Parametric level set reconstruction methods for hyperspectral diffuse optical tomography

Fridrik Larusson; Sergio Fantini; Eric L. Miller

doi:10.1364/BOE.3.001006

Parametric level set reconstruction methods for hyperspectral diffuse optical tomography

Fridrik Larusson, Sergio Fantini, and Eric L. Miller

Biomedical Optics Express
Vol. 3,
Issue 5,
pp. 1006-1024
(2012)
•https://doi.org/10.1364/BOE.3.001006

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Fridrik Larusson, Sergio Fantini, and Eric L. Miller, "Parametric level set reconstruction methods for hyperspectral diffuse optical tomography," Biomed. Opt. Express 3, 1006-1024 (2012)

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Related Topics
Table of Contents Category
- Image Reconstruction and Inverse Problems
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Inverse problems (100.3190)
- Image reconstruction techniques (170.3010)
- Light propagation in tissues (170.3660)
- Mammography (170.3830)
- Medical and biological imaging (170.3880)
- Photon migration (170.5280)
- Tomography (170.6960)
- Diffusion (290.1990)
- Turbid media (290.7050)

About this Article
History
- Original Manuscript: January 27, 2012
- Manuscript Accepted: March 15, 2012
- Published: April 18, 2012

Accessible

Open Access

Abstract

A parametric level set method (PaLS) is implemented for image reconstruction for hyperspectral diffuse optical tomography (DOT). Chromophore concentrations and diffusion amplitude are recovered using a linearized Born approximation model and employing data from over 100 wavelengths. The images to be recovered are taken to be piecewise constant and a newly introduced, shape-based model is used as the foundation for reconstruction. The PaLS method significantly reduces the number of unknowns relative to more traditional level-set reconstruction methods and has been show to be particularly well suited for ill-posed inverse problems such as the one of interest here. We report on reconstructions for multiple chromophores from simulated and experimental data where the PaLS method provides a more accurate estimation of chromophore concentrations compared to a pixel-based method.

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References

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[PubMed]
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 techniques for diffuse optical tomographic imaging of absorption coefficient,” Phys. Med. Biol 45, 1051–1069 (2000).
[Crossref] [PubMed]
D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18(6), 57–75 (2001).
[Crossref]
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]
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[Crossref] [PubMed]
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[Crossref]
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]
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[Crossref]
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[Crossref]
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[Crossref]
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[Crossref]
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[Crossref] [PubMed]
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[Crossref] [PubMed]
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[Crossref]
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[Crossref] [PubMed]
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 optical mammography,” J. Biomed. Opt. 9, 1137–1142 (2004).
[Crossref] [PubMed]
J. P. Culver, R. Choe, M. J. Holboke, L. Zubkov, T. Durduran, A. Slemp, V. Ntziachristos, B. Chance, 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 (2003).
[Crossref] [PubMed]
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[Crossref] [PubMed]
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[Crossref] [PubMed]
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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).
[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).
[Crossref]
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]
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, 054037 (2008).
[Crossref] [PubMed]

2011 (2)

F. Larusson, S. Fantini, and E. L. Miller, “Hyperspectral image reconstruction for diffuse optical tomography,” Biomed. Opt. Express 2, 947–965 (2011).
[Crossref]

A. Aghasi, M. Kilmer, and E. L. Miller “Parametric level set methods for inverse problems,” SIAM J. Imaging Sci. 4, 618–650 (2011).
[Crossref]

2010 (3)

S. Kukreti, A. E. Cerussi, W. Tanamai, D. Hsiang, B. J. Tromberg, and E. Gratton, “Characterization of metabolic differences between benign and malignant tumors: high-spectral-resolution diffuse optical spectroscopy,” Radiology 254, 277–284 (2010).
[Crossref]

H. Soliman, A. Gunasekara, M. Rycroft, J. Zubovits, R. Dent, J. Spayne, M. J. Yaffe, and G. Czarnota, “Functional imaging using diffuse optical spectroscopy of neoadjuvant chemotherapy response in women with locally advanced breast cancer,” Clin. Cancer Res. 16, 2605–2614 (2010).
[Crossref] [PubMed]

Q. Zhu, P. U. Hegde, A. Ricci, M. Kane, E. B. Cronin, Y. Ardeshirpour, C. Xu, A. Aguirre, S. H. Kurtzman, P. J. Deckers, and S. H. Tannenbaum, “Early-stage invasive breast cancers: potential role of optical tomography with US localization in assisting diagnosis,” Radiology 256, 367–378 (2010).
[Crossref] [PubMed]

2009 (1)

B. Brendel and T. Nielsen, “Selection of optimal wavelengths for spectral reconstruction in diffuse optical tomography,” J. Biomed. Opt. 14, 034041 (2009).
[Crossref] [PubMed]

2008 (5)

B. Brendel, R. Ziegler, and T. Nielsen “Algebraic reconstruction techniques for spectral reconstruction in diffuse optical tomography,” Appl. Opt. 47, 6392–6403 (2008).
[Crossref] [PubMed]

M. Schweiger, O. Dorn, and S. R. Arridge, “3-D shape and contrast reconstruction in optical tomography with level sets,” J. Phys.: Conf. Ser. 124, 012043 (2008).
[Crossref]

S. R. Arridge, O. Dorn, V. Kolehmainen, M. Schweiger, and A. Zacharopoulos, “Parameter and structure reconstruction in optical tomography,” J. Phys.: Conf. Ser. 135, 012001 (2008).
[Crossref]

T. T. Wu and K. Lange, “Coordinate descent algorithms for lasso penalized regression,” Ann. Appl. Stat. 2, 224–244 (2008).
[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, 054037 (2008).
[Crossref] [PubMed]

2007 (3)

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, 051602 (2007).
[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).
[Crossref] [PubMed]

A. Cerussi, D. Hsiang, N. Shah, R. Mehta, A. Durkin, J. Butler, and B. J. Tromberg “Predicting response to breast cancer neoadjuvant chemotherapy using diffuse optical spectroscopy,” Proc. Nat. Acad. Sci. U.S.A. 104, 4014–4019 (2007).
[Crossref]

2006 (2)

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]

O. Dorn and D. Lesselier, “Level set methods for inverse scattering,” Inverse Probl. 22, R67–R131 (2006).
[Crossref]

2005 (5)

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]

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).
[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]

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]

2004 (3)

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]

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 optical mammography,” J. Biomed. Opt. 9, 1137–1142 (2004).
[Crossref] [PubMed]

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]

2003 (4)

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

J. P. Culver, R. Choe, M. J. Holboke, L. Zubkov, T. Durduran, A. Slemp, V. Ntziachristos, B. Chance, 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 (2003).
[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]

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]

2002 (1)

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 (2002).
[Crossref] [PubMed]

2001 (3)

T. Chan and L. Vese “Active contours without edges,” IEEE Trans. Image Process. 10, 266–277 (2001).
[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]

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18(6), 57–75 (2001).
[Crossref]

2000 (3)

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 techniques for diffuse optical tomographic imaging of absorption coefficient,” Phys. Med. Biol 45, 1051–1069 (2000).
[Crossref] [PubMed]

H. K. Zhao, S. Osher, B. Merriman, and M. Kang, “Implicit, nonparametric shape reconstruction from unorganized points using a variational level set method,” Comput. Vision Image Understanding 80, 295–314 (2000).
[Crossref]

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 (3)

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]

1997 (2)

D. A. Boas, “A fundamental limitation of linearized algorithms for diffuse optical tomography,” Opt. Express 1, 404–413 (1997).
[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. Lond. B. Biol. Sci. 352, 727–735 (1997).
[Crossref] [PubMed]

1996 (1)

K. D. Paulsen and H. Jiang, “Enhanced frequency-domain optical image reconstruction in tissues through total-variation minimization,” Appl. Opt. 35, 3447–3458, (1996).
[Crossref] [PubMed]

1995 (2)

M. A. O’Leary, 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 (1)

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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Ann. Appl. Stat. (1)

T. T. Wu and K. Lange, “Coordinate descent algorithms for lasso penalized regression,” Ann. Appl. Stat. 2, 224–244 (2008).
[Crossref]

Appl. Opt. (7)

B. Brendel, R. Ziegler, and T. Nielsen “Algebraic reconstruction techniques for spectral reconstruction in diffuse optical tomography,” Appl. Opt. 47, 6392–6403 (2008).
[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).
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Biomed. Opt. Express (1)

F. Larusson, S. Fantini, and E. L. Miller, “Hyperspectral image reconstruction for diffuse optical tomography,” Biomed. Opt. Express 2, 947–965 (2011).
[Crossref]

Clin. Cancer Res. (1)

Comput. Vision Image Understanding (1)

IEEE Signal Process. Mag. (1)

IEEE Trans. Image Process. (1)

T. Chan and L. Vese “Active contours without edges,” IEEE Trans. Image Process. 10, 266–277 (2001).
[Crossref]

Inverse Probl. (1)

O. Dorn and D. Lesselier, “Level set methods for inverse scattering,” Inverse Probl. 22, R67–R131 (2006).
[Crossref]

J. Biomed. Opt. (5)

B. Brendel and T. Nielsen, “Selection of optimal wavelengths for spectral reconstruction in diffuse optical tomography,” J. Biomed. Opt. 14, 034041 (2009).
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J. Phys.: Conf. Ser. (2)

S. R. Arridge, O. Dorn, V. Kolehmainen, M. Schweiger, and A. Zacharopoulos, “Parameter and structure reconstruction in optical tomography,” J. Phys.: Conf. Ser. 135, 012001 (2008).
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M. Schweiger, O. Dorn, and S. R. Arridge, “3-D shape and contrast reconstruction in optical tomography with level sets,” J. Phys.: Conf. Ser. 124, 012043 (2008).
[Crossref]

Med. Phys. (1)

Opt. Eng. (1)

Opt. Express (2)

D. A. Boas, “A fundamental limitation of linearized algorithms for diffuse optical tomography,” Opt. Express 1, 404–413 (1997).
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Opt. Lett. (2)

Philos. Trans. R. Soc. Lond. B. Biol. Sci. (1)

Photochem. Photobiol. Sci. (1)

Phys. Med. Biol (1)

Phys. Med. Biol. (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).
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Proc. Nat. Acad. Sci. U.S.A. (1)

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

Radiology (2)

SIAM J. Imaging Sci. (1)

A. Aghasi, M. Kilmer, and E. L. Miller “Parametric level set methods for inverse problems,” SIAM J. Imaging Sci. 4, 618–650 (2011).
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Technol. Cancer Res. Treat. (1)

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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Other (10)

S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces (Springer, 2002)

K. Madsen, H. Bruun, and O. Tingleff “Methods for non-linear least squares problems,” lecture notes (2004).

C. R. Vogel, Computational Methods for Inverse Problems (SIAM, 2002)
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A. Mandelis, Diffusion-Wave Fields: Mathematical Methods and Green Functions (Springer, 2001).

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

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Fig. 1 (a) The setup of sources and detectors for simulation reconstructions. Same orientation of axes is used for experimental data. (b) Definition of domains used for the parametric level-set methods.

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Fig. 6 Reconstruction using the PaLS method. Middle column of images are generated with 8 wavelengths and rightmost images are generated with 176 wavelengths. From top to bottom the rows show HbO₂, HbR, lipid, water and diffusion amplitude, respectively. Concentration units are in mM.

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Fig. 2 (a) Order of the basis functions used for the PaLS method. (b) Absorption spectra of the ink and dye solutions chromophores used in experimental measurements. Specifically chosen wavelengths are marked with an asterisk.

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Fig. 3 (a) Absorption spectra for the background, μ_a, and the inclusion, μ_a + Δμ_a, in experimental set 1, containing 10% ink and 90% dye. (b) Contrast between the background and the inclusion for experimental set 1.

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Fig. 4 (a) Absorption spectra for the background, μ_a, and the inclusion, μ_a + Δμ_a, in experimental set 2, containing 70% ink and 30% dye. (b) Contrast between the background and the inclusion for experimental set 2.

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Fig. 5 Reconstruction using a pixel based method. Middle column of images are generated with 8 wavelengths and rightmost images are generated with 176 wavelengths. From top to bottom the rows show HbO₂, HbR, lipid, water and diffusion amplitude, respectively. Concentration units are in mM.

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Fig. 7 Pixel reconstruction from both experimental sets, set 1 containing 10% ink and 90% dye and set 2 70% ink and 30% dye.

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Fig. 8 PaLS Reconstruction from both experimental sets, set 1 containing 10% ink and 90% dye and set 2 70% ink and 30% dye.

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Tables (6)

Algorithm 1 Matlab-like code for estimating shape and concentration value

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Table 1 The MSE is compared for each chromphore for multiple wavelength choices. In each case the reconstructions are done with equally spaced wavelengths over the spectrum except for the 8 wavelength case.

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Table 2 D(S,G) is compared for each chromphore for multiple wavelength choices. In each case the reconstructions are done with equally spaced wavelengths over the spectrum except for the 8 wavelength case. D(S,G) is calculated comparing 80% of the target peak to the reconstruction.

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Table 3 D(S,G) is compared for each chromphore for multiple wavelength choices. In each case the reconstructions are done with equally spaced wavelengths over the spectrum except for the 6 wavelength case where we use optimally chosen wavelengths. D(S,G) is calculated comparing the half maximum of the target peak to the reconstruction.

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Table 4 Comparison of $\hat{c_{i}}$ and $\hat{c_{i}^{r}}$ to target concentration values for experimental results, for the pixel-based method. Best performance is highlighted in bold.

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Table 5 Comparison of $\hat{c_{i}}$ and $\hat{c_{i}^{r}}$ to target concentration values for experimental results, for the PaLS method. Best performance is highlighted in bold.

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Equations (27)

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\nabla \cdot D^{0} (r, λ) \nabla Φ (r, λ) + v μ_{a}^{0} (r, λ) Φ (r, λ) = - v S (r, λ)

Δ D (r, λ) = \frac{v}{3 Δ Ψ} {(\frac{λ}{λ_{0}})}^{b} = v Δ Ψ^{'} d (λ) .

[\nabla^{2} + k_{0}^{2} (λ)] Φ_{s} (r, λ) = - Δ k^{2} (r, λ) Φ (r, λ) - \frac{1}{D (λ)} \nabla \cdot Δ D (r, λ) \nabla Φ

\begin{array}{r} Φ_{s} (r, λ) = - \frac{v}{D (λ)} (\int Δ μ (r^{'}, λ) G (r, r^{'}, λ) Φ_{i} (r^{'}, λ) d r^{'} \\ + \int Δ D (r^{'}, λ) \nabla G (r_{d}, r_{j}, λ) \cdot \nabla Φ_{i} (r_{j}, r_{s}, λ) d r^{'}) \end{array}

Δ μ_{a} (r, λ) = \sum_{k = 1}^{N_{s}} ε_{k} (λ) c_{k} (r) .

c_{k} (r) = \sum_{j = 1}^{N_{v}} c_{k, j} φ (r)

φ (r) = {\begin{array}{l} 1, & if r \in V_{j} \\ 0, & if r \notin V_{j} . \end{array}

\begin{array}{l} Φ_{s} (λ) & = & a \frac{v}{D (λ)} \sum_{k, j = 1}^{N_{c}, N_{V}} (G (r_{d}, r_{j}, λ) Φ_{i} (r_{j}, r_{s}, λ) ε_{i} (λ) c_{k, j} \\ + & \nabla G (r_{d}, r_{j}, λ) \cdot \nabla Φ_{i} (r_{j}, r_{s}, λ) Δ D_{j} (λ)) . \end{array}

[\begin{matrix} Φ_{s} (λ_{1}) \\ Φ_{s} (λ_{2}) \\ ⋮ \\ Φ_{s} (λ_{N_{λ}}) \end{matrix}] = [\begin{matrix} ε_{1} (λ_{1}) K_{1}^{a} & ε_{2} (λ_{1}) K_{1}^{a} & \dots & ε_{N_{c}} (λ_{1}) K_{1}^{a} & K_{1}^{d} \\ ε_{1} (λ_{2}) K_{2}^{a} & ε_{2} (λ_{2}) K_{2}^{a} & \dots & ε_{N_{c}} (λ_{2}) K_{2}^{a} & K_{2}^{d} \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ ε_{1} (λ_{N_{λ}}) K_{N_{λ}}^{a} & ε_{2} (λ_{N_{λ}}) K_{N_{λ}}^{a} & \dots & ε_{N_{c}} (λ_{N_{λ}}) K_{N_{λ}}^{a} & K_{N_{λ}}^{d} \end{matrix}] [\begin{matrix} c_{1} \\ c_{2} \\ ⋮ \\ \begin{array}{l} c_{N_{c}} \\ Δ Ψ^{'} \end{array} \end{matrix}] \Leftrightarrow Φ_{s} = Kc

χ (x, y) = {\begin{array}{l} 1 & if (x, y) \in Ω, \\ 0 & if (x, y) \in ℱ \ Ω . \end{array}

c_{k} (x, y) = χ (x, y) c_{k}^{a} + [1 - χ (x, y)] c_{k}^{b}

χ (x, y) = H (𝒪 (x, y))

H_{ε} (x) = {\begin{array}{l} 1 & if x > ε, \\ 0 & if x < - ε . \\ \frac{1}{2} [1 + \frac{x}{ε} + \frac{1}{π} \sin (\frac{π x}{ε})] & if | x | \leq ε . \end{array}

𝒪 (x, y) = \sum_{i = 1}^{L} a_{i} p_{i} (x, y)

Φ_{s} = K (θ) = Kc (θ)

\hat{c} = arg min_{c} {‖ W (K (θ) - Φ_{s}) ‖}_{2}^{2}

σ_{m}^{2} = Ω (m) 10^{- \frac{{S N R}_{m}}{10}} .

{SNR}_{m} = 10 {log}_{10} (Ω (m) / \sqrt{Ω (m)}) .

ε = W (K (θ) - Φ) .

M (θ) = ε^{T} ε

J = [\frac{\partial ε (θ)}{\partial {c_{a}^{1}, \dots c_{b}^{l}, a}}]

(J^{T} J + ρ I) h = - J^{T} ε with ρ \geq 0

Φ = Kc + n

{M S E}_{k} = \frac{{‖ c_{k} - {\hat{c}}_{k} ‖}_{2}}{{‖ c_{k} ‖}_{2}}

D (S, G) = \frac{2 | S \cap G |}{| S | + | G |}

μ_{s}^{'} = Ψ {(\frac{λ}{λ_{0}})}^{- b} .

\hat{c_{ink}^{r}} = \hat{c_{ink}} / (\hat{c_{ink}} + \hat{c_{dye}})

Pixel based method
# λ	MSE HbO₂	MSE HbR	MSE Lipid	MSE H₂O	MSE D
8	0.075	0.030	0.048	0.010	0.052
176	0.062	0.021	0.034	0.015	0.030
PaLS method
# λ	MSE HbO₂	MSE HbR	MSE Lipid	MSE H₂O	MSE D
8	0.07	0.03	0.12	0.06	0.08
176	0.02	0.008	0.01	0.02	0.01

Pixel based method
# λ	D(S,G) HbO₂	D(S,G) HbR	D(S,G) Lipid	D(S,G) H₂O	D(S,G) D
8	0.12	0.088	0.089	0.65	0.8
176	0.554	0.1085	0.043	0.41	0.09

PaLS method
# λ	D(S,G)
8	0.60
126	0.99

Pixel based method
	D(S,G) Set 1		D(S,G) Set 2
# λ	Ink	Dye	Ink	Dye
6	0.143	0.113	0.139	0.145
126	0.142	0.114	0.145	0.140

	PaLS method
# λ	D(S,G) Set 1	D(S,G) Set 2
6	0.27	0.33
126	0.37	0.80

Experimental set 1, 10% ink and 90% dye
Fig.	# λ	Species	$\hat{c_{i}} [%]$	$\hat{c_{i}^{r}} [%]$	MSE
7(a)	6	Ink	1	4	1.8
7(a)	6	Dye	27	96	1.3
7(c)	126	Ink	17	16	2.8
7(c)	126	Dye	88	84	1.2