Abstract

The statistical independence between the distributions of different chromophores in tissue has previously been used for linear unmixing with independent component analysis (ICA). In this study, we propose exploiting this statistical property in a nonlinear model-based inversion method. The aim is to reduce the sensitivity of the inversion scheme to errors in the modelling of the fluence, and hence provide more accurate quantification of the concentration of independent chromophores. A gradient-based optimisation algorithm is used to minimise the error functional, which includes a term representing the mutual information between the chromophores in addition to the standard least-squares data error. Both numerical simulations and an experimental phantom study are conducted to demonstrate that, in the presence of experimental errors in the fluence model, the proposed inversion method results in more accurate estimation of the concentrations of independent chromophores compared to the standard model-based inversion.

Published by The Optical Society under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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References

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2017 (4)

F. M. Brochu, J. Brunker, J. Joseph, M. R. Tomaszewski, S. Morscher, and S. E. Bohndiek, “Towards quantitative evaluation of tissue absorption coefficients using light fluence correction in optoacoustic tomography,” IEEE Trans. Med. Imag. 36(1), 322–331 (2017).
[Crossref]

L. An, T. Saratoon, M. Fonseca, R. Ellwood, and B. Cox, “Exploiting statistical independence for quantitative photoacoustic tomography,” Proc. SPIE 10064, 1006419 (2017).
[Crossref]

R. Ellwood, O. Ogunlade, E. Zhang, P. Beard, and B. Cox, “Photoacoustic tomography using orthogonal Fabry-Pérot sensors,” J. Biomed. Opt. 22(4), 041009 (2017).
[Crossref]

M. Fonseca, E. Malone, F. Lucka, R. Ellwood, L. An, R. Arridge, P. Beard, and B. Cox, “Three-dimensional photoacoustic imaging and inversion for accurate quantification of chromophore distributions,” Proc. SPIE 10064, Photons Plus Ultrasound: Imaging and Sensing 2017, 1006415 (2017).
[Crossref]

2016 (3)

S. Arridge, P. Beard, M. Betcke, B. Cox, N. Huynh, F. Lucka, O. Ogunlade, and E. Zhang, “Accelerated high-resolution photoacoustic tomography via compressed sensing,” Phys. Med. Biol. 61(24), 8908 (2016).
[Crossref] [PubMed]

R. Ellwood, O. Ogunlade, E. Z. Zhang, P. C. Beard, and B. T. Cox, “Orthogonal Fabry-Pérot sensors for photoacoustic tomography,” Proc. SPIE 9708, 97082N (2016).

J. Weber, P. C. Beard, and S. E. Bohndiek, “Contrast agents for molecular photoacoustic imaging,” Nat. Methods 13, 639–650 (2016).
[Crossref] [PubMed]

2014 (2)

N. C. Deliolanis, A. Ale, S. Morscher, N. C. Burton, K. Schaefer, K. Radrich, D. Razansky, and V. Ntziachristos, “Deep-tissue reporter-gene imaging with fluorescence and optoacoustic tomography: A performance overview,” Mol. Imaging Biol. 16(5), 652–660 (2014).
[Crossref] [PubMed]

M. Schweiger and S. Arridge, “The Toast++ software suite for forward and inverse modeling in optical tomography,” J. Biomed. Opt. 19(4), 040801 (2014).
[Crossref] [PubMed]

2013 (4)

C. Huang, K. Wang, L. Nie, L. Wang, and M. Anastasio, “Full-wave iterative image reconstruction in photoacoustic tomography with acoustically inhomogeneous media,” IEEE Trans. Med. Imaging. 32(6), 1097–1110 (2013).
[Crossref] [PubMed]

T. Saratoon, T. Tarvainen, B. T. Cox, and S. R. Arridge, “A gradient-based method for quantitative photoacoustic tomography using the radiative transfer equation,” Invserse Prob. 29(7), 075006 (2013).
[Crossref]

A. Taruttis, M. Wildgruber, K. Kosanke, N. Beziere, K. Licha, R. Haag, M. Aichler, A. Walch, E. Rummeny, and V. Ntziachristos, “Multispectral optoacoustic tomography of myocardial infarction,” Photoacoustics 1(1), 3–8 (2013).
[Crossref] [PubMed]

S. L. Jacques, “Optical properties of biological tissues: a review,” Phys. Med. Biol. 58, R37–R61 (2013).
[Crossref] [PubMed]

2012 (3)

A. Buehler, E. Herzog, A. Ale, B. D. Smith, V. Ntziachristos, and D. Razansky, “High resolution tumor targeting in living mice by means of multispectral optoacoustic tomography,” EJNMMI Research 2(1), 1–6 (2012).
[Crossref]

G. Bal and K. Ren, “On multi-spectral quantitative photoacoustic tomography in diffusive regime,” Inverse Probl. 28(2), 025010 (2012).
[Crossref]

T. Tarvainen, B. T. Cox, J. P. Kaipio, and S. R. Arridge, “Reconstructing absorption and scattering distributions in quantitative photoacoustic tomography,” Invserse Prob. 28(8), 084009 (2012).
[Crossref]

2011 (2)

2010 (2)

J. Laufer, E. Zhang, and P. Beard, “Evaluation of absorbing chromophores used in tissue phantoms for quantitative photoacoustic spectroscopy and imaging,” IEEE J. Sel. Top. Quantum Electron. 16(3), 600–607 (2010).
[Crossref]

J. Laufer, B. Cox, E. Zhang, and P. Beard, “Quantitative determination of chromophore concentrations from 2D photoacoustic images using a nonlinear model-based inversion scheme,” Appl. Opt. 49(8), 1219–1233 (2010).
[Crossref] [PubMed]

2009 (5)

B. T. Cox, S. R. Arridge, and P. C. Beard, “Estimating chromophore distributions from multiwavelength photoacoustic images,” J. Opt. Soc. Am. A 26(2), 443–455 (2009).
[Crossref]

Y Sun, E Sobel, and H Jiang, “Quantitative three-dimensional photoacoustic tomography of the finger joints: an in vivo study,” J. Biomed. Opt. 14(6), 064002 (2009).
[Crossref]

V. D. Calhoun, J. Liu, and T. Adal, “A review of group ICA for fMRI data and ICA for joint inference of imaging, genetic, and ERP data,” NeuroImage 45(1, Supplement 1), S163–S172 (2009).
[Crossref]

C. Panagiotou, S. Somayajula, A. P. Gibson, M. Schweiger, R. M. Leahy, and S. R. Arridge, “Information theoretic regularization in diffuse optical tomography,” J. Opt. Soc. Am. 26(5), 1277–1290 (2009).
[Crossref]

P. Stefanov and G. Uhlmann, “Thermoacoustic tomography with variable sound speed,” Inverse Probl. 25(7), 075011 (2009).
[Crossref]

2008 (1)

2006 (1)

J. Wang and C. I. Chang, “Applications of independent component analysis in endmember extraction and abundance quantification for hyperspectral imagery,” IEEE Trans. Geosci. Remote Sens. 44(9), 2601–2616 (2006).
[Crossref]

2002 (1)

G. Paltauf, J. A. Viator, S. A. Prahl, and S. L. Jacques, “Iterative reconstruction algorithm for optoacoustic imaging,” J. Acoust. Soc. Am. 112(4), 1536–1544 (2002).
[Crossref] [PubMed]

2000 (1)

A. Hyvarinen and E. Oja, “Independent component analysis: algorithms and applications,” Neural Netw. 13, 411–430 (2000).
[Crossref] [PubMed]

1993 (1)

1992 (2)

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D. 60(1–4), 259–268 (1992).
[Crossref]

S. T. Flock, S. L. Jacques, B. C. Wilson, W. M. Star, and M. J. van Gement, “Optical properties of intralipid: A phantom medium for light propagation studies,” Lasers. Surg. Med. 12(5), 510–519 (1992).
[Crossref] [PubMed]

1991 (2)

A. M. Thompson, J. C. Brown, J. W. Kay, and D. M. Titterington, “A study of methods of choosing the smoothing parameter in image restoration by regularization,” IEEE Trans. Pattern Anal. Mach. Intell. 13(13),326–339 (1991).
[Crossref]

H. J. Van Staveren, C. J. Moes, J. van Marie, S. A. Prahl, and M. J. C. van Gemert, “Light scattering in lntralipid-10% in the wavelength range of 400–1100nm,” Appl. Opt. 30(31), 4507–4514 (1991).
[Crossref] [PubMed]

Adal, T.

V. D. Calhoun, J. Liu, and T. Adal, “A review of group ICA for fMRI data and ICA for joint inference of imaging, genetic, and ERP data,” NeuroImage 45(1, Supplement 1), S163–S172 (2009).
[Crossref]

Aichler, M.

A. Taruttis, M. Wildgruber, K. Kosanke, N. Beziere, K. Licha, R. Haag, M. Aichler, A. Walch, E. Rummeny, and V. Ntziachristos, “Multispectral optoacoustic tomography of myocardial infarction,” Photoacoustics 1(1), 3–8 (2013).
[Crossref] [PubMed]

Ale, A.

N. C. Deliolanis, A. Ale, S. Morscher, N. C. Burton, K. Schaefer, K. Radrich, D. Razansky, and V. Ntziachristos, “Deep-tissue reporter-gene imaging with fluorescence and optoacoustic tomography: A performance overview,” Mol. Imaging Biol. 16(5), 652–660 (2014).
[Crossref] [PubMed]

A. Buehler, E. Herzog, A. Ale, B. D. Smith, V. Ntziachristos, and D. Razansky, “High resolution tumor targeting in living mice by means of multispectral optoacoustic tomography,” EJNMMI Research 2(1), 1–6 (2012).
[Crossref]

An, L.

L. An, T. Saratoon, M. Fonseca, R. Ellwood, and B. Cox, “Exploiting statistical independence for quantitative photoacoustic tomography,” Proc. SPIE 10064, 1006419 (2017).
[Crossref]

M. Fonseca, E. Malone, F. Lucka, R. Ellwood, L. An, R. Arridge, P. Beard, and B. Cox, “Three-dimensional photoacoustic imaging and inversion for accurate quantification of chromophore distributions,” Proc. SPIE 10064, Photons Plus Ultrasound: Imaging and Sensing 2017, 1006415 (2017).
[Crossref]

Anastasio, M.

C. Huang, K. Wang, L. Nie, L. Wang, and M. Anastasio, “Full-wave iterative image reconstruction in photoacoustic tomography with acoustically inhomogeneous media,” IEEE Trans. Med. Imaging. 32(6), 1097–1110 (2013).
[Crossref] [PubMed]

Arridge, R.

M. Fonseca, E. Malone, F. Lucka, R. Ellwood, L. An, R. Arridge, P. Beard, and B. Cox, “Three-dimensional photoacoustic imaging and inversion for accurate quantification of chromophore distributions,” Proc. SPIE 10064, Photons Plus Ultrasound: Imaging and Sensing 2017, 1006415 (2017).
[Crossref]

Arridge, S.

S. Arridge, P. Beard, M. Betcke, B. Cox, N. Huynh, F. Lucka, O. Ogunlade, and E. Zhang, “Accelerated high-resolution photoacoustic tomography via compressed sensing,” Phys. Med. Biol. 61(24), 8908 (2016).
[Crossref] [PubMed]

M. Schweiger and S. Arridge, “The Toast++ software suite for forward and inverse modeling in optical tomography,” J. Biomed. Opt. 19(4), 040801 (2014).
[Crossref] [PubMed]

Arridge, S. R.

T. Saratoon, T. Tarvainen, B. T. Cox, and S. R. Arridge, “A gradient-based method for quantitative photoacoustic tomography using the radiative transfer equation,” Invserse Prob. 29(7), 075006 (2013).
[Crossref]

T. Tarvainen, B. T. Cox, J. P. Kaipio, and S. R. Arridge, “Reconstructing absorption and scattering distributions in quantitative photoacoustic tomography,” Invserse Prob. 28(8), 084009 (2012).
[Crossref]

B. T. Cox, S. R. Arridge, and P. C. Beard, “Estimating chromophore distributions from multiwavelength photoacoustic images,” J. Opt. Soc. Am. A 26(2), 443–455 (2009).
[Crossref]

C. Panagiotou, S. Somayajula, A. P. Gibson, M. Schweiger, R. M. Leahy, and S. R. Arridge, “Information theoretic regularization in diffuse optical tomography,” J. Opt. Soc. Am. 26(5), 1277–1290 (2009).
[Crossref]

Bal, G.

G. Bal and K. Ren, “On multi-spectral quantitative photoacoustic tomography in diffusive regime,” Inverse Probl. 28(2), 025010 (2012).
[Crossref]

Beard, P.

M. Fonseca, E. Malone, F. Lucka, R. Ellwood, L. An, R. Arridge, P. Beard, and B. Cox, “Three-dimensional photoacoustic imaging and inversion for accurate quantification of chromophore distributions,” Proc. SPIE 10064, Photons Plus Ultrasound: Imaging and Sensing 2017, 1006415 (2017).
[Crossref]

R. Ellwood, O. Ogunlade, E. Zhang, P. Beard, and B. Cox, “Photoacoustic tomography using orthogonal Fabry-Pérot sensors,” J. Biomed. Opt. 22(4), 041009 (2017).
[Crossref]

S. Arridge, P. Beard, M. Betcke, B. Cox, N. Huynh, F. Lucka, O. Ogunlade, and E. Zhang, “Accelerated high-resolution photoacoustic tomography via compressed sensing,” Phys. Med. Biol. 61(24), 8908 (2016).
[Crossref] [PubMed]

P. Beard, “Biomedical photoacoustic imaging,” Interface Focus 1, 602–631 (2011).
[Crossref]

J. Laufer, B. Cox, E. Zhang, and P. Beard, “Quantitative determination of chromophore concentrations from 2D photoacoustic images using a nonlinear model-based inversion scheme,” Appl. Opt. 49(8), 1219–1233 (2010).
[Crossref] [PubMed]

J. Laufer, E. Zhang, and P. Beard, “Evaluation of absorbing chromophores used in tissue phantoms for quantitative photoacoustic spectroscopy and imaging,” IEEE J. Sel. Top. Quantum Electron. 16(3), 600–607 (2010).
[Crossref]

E. Zhang, J. Laufer, and P. Beard, “Backward-mode multiwavelength photoacoustic scanner using a planar Fabry-Perot polymer film ultrasound sensor for high-resolution threedimensional imaging of biological tissues,” Appl. Opt. 47, 561–577 (2008).
[Crossref] [PubMed]

Beard, P. C.

J. Weber, P. C. Beard, and S. E. Bohndiek, “Contrast agents for molecular photoacoustic imaging,” Nat. Methods 13, 639–650 (2016).
[Crossref] [PubMed]

R. Ellwood, O. Ogunlade, E. Z. Zhang, P. C. Beard, and B. T. Cox, “Orthogonal Fabry-Pérot sensors for photoacoustic tomography,” Proc. SPIE 9708, 97082N (2016).

B. T. Cox, S. R. Arridge, and P. C. Beard, “Estimating chromophore distributions from multiwavelength photoacoustic images,” J. Opt. Soc. Am. A 26(2), 443–455 (2009).
[Crossref]

Betcke, M.

S. Arridge, P. Beard, M. Betcke, B. Cox, N. Huynh, F. Lucka, O. Ogunlade, and E. Zhang, “Accelerated high-resolution photoacoustic tomography via compressed sensing,” Phys. Med. Biol. 61(24), 8908 (2016).
[Crossref] [PubMed]

Beziere, N.

A. Taruttis, M. Wildgruber, K. Kosanke, N. Beziere, K. Licha, R. Haag, M. Aichler, A. Walch, E. Rummeny, and V. Ntziachristos, “Multispectral optoacoustic tomography of myocardial infarction,” Photoacoustics 1(1), 3–8 (2013).
[Crossref] [PubMed]

Bohndiek, S. E.

F. M. Brochu, J. Brunker, J. Joseph, M. R. Tomaszewski, S. Morscher, and S. E. Bohndiek, “Towards quantitative evaluation of tissue absorption coefficients using light fluence correction in optoacoustic tomography,” IEEE Trans. Med. Imag. 36(1), 322–331 (2017).
[Crossref]

J. Weber, P. C. Beard, and S. E. Bohndiek, “Contrast agents for molecular photoacoustic imaging,” Nat. Methods 13, 639–650 (2016).
[Crossref] [PubMed]

Brochu, F. M.

F. M. Brochu, J. Brunker, J. Joseph, M. R. Tomaszewski, S. Morscher, and S. E. Bohndiek, “Towards quantitative evaluation of tissue absorption coefficients using light fluence correction in optoacoustic tomography,” IEEE Trans. Med. Imag. 36(1), 322–331 (2017).
[Crossref]

Brown, J. C.

A. M. Thompson, J. C. Brown, J. W. Kay, and D. M. Titterington, “A study of methods of choosing the smoothing parameter in image restoration by regularization,” IEEE Trans. Pattern Anal. Mach. Intell. 13(13),326–339 (1991).
[Crossref]

Brunker, J.

F. M. Brochu, J. Brunker, J. Joseph, M. R. Tomaszewski, S. Morscher, and S. E. Bohndiek, “Towards quantitative evaluation of tissue absorption coefficients using light fluence correction in optoacoustic tomography,” IEEE Trans. Med. Imag. 36(1), 322–331 (2017).
[Crossref]

Buehler, A.

A. Buehler, E. Herzog, A. Ale, B. D. Smith, V. Ntziachristos, and D. Razansky, “High resolution tumor targeting in living mice by means of multispectral optoacoustic tomography,” EJNMMI Research 2(1), 1–6 (2012).
[Crossref]

J. Glatz, N. C. Deliolanis, A. Buehler, D. Razansky, and V. Ntziachristos, “Blind source unmixing in multi-spectral optoacoustic tomography,” Opt. Express 19(4), 3175–3184 (2011).
[Crossref] [PubMed]

Burton, N. C.

N. C. Deliolanis, A. Ale, S. Morscher, N. C. Burton, K. Schaefer, K. Radrich, D. Razansky, and V. Ntziachristos, “Deep-tissue reporter-gene imaging with fluorescence and optoacoustic tomography: A performance overview,” Mol. Imaging Biol. 16(5), 652–660 (2014).
[Crossref] [PubMed]

Calhoun, V. D.

V. D. Calhoun, J. Liu, and T. Adal, “A review of group ICA for fMRI data and ICA for joint inference of imaging, genetic, and ERP data,” NeuroImage 45(1, Supplement 1), S163–S172 (2009).
[Crossref]

Chang, C. I.

J. Wang and C. I. Chang, “Applications of independent component analysis in endmember extraction and abundance quantification for hyperspectral imagery,” IEEE Trans. Geosci. Remote Sens. 44(9), 2601–2616 (2006).
[Crossref]

Chylek, P.

Cox, B.

L. An, T. Saratoon, M. Fonseca, R. Ellwood, and B. Cox, “Exploiting statistical independence for quantitative photoacoustic tomography,” Proc. SPIE 10064, 1006419 (2017).
[Crossref]

R. Ellwood, O. Ogunlade, E. Zhang, P. Beard, and B. Cox, “Photoacoustic tomography using orthogonal Fabry-Pérot sensors,” J. Biomed. Opt. 22(4), 041009 (2017).
[Crossref]

M. Fonseca, E. Malone, F. Lucka, R. Ellwood, L. An, R. Arridge, P. Beard, and B. Cox, “Three-dimensional photoacoustic imaging and inversion for accurate quantification of chromophore distributions,” Proc. SPIE 10064, Photons Plus Ultrasound: Imaging and Sensing 2017, 1006415 (2017).
[Crossref]

S. Arridge, P. Beard, M. Betcke, B. Cox, N. Huynh, F. Lucka, O. Ogunlade, and E. Zhang, “Accelerated high-resolution photoacoustic tomography via compressed sensing,” Phys. Med. Biol. 61(24), 8908 (2016).
[Crossref] [PubMed]

J. Laufer, B. Cox, E. Zhang, and P. Beard, “Quantitative determination of chromophore concentrations from 2D photoacoustic images using a nonlinear model-based inversion scheme,” Appl. Opt. 49(8), 1219–1233 (2010).
[Crossref] [PubMed]

Cox, B. T.

R. Ellwood, O. Ogunlade, E. Z. Zhang, P. C. Beard, and B. T. Cox, “Orthogonal Fabry-Pérot sensors for photoacoustic tomography,” Proc. SPIE 9708, 97082N (2016).

T. Saratoon, T. Tarvainen, B. T. Cox, and S. R. Arridge, “A gradient-based method for quantitative photoacoustic tomography using the radiative transfer equation,” Invserse Prob. 29(7), 075006 (2013).
[Crossref]

T. Tarvainen, B. T. Cox, J. P. Kaipio, and S. R. Arridge, “Reconstructing absorption and scattering distributions in quantitative photoacoustic tomography,” Invserse Prob. 28(8), 084009 (2012).
[Crossref]

B. T. Cox, S. R. Arridge, and P. C. Beard, “Estimating chromophore distributions from multiwavelength photoacoustic images,” J. Opt. Soc. Am. A 26(2), 443–455 (2009).
[Crossref]

Deliolanis, N. C.

N. C. Deliolanis, A. Ale, S. Morscher, N. C. Burton, K. Schaefer, K. Radrich, D. Razansky, and V. Ntziachristos, “Deep-tissue reporter-gene imaging with fluorescence and optoacoustic tomography: A performance overview,” Mol. Imaging Biol. 16(5), 652–660 (2014).
[Crossref] [PubMed]

J. Glatz, N. C. Deliolanis, A. Buehler, D. Razansky, and V. Ntziachristos, “Blind source unmixing in multi-spectral optoacoustic tomography,” Opt. Express 19(4), 3175–3184 (2011).
[Crossref] [PubMed]

Ellwood, R.

M. Fonseca, E. Malone, F. Lucka, R. Ellwood, L. An, R. Arridge, P. Beard, and B. Cox, “Three-dimensional photoacoustic imaging and inversion for accurate quantification of chromophore distributions,” Proc. SPIE 10064, Photons Plus Ultrasound: Imaging and Sensing 2017, 1006415 (2017).
[Crossref]

R. Ellwood, O. Ogunlade, E. Zhang, P. Beard, and B. Cox, “Photoacoustic tomography using orthogonal Fabry-Pérot sensors,” J. Biomed. Opt. 22(4), 041009 (2017).
[Crossref]

L. An, T. Saratoon, M. Fonseca, R. Ellwood, and B. Cox, “Exploiting statistical independence for quantitative photoacoustic tomography,” Proc. SPIE 10064, 1006419 (2017).
[Crossref]

R. Ellwood, O. Ogunlade, E. Z. Zhang, P. C. Beard, and B. T. Cox, “Orthogonal Fabry-Pérot sensors for photoacoustic tomography,” Proc. SPIE 9708, 97082N (2016).

Fatemi, E.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D. 60(1–4), 259–268 (1992).
[Crossref]

Flock, S. T.

S. T. Flock, S. L. Jacques, B. C. Wilson, W. M. Star, and M. J. van Gement, “Optical properties of intralipid: A phantom medium for light propagation studies,” Lasers. Surg. Med. 12(5), 510–519 (1992).
[Crossref] [PubMed]

Fonseca, M.

M. Fonseca, E. Malone, F. Lucka, R. Ellwood, L. An, R. Arridge, P. Beard, and B. Cox, “Three-dimensional photoacoustic imaging and inversion for accurate quantification of chromophore distributions,” Proc. SPIE 10064, Photons Plus Ultrasound: Imaging and Sensing 2017, 1006415 (2017).
[Crossref]

L. An, T. Saratoon, M. Fonseca, R. Ellwood, and B. Cox, “Exploiting statistical independence for quantitative photoacoustic tomography,” Proc. SPIE 10064, 1006419 (2017).
[Crossref]

Gibson, A. P.

C. Panagiotou, S. Somayajula, A. P. Gibson, M. Schweiger, R. M. Leahy, and S. R. Arridge, “Information theoretic regularization in diffuse optical tomography,” J. Opt. Soc. Am. 26(5), 1277–1290 (2009).
[Crossref]

Glatz, J.

Haag, R.

A. Taruttis, M. Wildgruber, K. Kosanke, N. Beziere, K. Licha, R. Haag, M. Aichler, A. Walch, E. Rummeny, and V. Ntziachristos, “Multispectral optoacoustic tomography of myocardial infarction,” Photoacoustics 1(1), 3–8 (2013).
[Crossref] [PubMed]

Herzog, E.

A. Buehler, E. Herzog, A. Ale, B. D. Smith, V. Ntziachristos, and D. Razansky, “High resolution tumor targeting in living mice by means of multispectral optoacoustic tomography,” EJNMMI Research 2(1), 1–6 (2012).
[Crossref]

Huang, C.

C. Huang, K. Wang, L. Nie, L. Wang, and M. Anastasio, “Full-wave iterative image reconstruction in photoacoustic tomography with acoustically inhomogeneous media,” IEEE Trans. Med. Imaging. 32(6), 1097–1110 (2013).
[Crossref] [PubMed]

Huynh, N.

S. Arridge, P. Beard, M. Betcke, B. Cox, N. Huynh, F. Lucka, O. Ogunlade, and E. Zhang, “Accelerated high-resolution photoacoustic tomography via compressed sensing,” Phys. Med. Biol. 61(24), 8908 (2016).
[Crossref] [PubMed]

Hyvarinen, A.

A. Hyvarinen and E. Oja, “Independent component analysis: algorithms and applications,” Neural Netw. 13, 411–430 (2000).
[Crossref] [PubMed]

A. Hyvarinen, J. Karhunen, and E. Oja, Independent Component Analysis (John Wiley & Sons, 2004), Chap. 2.

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S. L. Jacques, “Optical properties of biological tissues: a review,” Phys. Med. Biol. 58, R37–R61 (2013).
[Crossref] [PubMed]

G. Paltauf, J. A. Viator, S. A. Prahl, and S. L. Jacques, “Iterative reconstruction algorithm for optoacoustic imaging,” J. Acoust. Soc. Am. 112(4), 1536–1544 (2002).
[Crossref] [PubMed]

S. T. Flock, S. L. Jacques, B. C. Wilson, W. M. Star, and M. J. van Gement, “Optical properties of intralipid: A phantom medium for light propagation studies,” Lasers. Surg. Med. 12(5), 510–519 (1992).
[Crossref] [PubMed]

Jiang, H

Y Sun, E Sobel, and H Jiang, “Quantitative three-dimensional photoacoustic tomography of the finger joints: an in vivo study,” J. Biomed. Opt. 14(6), 064002 (2009).
[Crossref]

Joseph, J.

F. M. Brochu, J. Brunker, J. Joseph, M. R. Tomaszewski, S. Morscher, and S. E. Bohndiek, “Towards quantitative evaluation of tissue absorption coefficients using light fluence correction in optoacoustic tomography,” IEEE Trans. Med. Imag. 36(1), 322–331 (2017).
[Crossref]

Kaipio, J. P.

T. Tarvainen, B. T. Cox, J. P. Kaipio, and S. R. Arridge, “Reconstructing absorption and scattering distributions in quantitative photoacoustic tomography,” Invserse Prob. 28(8), 084009 (2012).
[Crossref]

Karhunen, J.

A. Hyvarinen, J. Karhunen, and E. Oja, Independent Component Analysis (John Wiley & Sons, 2004), Chap. 2.

Kay, J. W.

A. M. Thompson, J. C. Brown, J. W. Kay, and D. M. Titterington, “A study of methods of choosing the smoothing parameter in image restoration by regularization,” IEEE Trans. Pattern Anal. Mach. Intell. 13(13),326–339 (1991).
[Crossref]

Kosanke, K.

A. Taruttis, M. Wildgruber, K. Kosanke, N. Beziere, K. Licha, R. Haag, M. Aichler, A. Walch, E. Rummeny, and V. Ntziachristos, “Multispectral optoacoustic tomography of myocardial infarction,” Photoacoustics 1(1), 3–8 (2013).
[Crossref] [PubMed]

Kou, L.

Labrie, D.

Laufer, J.

Leahy, R. M.

C. Panagiotou, S. Somayajula, A. P. Gibson, M. Schweiger, R. M. Leahy, and S. R. Arridge, “Information theoretic regularization in diffuse optical tomography,” J. Opt. Soc. Am. 26(5), 1277–1290 (2009).
[Crossref]

Licha, K.

A. Taruttis, M. Wildgruber, K. Kosanke, N. Beziere, K. Licha, R. Haag, M. Aichler, A. Walch, E. Rummeny, and V. Ntziachristos, “Multispectral optoacoustic tomography of myocardial infarction,” Photoacoustics 1(1), 3–8 (2013).
[Crossref] [PubMed]

Liu, J.

V. D. Calhoun, J. Liu, and T. Adal, “A review of group ICA for fMRI data and ICA for joint inference of imaging, genetic, and ERP data,” NeuroImage 45(1, Supplement 1), S163–S172 (2009).
[Crossref]

Lucka, F.

M. Fonseca, E. Malone, F. Lucka, R. Ellwood, L. An, R. Arridge, P. Beard, and B. Cox, “Three-dimensional photoacoustic imaging and inversion for accurate quantification of chromophore distributions,” Proc. SPIE 10064, Photons Plus Ultrasound: Imaging and Sensing 2017, 1006415 (2017).
[Crossref]

S. Arridge, P. Beard, M. Betcke, B. Cox, N. Huynh, F. Lucka, O. Ogunlade, and E. Zhang, “Accelerated high-resolution photoacoustic tomography via compressed sensing,” Phys. Med. Biol. 61(24), 8908 (2016).
[Crossref] [PubMed]

Malone, E.

M. Fonseca, E. Malone, F. Lucka, R. Ellwood, L. An, R. Arridge, P. Beard, and B. Cox, “Three-dimensional photoacoustic imaging and inversion for accurate quantification of chromophore distributions,” Proc. SPIE 10064, Photons Plus Ultrasound: Imaging and Sensing 2017, 1006415 (2017).
[Crossref]

Moes, C. J.

Morscher, S.

F. M. Brochu, J. Brunker, J. Joseph, M. R. Tomaszewski, S. Morscher, and S. E. Bohndiek, “Towards quantitative evaluation of tissue absorption coefficients using light fluence correction in optoacoustic tomography,” IEEE Trans. Med. Imag. 36(1), 322–331 (2017).
[Crossref]

N. C. Deliolanis, A. Ale, S. Morscher, N. C. Burton, K. Schaefer, K. Radrich, D. Razansky, and V. Ntziachristos, “Deep-tissue reporter-gene imaging with fluorescence and optoacoustic tomography: A performance overview,” Mol. Imaging Biol. 16(5), 652–660 (2014).
[Crossref] [PubMed]

Nie, L.

C. Huang, K. Wang, L. Nie, L. Wang, and M. Anastasio, “Full-wave iterative image reconstruction in photoacoustic tomography with acoustically inhomogeneous media,” IEEE Trans. Med. Imaging. 32(6), 1097–1110 (2013).
[Crossref] [PubMed]

Nocedal, J.

J. Nocedal and S. Wright, Numerical Optimization (Springer Science & Business Media, 2006).

Ntziachristos, V.

N. C. Deliolanis, A. Ale, S. Morscher, N. C. Burton, K. Schaefer, K. Radrich, D. Razansky, and V. Ntziachristos, “Deep-tissue reporter-gene imaging with fluorescence and optoacoustic tomography: A performance overview,” Mol. Imaging Biol. 16(5), 652–660 (2014).
[Crossref] [PubMed]

A. Taruttis, M. Wildgruber, K. Kosanke, N. Beziere, K. Licha, R. Haag, M. Aichler, A. Walch, E. Rummeny, and V. Ntziachristos, “Multispectral optoacoustic tomography of myocardial infarction,” Photoacoustics 1(1), 3–8 (2013).
[Crossref] [PubMed]

A. Buehler, E. Herzog, A. Ale, B. D. Smith, V. Ntziachristos, and D. Razansky, “High resolution tumor targeting in living mice by means of multispectral optoacoustic tomography,” EJNMMI Research 2(1), 1–6 (2012).
[Crossref]

J. Glatz, N. C. Deliolanis, A. Buehler, D. Razansky, and V. Ntziachristos, “Blind source unmixing in multi-spectral optoacoustic tomography,” Opt. Express 19(4), 3175–3184 (2011).
[Crossref] [PubMed]

Ogunlade, O.

R. Ellwood, O. Ogunlade, E. Zhang, P. Beard, and B. Cox, “Photoacoustic tomography using orthogonal Fabry-Pérot sensors,” J. Biomed. Opt. 22(4), 041009 (2017).
[Crossref]

R. Ellwood, O. Ogunlade, E. Z. Zhang, P. C. Beard, and B. T. Cox, “Orthogonal Fabry-Pérot sensors for photoacoustic tomography,” Proc. SPIE 9708, 97082N (2016).

S. Arridge, P. Beard, M. Betcke, B. Cox, N. Huynh, F. Lucka, O. Ogunlade, and E. Zhang, “Accelerated high-resolution photoacoustic tomography via compressed sensing,” Phys. Med. Biol. 61(24), 8908 (2016).
[Crossref] [PubMed]

Oja, E.

A. Hyvarinen and E. Oja, “Independent component analysis: algorithms and applications,” Neural Netw. 13, 411–430 (2000).
[Crossref] [PubMed]

A. Hyvarinen, J. Karhunen, and E. Oja, Independent Component Analysis (John Wiley & Sons, 2004), Chap. 2.

Osher, S.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D. 60(1–4), 259–268 (1992).
[Crossref]

Paltauf, G.

G. Paltauf, J. A. Viator, S. A. Prahl, and S. L. Jacques, “Iterative reconstruction algorithm for optoacoustic imaging,” J. Acoust. Soc. Am. 112(4), 1536–1544 (2002).
[Crossref] [PubMed]

Panagiotou, C.

C. Panagiotou, S. Somayajula, A. P. Gibson, M. Schweiger, R. M. Leahy, and S. R. Arridge, “Information theoretic regularization in diffuse optical tomography,” J. Opt. Soc. Am. 26(5), 1277–1290 (2009).
[Crossref]

Prahl, S. A.

G. Paltauf, J. A. Viator, S. A. Prahl, and S. L. Jacques, “Iterative reconstruction algorithm for optoacoustic imaging,” J. Acoust. Soc. Am. 112(4), 1536–1544 (2002).
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H. J. Van Staveren, C. J. Moes, J. van Marie, S. A. Prahl, and M. J. C. van Gemert, “Light scattering in lntralipid-10% in the wavelength range of 400–1100nm,” Appl. Opt. 30(31), 4507–4514 (1991).
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Radrich, K.

N. C. Deliolanis, A. Ale, S. Morscher, N. C. Burton, K. Schaefer, K. Radrich, D. Razansky, and V. Ntziachristos, “Deep-tissue reporter-gene imaging with fluorescence and optoacoustic tomography: A performance overview,” Mol. Imaging Biol. 16(5), 652–660 (2014).
[Crossref] [PubMed]

Razansky, D.

N. C. Deliolanis, A. Ale, S. Morscher, N. C. Burton, K. Schaefer, K. Radrich, D. Razansky, and V. Ntziachristos, “Deep-tissue reporter-gene imaging with fluorescence and optoacoustic tomography: A performance overview,” Mol. Imaging Biol. 16(5), 652–660 (2014).
[Crossref] [PubMed]

A. Buehler, E. Herzog, A. Ale, B. D. Smith, V. Ntziachristos, and D. Razansky, “High resolution tumor targeting in living mice by means of multispectral optoacoustic tomography,” EJNMMI Research 2(1), 1–6 (2012).
[Crossref]

J. Glatz, N. C. Deliolanis, A. Buehler, D. Razansky, and V. Ntziachristos, “Blind source unmixing in multi-spectral optoacoustic tomography,” Opt. Express 19(4), 3175–3184 (2011).
[Crossref] [PubMed]

Ren, K.

G. Bal and K. Ren, “On multi-spectral quantitative photoacoustic tomography in diffusive regime,” Inverse Probl. 28(2), 025010 (2012).
[Crossref]

Rudin, L. I.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D. 60(1–4), 259–268 (1992).
[Crossref]

Rummeny, E.

A. Taruttis, M. Wildgruber, K. Kosanke, N. Beziere, K. Licha, R. Haag, M. Aichler, A. Walch, E. Rummeny, and V. Ntziachristos, “Multispectral optoacoustic tomography of myocardial infarction,” Photoacoustics 1(1), 3–8 (2013).
[Crossref] [PubMed]

Saratoon, T.

L. An, T. Saratoon, M. Fonseca, R. Ellwood, and B. Cox, “Exploiting statistical independence for quantitative photoacoustic tomography,” Proc. SPIE 10064, 1006419 (2017).
[Crossref]

T. Saratoon, T. Tarvainen, B. T. Cox, and S. R. Arridge, “A gradient-based method for quantitative photoacoustic tomography using the radiative transfer equation,” Invserse Prob. 29(7), 075006 (2013).
[Crossref]

Schaefer, K.

N. C. Deliolanis, A. Ale, S. Morscher, N. C. Burton, K. Schaefer, K. Radrich, D. Razansky, and V. Ntziachristos, “Deep-tissue reporter-gene imaging with fluorescence and optoacoustic tomography: A performance overview,” Mol. Imaging Biol. 16(5), 652–660 (2014).
[Crossref] [PubMed]

Schechner, Y. Y.

S. Shwartz, M. Zibulevsky, and Y. Y. Schechner, “ICA using kernel entropy estimation with NlogN complexity,” in Independent Component Analysis and Blind Signal Separation: Fifth International Conference, ICA 2004, Granada, Spain, September 22–24, 2004. Proceedings, C. G. Puntonet and A. Prieto, eds. (Springer, 2004), pp. 422–429.
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Schweiger, M.

M. Schweiger and S. Arridge, “The Toast++ software suite for forward and inverse modeling in optical tomography,” J. Biomed. Opt. 19(4), 040801 (2014).
[Crossref] [PubMed]

C. Panagiotou, S. Somayajula, A. P. Gibson, M. Schweiger, R. M. Leahy, and S. R. Arridge, “Information theoretic regularization in diffuse optical tomography,” J. Opt. Soc. Am. 26(5), 1277–1290 (2009).
[Crossref]

Shwartz, S.

S. Shwartz, M. Zibulevsky, and Y. Y. Schechner, “ICA using kernel entropy estimation with NlogN complexity,” in Independent Component Analysis and Blind Signal Separation: Fifth International Conference, ICA 2004, Granada, Spain, September 22–24, 2004. Proceedings, C. G. Puntonet and A. Prieto, eds. (Springer, 2004), pp. 422–429.
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B. W. Silverman, Density Estimation for Statistics and Data Analysis (Chapman and Hall1986).
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Smith, B. D.

A. Buehler, E. Herzog, A. Ale, B. D. Smith, V. Ntziachristos, and D. Razansky, “High resolution tumor targeting in living mice by means of multispectral optoacoustic tomography,” EJNMMI Research 2(1), 1–6 (2012).
[Crossref]

Sobel, E

Y Sun, E Sobel, and H Jiang, “Quantitative three-dimensional photoacoustic tomography of the finger joints: an in vivo study,” J. Biomed. Opt. 14(6), 064002 (2009).
[Crossref]

Somayajula, S.

C. Panagiotou, S. Somayajula, A. P. Gibson, M. Schweiger, R. M. Leahy, and S. R. Arridge, “Information theoretic regularization in diffuse optical tomography,” J. Opt. Soc. Am. 26(5), 1277–1290 (2009).
[Crossref]

Star, W. M.

S. T. Flock, S. L. Jacques, B. C. Wilson, W. M. Star, and M. J. van Gement, “Optical properties of intralipid: A phantom medium for light propagation studies,” Lasers. Surg. Med. 12(5), 510–519 (1992).
[Crossref] [PubMed]

Stefanov, P.

P. Stefanov and G. Uhlmann, “Thermoacoustic tomography with variable sound speed,” Inverse Probl. 25(7), 075011 (2009).
[Crossref]

Sun, Y

Y Sun, E Sobel, and H Jiang, “Quantitative three-dimensional photoacoustic tomography of the finger joints: an in vivo study,” J. Biomed. Opt. 14(6), 064002 (2009).
[Crossref]

Taruttis, A.

A. Taruttis, M. Wildgruber, K. Kosanke, N. Beziere, K. Licha, R. Haag, M. Aichler, A. Walch, E. Rummeny, and V. Ntziachristos, “Multispectral optoacoustic tomography of myocardial infarction,” Photoacoustics 1(1), 3–8 (2013).
[Crossref] [PubMed]

Tarvainen, T.

T. Saratoon, T. Tarvainen, B. T. Cox, and S. R. Arridge, “A gradient-based method for quantitative photoacoustic tomography using the radiative transfer equation,” Invserse Prob. 29(7), 075006 (2013).
[Crossref]

T. Tarvainen, B. T. Cox, J. P. Kaipio, and S. R. Arridge, “Reconstructing absorption and scattering distributions in quantitative photoacoustic tomography,” Invserse Prob. 28(8), 084009 (2012).
[Crossref]

Thompson, A. M.

A. M. Thompson, J. C. Brown, J. W. Kay, and D. M. Titterington, “A study of methods of choosing the smoothing parameter in image restoration by regularization,” IEEE Trans. Pattern Anal. Mach. Intell. 13(13),326–339 (1991).
[Crossref]

Titterington, D. M.

A. M. Thompson, J. C. Brown, J. W. Kay, and D. M. Titterington, “A study of methods of choosing the smoothing parameter in image restoration by regularization,” IEEE Trans. Pattern Anal. Mach. Intell. 13(13),326–339 (1991).
[Crossref]

Tomaszewski, M. R.

F. M. Brochu, J. Brunker, J. Joseph, M. R. Tomaszewski, S. Morscher, and S. E. Bohndiek, “Towards quantitative evaluation of tissue absorption coefficients using light fluence correction in optoacoustic tomography,” IEEE Trans. Med. Imag. 36(1), 322–331 (2017).
[Crossref]

Uhlmann, G.

P. Stefanov and G. Uhlmann, “Thermoacoustic tomography with variable sound speed,” Inverse Probl. 25(7), 075011 (2009).
[Crossref]

van Gement, M. J.

S. T. Flock, S. L. Jacques, B. C. Wilson, W. M. Star, and M. J. van Gement, “Optical properties of intralipid: A phantom medium for light propagation studies,” Lasers. Surg. Med. 12(5), 510–519 (1992).
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van Gemert, M. J. C.

van Marie, J.

Van Staveren, H. J.

Viator, J. A.

G. Paltauf, J. A. Viator, S. A. Prahl, and S. L. Jacques, “Iterative reconstruction algorithm for optoacoustic imaging,” J. Acoust. Soc. Am. 112(4), 1536–1544 (2002).
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C. R. Vogel, Computational Methods for Inverse Problems (Society for Industrial and Applied Mathematics (SIAM): Philadelphia, PA: 2002). Chap. 7.
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A. Taruttis, M. Wildgruber, K. Kosanke, N. Beziere, K. Licha, R. Haag, M. Aichler, A. Walch, E. Rummeny, and V. Ntziachristos, “Multispectral optoacoustic tomography of myocardial infarction,” Photoacoustics 1(1), 3–8 (2013).
[Crossref] [PubMed]

Wang, J.

J. Wang and C. I. Chang, “Applications of independent component analysis in endmember extraction and abundance quantification for hyperspectral imagery,” IEEE Trans. Geosci. Remote Sens. 44(9), 2601–2616 (2006).
[Crossref]

Wang, K.

C. Huang, K. Wang, L. Nie, L. Wang, and M. Anastasio, “Full-wave iterative image reconstruction in photoacoustic tomography with acoustically inhomogeneous media,” IEEE Trans. Med. Imaging. 32(6), 1097–1110 (2013).
[Crossref] [PubMed]

Wang, L.

C. Huang, K. Wang, L. Nie, L. Wang, and M. Anastasio, “Full-wave iterative image reconstruction in photoacoustic tomography with acoustically inhomogeneous media,” IEEE Trans. Med. Imaging. 32(6), 1097–1110 (2013).
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J. Weber, P. C. Beard, and S. E. Bohndiek, “Contrast agents for molecular photoacoustic imaging,” Nat. Methods 13, 639–650 (2016).
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A. Taruttis, M. Wildgruber, K. Kosanke, N. Beziere, K. Licha, R. Haag, M. Aichler, A. Walch, E. Rummeny, and V. Ntziachristos, “Multispectral optoacoustic tomography of myocardial infarction,” Photoacoustics 1(1), 3–8 (2013).
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Wilson, B. C.

S. T. Flock, S. L. Jacques, B. C. Wilson, W. M. Star, and M. J. van Gement, “Optical properties of intralipid: A phantom medium for light propagation studies,” Lasers. Surg. Med. 12(5), 510–519 (1992).
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Wright, S.

J. Nocedal and S. Wright, Numerical Optimization (Springer Science & Business Media, 2006).

Zhang, E.

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Figures (6)

Fig. 1
Fig. 1 Experimental set-up and phantom structure. The four tubes containing CuCl2 or NiCl2 are fixed in a vertical column and submerged in the India ink and Intralipid solution. Two orthogonal Fabry-Perot interferometer sensors are used for increased detection aperture. The fibre tip at the top of the phantom delivers the pulsed excitation beam.
Fig. 2
Fig. 2 (a) The absorption coefficients of 36gL−1 CuCl2 (squares) and 399gL−1 NiCl2 (circles). (b) The absorption coefficient of the background solution (crosses), which is a sum of the absorption of water [20] (dotted) and the India ink (dashed). (c) The scattering amplitude of 1% Intralipid as a function of wavelength [21]. A spectrophotometer (Lambda 750S, Perkin Elmer) was used to measure the transmittance of CuCl2, NiCl2 and India ink in order to determine their absorption spectra. Reprinted from [22].
Fig. 3
Fig. 3 The 2D cross-sectional slices of the 3D reconstructed photoacoustic images which are used for the optical inversion at wavelengths 750, 830 and 890nm. The size of this region of interest is 12×12mm2 and the element spacing is 166μm. As expected, the intensity of the tubes decreases with depth for all wavelengths due to the decay of the fluence.
Fig. 4
Fig. 4 Diagram of the 2D numerical phantom. The phantom contains regions with different concentrations of CuCl2 and NiCl2. The background region contains India ink and Intralipid.
Fig. 5
Fig. 5 The average errors of the estimated concentrations of CuCl2 and NiCl2 at the insertions as a function of errors in (a) the beam diameter or (b) the scattering amplitude in the inversion. As expected, the quantification errors increase for larger errors in the beam diameter or scattering amplitude. However, the inversions using εd+MI (asterisks) result in smaller errors compared to using εd (circles) for all data points. The individual errors for CuCl2 and NiCl2 show similar general trends as the average of the two. The average errors outside the ROI are <2% for inversions using both εd and εd+MI. Reprinted from [22].
Fig. 6
Fig. 6 (a) The estimated concentrations of CuCl2 (top row) and NiCl2 (bottom row) in units of gL−1. The results from the inversions using εd (left column) show overestimation of the the upper tubes and large cross-talk errors, while using εd+MI results in more accurate quantification without cross-talk errors. The true concentrations are shown in the column to the right for comparison. (b) The estimated concentration of the CuCl2 (top) and NiCl2 (bottom) using εd (crosses) and εd+MI (circles) along a line across the tubes. The solid curves represent the true concentrations.

Tables (1)

Tables Icon

Table 1 The average estimated and true concentrations of CuCl2 (left) and NiCl2 (right) in gL−1 for each tube. The largest improvements using εd+MI are mostly seen for the tubes that are not expected to contain the relevant chromophore, as they suffer from significant cross-talk errors when εd is used. The average expected concentrations are lower than the true concentrations in the solutions due to the interpolation from the original images.

Equations (24)

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ρ y 1 , y 2 ( y 1 , y 2 ) = ρ y 1 ( y 1 ) ρ y 2 ( y 2 ) ,
I ( y 1 , y 2 ) = ( y 1 ) + ( y 2 ) ( y 1 , y 2 ) ,
( y k ) = y k ρ y k ( y k ) log ρ y k ( y k ) d y k ,
( y 1 , y 2 ) = y 1 y 2 ρ y 1 , y 2 ( y 1 , y 2 ) log ρ y 1 , y 2 ( y 1 , y 2 ) d y 1 d y 2 .
I ( y 1 , , y K ) = k = 1 K ( y k ) ( y 1 , , y K ) .
ρ c k ( ξ k ) = 1 M m = 1 M κ ( ξ k c k , m )
ρ c 1 , , c K ( ξ 1 , , ξ K ) = 1 M m = 1 M k = 1 K κ ( ξ k c k , m ) ,
κ ( x ) = 1 h 2 π exp ( x 2 / 2 h 2 ) ,
h = ( 4 3 M ) 1 5 σ 1.06 σ M 1 5 ,
¯ ( c k ) = q k = 1 Q ρ c k ( ξ k , q k ) log ρ c k ( ξ k , q k ) Δ ξ k
( c 1 , , c K ) = q 1 , , q K = 1 Q ρ c 1 , , c K ( ξ 1 , q 1 , , ξ K , q K ) log ρ c 1 , , c K ( ξ 1 , q 1 , , ξ K , q K ) k = 1 K Δ ξ k
I ( c 1 , , c K ) = k = 1 K ( c k ) ( c 1 , , c K ) .
I ( c 1 , , c K ) c k , m = ( c k ) c k , m ( c 1 , , c K ) c k , m .
( c k ) c k , m = ( c k ) ρ c k ρ c k c k , m ,
c k ρ c k = q k = 1 Q [ 1 + log ρ c k ( ξ k , q k ) ] Δ ξ k
ρ c k c k , m = κ ( ξ k , q k c k , m ) ,
κ ( ξ k , q k c k , m ) = ξ k , q k c k , m h 2 κ ( ξ k , q k c k , m )
( c 1 , , c K ) c k , m = ( c 1 , , c K ) ρ c 1 , , c K ρ c 1 , , c K c k , m ,
( c 1 , , c K ) ρ c 1 , , c K = q 1 , q K = 1 Q [ 1 + log ρ c 1 , , c K ( ξ 1 , q 1 , , ξ K , q K ) ] i = 1 K Δ ξ i
ρ c 1 , , c K c k , m = κ ( ξ k c k , m ) i = 1 , i k K κ ( ξ i c i , m ) .
p m , λ n model = S Γ m Φ m , λ n μ a _ m , λ n ,
argmin c 1 , , c K t ( c 1 , , c K t ) = n = 1 N m = 1 M [ p m , λ n model ( c 1 , , c K t ) p m , λ n meas ] 2 ,
argmin c 1 , , c K t ε d + M I ( c 1 , , c K t ) = n = 1 N m = 1 M [ p m , λ n model ( c 1 , , c K t ) p m , λ n meas ] 2 + γ I ( c 1 , , c K ) ,
Γ i = Γ H 2 O ( 1 + β i c i ) , i = CuCl 2 or NiCl 2

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