Abstract

Reconstruction algorithms for diffuse optical tomography (DOT) and bioluminescence tomography (BLT) have been developed based on diffusion theory. The algorithms numerically solve the diffusion equation using the finite element method. The direct measurements of the uncalibrated light fluence rates by a camera are used for the reconstructions. The DOT is self-calibrated by using all possible pairs of transmission images obtained with external sources along with the relative values of the simulated data and the calculated Jacobian. The reconstruction is done in the relative domain with the cancelation of any geometrical or optical factors. The transmission measurements for the DOT are used for calibrating the bioluminescence measurements at each wavelength and then a normalized system of equations is built up which is self-calibrated for the BLT. The algorithms have been applied to a three dimensional model of the mouse (MOBY) segmented into tissue regions which are assumed to have uniform optical properties. The DOT uses the direct method for calculating the Jacobian. The BLT uses a reduced space of eigenvectors of the Green's function with iterative shrinking of the permissible source region. The reconstruction results of the DOT and BLT algorithms show good agreement with the actual values when using either absolute or relative data. Even a small calibration error causes significant degradation of the reconstructions based on absolute data.

© 2012 OSA

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2011 (3)

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M. A. Naser and M. S. Patterson, “Algorithms for bioluminescence tomography incorporating anatomical information and reconstruction of tissue optical properties,” Biomed. Opt. Express1(2), 512–526 (2010).
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2009 (3)

2008 (4)

J. Feng, K. Jia, G. Yan, S. Zhu, C. Qin, Y. Lv, and J. Tian, “An optimal permissible source region strategy for multispectral bioluminescence tomography,” Opt. Express16(20), 15640–15654 (2008).
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[CrossRef] [PubMed]

2007 (7)

M. A. J. Chaudhari, A. A. Joshi, F. Darvas, and R. M. Leahy, “A method for atlas-based volumetric registration with surface constraints for optical bioluminescence tomography in small animal imaging,” Proc. SPIE6510, 651024, 651024–10 (2007).
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[CrossRef] [PubMed]

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[CrossRef]

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N. Cao, A. Nehorai, and M. Jacobs, “Image reconstruction for diffuse optical tomography using sparsity regularization and expectation-maximization algorithm,” Opt. Express15(21), 13695–13708 (2007).
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2006 (3)

H. Dehghani, S. C. Davis, S. Jiang, B. W. Pogue, K. D. Paulsen, and M. S. Patterson, “Spectrally resolved bioluminescence optical tomography,” Opt. Lett.31(3), 365–367 (2006).
[CrossRef] [PubMed]

D. C. Comsa, T. J. Farrell, and M. S. Patterson, “Quantification of bioluminescence images of point source objects using diffusion theory models,” Phys. Med. Biol.51(15), 3733–3746 (2006).
[CrossRef] [PubMed]

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[CrossRef] [PubMed]

2005 (4)

G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol.50(17), 4225–4241 (2005).
[CrossRef] [PubMed]

A. J. Chaudhari, F. Darvas, J. R. Bading, R. A. Moats, P. S. Conti, D. J. Smith, S. R. Cherry, and R. M. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol.50(23), 5421–5441 (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]

W. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. V. Wang, E. A. Hoffman, G. McLennan, P. B. McCray, J. Zabner, and A. Cong, “Practical reconstruction method for bioluminescence tomography,” Opt. Express13(18), 6756–6771 (2005).
[CrossRef] [PubMed]

2004 (2)

X. Gu, Q. Zhang, L. Larcom, and H. Jiang, “Three-dimensional bioluminescence tomography with model-based reconstruction,” Opt. Express12(17), 3996–4000 (2004).
[CrossRef] [PubMed]

W. P. Segars, B. M. Tsui, E. C. Frey, G. A. Johnson, and S. S. Berr, “Development of a 4-D digital mouse phantom for molecular imaging research,” Mol. Imaging Biol.6(3), 149–159 (2004).
[CrossRef] [PubMed]

2001 (1)

R. Weissleder and U. Mahmood, “Molecular imaging,” Radiology219(2), 316–333 (2001).
[PubMed]

1999 (1)

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

1998 (1)

1995 (1)

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

1993 (1)

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys.20(2), 299–309 (1993).
[CrossRef] [PubMed]

Adibi, A.

Ahn, S.

S. Ahn, A. J. Chaudhari, F. Darvas, C. A. Bouman, and R. M. Leahy, “Fast iterative image reconstruction methods for fully 3D multispectral bioluminescence tomography,” Phys. Med. Biol.53(14), 3921–3942 (2008).
[CrossRef] [PubMed]

Alexandrakis, G.

G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol.50(17), 4225–4241 (2005).
[CrossRef] [PubMed]

Antich, P. P.

N. V. Slavine, M. A. Lewis, E. Richer, and P. P. Antich, “Iterative reconstruction method for light emitting sources based on the diffusion equation,” Med. Phys.33(1), 61–68 (2006).
[CrossRef] [PubMed]

Arridge, S. R.

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

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

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys.20(2), 299–309 (1993).
[CrossRef] [PubMed]

Bading, J. R.

A. J. Chaudhari, F. Darvas, J. R. Bading, R. A. Moats, P. S. Conti, D. J. Smith, S. R. Cherry, and R. M. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol.50(23), 5421–5441 (2005).
[CrossRef] [PubMed]

Bai, J.

J. Tian, J. Bai, X. P. Yan, S. Bao, Y. Li, W. Liang, and X. Yang, “Multimodality molecular imaging,” IEEE Eng. Med. Biol. Mag.27(5), 48–57 (2008).
[CrossRef] [PubMed]

Bao, S.

J. Tian, J. Bai, X. P. Yan, S. Bao, Y. Li, W. Liang, and X. Yang, “Multimodality molecular imaging,” IEEE Eng. Med. Biol. Mag.27(5), 48–57 (2008).
[CrossRef] [PubMed]

Berger, M.

Berr, S. S.

W. P. Segars, B. M. Tsui, E. C. Frey, G. A. Johnson, and S. S. Berr, “Development of a 4-D digital mouse phantom for molecular imaging research,” Mol. Imaging Biol.6(3), 149–159 (2004).
[CrossRef] [PubMed]

Bordy, T.

Bouman, C. A.

S. Ahn, A. J. Chaudhari, F. Darvas, C. A. Bouman, and R. M. Leahy, “Fast iterative image reconstruction methods for fully 3D multispectral bioluminescence tomography,” Phys. Med. Biol.53(14), 3921–3942 (2008).
[CrossRef] [PubMed]

Boutet, J.

Cao, N.

Carpenter, C. M.

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

Chan, T. F.

Chatziioannou, A. F.

Y. Lu, X. Zhang, A. Douraghy, D. Stout, J. Tian, T. F. Chan, and A. F. Chatziioannou, “Source reconstruction for spectrally-resolved bioluminescence tomography with sparse a priori information,” Opt. Express17(10), 8062–8080 (2009).
[CrossRef] [PubMed]

G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol.50(17), 4225–4241 (2005).
[CrossRef] [PubMed]

Chaudhari, A. J.

S. Ahn, A. J. Chaudhari, F. Darvas, C. A. Bouman, and R. M. Leahy, “Fast iterative image reconstruction methods for fully 3D multispectral bioluminescence tomography,” Phys. Med. Biol.53(14), 3921–3942 (2008).
[CrossRef] [PubMed]

A. J. Chaudhari, F. Darvas, J. R. Bading, R. A. Moats, P. S. Conti, D. J. Smith, S. R. Cherry, and R. M. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol.50(23), 5421–5441 (2005).
[CrossRef] [PubMed]

Chaudhari, M. A. J.

M. A. J. Chaudhari, A. A. Joshi, F. Darvas, and R. M. Leahy, “A method for atlas-based volumetric registration with surface constraints for optical bioluminescence tomography in small animal imaging,” Proc. SPIE6510, 651024, 651024–10 (2007).
[CrossRef]

Chen, D.

Chen, X.

X. Chen, X. Gao, D. Chen, X. Ma, X. Zhao, M. Shen, X. Li, X. Qu, J. Liang, J. Ripoll, and J. Tian, “3D reconstruction of light flux distribution on arbitrary surfaces from 2D multi-photographic images,” Opt. Express18(19), 19876–19893 (2010).
[CrossRef] [PubMed]

H. Huang, X. Qu, J. Liang, X. He, X. Chen, D. Yang, and J. Tian, “A multi-phase level set framework for source reconstruction in bioluminescence tomography,” J. Comput. Phys.229(13), 5246–5256 (2010).
[CrossRef]

Cherry, S. R.

A. J. Chaudhari, F. Darvas, J. R. Bading, R. A. Moats, P. S. Conti, D. J. Smith, S. R. Cherry, and R. M. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol.50(23), 5421–5441 (2005).
[CrossRef] [PubMed]

Comsa, D. C.

D. C. Comsa, T. J. Farrell, and M. S. Patterson, “Quantification of bioluminescence images of point source objects using diffusion theory models,” Phys. Med. Biol.51(15), 3733–3746 (2006).
[CrossRef] [PubMed]

Cong, A.

Cong, W.

Conti, P. S.

A. J. Chaudhari, F. Darvas, J. R. Bading, R. A. Moats, P. S. Conti, D. J. Smith, S. R. Cherry, and R. M. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol.50(23), 5421–5441 (2005).
[CrossRef] [PubMed]

Coquoz, O.

C. Kuo, O. Coquoz, T. L. Troy, H. Xu, and B. W. Rice, “Three-dimensional reconstruction of in vivo bioluminescent sources based on multispectral imaging,” J. Biomed. Opt.12(2), 024007 (2007).
[CrossRef] [PubMed]

Da Silva, A.

Darvas, F.

S. Ahn, A. J. Chaudhari, F. Darvas, C. A. Bouman, and R. M. Leahy, “Fast iterative image reconstruction methods for fully 3D multispectral bioluminescence tomography,” Phys. Med. Biol.53(14), 3921–3942 (2008).
[CrossRef] [PubMed]

M. A. J. Chaudhari, A. A. Joshi, F. Darvas, and R. M. Leahy, “A method for atlas-based volumetric registration with surface constraints for optical bioluminescence tomography in small animal imaging,” Proc. SPIE6510, 651024, 651024–10 (2007).
[CrossRef]

A. J. Chaudhari, F. Darvas, J. R. Bading, R. A. Moats, P. S. Conti, D. J. Smith, S. R. Cherry, and R. M. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol.50(23), 5421–5441 (2005).
[CrossRef] [PubMed]

Davis, S. C.

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

H. Dehghani, B. W. Pogue, S. C. Davis, and M. S. Patterson, “Modeling and image reconstruction in spectrally resolved bioluminescence tomography,” Proc. SPIE6434, 64340V, 64340V–9 (2007).
[CrossRef]

H. Dehghani, S. C. Davis, S. Jiang, B. W. Pogue, K. D. Paulsen, and M. S. Patterson, “Spectrally resolved bioluminescence optical tomography,” Opt. Lett.31(3), 365–367 (2006).
[CrossRef] [PubMed]

Debourdeau, M.

Dehghani, H.

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

H. Dehghani, B. W. Pogue, S. C. Davis, and M. S. Patterson, “Modeling and image reconstruction in spectrally resolved bioluminescence tomography,” Proc. SPIE6434, 64340V, 64340V–9 (2007).
[CrossRef]

H. Dehghani, S. C. Davis, S. Jiang, B. W. Pogue, K. D. Paulsen, and M. S. Patterson, “Spectrally resolved bioluminescence optical tomography,” Opt. Lett.31(3), 365–367 (2006).
[CrossRef] [PubMed]

Delpy, D. T.

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

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys.20(2), 299–309 (1993).
[CrossRef] [PubMed]

Dinkelborg, L. M.

J. K. Willmann, N. van Bruggen, L. M. Dinkelborg, and S. S. Gambhir, “Molecular imaging in drug development,” Nat. Rev. Drug Discov.7(7), 591–607 (2008).
[CrossRef] [PubMed]

Dinten, J. M.

Douraghy, A.

Driol, C.

Eames, M. E.

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

Eftekhar, A. A.

Farrell, T. J.

D. C. Comsa, T. J. Farrell, and M. S. Patterson, “Quantification of bioluminescence images of point source objects using diffusion theory models,” Phys. Med. Biol.51(15), 3733–3746 (2006).
[CrossRef] [PubMed]

Feng, J.

Foschum, F.

Frey, E. C.

W. P. Segars, B. M. Tsui, E. C. Frey, G. A. Johnson, and S. S. Berr, “Development of a 4-D digital mouse phantom for molecular imaging research,” Mol. Imaging Biol.6(3), 149–159 (2004).
[CrossRef] [PubMed]

Gambhir, S. S.

J. K. Willmann, N. van Bruggen, L. M. Dinkelborg, and S. S. Gambhir, “Molecular imaging in drug development,” Nat. Rev. Drug Discov.7(7), 591–607 (2008).
[CrossRef] [PubMed]

Gao, X.

Gu, X.

He, X.

X. He, J. Liang, X. Qu, H. Huang, Y. Hou, and J. Tian, “Truncated total least squares method with a practical truncation parameter choice scheme for bioluminescence tomography inverse problem,” Int. J. Biomed. Imaging2010, 12 (2010).
[CrossRef] [PubMed]

H. Huang, X. Qu, J. Liang, X. He, X. Chen, D. Yang, and J. Tian, “A multi-phase level set framework for source reconstruction in bioluminescence tomography,” J. Comput. Phys.229(13), 5246–5256 (2010).
[CrossRef]

Hervé, L.

Hiraoka, M.

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

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys.20(2), 299–309 (1993).
[CrossRef] [PubMed]

Hoffman, E. A.

Hou, Y.

X. He, J. Liang, X. Qu, H. Huang, Y. Hou, and J. Tian, “Truncated total least squares method with a practical truncation parameter choice scheme for bioluminescence tomography inverse problem,” Int. J. Biomed. Imaging2010, 12 (2010).
[CrossRef] [PubMed]

Huang, H.

X. He, J. Liang, X. Qu, H. Huang, Y. Hou, and J. Tian, “Truncated total least squares method with a practical truncation parameter choice scheme for bioluminescence tomography inverse problem,” Int. J. Biomed. Imaging2010, 12 (2010).
[CrossRef] [PubMed]

H. Huang, X. Qu, J. Liang, X. He, X. Chen, D. Yang, and J. Tian, “A multi-phase level set framework for source reconstruction in bioluminescence tomography,” J. Comput. Phys.229(13), 5246–5256 (2010).
[CrossRef]

Huang, J.

Jacobs, M.

Jia, K.

Jiang, H.

Jiang, M.

Jiang, S.

Johnson, G. A.

W. P. Segars, B. M. Tsui, E. C. Frey, G. A. Johnson, and S. S. Berr, “Development of a 4-D digital mouse phantom for molecular imaging research,” Mol. Imaging Biol.6(3), 149–159 (2004).
[CrossRef] [PubMed]

Joshi, A. A.

M. A. J. Chaudhari, A. A. Joshi, F. Darvas, and R. M. Leahy, “A method for atlas-based volumetric registration with surface constraints for optical bioluminescence tomography in small animal imaging,” Proc. SPIE6510, 651024, 651024–10 (2007).
[CrossRef]

Kienle, A.

Klose, A. D.

Koenig, A.

Kumar, D.

Kuo, C.

C. Kuo, O. Coquoz, T. L. Troy, H. Xu, and B. W. Rice, “Three-dimensional reconstruction of in vivo bioluminescent sources based on multispectral imaging,” J. Biomed. Opt.12(2), 024007 (2007).
[CrossRef] [PubMed]

Larcom, L.

Leabad, M.

Leahy, R. M.

S. Ahn, A. J. Chaudhari, F. Darvas, C. A. Bouman, and R. M. Leahy, “Fast iterative image reconstruction methods for fully 3D multispectral bioluminescence tomography,” Phys. Med. Biol.53(14), 3921–3942 (2008).
[CrossRef] [PubMed]

M. A. J. Chaudhari, A. A. Joshi, F. Darvas, and R. M. Leahy, “A method for atlas-based volumetric registration with surface constraints for optical bioluminescence tomography in small animal imaging,” Proc. SPIE6510, 651024, 651024–10 (2007).
[CrossRef]

A. J. Chaudhari, F. Darvas, J. R. Bading, R. A. Moats, P. S. Conti, D. J. Smith, S. R. Cherry, and R. M. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol.50(23), 5421–5441 (2005).
[CrossRef] [PubMed]

Lewis, M. A.

N. V. Slavine, M. A. Lewis, E. Richer, and P. P. Antich, “Iterative reconstruction method for light emitting sources based on the diffusion equation,” Med. Phys.33(1), 61–68 (2006).
[CrossRef] [PubMed]

Li, X.

Li, Y.

J. Tian, J. Bai, X. P. Yan, S. Bao, Y. Li, W. Liang, and X. Yang, “Multimodality molecular imaging,” IEEE Eng. Med. Biol. Mag.27(5), 48–57 (2008).
[CrossRef] [PubMed]

Liang, J.

X. He, J. Liang, X. Qu, H. Huang, Y. Hou, and J. Tian, “Truncated total least squares method with a practical truncation parameter choice scheme for bioluminescence tomography inverse problem,” Int. J. Biomed. Imaging2010, 12 (2010).
[CrossRef] [PubMed]

X. Chen, X. Gao, D. Chen, X. Ma, X. Zhao, M. Shen, X. Li, X. Qu, J. Liang, J. Ripoll, and J. Tian, “3D reconstruction of light flux distribution on arbitrary surfaces from 2D multi-photographic images,” Opt. Express18(19), 19876–19893 (2010).
[CrossRef] [PubMed]

H. Huang, X. Qu, J. Liang, X. He, X. Chen, D. Yang, and J. Tian, “A multi-phase level set framework for source reconstruction in bioluminescence tomography,” J. Comput. Phys.229(13), 5246–5256 (2010).
[CrossRef]

Liang, W.

J. Tian, J. Bai, X. P. Yan, S. Bao, Y. Li, W. Liang, and X. Yang, “Multimodality molecular imaging,” IEEE Eng. Med. Biol. Mag.27(5), 48–57 (2008).
[CrossRef] [PubMed]

Liu, Y.

Lu, Y.

Lv, Y.

Ma, X.

Mahmood, U.

R. Weissleder and U. Mahmood, “Molecular imaging,” Radiology219(2), 316–333 (2001).
[PubMed]

McCray, P. B.

McLennan, G.

Moats, R. A.

A. J. Chaudhari, F. Darvas, J. R. Bading, R. A. Moats, P. S. Conti, D. J. Smith, S. R. Cherry, and R. M. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol.50(23), 5421–5441 (2005).
[CrossRef] [PubMed]

Mohajerani, P.

Naser, M. A.

Nehorai, A.

Ntziachristos, 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]

Patterson, M. S.

Paulsen, K. D.

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

H. Dehghani, S. C. Davis, S. Jiang, B. W. Pogue, K. D. Paulsen, and M. S. Patterson, “Spectrally resolved bioluminescence optical tomography,” Opt. Lett.31(3), 365–367 (2006).
[CrossRef] [PubMed]

Peltié, P.

Pogue, B. W.

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

H. Dehghani, B. W. Pogue, S. C. Davis, and M. S. Patterson, “Modeling and image reconstruction in spectrally resolved bioluminescence tomography,” Proc. SPIE6434, 64340V, 64340V–9 (2007).
[CrossRef]

H. Dehghani, S. C. Davis, S. Jiang, B. W. Pogue, K. D. Paulsen, and M. S. Patterson, “Spectrally resolved bioluminescence optical tomography,” Opt. Lett.31(3), 365–367 (2006).
[CrossRef] [PubMed]

Qin, C.

Qu, X.

X. Chen, X. Gao, D. Chen, X. Ma, X. Zhao, M. Shen, X. Li, X. Qu, J. Liang, J. Ripoll, and J. Tian, “3D reconstruction of light flux distribution on arbitrary surfaces from 2D multi-photographic images,” Opt. Express18(19), 19876–19893 (2010).
[CrossRef] [PubMed]

X. He, J. Liang, X. Qu, H. Huang, Y. Hou, and J. Tian, “Truncated total least squares method with a practical truncation parameter choice scheme for bioluminescence tomography inverse problem,” Int. J. Biomed. Imaging2010, 12 (2010).
[CrossRef] [PubMed]

H. Huang, X. Qu, J. Liang, X. He, X. Chen, D. Yang, and J. Tian, “A multi-phase level set framework for source reconstruction in bioluminescence tomography,” J. Comput. Phys.229(13), 5246–5256 (2010).
[CrossRef]

Rannou, F. R.

G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol.50(17), 4225–4241 (2005).
[CrossRef] [PubMed]

Rice, B. W.

C. Kuo, O. Coquoz, T. L. Troy, H. Xu, and B. W. Rice, “Three-dimensional reconstruction of in vivo bioluminescent sources based on multispectral imaging,” J. Biomed. Opt.12(2), 024007 (2007).
[CrossRef] [PubMed]

Richer, E.

N. V. Slavine, M. A. Lewis, E. Richer, and P. P. Antich, “Iterative reconstruction method for light emitting sources based on the diffusion equation,” Med. Phys.33(1), 61–68 (2006).
[CrossRef] [PubMed]

Ripoll, J.

Rizo, P.

Schweiger, M.

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

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys.20(2), 299–309 (1993).
[CrossRef] [PubMed]

Segars, W. P.

W. P. Segars, B. M. Tsui, E. C. Frey, G. A. Johnson, and S. S. Berr, “Development of a 4-D digital mouse phantom for molecular imaging research,” Mol. Imaging Biol.6(3), 149–159 (2004).
[CrossRef] [PubMed]

Shen, M.

Slavine, N. V.

N. V. Slavine, M. A. Lewis, E. Richer, and P. P. Antich, “Iterative reconstruction method for light emitting sources based on the diffusion equation,” Med. Phys.33(1), 61–68 (2006).
[CrossRef] [PubMed]

Smith, D. J.

A. J. Chaudhari, F. Darvas, J. R. Bading, R. A. Moats, P. S. Conti, D. J. Smith, S. R. Cherry, and R. M. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol.50(23), 5421–5441 (2005).
[CrossRef] [PubMed]

Srinivasan, S.

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

Stout, D.

Tian, J.

H. Huang, X. Qu, J. Liang, X. He, X. Chen, D. Yang, and J. Tian, “A multi-phase level set framework for source reconstruction in bioluminescence tomography,” J. Comput. Phys.229(13), 5246–5256 (2010).
[CrossRef]

X. He, J. Liang, X. Qu, H. Huang, Y. Hou, and J. Tian, “Truncated total least squares method with a practical truncation parameter choice scheme for bioluminescence tomography inverse problem,” Int. J. Biomed. Imaging2010, 12 (2010).
[CrossRef] [PubMed]

X. Chen, X. Gao, D. Chen, X. Ma, X. Zhao, M. Shen, X. Li, X. Qu, J. Liang, J. Ripoll, and J. Tian, “3D reconstruction of light flux distribution on arbitrary surfaces from 2D multi-photographic images,” Opt. Express18(19), 19876–19893 (2010).
[CrossRef] [PubMed]

Y. Lu, X. Zhang, A. Douraghy, D. Stout, J. Tian, T. F. Chan, and A. F. Chatziioannou, “Source reconstruction for spectrally-resolved bioluminescence tomography with sparse a priori information,” Opt. Express17(10), 8062–8080 (2009).
[CrossRef] [PubMed]

J. Feng, K. Jia, G. Yan, S. Zhu, C. Qin, Y. Lv, and J. Tian, “An optimal permissible source region strategy for multispectral bioluminescence tomography,” Opt. Express16(20), 15640–15654 (2008).
[CrossRef] [PubMed]

J. Tian, J. Bai, X. P. Yan, S. Bao, Y. Li, W. Liang, and X. Yang, “Multimodality molecular imaging,” IEEE Eng. Med. Biol. Mag.27(5), 48–57 (2008).
[CrossRef] [PubMed]

Troy, T. L.

C. Kuo, O. Coquoz, T. L. Troy, H. Xu, and B. W. Rice, “Three-dimensional reconstruction of in vivo bioluminescent sources based on multispectral imaging,” J. Biomed. Opt.12(2), 024007 (2007).
[CrossRef] [PubMed]

Tsui, B. M.

W. P. Segars, B. M. Tsui, E. C. Frey, G. A. Johnson, and S. S. Berr, “Development of a 4-D digital mouse phantom for molecular imaging research,” Mol. Imaging Biol.6(3), 149–159 (2004).
[CrossRef] [PubMed]

van Bruggen, N.

J. K. Willmann, N. van Bruggen, L. M. Dinkelborg, and S. S. Gambhir, “Molecular imaging in drug development,” Nat. Rev. Drug Discov.7(7), 591–607 (2008).
[CrossRef] [PubMed]

Wang, G.

Wang, L. V.

W. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. V. Wang, E. A. Hoffman, G. McLennan, P. B. McCray, J. Zabner, and A. Cong, “Practical reconstruction method for bioluminescence tomography,” Opt. Express13(18), 6756–6771 (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]

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]

R. Weissleder and U. Mahmood, “Molecular imaging,” Radiology219(2), 316–333 (2001).
[PubMed]

Willmann, J. K.

J. K. Willmann, N. van Bruggen, L. M. Dinkelborg, and S. S. Gambhir, “Molecular imaging in drug development,” Nat. Rev. Drug Discov.7(7), 591–607 (2008).
[CrossRef] [PubMed]

Xu, H.

C. Kuo, O. Coquoz, T. L. Troy, H. Xu, and B. W. Rice, “Three-dimensional reconstruction of in vivo bioluminescent sources based on multispectral imaging,” J. Biomed. Opt.12(2), 024007 (2007).
[CrossRef] [PubMed]

Yalavarthy, P. K.

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

Yan, G.

Yan, X. P.

J. Tian, J. Bai, X. P. Yan, S. Bao, Y. Li, W. Liang, and X. Yang, “Multimodality molecular imaging,” IEEE Eng. Med. Biol. Mag.27(5), 48–57 (2008).
[CrossRef] [PubMed]

Yang, D.

H. Huang, X. Qu, J. Liang, X. He, X. Chen, D. Yang, and J. Tian, “A multi-phase level set framework for source reconstruction in bioluminescence tomography,” J. Comput. Phys.229(13), 5246–5256 (2010).
[CrossRef]

Yang, X.

J. Tian, J. Bai, X. P. Yan, S. Bao, Y. Li, W. Liang, and X. Yang, “Multimodality molecular imaging,” IEEE Eng. Med. Biol. Mag.27(5), 48–57 (2008).
[CrossRef] [PubMed]

Zabner, J.

Zhang, Q.

Zhang, X.

Zhao, X.

Zhu, S.

Appl. Opt. (4)

Biomed. Opt. Express (3)

Commun. Numer. Methods Eng. (1)

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

IEEE Eng. Med. Biol. Mag. (1)

J. Tian, J. Bai, X. P. Yan, S. Bao, Y. Li, W. Liang, and X. Yang, “Multimodality molecular imaging,” IEEE Eng. Med. Biol. Mag.27(5), 48–57 (2008).
[CrossRef] [PubMed]

Int. J. Biomed. Imaging (1)

X. He, J. Liang, X. Qu, H. Huang, Y. Hou, and J. Tian, “Truncated total least squares method with a practical truncation parameter choice scheme for bioluminescence tomography inverse problem,” Int. J. Biomed. Imaging2010, 12 (2010).
[CrossRef] [PubMed]

Inverse Probl. (1)

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

J. Biomed. Opt. (1)

C. Kuo, O. Coquoz, T. L. Troy, H. Xu, and B. W. Rice, “Three-dimensional reconstruction of in vivo bioluminescent sources based on multispectral imaging,” J. Biomed. Opt.12(2), 024007 (2007).
[CrossRef] [PubMed]

J. Comput. Phys. (1)

H. Huang, X. Qu, J. Liang, X. He, X. Chen, D. Yang, and J. Tian, “A multi-phase level set framework for source reconstruction in bioluminescence tomography,” J. Comput. Phys.229(13), 5246–5256 (2010).
[CrossRef]

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

Med. Phys. (3)

N. V. Slavine, M. A. Lewis, E. Richer, and P. P. Antich, “Iterative reconstruction method for light emitting sources based on the diffusion equation,” Med. Phys.33(1), 61–68 (2006).
[CrossRef] [PubMed]

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys.20(2), 299–309 (1993).
[CrossRef] [PubMed]

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

Mol. Imaging Biol. (1)

W. P. Segars, B. M. Tsui, E. C. Frey, G. A. Johnson, and S. S. Berr, “Development of a 4-D digital mouse phantom for molecular imaging research,” Mol. Imaging Biol.6(3), 149–159 (2004).
[CrossRef] [PubMed]

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]

Nat. Rev. Drug Discov. (1)

J. K. Willmann, N. van Bruggen, L. M. Dinkelborg, and S. S. Gambhir, “Molecular imaging in drug development,” Nat. Rev. Drug Discov.7(7), 591–607 (2008).
[CrossRef] [PubMed]

Opt. Express (7)

X. Gu, Q. Zhang, L. Larcom, and H. Jiang, “Three-dimensional bioluminescence tomography with model-based reconstruction,” Opt. Express12(17), 3996–4000 (2004).
[CrossRef] [PubMed]

W. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. V. Wang, E. A. Hoffman, G. McLennan, P. B. McCray, J. Zabner, and A. Cong, “Practical reconstruction method for bioluminescence tomography,” Opt. Express13(18), 6756–6771 (2005).
[CrossRef] [PubMed]

A. Kienle and F. Foschum, “250 years Lambert surface: does it really exist?” Opt. Express19(5), 3881–3889 (2011).
[CrossRef] [PubMed]

X. Chen, X. Gao, D. Chen, X. Ma, X. Zhao, M. Shen, X. Li, X. Qu, J. Liang, J. Ripoll, and J. Tian, “3D reconstruction of light flux distribution on arbitrary surfaces from 2D multi-photographic images,” Opt. Express18(19), 19876–19893 (2010).
[CrossRef] [PubMed]

Y. Lu, X. Zhang, A. Douraghy, D. Stout, J. Tian, T. F. Chan, and A. F. Chatziioannou, “Source reconstruction for spectrally-resolved bioluminescence tomography with sparse a priori information,” Opt. Express17(10), 8062–8080 (2009).
[CrossRef] [PubMed]

N. Cao, A. Nehorai, and M. Jacobs, “Image reconstruction for diffuse optical tomography using sparsity regularization and expectation-maximization algorithm,” Opt. Express15(21), 13695–13708 (2007).
[CrossRef] [PubMed]

J. Feng, K. Jia, G. Yan, S. Zhu, C. Qin, Y. Lv, and J. Tian, “An optimal permissible source region strategy for multispectral bioluminescence tomography,” Opt. Express16(20), 15640–15654 (2008).
[CrossRef] [PubMed]

Opt. Lett. (1)

Phys. Med. Biol. (4)

G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol.50(17), 4225–4241 (2005).
[CrossRef] [PubMed]

A. J. Chaudhari, F. Darvas, J. R. Bading, R. A. Moats, P. S. Conti, D. J. Smith, S. R. Cherry, and R. M. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol.50(23), 5421–5441 (2005).
[CrossRef] [PubMed]

S. Ahn, A. J. Chaudhari, F. Darvas, C. A. Bouman, and R. M. Leahy, “Fast iterative image reconstruction methods for fully 3D multispectral bioluminescence tomography,” Phys. Med. Biol.53(14), 3921–3942 (2008).
[CrossRef] [PubMed]

D. C. Comsa, T. J. Farrell, and M. S. Patterson, “Quantification of bioluminescence images of point source objects using diffusion theory models,” Phys. Med. Biol.51(15), 3733–3746 (2006).
[CrossRef] [PubMed]

Proc. SPIE (2)

M. A. J. Chaudhari, A. A. Joshi, F. Darvas, and R. M. Leahy, “A method for atlas-based volumetric registration with surface constraints for optical bioluminescence tomography in small animal imaging,” Proc. SPIE6510, 651024, 651024–10 (2007).
[CrossRef]

H. Dehghani, B. W. Pogue, S. C. Davis, and M. S. Patterson, “Modeling and image reconstruction in spectrally resolved bioluminescence tomography,” Proc. SPIE6434, 64340V, 64340V–9 (2007).
[CrossRef]

Radiology (1)

R. Weissleder and U. Mahmood, “Molecular imaging,” Radiology219(2), 316–333 (2001).
[PubMed]

Other (3)

J. Pekar, “Multispectral bioluminescence tomography with x-ray CT spatial priors” Ph.D. thesis (McMaster University, Hamilton, 2011), Open Access Dissertations and Theses, Paper 4329, http://digitalcommons.mcmaster.ca/opendissertations/4329 .

Q. Fang and D. Boas, “Tetrahedral mesh generation from volumetric binary and gray-scale images,” in IEEE International Symposium on Biomedical Imaging: from Nano to Macro,2009. ISBI '09 (IEEE, 2009), pp. 1142–1145.

S. A. Prahl, “Optical properties spectra,” (Oregon Medical Laser Clinic, 2001), http://omlc.ogi.edu/spectra/index.html .

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Figures (5)

Fig. 1
Fig. 1

(a) A 3D finite element mesh for the object used in the simulation. The mesh is a regular Delaunay tetrahedral with 17966 nodes and 99140 tetrahedrons generated by iso2mesh [36]. The object represents a section of a mouse abdomen which illustrates 5 different tissue regions: (1) adipose, (2) liver and spleen, (3) bowel and intestine, (4) kidneys, and (5) bone. (b) A schematic of the object showing the sources and detectors set up and the coordinate system.

Fig. 2
Fig. 2

The relative percent error in the reconstructions of scattering μ s , absorption μ a ,and effective attenuation μ e f f , for the 5 tissue regions (1) adipose, (2) liver, (3) bowel, (4) kidney, and (5) bone at 590 nm using absolute, relative, and imperfect calibrations. Note the split vertical scale.

Fig. 3
Fig. 3

BLT reconstruction of two uniform bioluminescence sources localized in the left and right kidneys (4 mm diameters). The “anatomy” is shown in Fig. 1. The actual sources in transverse section (a) and coronal section (b) have magnitudes 2 nW/ mm3 and 1 nW/ mm3 for the left and right source respectively. The reconstructed sources are shown in (c) and (d) using absolute data, (e) and (f) using relative data, and (g) and (h) using imperfect calibrated data. The total powers of the actual and reconstructed sources and the normalized magnitude errors of the reconstructed sources using different reconstruction methods are shown in the figure.

Fig. 4
Fig. 4

BLT reconstruction of large distributed sources that fill the entire two kidneys. The magnitude of the actual sources in (a) and (b) is 1 nW/mm3, and the reconstructed sources are shown in (c) and (d) using absolute data, (e) and (f) using relative data, and (g) and (h) using imperfect calibrated data. The total powers of the actual and reconstructed sources and the normalized magnitude errors of the reconstructed sources using different reconstruction methods are shown in the figure.

Fig. 5
Fig. 5

BLT reconstruction of a large non-uniform source distribution that fills the entire right kidney. The actual source in (a) and (b) has a hot spot of 4 mm diameter with magnitude 2 nW/mm3 which is twice the magnitude of the background. The reconstructed source in (c) and (d) using the absolute data, (e) and (f) using the relative data, and (g) and (g) using the imperfect calibrated data. The total powers of the actual and reconstructed sources and the normalized magnitude errors of the reconstructed sources using different reconstruction methods are shown in the figure.

Tables (2)

Tables Icon

Table 1 True [24,37] reduced scattering coefficients (mm−1) for each region in the heterogeneous object at six different wavelengths

Tables Icon

Table 2 True [24,37] absorption coefficients (mm−1) for each region in the heterogeneous object at six different wavelengths

Equations (26)

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[ κ ( r , λ ) + ( μ a ( r , λ ) + i ω c ) ] ϕ ( r , ω , λ ) = s ( r , ω , λ )        ( r Ω ) ,
[ 1 + 2 κ ( r , λ ) ξ n ^ ( r ) μ a ( r , λ ) ] ϕ ( r , λ , ω ) = 0           ( r Ω ) ,
A ( λ ) ϕ ( λ ) = s ( λ ) ,
A μ n ϕ + A ϕ μ n = s μ n = 0 ,
ϕ μ n = A 1 A μ n ϕ .
ϕ d μ n = ϕ ( d ) μ n = A 1 ( d , : ) A μ n ϕ ,
     ϕ d μ n = φ d μ n e i θ d + i φ d e i θ d θ d μ n ϕ d μ n / ϕ d = φ d μ n / φ d + i θ d μ n = log ( φ d ) μ n + i θ d μ n .
log ( φ d ) μ n = real ( ϕ d μ n / ϕ d )  , θ d μ n = imag ( ϕ d μ n / ϕ d ) .
log ( φ d ) = log ( φ d 0 ) + n log ( φ d ) μ n δ μ n , θ d = θ d 0 + n θ d μ n δ μ n ,
( log ( φ d ) θ d ) = ( log ( φ d 0 ) θ d 0 ) + ( log ( φ d ) μ 1 log ( φ d ) μ 2 N θ d μ 1 θ d μ 2 N ) ( μ 1 μ 2 N )
Φ d = Φ d 0 + J d μ .
( Φ d 1 Φ d N s ) = ( Φ d 0 1 Φ d 0 N s ) + ( J 1 J N s ) d μ ,
d μ = ( J ¯ T J ¯ ) 1 ( J ¯ T ( Φ ¯ d Φ ¯ d 0 ) ) .
      ϕ m = φ m e i θ m = α φ d e i ( θ d + β ) ( log ( φ m ) θ m ) = ( log ( φ d ) θ d ) + ( log ( α ) β ) , or  Φ m = Φ d + ( log ( α ) β ) ,
( Φ m 1 Φ m N s ) = ( Φ d 1 + ( log ( α ) β ) Φ d N s + ( log ( α ) β ) ) .
( Φ m 2 Φ m 1 Φ m 3 Φ m 1 Φ m N s Φ m 1 Φ m 3 Φ m 2 Φ m 4 Φ m 2 Φ m N s Φ m 2 Φ m N s Φ m N s 1 ) = ( Φ d 2 Φ d 1 Φ d 3 Φ d 1 Φ d N s Φ d 1 Φ d 3 Φ d 2 Φ d 4 Φ d 2 Φ d N s Φ d 2 Φ d N s Φ d N s 1 ) = ( Φ d 0 2 Φ d 0 1 Φ d 0 3 Φ d 0 1 Φ d 0 N s Φ d 0 1 Φ d 0 3 Φ d 0 2 Φ d 0 4 Φ d 0 2 Φ d 0 N s Φ d 0 2 Φ d 0 N s Φ d 0 N s 1 ) + ( J 2 J 1 J 3 J 1 J N s J 1 J 3 J 2 J 4 J 2 J N s J 2 J N s J N s 1 )   d μ ,
d μ = ( J ¯ ¯ T J ¯ ¯ ) 1 ( J ¯ ¯ T ( Φ ¯ ¯ m Φ ¯ ¯ d 0 ) ) .
μ i = μ i 1 + d μ i .
φ ( λ ) = G ( λ ) s ( λ ) ,
φ ( d ; λ ) = G ( d , R ; λ ) s ( R )  or  φ d ( λ ) = G d R ( λ ) s R ,
φ m = α φ d      φ m = α G d R s R ,
log φ m s R = log α + log ( G d R s R ) log φ m s 1 = log α + log ( G d s 1 ) log φ m s 2 = log α + log ( G d s 2 )             log φ m s N s = log α + log ( G d s N s )
G d R s R = exp ( log φ m s R 1 N s i = 1 N s ( log φ m s i log ( G d s i ) ) ) G d R s R = φ ¯ m s R ,
( G d R ( λ 1 ) / max ( φ ¯ m s R ( λ 1 ) ) G d R ( λ 2 ) / max ( φ ¯ m s R ( λ 2 ) ) G d R ( λ N ) / max ( φ ¯ m s R ( λ N ) ) ) s R = ( φ ¯ m s R ( λ 1 ) / max ( φ ¯ m s R ( λ 1 ) ) φ ¯ m s R ( λ 2 ) / max ( φ ¯ m s R ( λ 2 ) ) φ ¯ m s R ( λ N ) / max ( φ ¯ m s R ( λ N ) ) ) ,
( G ¯ d R T G ¯ d R ) s R = G ¯ d R T φ ¯ m .
i | s actual ( i ) s reconst ( i ) | V ( i ) / i s actual ( i ) V ( i ) .

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