Abstract

When dealing with ill-posed problems such as fluorescence diffuse optical tomography (fDOT) the choice of the regularization parameter is extremely important for computing a reliable reconstruction. Several automatic methods for the selection of the regularization parameter have been introduced over the years and their performance depends on the particular inverse problem. Herein a U-curve-based algorithm for the selection of regularization parameter has been applied for the first time to fDOT. To increase the computational efficiency for large systems an interval of the regularization parameter is desirable. The U-curve provided a suitable selection of the regularization parameter in terms of Picard’s condition, image resolution and image noise. Results are shown both on phantom and mouse data.

© 2011 OSA

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2010 (1)

L. Z. Y. Qiangquiang, S. Huanfeng, and L. Pingxiang, “Adaptative multi-frame image super-resolution based on U-curve,” IEEE Trans. Image Process. ; e-pub ahead of print (2010).
[CrossRef]

2009 (3)

N. Ducros, A. Da Silva, L. Hervé, J.-M. Dinten, and F. Peyrin, “A comprehensive study of the use of temporal moments in time-resolved diffuse optical tomography: part II. three-dimensional reconstructions,” Phys. Med. Biol. 54(23), 7107–7119 (2009).
[CrossRef] [PubMed]

A. J. Chaudhari, S. Ahn, R. Levenson, R. D. Badawi, S. R. Cherry, and R. M. Leahy, “Excitation spectroscopy in multispectral optical fluorescence tomography: methodology, feasibility and computer simulation studies,” Phys. Med. Biol. 54(15), 4687–4704 (2009).
[CrossRef] [PubMed]

T. Correia, A. Gibson, M. Schweiger, and J. Hebden, “Selection of regularization parameter for optical topography,” J. Biomed. Opt. 14(3), 034044 (2009).
[CrossRef] [PubMed]

2008 (3)

A. Martin, J. Aguirre, A. Sarasa-Renedo, D. Tsoukatou, A. Garofalakis, H. Meyer, C. Mamalaki, J. Ripoll, and A. M. Planas, “Imaging changes in lymphoid organs in vivo after brain ischemia with three-dimensional fluorescence molecular tomography in transgenic mice expressing green fluorescent protein in T lymphocytes,” Mol. Imaging 7(4), 157–167 (2008).

J. F. Abascal, S. R. Arridge, R. H. Bayford, and D. S. Holder, “Comparison of methods for optimal choice of the regularization parameter for linear electrical impedance tomography of brain function,” Physiol. Meas. 29(11), 1319–1334 (2008).
[CrossRef] [PubMed]

D. Krawczyk-Stańdo and M. Rudnicki, “The use of L-curve and U-curve in inverse electromagnetic modelling,” Intell. Comput. Tech. Appl. Electromagn. 119, 73–82 (2008).
[CrossRef]

2007 (3)

D. Krawczyk-Stańdo and M. Rudnicki, “Regularization parameter selection in discrete ill-posed problems-the use of the U-curve,” Int. J. Appl. Math. Comput. Sci. 17(2), 157–164 (2007).
[CrossRef]

T. Lasser and V. Ntziachristos, “Optimization of 360° projection fluorescence molecular tomography,” Med. Image Anal. 11(4), 389–399 (2007).
[CrossRef] [PubMed]

A. Corlu, R. Choe, T. Durduran, M. A. Rosen, M. Schweiger, S. R. Arridge, M. D. Schnall, and A. G. Yodh, “Three-dimensional in vivo fluorescence diffuse optical tomography of breast cancer in humans,” Opt. Express 15(11), 6696–6716 (2007).
[CrossRef] [PubMed]

2005 (2)

V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol. 23(3), 313–320 (2005).
[CrossRef] [PubMed]

A. Soubret, J. Ripoll, and V. Ntziachristos, “Accuracy of fluorescent tomography in the presence of heterogeneities: study of the normalized Born ratio,” IEEE Trans. Med. Imaging 24(10), 1377–1386 (2005).
[CrossRef] [PubMed]

2004 (1)

2003 (5)

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(2), 235–247 (2003).
[CrossRef] [PubMed]

X. Intes, J. Ripoll, Y. Chen, S. Nioka, A. G. Yodh, and B. Chance, “In vivo continuous-wave optical breast imaging enhanced with Indocyanine Green,” Med. Phys. 30(6), 1039–1047 (2003).
[CrossRef] [PubMed]

A. Li, E. L. Miller, M. E. Kilmer, T. J. Brukilacchio, T. Chaves, J. Stott, Q. Zhang, T. Wu, M. Chorlton, R. H. Moore, D. B. Kopans, and D. A. Boas, “Tomographic optical breast imaging guided by three-dimensional mammography,” Appl. Opt. 42(25), 5181–5190 (2003).
[CrossRef] [PubMed]

Y. Xu, X. J. Gu, L. L. Fajardo, and H. B. Jiang, “In vivo breast imaging with diffuse optical tomography based on higher-order diffusion equations,” Appl. Opt. 42(16), 3163–3169 (2003).
[CrossRef] [PubMed]

J. Ripoll, R. B. Schulz, and V. Ntziachristos, “Free-space propagation of diffuse light: theory and experiments,” Phys. Rev. Lett. 91(10), 103901 (2003).
[CrossRef] [PubMed]

2002 (2)

V. A. Markel and J. C. Schotland, “Inverse problem in optical diffusion tomography. II. role of boundary conditions,” J. Opt. Soc. Am. A 19(3), 558–566 (2002).
[CrossRef]

V. Ntziachristos, C. Bremer, E. E. Graves, J. Ripoll, and R. Weissleder, “In vivo tomographic imaging of near-infrared fluorescent probes,” Mol. Imaging 1(2), 82–88 (2002).
[CrossRef]

2001 (1)

1999 (2)

1997 (2)

H. R. Busby and D. M. Trujillo, “Optimal regularization of an inverse dynamics problem,” Comput. Struc. 63(2), 243–248 (1997).
[CrossRef]

R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, and G. Valentini, “A solid tissue phantom for photon migration studies,” Phys. Med. Biol. 42(10), 1971–1979 (1997).
[CrossRef] [PubMed]

1996 (2)

M. Hanke, “Limitations of the L-curve method in ill-posed problems,” BIT 36(2), 287–301 (1996).
[CrossRef]

C. R. Vogel, “Non-convergence of the L-curve regularization parameter selection method,” Inverse Probl. 12(4), 535–548 (1996).
[CrossRef]

1995 (1)

1993 (2)

P. C. Hansen and D. P. O’Leary, “The use of L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput. 14(6), 1487–1503 (1993).
[CrossRef]

Hanke and Hansen, “Regularization methods for large scale problems,” Surv. Math. Ind. 3, 253–315 (1993).

1992 (1)

P. C. Hansen, “Analysis of discrete ill-posed problems by means of the L-curve,” SIAM Rev. 34(4), 561–580 (1992).
[CrossRef]

1990 (1)

P. C. Hansen, “The discrete Picard condition for discrete ill-posed problems,” BIT 30(4), 658–672 (1990).
[CrossRef]

1988 (1)

I. Freund, M. Kaveh, and M. Rosenbluh, “Dynamic multiple scattering: Ballistic photons and the breakdown of the photon-diffusion approximation,” Phys. Rev. Lett. 60(12), 1130–1133 (1988).
[CrossRef] [PubMed]

1987 (1)

P. C. Hansen, “The truncated SVD as a method for regularization,” BIT Num. Math. 27(4), 534–553 (1987).
[CrossRef]

Abascal, J. F.

J. F. Abascal, S. R. Arridge, R. H. Bayford, and D. S. Holder, “Comparison of methods for optimal choice of the regularization parameter for linear electrical impedance tomography of brain function,” Physiol. Meas. 29(11), 1319–1334 (2008).
[CrossRef] [PubMed]

Aguirre, J.

A. Martin, J. Aguirre, A. Sarasa-Renedo, D. Tsoukatou, A. Garofalakis, H. Meyer, C. Mamalaki, J. Ripoll, and A. M. Planas, “Imaging changes in lymphoid organs in vivo after brain ischemia with three-dimensional fluorescence molecular tomography in transgenic mice expressing green fluorescent protein in T lymphocytes,” Mol. Imaging 7(4), 157–167 (2008).

Ahn, S.

A. J. Chaudhari, S. Ahn, R. Levenson, R. D. Badawi, S. R. Cherry, and R. M. Leahy, “Excitation spectroscopy in multispectral optical fluorescence tomography: methodology, feasibility and computer simulation studies,” Phys. Med. Biol. 54(15), 4687–4704 (2009).
[CrossRef] [PubMed]

Arridge, S. R.

J. F. Abascal, S. R. Arridge, R. H. Bayford, and D. S. Holder, “Comparison of methods for optimal choice of the regularization parameter for linear electrical impedance tomography of brain function,” Physiol. Meas. 29(11), 1319–1334 (2008).
[CrossRef] [PubMed]

A. Corlu, R. Choe, T. Durduran, M. A. Rosen, M. Schweiger, S. R. Arridge, M. D. Schnall, and A. G. Yodh, “Three-dimensional in vivo fluorescence diffuse optical tomography of breast cancer in humans,” Opt. Express 15(11), 6696–6716 (2007).
[CrossRef] [PubMed]

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

S. R. Arridge, “Photon-measurement density functions. part I: analytical forms,” Appl. Opt. 34(31), 7395–7409 (1995).
[CrossRef] [PubMed]

Badawi, R. D.

A. J. Chaudhari, S. Ahn, R. Levenson, R. D. Badawi, S. R. Cherry, and R. M. Leahy, “Excitation spectroscopy in multispectral optical fluorescence tomography: methodology, feasibility and computer simulation studies,” Phys. Med. Biol. 54(15), 4687–4704 (2009).
[CrossRef] [PubMed]

Bayford, R. H.

J. F. Abascal, S. R. Arridge, R. H. Bayford, and D. S. Holder, “Comparison of methods for optimal choice of the regularization parameter for linear electrical impedance tomography of brain function,” Physiol. Meas. 29(11), 1319–1334 (2008).
[CrossRef] [PubMed]

Boas, D. A.

Bremer, C.

V. Ntziachristos, C. Bremer, E. E. Graves, J. Ripoll, and R. Weissleder, “In vivo tomographic imaging of near-infrared fluorescent probes,” Mol. Imaging 1(2), 82–88 (2002).
[CrossRef]

Brukilacchio, T. J.

Busby, H. R.

H. R. Busby and D. M. Trujillo, “Optimal regularization of an inverse dynamics problem,” Comput. Struc. 63(2), 243–248 (1997).
[CrossRef]

Chance, B.

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(2), 235–247 (2003).
[CrossRef] [PubMed]

X. Intes, J. Ripoll, Y. Chen, S. Nioka, A. G. Yodh, and B. Chance, “In vivo continuous-wave optical breast imaging enhanced with Indocyanine Green,” Med. Phys. 30(6), 1039–1047 (2003).
[CrossRef] [PubMed]

Chaudhari, A. J.

A. J. Chaudhari, S. Ahn, R. Levenson, R. D. Badawi, S. R. Cherry, and R. M. Leahy, “Excitation spectroscopy in multispectral optical fluorescence tomography: methodology, feasibility and computer simulation studies,” Phys. Med. Biol. 54(15), 4687–4704 (2009).
[CrossRef] [PubMed]

Chaves, T.

Chen, Y.

X. Intes, J. Ripoll, Y. Chen, S. Nioka, A. G. Yodh, and B. Chance, “In vivo continuous-wave optical breast imaging enhanced with Indocyanine Green,” Med. Phys. 30(6), 1039–1047 (2003).
[CrossRef] [PubMed]

Cherry, S. R.

A. J. Chaudhari, S. Ahn, R. Levenson, R. D. Badawi, S. R. Cherry, and R. M. Leahy, “Excitation spectroscopy in multispectral optical fluorescence tomography: methodology, feasibility and computer simulation studies,” Phys. Med. Biol. 54(15), 4687–4704 (2009).
[CrossRef] [PubMed]

Choe, R.

A. Corlu, R. Choe, T. Durduran, M. A. Rosen, M. Schweiger, S. R. Arridge, M. D. Schnall, and A. G. Yodh, “Three-dimensional in vivo fluorescence diffuse optical tomography of breast cancer in humans,” Opt. Express 15(11), 6696–6716 (2007).
[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(2), 235–247 (2003).
[CrossRef] [PubMed]

Chorlton, M.

Corlu, A.

Correia, T.

T. Correia, A. Gibson, M. Schweiger, and J. Hebden, “Selection of regularization parameter for optical topography,” J. Biomed. Opt. 14(3), 034044 (2009).
[CrossRef] [PubMed]

Cubeddu, R.

R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, and G. Valentini, “A solid tissue phantom for photon migration studies,” Phys. Med. Biol. 42(10), 1971–1979 (1997).
[CrossRef] [PubMed]

Culver, J. P.

E. E. Graves, J. P. Culver, J. Ripoll, R. Weissleder, and V. Ntziachristos, “Singular-value analysis and optimization of experimental parameters in fluorescence molecular tomography,” J. Opt. Soc. Am. A 21(2), 231–241 (2004).
[CrossRef]

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(2), 235–247 (2003).
[CrossRef] [PubMed]

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(10), 701–703 (2001).
[CrossRef]

Da Silva, A.

N. Ducros, A. Da Silva, L. Hervé, J.-M. Dinten, and F. Peyrin, “A comprehensive study of the use of temporal moments in time-resolved diffuse optical tomography: part II. three-dimensional reconstructions,” Phys. Med. Biol. 54(23), 7107–7119 (2009).
[CrossRef] [PubMed]

Dinten, J.-M.

N. Ducros, A. Da Silva, L. Hervé, J.-M. Dinten, and F. Peyrin, “A comprehensive study of the use of temporal moments in time-resolved diffuse optical tomography: part II. three-dimensional reconstructions,” Phys. Med. Biol. 54(23), 7107–7119 (2009).
[CrossRef] [PubMed]

Ducros, N.

N. Ducros, A. Da Silva, L. Hervé, J.-M. Dinten, and F. Peyrin, “A comprehensive study of the use of temporal moments in time-resolved diffuse optical tomography: part II. three-dimensional reconstructions,” Phys. Med. Biol. 54(23), 7107–7119 (2009).
[CrossRef] [PubMed]

Durduran, T.

A. Corlu, R. Choe, T. Durduran, M. A. Rosen, M. Schweiger, S. R. Arridge, M. D. Schnall, and A. G. Yodh, “Three-dimensional in vivo fluorescence diffuse optical tomography of breast cancer in humans,” Opt. Express 15(11), 6696–6716 (2007).
[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(2), 235–247 (2003).
[CrossRef] [PubMed]

Fajardo, L. L.

Freund, I.

I. Freund, M. Kaveh, and M. Rosenbluh, “Dynamic multiple scattering: Ballistic photons and the breakdown of the photon-diffusion approximation,” Phys. Rev. Lett. 60(12), 1130–1133 (1988).
[CrossRef] [PubMed]

Garofalakis, A.

A. Martin, J. Aguirre, A. Sarasa-Renedo, D. Tsoukatou, A. Garofalakis, H. Meyer, C. Mamalaki, J. Ripoll, and A. M. Planas, “Imaging changes in lymphoid organs in vivo after brain ischemia with three-dimensional fluorescence molecular tomography in transgenic mice expressing green fluorescent protein in T lymphocytes,” Mol. Imaging 7(4), 157–167 (2008).

Gibson, A.

T. Correia, A. Gibson, M. Schweiger, and J. Hebden, “Selection of regularization parameter for optical topography,” J. Biomed. Opt. 14(3), 034044 (2009).
[CrossRef] [PubMed]

Graves, E. E.

E. E. Graves, J. P. Culver, J. Ripoll, R. Weissleder, and V. Ntziachristos, “Singular-value analysis and optimization of experimental parameters in fluorescence molecular tomography,” J. Opt. Soc. Am. A 21(2), 231–241 (2004).
[CrossRef]

V. Ntziachristos, C. Bremer, E. E. Graves, J. Ripoll, and R. Weissleder, “In vivo tomographic imaging of near-infrared fluorescent probes,” Mol. Imaging 1(2), 82–88 (2002).
[CrossRef]

Gu, X. J.

Hanke,

Hanke and Hansen, “Regularization methods for large scale problems,” Surv. Math. Ind. 3, 253–315 (1993).

Hanke, M.

M. Hanke, “Limitations of the L-curve method in ill-posed problems,” BIT 36(2), 287–301 (1996).
[CrossRef]

Hansen,

Hanke and Hansen, “Regularization methods for large scale problems,” Surv. Math. Ind. 3, 253–315 (1993).

Hansen, P. C.

P. C. Hansen and D. P. O’Leary, “The use of L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput. 14(6), 1487–1503 (1993).
[CrossRef]

P. C. Hansen, “Analysis of discrete ill-posed problems by means of the L-curve,” SIAM Rev. 34(4), 561–580 (1992).
[CrossRef]

P. C. Hansen, “The discrete Picard condition for discrete ill-posed problems,” BIT 30(4), 658–672 (1990).
[CrossRef]

P. C. Hansen, “The truncated SVD as a method for regularization,” BIT Num. Math. 27(4), 534–553 (1987).
[CrossRef]

Hebden, J.

T. Correia, A. Gibson, M. Schweiger, and J. Hebden, “Selection of regularization parameter for optical topography,” J. Biomed. Opt. 14(3), 034044 (2009).
[CrossRef] [PubMed]

Hervé, L.

N. Ducros, A. Da Silva, L. Hervé, J.-M. Dinten, and F. Peyrin, “A comprehensive study of the use of temporal moments in time-resolved diffuse optical tomography: part II. three-dimensional reconstructions,” Phys. Med. Biol. 54(23), 7107–7119 (2009).
[CrossRef] [PubMed]

Holboke, M. J.

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(2), 235–247 (2003).
[CrossRef] [PubMed]

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(10), 701–703 (2001).
[CrossRef]

Holder, D. S.

J. F. Abascal, S. R. Arridge, R. H. Bayford, and D. S. Holder, “Comparison of methods for optimal choice of the regularization parameter for linear electrical impedance tomography of brain function,” Physiol. Meas. 29(11), 1319–1334 (2008).
[CrossRef] [PubMed]

Huanfeng, S.

L. Z. Y. Qiangquiang, S. Huanfeng, and L. Pingxiang, “Adaptative multi-frame image super-resolution based on U-curve,” IEEE Trans. Image Process. ; e-pub ahead of print (2010).
[CrossRef]

Intes, X.

X. Intes, J. Ripoll, Y. Chen, S. Nioka, A. G. Yodh, and B. Chance, “In vivo continuous-wave optical breast imaging enhanced with Indocyanine Green,” Med. Phys. 30(6), 1039–1047 (2003).
[CrossRef] [PubMed]

Jiang, H. B.

Kaveh, M.

I. Freund, M. Kaveh, and M. Rosenbluh, “Dynamic multiple scattering: Ballistic photons and the breakdown of the photon-diffusion approximation,” Phys. Rev. Lett. 60(12), 1130–1133 (1988).
[CrossRef] [PubMed]

Kilmer, M. E.

Kopans, D. B.

Krawczyk-Stando, D.

D. Krawczyk-Stańdo and M. Rudnicki, “The use of L-curve and U-curve in inverse electromagnetic modelling,” Intell. Comput. Tech. Appl. Electromagn. 119, 73–82 (2008).
[CrossRef]

D. Krawczyk-Stańdo and M. Rudnicki, “Regularization parameter selection in discrete ill-posed problems-the use of the U-curve,” Int. J. Appl. Math. Comput. Sci. 17(2), 157–164 (2007).
[CrossRef]

Lasser, T.

T. Lasser and V. Ntziachristos, “Optimization of 360° projection fluorescence molecular tomography,” Med. Image Anal. 11(4), 389–399 (2007).
[CrossRef] [PubMed]

Leahy, R. M.

A. J. Chaudhari, S. Ahn, R. Levenson, R. D. Badawi, S. R. Cherry, and R. M. Leahy, “Excitation spectroscopy in multispectral optical fluorescence tomography: methodology, feasibility and computer simulation studies,” Phys. Med. Biol. 54(15), 4687–4704 (2009).
[CrossRef] [PubMed]

Levenson, R.

A. J. Chaudhari, S. Ahn, R. Levenson, R. D. Badawi, S. R. Cherry, and R. M. Leahy, “Excitation spectroscopy in multispectral optical fluorescence tomography: methodology, feasibility and computer simulation studies,” Phys. Med. Biol. 54(15), 4687–4704 (2009).
[CrossRef] [PubMed]

Li, A.

Mamalaki, C.

A. Martin, J. Aguirre, A. Sarasa-Renedo, D. Tsoukatou, A. Garofalakis, H. Meyer, C. Mamalaki, J. Ripoll, and A. M. Planas, “Imaging changes in lymphoid organs in vivo after brain ischemia with three-dimensional fluorescence molecular tomography in transgenic mice expressing green fluorescent protein in T lymphocytes,” Mol. Imaging 7(4), 157–167 (2008).

Markel, V. A.

Martin, A.

A. Martin, J. Aguirre, A. Sarasa-Renedo, D. Tsoukatou, A. Garofalakis, H. Meyer, C. Mamalaki, J. Ripoll, and A. M. Planas, “Imaging changes in lymphoid organs in vivo after brain ischemia with three-dimensional fluorescence molecular tomography in transgenic mice expressing green fluorescent protein in T lymphocytes,” Mol. Imaging 7(4), 157–167 (2008).

McBride, T. O.

Meyer, H.

A. Martin, J. Aguirre, A. Sarasa-Renedo, D. Tsoukatou, A. Garofalakis, H. Meyer, C. Mamalaki, J. Ripoll, and A. M. Planas, “Imaging changes in lymphoid organs in vivo after brain ischemia with three-dimensional fluorescence molecular tomography in transgenic mice expressing green fluorescent protein in T lymphocytes,” Mol. Imaging 7(4), 157–167 (2008).

Miller, E. L.

Moore, R. H.

Nioka, S.

X. Intes, J. Ripoll, Y. Chen, S. Nioka, A. G. Yodh, and B. Chance, “In vivo continuous-wave optical breast imaging enhanced with Indocyanine Green,” Med. Phys. 30(6), 1039–1047 (2003).
[CrossRef] [PubMed]

Ntziachristos, V.

T. Lasser and V. Ntziachristos, “Optimization of 360° projection fluorescence molecular tomography,” Med. Image Anal. 11(4), 389–399 (2007).
[CrossRef] [PubMed]

A. Soubret, J. Ripoll, and V. Ntziachristos, “Accuracy of fluorescent tomography in the presence of heterogeneities: study of the normalized Born ratio,” IEEE Trans. Med. Imaging 24(10), 1377–1386 (2005).
[CrossRef] [PubMed]

V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol. 23(3), 313–320 (2005).
[CrossRef] [PubMed]

E. E. Graves, J. P. Culver, J. Ripoll, R. Weissleder, and V. Ntziachristos, “Singular-value analysis and optimization of experimental parameters in fluorescence molecular tomography,” J. Opt. Soc. Am. A 21(2), 231–241 (2004).
[CrossRef]

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(2), 235–247 (2003).
[CrossRef] [PubMed]

J. Ripoll, R. B. Schulz, and V. Ntziachristos, “Free-space propagation of diffuse light: theory and experiments,” Phys. Rev. Lett. 91(10), 103901 (2003).
[CrossRef] [PubMed]

V. Ntziachristos, C. Bremer, E. E. Graves, J. Ripoll, and R. Weissleder, “In vivo tomographic imaging of near-infrared fluorescent probes,” Mol. Imaging 1(2), 82–88 (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(10), 701–703 (2001).
[CrossRef]

O’Leary, D. P.

P. C. Hansen and D. P. O’Leary, “The use of L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput. 14(6), 1487–1503 (1993).
[CrossRef]

Osterberg, U. L.

Paulsen, K. D.

Peyrin, F.

N. Ducros, A. Da Silva, L. Hervé, J.-M. Dinten, and F. Peyrin, “A comprehensive study of the use of temporal moments in time-resolved diffuse optical tomography: part II. three-dimensional reconstructions,” Phys. Med. Biol. 54(23), 7107–7119 (2009).
[CrossRef] [PubMed]

Pifferi, A.

R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, and G. Valentini, “A solid tissue phantom for photon migration studies,” Phys. Med. Biol. 42(10), 1971–1979 (1997).
[CrossRef] [PubMed]

Pingxiang, L.

L. Z. Y. Qiangquiang, S. Huanfeng, and L. Pingxiang, “Adaptative multi-frame image super-resolution based on U-curve,” IEEE Trans. Image Process. ; e-pub ahead of print (2010).
[CrossRef]

Planas, A. M.

A. Martin, J. Aguirre, A. Sarasa-Renedo, D. Tsoukatou, A. Garofalakis, H. Meyer, C. Mamalaki, J. Ripoll, and A. M. Planas, “Imaging changes in lymphoid organs in vivo after brain ischemia with three-dimensional fluorescence molecular tomography in transgenic mice expressing green fluorescent protein in T lymphocytes,” Mol. Imaging 7(4), 157–167 (2008).

Pogue, B. W.

Prewitt, J.

Qiangquiang, L. Z. Y.

L. Z. Y. Qiangquiang, S. Huanfeng, and L. Pingxiang, “Adaptative multi-frame image super-resolution based on U-curve,” IEEE Trans. Image Process. ; e-pub ahead of print (2010).
[CrossRef]

Ripoll, J.

A. Martin, J. Aguirre, A. Sarasa-Renedo, D. Tsoukatou, A. Garofalakis, H. Meyer, C. Mamalaki, J. Ripoll, and A. M. Planas, “Imaging changes in lymphoid organs in vivo after brain ischemia with three-dimensional fluorescence molecular tomography in transgenic mice expressing green fluorescent protein in T lymphocytes,” Mol. Imaging 7(4), 157–167 (2008).

V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol. 23(3), 313–320 (2005).
[CrossRef] [PubMed]

A. Soubret, J. Ripoll, and V. Ntziachristos, “Accuracy of fluorescent tomography in the presence of heterogeneities: study of the normalized Born ratio,” IEEE Trans. Med. Imaging 24(10), 1377–1386 (2005).
[CrossRef] [PubMed]

E. E. Graves, J. P. Culver, J. Ripoll, R. Weissleder, and V. Ntziachristos, “Singular-value analysis and optimization of experimental parameters in fluorescence molecular tomography,” J. Opt. Soc. Am. A 21(2), 231–241 (2004).
[CrossRef]

X. Intes, J. Ripoll, Y. Chen, S. Nioka, A. G. Yodh, and B. Chance, “In vivo continuous-wave optical breast imaging enhanced with Indocyanine Green,” Med. Phys. 30(6), 1039–1047 (2003).
[CrossRef] [PubMed]

J. Ripoll, R. B. Schulz, and V. Ntziachristos, “Free-space propagation of diffuse light: theory and experiments,” Phys. Rev. Lett. 91(10), 103901 (2003).
[CrossRef] [PubMed]

V. Ntziachristos, C. Bremer, E. E. Graves, J. Ripoll, and R. Weissleder, “In vivo tomographic imaging of near-infrared fluorescent probes,” Mol. Imaging 1(2), 82–88 (2002).
[CrossRef]

Rosen, M. A.

Rosenbluh, M.

I. Freund, M. Kaveh, and M. Rosenbluh, “Dynamic multiple scattering: Ballistic photons and the breakdown of the photon-diffusion approximation,” Phys. Rev. Lett. 60(12), 1130–1133 (1988).
[CrossRef] [PubMed]

Rudnicki, M.

D. Krawczyk-Stańdo and M. Rudnicki, “The use of L-curve and U-curve in inverse electromagnetic modelling,” Intell. Comput. Tech. Appl. Electromagn. 119, 73–82 (2008).
[CrossRef]

D. Krawczyk-Stańdo and M. Rudnicki, “Regularization parameter selection in discrete ill-posed problems-the use of the U-curve,” Int. J. Appl. Math. Comput. Sci. 17(2), 157–164 (2007).
[CrossRef]

Sarasa-Renedo, A.

A. Martin, J. Aguirre, A. Sarasa-Renedo, D. Tsoukatou, A. Garofalakis, H. Meyer, C. Mamalaki, J. Ripoll, and A. M. Planas, “Imaging changes in lymphoid organs in vivo after brain ischemia with three-dimensional fluorescence molecular tomography in transgenic mice expressing green fluorescent protein in T lymphocytes,” Mol. Imaging 7(4), 157–167 (2008).

Schnall, M. D.

Schotland, J. C.

Schulz, R. B.

J. Ripoll, R. B. Schulz, and V. Ntziachristos, “Free-space propagation of diffuse light: theory and experiments,” Phys. Rev. Lett. 91(10), 103901 (2003).
[CrossRef] [PubMed]

Schweiger, M.

Slemp, A.

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(2), 235–247 (2003).
[CrossRef] [PubMed]

Soubret, A.

A. Soubret, J. Ripoll, and V. Ntziachristos, “Accuracy of fluorescent tomography in the presence of heterogeneities: study of the normalized Born ratio,” IEEE Trans. Med. Imaging 24(10), 1377–1386 (2005).
[CrossRef] [PubMed]

Stott, J.

Taroni, P.

R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, and G. Valentini, “A solid tissue phantom for photon migration studies,” Phys. Med. Biol. 42(10), 1971–1979 (1997).
[CrossRef] [PubMed]

Torricelli, A.

R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, and G. Valentini, “A solid tissue phantom for photon migration studies,” Phys. Med. Biol. 42(10), 1971–1979 (1997).
[CrossRef] [PubMed]

Trujillo, D. M.

H. R. Busby and D. M. Trujillo, “Optimal regularization of an inverse dynamics problem,” Comput. Struc. 63(2), 243–248 (1997).
[CrossRef]

Tsoukatou, D.

A. Martin, J. Aguirre, A. Sarasa-Renedo, D. Tsoukatou, A. Garofalakis, H. Meyer, C. Mamalaki, J. Ripoll, and A. M. Planas, “Imaging changes in lymphoid organs in vivo after brain ischemia with three-dimensional fluorescence molecular tomography in transgenic mice expressing green fluorescent protein in T lymphocytes,” Mol. Imaging 7(4), 157–167 (2008).

Valentini, G.

R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, and G. Valentini, “A solid tissue phantom for photon migration studies,” Phys. Med. Biol. 42(10), 1971–1979 (1997).
[CrossRef] [PubMed]

Vogel, C. R.

C. R. Vogel, “Non-convergence of the L-curve regularization parameter selection method,” Inverse Probl. 12(4), 535–548 (1996).
[CrossRef]

Wang, L. V.

V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol. 23(3), 313–320 (2005).
[CrossRef] [PubMed]

Weissleder, R.

V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol. 23(3), 313–320 (2005).
[CrossRef] [PubMed]

E. E. Graves, J. P. Culver, J. Ripoll, R. Weissleder, and V. Ntziachristos, “Singular-value analysis and optimization of experimental parameters in fluorescence molecular tomography,” J. Opt. Soc. Am. A 21(2), 231–241 (2004).
[CrossRef]

V. Ntziachristos, C. Bremer, E. E. Graves, J. Ripoll, and R. Weissleder, “In vivo tomographic imaging of near-infrared fluorescent probes,” Mol. Imaging 1(2), 82–88 (2002).
[CrossRef]

Wu, T.

Xu, Y.

Yodh, A. G.

A. Corlu, R. Choe, T. Durduran, M. A. Rosen, M. Schweiger, S. R. Arridge, M. D. Schnall, and A. G. Yodh, “Three-dimensional in vivo fluorescence diffuse optical tomography of breast cancer in humans,” Opt. Express 15(11), 6696–6716 (2007).
[CrossRef] [PubMed]

X. Intes, J. Ripoll, Y. Chen, S. Nioka, A. G. Yodh, and B. Chance, “In vivo continuous-wave optical breast imaging enhanced with Indocyanine Green,” Med. Phys. 30(6), 1039–1047 (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(2), 235–247 (2003).
[CrossRef] [PubMed]

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(10), 701–703 (2001).
[CrossRef]

Zhang, Q.

Zubkov, L.

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(2), 235–247 (2003).
[CrossRef] [PubMed]

Appl. Opt. (4)

BIT (2)

M. Hanke, “Limitations of the L-curve method in ill-posed problems,” BIT 36(2), 287–301 (1996).
[CrossRef]

P. C. Hansen, “The discrete Picard condition for discrete ill-posed problems,” BIT 30(4), 658–672 (1990).
[CrossRef]

BIT Num. Math. (1)

P. C. Hansen, “The truncated SVD as a method for regularization,” BIT Num. Math. 27(4), 534–553 (1987).
[CrossRef]

Comput. Struc. (1)

H. R. Busby and D. M. Trujillo, “Optimal regularization of an inverse dynamics problem,” Comput. Struc. 63(2), 243–248 (1997).
[CrossRef]

IEEE Trans. Image Process. (1)

L. Z. Y. Qiangquiang, S. Huanfeng, and L. Pingxiang, “Adaptative multi-frame image super-resolution based on U-curve,” IEEE Trans. Image Process. ; e-pub ahead of print (2010).
[CrossRef]

IEEE Trans. Med. Imaging (1)

A. Soubret, J. Ripoll, and V. Ntziachristos, “Accuracy of fluorescent tomography in the presence of heterogeneities: study of the normalized Born ratio,” IEEE Trans. Med. Imaging 24(10), 1377–1386 (2005).
[CrossRef] [PubMed]

Int. J. Appl. Math. Comput. Sci. (1)

D. Krawczyk-Stańdo and M. Rudnicki, “Regularization parameter selection in discrete ill-posed problems-the use of the U-curve,” Int. J. Appl. Math. Comput. Sci. 17(2), 157–164 (2007).
[CrossRef]

Intell. Comput. Tech. Appl. Electromagn. (1)

D. Krawczyk-Stańdo and M. Rudnicki, “The use of L-curve and U-curve in inverse electromagnetic modelling,” Intell. Comput. Tech. Appl. Electromagn. 119, 73–82 (2008).
[CrossRef]

Inverse Probl. (2)

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

C. R. Vogel, “Non-convergence of the L-curve regularization parameter selection method,” Inverse Probl. 12(4), 535–548 (1996).
[CrossRef]

J. Biomed. Opt. (1)

T. Correia, A. Gibson, M. Schweiger, and J. Hebden, “Selection of regularization parameter for optical topography,” J. Biomed. Opt. 14(3), 034044 (2009).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (2)

Med. Image Anal. (1)

T. Lasser and V. Ntziachristos, “Optimization of 360° projection fluorescence molecular tomography,” Med. Image Anal. 11(4), 389–399 (2007).
[CrossRef] [PubMed]

Med. Phys. (2)

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(2), 235–247 (2003).
[CrossRef] [PubMed]

X. Intes, J. Ripoll, Y. Chen, S. Nioka, A. G. Yodh, and B. Chance, “In vivo continuous-wave optical breast imaging enhanced with Indocyanine Green,” Med. Phys. 30(6), 1039–1047 (2003).
[CrossRef] [PubMed]

Mol. Imaging (2)

A. Martin, J. Aguirre, A. Sarasa-Renedo, D. Tsoukatou, A. Garofalakis, H. Meyer, C. Mamalaki, J. Ripoll, and A. M. Planas, “Imaging changes in lymphoid organs in vivo after brain ischemia with three-dimensional fluorescence molecular tomography in transgenic mice expressing green fluorescent protein in T lymphocytes,” Mol. Imaging 7(4), 157–167 (2008).

V. Ntziachristos, C. Bremer, E. E. Graves, J. Ripoll, and R. Weissleder, “In vivo tomographic imaging of near-infrared fluorescent probes,” Mol. Imaging 1(2), 82–88 (2002).
[CrossRef]

Nat. Biotechnol. (1)

V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol. 23(3), 313–320 (2005).
[CrossRef] [PubMed]

Opt. Express (1)

Opt. Lett. (1)

Phys. Med. Biol. (3)

R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, and G. Valentini, “A solid tissue phantom for photon migration studies,” Phys. Med. Biol. 42(10), 1971–1979 (1997).
[CrossRef] [PubMed]

N. Ducros, A. Da Silva, L. Hervé, J.-M. Dinten, and F. Peyrin, “A comprehensive study of the use of temporal moments in time-resolved diffuse optical tomography: part II. three-dimensional reconstructions,” Phys. Med. Biol. 54(23), 7107–7119 (2009).
[CrossRef] [PubMed]

A. J. Chaudhari, S. Ahn, R. Levenson, R. D. Badawi, S. R. Cherry, and R. M. Leahy, “Excitation spectroscopy in multispectral optical fluorescence tomography: methodology, feasibility and computer simulation studies,” Phys. Med. Biol. 54(15), 4687–4704 (2009).
[CrossRef] [PubMed]

Phys. Rev. Lett. (2)

I. Freund, M. Kaveh, and M. Rosenbluh, “Dynamic multiple scattering: Ballistic photons and the breakdown of the photon-diffusion approximation,” Phys. Rev. Lett. 60(12), 1130–1133 (1988).
[CrossRef] [PubMed]

J. Ripoll, R. B. Schulz, and V. Ntziachristos, “Free-space propagation of diffuse light: theory and experiments,” Phys. Rev. Lett. 91(10), 103901 (2003).
[CrossRef] [PubMed]

Physiol. Meas. (1)

J. F. Abascal, S. R. Arridge, R. H. Bayford, and D. S. Holder, “Comparison of methods for optimal choice of the regularization parameter for linear electrical impedance tomography of brain function,” Physiol. Meas. 29(11), 1319–1334 (2008).
[CrossRef] [PubMed]

SIAM J. Sci. Comput. (1)

P. C. Hansen and D. P. O’Leary, “The use of L-curve in the regularization of discrete ill-posed problems,” SIAM J. Sci. Comput. 14(6), 1487–1503 (1993).
[CrossRef]

SIAM Rev. (1)

P. C. Hansen, “Analysis of discrete ill-posed problems by means of the L-curve,” SIAM Rev. 34(4), 561–580 (1992).
[CrossRef]

Surv. Math. Ind. (1)

Hanke and Hansen, “Regularization methods for large scale problems,” Surv. Math. Ind. 3, 253–315 (1993).

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A. Serdaroglu, B. Yazici, and V. Ntziachristos, "Fluorescence molecular tomography based on a priori information," in Biomedical Optics, Technical Digest (CD) (Optical Society of America, 2006), paper SH46, http://www.opticsinfobase.org/abstract.cfm?URI=BIO-2006-SH46 .

Z. Xu, J. Yan, and J. Bai, “Determining the regularization parameter: a hybrid reconstruction technique in fluorescence molecular tomography,” in Communications and Photonics Conference and Exhibition (ACP),2009Asia 2009), 1 - 2.

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A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978) vol. 1.

J. Aguirre, A. Sisniega, J. Ripoll, M. Desco, and J. J. Vaquero, “Co-planar FMT-CT,” in 2008 World Molecular Imaging Congress (WMIC), (2008).

J. Aguirre, A. Sisniega, J. Ripoll, M. Desco, and J. J. Vaquero, “Design and development of a co-planar fluorescence and X-ray tomograph,” 2008 IEEE Nuclear Science Symposium Conference Record, 5412–5413 (2008).

M. Born and E. Wolf, Principles of Optics (University Press, 1999).

G. H. Golub and C. F. Van Loan, Matrix computations (Johns Hopkins University Press, 1996), p. 694.

P. C. Hansen, Regularization Tools Version 4.0 for MATLAB 7.3 (Springer, 2007), pp. 189–194.

J. Chamorro, J. Aguirre, J. Ripoll, J. J. Vaquero, and M. Desco, “FDOT setting optimization and reconstruction using singular value analysis with automatic thresholding,” in Nuclear Science Symposium and Medical Imaging Conference (IEEE), (2009).

J. Chamorro-Servent, J. Aguirre, J. Ripoll, J. J. Vaquero, and M. Desco, “Maximizing the information content in acquired measurements of a parallel plate non-contact FDOT while minimizing the computational cost: singular value analysis,” in 4th European Molecular Imaging Meeting (ESMI), (2009).

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Figures (12)

Fig. 1
Fig. 1

Experimental setup.

Fig. 2
Fig. 2

(a) Geometrical configuration of the slab (black) and the mesh FOV (red). The capillary tip is represented by the black sphere. (b) Detail of the mesh FOV. The grey dots represent the positions of the voxels; the red dot represents the capillary tip. (c) Geometrical configuration of the sources and detectors for the slice corresponding to y=0.75; the rectangles represent the reconstructed image voxels.

Fig. 6
Fig. 6

(a) Profiles taken in the x direction, corresponding to the line z=y=7.5 mm for each regularization parameter. FWHM improves when the regularization parameter decreases. b) Resolution vs. noise plot.

Fig. 11
Fig. 11

(a) Profiles taken in the x direction, corresponding to the line z=8 mm, y=0.7 mm for each regularization parameter. While the regularization parameter decreases, the FWHM improves. (b) Resolution vs. noise plot.

Fig. 3
Fig. 3

U-curve plots on log-log scale. (Minimum which corresponds to α u = 4.38*10−2).

Fig. 4
Fig. 4

L-curve plots on log-log scale. (Minimum which corresponds to α l = 5.65*10−2).

Fig. 5
Fig. 5

Slices in the y-z plane of the reconstructions obtained for the α parameter in the 10−1 to 10−6 range. The result for α u =4.38*10−2 (obtained from the U-curve) is showed at the bottom center. At the bottom right: slice indicating the phantom fluorescence concentration.

Fig. 7
Fig. 7

Plots of the SVD components of the phantom and Picard’s plot. (a) Noisy SVD components of the solution. (b) Noisy SVD components of the right-hand side. (c) Decay of the singular values. The regularization parameter provided by U-curve method ( α u =4.38*10−2) is plotted as a horizontal dashed blue line. (d) Picard’s plot with the maximum and minimum parameter of the heuristic acceptable range plotted as two horizontal dashed black lines (10−1 and 10−3).

Fig. 8
Fig. 8

U-curve plot on the log-log scale. (Minimum which corresponds to α u = 5.72*10−2).

Fig. 9
Fig. 9

Slices in the y-z plane of the 3D render of the reconstructions obtained for the α parameter in the 10-1 to 10-6 range. The result for α u = 5.72*10-2 (obtained from the U-curve) is showed at the bottom center. Dye concentration and volume are indicated at the bottom right.

Fig. 10
Fig. 10

Coronal view of a 3D render of the reconstruction for α u = 5.72*10−2 (obtained from U-curve). The white light image is shown behind as a reference image. Dye concentration and volume are indicated on the image.

Fig. 12
Fig. 12

Picard’s plot with the maximum and minimum parameter of the heuristic acceptable range plotted as two horizontal dashed black lines (10−1 and 10−2).

Equations (9)

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d = W f
min f W f d 2
min f { W f d 2 2 + α f 2 2 }
d n B ( r s , r d ) = S 0 U ( r s , r d , k λ 1 ) ( U ( r s , r d , k λ 1 ) f ( r ) D λ 2 G ( r , r d , k λ 2 ) ) d 3 r
( d n B ( r s 1 , r d 1 ) · · · d n B ( r s M , r d M ) ) = ( w 11 · · · w 1 N · · · w M 1 · · · w M N ) ( f ( r 1 ) · · · f ( r N ) )
W i j = k = 1 r a n k ( W ) u i k Σ k k v k j W = U Σ V T
f = i = 1 r a n k ( W ) u i T d σ i v i
f α = i = 1 r a n k ( W ) u i T σ i d σ i 2 + α 2 v i
U c u r v e ( α ) = 1 W f α d 2 + 1 f α 2

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