical tomography,” M.S. Thesis (Department of Electrical Engineering, Tufts University, Medford, 2009).

A. Mandelis, Diffusion-Wave Fields: Mathematical Methods and Green Functions , 1st ed.(Springer, 2001).

A. J. Laub, Matrix Analysis for Scientists and Engineers , 1st ed. (SIAM: Society for industrial and applied mathematics, 2004).

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Figures (10)

Fig. 1
Fig. 1

(a) The setup of sources and detectors for simulation reconstructions. Same orientation of axes is used for experimental data. (b) Molar extinction coefficients used in simulations plotted as a function of wavelength.

Fig. 5
Fig. 5

Reconstruction for first set. Middle images are generated with 6 wavelengths and rightmost images are done with 126 wavelengths. Upper row is for the HbO2 chromophore and the lower for HbR. Concentration units are in mM.

Fig. 7
Fig. 7

Reconstruction for second set. Middle images are generated with 6 wavelengths and rightmost images are done with 126 wavelengths. Upper row is for the HbO2 chromophore and the lower for HbR. Concentration units are in mM.

Fig. 6
Fig. 6

(a) L-hypersurface, defined by (23) plotted against regularization parameters. (b) H curvature, defined by (24), computed for the L-hypersurface. (c) Error estimation surface, defined by (22), plotted against regularization parameters. The optimal regularization parameters are marked in each case with a red arrow.

Fig. 2
Fig. 2

Absorption spectra of the ink and dye solutions chromophores used in experimental measurements. Specifically chosen wavelengths are marked with an asterisk.

Fig. 3
Fig. 3

(a) Absorption spectra for the background, μa , and the inclusion, μa + Δμa , in experimental set 1, containing 10% ink and 90% dye. (b) Contrast between the background and the inclusion for experimental set 1.

Fig. 4
Fig. 4

(a) Absorption spectra for the background, μa , and the inclusion, μa + Δμa , in experimental set 2, containing 70% ink and 30% dye. (b) Contrast between the background and the inclusion for experimental set 2.

Fig. 8
Fig. 8

(a) Dice coefficients for the first simulation set as a function of threshold. (b) Dice coefficients for the second simulation set as a function of threshold.

Fig. 9
Fig. 9

(a) Dice coefficients for the first experimental set as a function of threshold. (b) Dice coefficients for the second experimental set as a function of threshold.

Fig. 10
Fig. 10

Reconstruction from both experimental sets, set 1 containing 10% ink and 90% dye and set 2 70% ink and 30% dye.

Tables (2)

Tables Icon

Table 1 MSE Compared to the Number of Wavelengths used in the Reconstruction a

Tables Icon

Table 2 Comparison of c i ^ and c i r ^ to Target Values for Experimental Results a

Equations (28)

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( 2 + ν μ a 0 ( r , λ ) D ( λ ) ) Φ ( r , λ ) = v D ( λ ) S ( r , λ )
( 2 v μ a ( λ ) D ( λ ) v Δ μ a ( r , λ ) D ( λ ) ) ( Φ i ( r , λ ) + Φ s ( r , λ ) ) = v D ( λ ) S ( r , λ ) .
[ 2 + k 0 2 ( λ ) ] Φ s ( r , λ ) = Δ k 2 ( r , λ ) Φ ( r , λ )
Φ s ( r , λ ) = G ( r , r , λ ) Φ ( r , λ ) Δ k 2 ( r , λ ) d r .
( 2 v μ a ( λ ) D ( λ ) ) Φ i ( r , λ ) = v D ( λ ) S ( r , λ )
Φ s ( r d , λ ) v D ( λ ) G ( r d , r , λ ) Φ i ( r , r s , λ ) Δ μ a ( r , λ ) d r .
Δ μ a ( r , λ ) = k = 1 N s ɛ k ( λ ) c k ( r ) .
c k ( r ) = j = 1 N v c k , j φ ( r )
φ ( r ) = { 1 , if r V j 0 , if r V j .
Φ s ( r d , λ ) = a k = 1 N c j = 1 N v v D ( λ ) G ( r d , r j , λ ) Φ i ( r j , r s , λ ) ɛ i ( λ ) c k , j .
[ Φ s ( λ 1 ) Φ s ( λ 2 ) Φ s ( λ N λ ) ] = [ ɛ 1 ( λ 1 ) K 1 ɛ 2 ( λ 1 ) K 1 ɛ N c ( λ 1 ) K 1 ɛ 1 ( λ 2 ) K 2 ɛ 2 ( λ 2 ) K 2 ɛ N c ( λ 2 ) K 2 ɛ 1 ( λ N λ ) K N λ ɛ 2 ( λ N λ ) K N λ ɛ N c ( λ N λ ) K N λ ] [ c 1 c 2 c N c ] Φ s = Kc
c ^ = arg min c 0 W ( Φ s Kc ) 2 2 + Lc 2 2
σ m 2 = Ω ( m ) 10 SNR m 10 .
SNR m = 10 log 10 ( Ω ( m ) / Ω ( m ) ) .
L = 𝒟 ( α ) [ x y ]
G ( r , r , λ ) = 1 4 π | r r | e j k 0 ( λ ) | r r | .
Φ s ( r d , r s , r j , λ ) = r j e j k 0 ( λ ) | r d r j | 4 π | r d r j | e j k 0 ( λ ) | r j r s | 4 π | r j r s | v D ( λ ) Δ μ a ( r j , λ )
= v 16 π 2 D ( λ ) r j e j k 0 ( λ ) ( | r d r j | + | r j r s | ) | r d r j | | r j r s | Δ μ a ( r j , λ )
μ s = Ψ ( λ λ 0 ) b .
Φ = K [ c 1 c 2 ] + n
MSE k = c k c ^ k 2 c k 2
e = c c ^ 2 2
z ( α ) = Φ K c ^ ( α ) 2 2
H = rt s 2 w 4
w 2 = 1 + p 2 + q 2 .
p = z α 1 , q = z α 2 , r = 2 z α 1 2 , t = 2 z α 2 2 , s = 2 z α 1 α 2
c ink r ^ = c ink ^ / ( c ink ^ + c dye ^ )
D ( S , G ) = 2 | S G | | S | + | G |

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