Abstract

We present a finite-element-based algorithm for reconstruction of fluorescence lifetime and yield in turbid media, using frequency-domain data. The algorithm is based on a set of coupled diffusion equations that describe the propagation of both excitation and fluorescent emission light in multiply scattering media. Centered on Newton’s iterative method, we implemented our algorithm by using a synthesized scheme of Marquardt and Tikhonov regularizations. A low-pass spatial filter is also incorporated into the algorithm for enhancing image reconstruction. Simulation studies using both noise-free and noisy data have been performed with the nonzero photon density boundary conditions. Our results suggest that quantitative images can be produced in terms of fluorescent lifetime and yield values and location, size, and shape of heterogeneities within a circular background region.

© 1998 Optical Society of America

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1998

H. Jiang, K. D. Paulsen, U. L. Osterberg, M. S. Patterson, “Frequency-domain optical image reconstruction for breast imaging: initial evaluation in multitarget tissuelike phantoms,” Med. Phys. 25, 183–193 (1998).
[CrossRef] [PubMed]

1997

H. Jiang, K. D. Paulsen, U. L. Osterberg, M. S. Patterson, “Frequency-domain optical image reconstruction in heterogeneous media: an experimental study of single-target detectability,” Appl. Opt. 36, 52–63 (1997).
[CrossRef] [PubMed]

M. Schweijer, S. R. Arridge, “Direct calculation with a finite-element method of the Laplace transform of the distribution of photon time of flight in tissue,” Appl. Opt. 36, 9042–9049 (1997).
[CrossRef]

J. Chang, H. L. Graber, R. L. Barbour, “Luminescence optical tomography of dense scattering media,” J. Opt. Soc. Am. A 14, 288–299 (1997).
[CrossRef]

J. Chang, H. Graber, R. L. Barbour, “Imaging of fluorescence in highly scattering media,” IEEE Trans. Biomed. Eng. 44, 810–822 (1997).
[CrossRef] [PubMed]

D. Y. Paithankar, A. U. Chen, B. W. Pogue, M. S. Patterson, E. M. Sevick-Muraca, “Imaging of fluorescent lifetime and yield from multiple scattered light reemitted from tissues and other random media,” Appl. Opt. 36, 2260–2272 (1997).
[CrossRef] [PubMed]

E. Sevick-Muraca, G. Lopez, T. Troy, J. Reynolds, C. Hutchinson, “Fluorescence and absorption contrast mechanisms for biomedical optical imaging using frequency-domain techniques,” Photochem. Photobiol. 66, 55–64 (1997).
[CrossRef] [PubMed]

B. J. Tromberg, O. Coquoz, J. Fishkin, T. Pham, E. Anderson, J. Butler, M. Cahn, J. Gross, V. Venugopalan, D. Pham, “Noninvasive measurements of breast tissue optical properties using frequency-domain photon migration,” Philos. Trans. R. Soc. London B 352, 661–668 (1997).
[CrossRef]

A. E. Cerussi, J. S. Maier, S. Fantini, M. A. Franceschini, W. W. Mantulin, E. Gratton, “Experimental verification of a theory for the time-resolved fluorescence spectroscopy of thick tissues,” Appl. Opt. 36, 116–124 (1997).
[CrossRef] [PubMed]

1996

1995

1994

1993

1992

B. C. Wilson, E. M. Sevick, M. S. Patterson, B. Chance, “Time-dependent optical spectroscopy and imaging for biomedical applications,” Proc. IEEE 80, 918–930 (1992).
[CrossRef]

1991

E. S. Andersson, J. Johansson, K. Svanberg, S. Svanberg, “Fluorescence imaging and point measurements of tissue: applications to the determination of malignant tumors and atherosclerotic lesions from normal tissue,” Photochem. Photobiol. 53, 807–814 (1991).

1990

V. G. Peters, D. R. Wyman, M. S. Patterson, G. L. Frank, “Optical properties of normal and diseased human breast tissue in the visible and near infrared,” Phys. Med. Biol. 35, 1317–1334 (1990).
[CrossRef] [PubMed]

1989

R. Richards-Kortum, R. P. Rava, M. Fitzmaurice, L. L. Tong, N. Ratliff, J. Kramer, M. S. Feld, “A one layer model of laser-induced fluorescence for diagnosis of disease in human tissue: applications to atherosclerosis,” IEEE Trans. Biomed. Eng. 36, 1222–1232 (1989).
[CrossRef] [PubMed]

1987

R. R. Alfano, G. C. Tang, A. Pradhan, W. Lam, D. S. Choy, E. Opher, “Fluorescence spectra from cancerous and normal human breast and lung tissues,” IEEE J. Quantum Electron. QE-23, 1806–1811 (1987).
[CrossRef]

Alfano, R. R.

B. B. Das, K. M. Yoo, R. R. Alfano, “Ultrafast time-gated imaging in thick tissues—a step toward optical mammography,” Opt. Lett. 18, 1092–1094 (1993).
[CrossRef] [PubMed]

R. R. Alfano, G. C. Tang, A. Pradhan, W. Lam, D. S. Choy, E. Opher, “Fluorescence spectra from cancerous and normal human breast and lung tissues,” IEEE J. Quantum Electron. QE-23, 1806–1811 (1987).
[CrossRef]

Anderson, E.

B. J. Tromberg, O. Coquoz, J. Fishkin, T. Pham, E. Anderson, J. Butler, M. Cahn, J. Gross, V. Venugopalan, D. Pham, “Noninvasive measurements of breast tissue optical properties using frequency-domain photon migration,” Philos. Trans. R. Soc. London B 352, 661–668 (1997).
[CrossRef]

Anderson-Engels, S.

R. Berg, S. Anderson-Engels, O. Jarlman, S. Svanberg, “Tumor detection using time-resolved light transillumination,” in Future Trends in Biomedical Applications of Lasers, L. O. Svaasand, ed., Proc. SPIE1525, 59–67 (1991).
[CrossRef]

Andersson, E. S.

E. S. Andersson, J. Johansson, K. Svanberg, S. Svanberg, “Fluorescence imaging and point measurements of tissue: applications to the determination of malignant tumors and atherosclerotic lesions from normal tissue,” Photochem. Photobiol. 53, 807–814 (1991).

Aronson, R.

R. L. Barbour, H. Graber, Y. Wang, J. Chang, R. Aronson, “Perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Muller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. Van der Zee, eds. (SPIE Press, Bellingham, Wash., 1993), pp. 87–120.

Arridge, S. R.

M. Schweijer, S. R. Arridge, “Direct calculation with a finite-element method of the Laplace transform of the distribution of photon time of flight in tissue,” Appl. Opt. 36, 9042–9049 (1997).
[CrossRef]

S. R. Arridge, M. Schweijer, M. Hiraoka, D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med Phys. 20, 299–308 (1993).
[CrossRef] [PubMed]

Barbour, R. L.

J. Chang, H. L. Graber, R. L. Barbour, “Luminescence optical tomography of dense scattering media,” J. Opt. Soc. Am. A 14, 288–299 (1997).
[CrossRef]

J. Chang, H. Graber, R. L. Barbour, “Imaging of fluorescence in highly scattering media,” IEEE Trans. Biomed. Eng. 44, 810–822 (1997).
[CrossRef] [PubMed]

R. L. Barbour, H. Graber, Y. Wang, J. Chang, R. Aronson, “Perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Muller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. Van der Zee, eds. (SPIE Press, Bellingham, Wash., 1993), pp. 87–120.

Berg, R.

R. Berg, S. Anderson-Engels, O. Jarlman, S. Svanberg, “Tumor detection using time-resolved light transillumination,” in Future Trends in Biomedical Applications of Lasers, L. O. Svaasand, ed., Proc. SPIE1525, 59–67 (1991).
[CrossRef]

Boas, D.

Boas, D. A.

Burch, C. L.

Butler, J.

B. J. Tromberg, O. Coquoz, J. Fishkin, T. Pham, E. Anderson, J. Butler, M. Cahn, J. Gross, V. Venugopalan, D. Pham, “Noninvasive measurements of breast tissue optical properties using frequency-domain photon migration,” Philos. Trans. R. Soc. London B 352, 661–668 (1997).
[CrossRef]

Cahn, M.

B. J. Tromberg, O. Coquoz, J. Fishkin, T. Pham, E. Anderson, J. Butler, M. Cahn, J. Gross, V. Venugopalan, D. Pham, “Noninvasive measurements of breast tissue optical properties using frequency-domain photon migration,” Philos. Trans. R. Soc. London B 352, 661–668 (1997).
[CrossRef]

Cerussi, A. E.

Chance, B.

X. Li, M. O’Leary, D. Boas, B. Chance, A. Yodh, “Fluorescent diffuse photon density waves in homogeneous and heterogeneous turbid media: analytic solutions and applications,” Appl. Opt. 35, 3746–3758 (1996).
[CrossRef] [PubMed]

M. A. O’Leary, D. A. Boas, X. D. Li, B. Chance, A. G. Yodh, “Fluorescence lifetime imaging in turbid media,” Opt. Lett. 21, 158–160 (1996).
[CrossRef] [PubMed]

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

E. Gratton, W. Mantulin, M. J. van de Ven, J. Fishkin, M. Maris, B. Chance, “A novel approach to laser tomography,” Bioimaging 1, 40–46 (1993).
[CrossRef]

B. C. Wilson, E. M. Sevick, M. S. Patterson, B. Chance, “Time-dependent optical spectroscopy and imaging for biomedical applications,” Proc. IEEE 80, 918–930 (1992).
[CrossRef]

S. Zhao, Y. Yang, S. Nioka, B. Chance, “Human breast tumor detection using contrast agent,” in Integration of Medical Optical Imaging and Spectroscopy and Magnetic Resonance Imaging Symposium Abstracts (University of Pennsylvania, Philadelphia, Pa., 1994).

Chang, J.

J. Chang, H. Graber, R. L. Barbour, “Imaging of fluorescence in highly scattering media,” IEEE Trans. Biomed. Eng. 44, 810–822 (1997).
[CrossRef] [PubMed]

J. Chang, H. L. Graber, R. L. Barbour, “Luminescence optical tomography of dense scattering media,” J. Opt. Soc. Am. A 14, 288–299 (1997).
[CrossRef]

R. L. Barbour, H. Graber, Y. Wang, J. Chang, R. Aronson, “Perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Muller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. Van der Zee, eds. (SPIE Press, Bellingham, Wash., 1993), pp. 87–120.

Chapman, C.

B. J. Tromberg, S. Madsen, C. Chapman, L. Svaasand, R. Haskell, “Fluorescence energy transfer studies on the macrophage scavenger receptor,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of 1994 OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), p. 93.

Chen, A. U.

Choy, D. S.

R. R. Alfano, G. C. Tang, A. Pradhan, W. Lam, D. S. Choy, E. Opher, “Fluorescence spectra from cancerous and normal human breast and lung tissues,” IEEE J. Quantum Electron. QE-23, 1806–1811 (1987).
[CrossRef]

Coquoz, O.

B. J. Tromberg, O. Coquoz, J. Fishkin, T. Pham, E. Anderson, J. Butler, M. Cahn, J. Gross, V. Venugopalan, D. Pham, “Noninvasive measurements of breast tissue optical properties using frequency-domain photon migration,” Philos. Trans. R. Soc. London B 352, 661–668 (1997).
[CrossRef]

Das, B. B.

Dasari, R. R.

Delpy, D. T.

S. R. Arridge, M. Schweijer, M. Hiraoka, D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med Phys. 20, 299–308 (1993).
[CrossRef] [PubMed]

Durkin, A. J.

Fantini, S.

Feld, M. S.

J. Wu, Y. Wang, L. Perelman, I. Itzkan, R. R. Dasari, M. S. Feld, “Time-resolved multichannel imaging of fluorescent objects embedded in turbid media,” Opt. Lett. 20, 489–491 (1995).
[CrossRef] [PubMed]

J. Wu, M. S. Feld, R. P. Rava, “Analytical model for extracting intrinsic fluorescence in turbid media,” Appl. Opt. 32, 3585–3595 (1993).
[CrossRef] [PubMed]

R. Richards-Kortum, R. P. Rava, M. Fitzmaurice, L. L. Tong, N. Ratliff, J. Kramer, M. S. Feld, “A one layer model of laser-induced fluorescence for diagnosis of disease in human tissue: applications to atherosclerosis,” IEEE Trans. Biomed. Eng. 36, 1222–1232 (1989).
[CrossRef] [PubMed]

Feng, T. C.

Fishkin, J.

B. J. Tromberg, O. Coquoz, J. Fishkin, T. Pham, E. Anderson, J. Butler, M. Cahn, J. Gross, V. Venugopalan, D. Pham, “Noninvasive measurements of breast tissue optical properties using frequency-domain photon migration,” Philos. Trans. R. Soc. London B 352, 661–668 (1997).
[CrossRef]

E. Gratton, W. Mantulin, M. J. van de Ven, J. Fishkin, M. Maris, B. Chance, “A novel approach to laser tomography,” Bioimaging 1, 40–46 (1993).
[CrossRef]

Fitzmaurice, M.

R. Richards-Kortum, R. P. Rava, M. Fitzmaurice, L. L. Tong, N. Ratliff, J. Kramer, M. S. Feld, “A one layer model of laser-induced fluorescence for diagnosis of disease in human tissue: applications to atherosclerosis,” IEEE Trans. Biomed. Eng. 36, 1222–1232 (1989).
[CrossRef] [PubMed]

Franceschini, M. A.

Frank, G. L.

V. G. Peters, D. R. Wyman, M. S. Patterson, G. L. Frank, “Optical properties of normal and diseased human breast tissue in the visible and near infrared,” Phys. Med. Biol. 35, 1317–1334 (1990).
[CrossRef] [PubMed]

Frisoli, J. K.

Graber, H.

J. Chang, H. Graber, R. L. Barbour, “Imaging of fluorescence in highly scattering media,” IEEE Trans. Biomed. Eng. 44, 810–822 (1997).
[CrossRef] [PubMed]

R. L. Barbour, H. Graber, Y. Wang, J. Chang, R. Aronson, “Perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Muller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. Van der Zee, eds. (SPIE Press, Bellingham, Wash., 1993), pp. 87–120.

Graber, H. L.

Gratton, E.

Gross, J.

B. J. Tromberg, O. Coquoz, J. Fishkin, T. Pham, E. Anderson, J. Butler, M. Cahn, J. Gross, V. Venugopalan, D. Pham, “Noninvasive measurements of breast tissue optical properties using frequency-domain photon migration,” Philos. Trans. R. Soc. London B 352, 661–668 (1997).
[CrossRef]

Hasegawa, Y.

Y. Yamada, Y. Hasegawa, “Time-dependent FEM analysis of photon migration in random media,” in Optical Tomography and Spectroscopy of Tissue: Theory, Instrumentation, Model and Human Studies, B. Chance, R. Alfano, eds., Proc. SPIE1888, 167–178 (1993).

Haskell, R.

B. J. Tromberg, S. Madsen, C. Chapman, L. Svaasand, R. Haskell, “Fluorescence energy transfer studies on the macrophage scavenger receptor,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of 1994 OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), p. 93.

Haskell, R. C.

Hebden, J. C.

Hiraoka, M.

S. R. Arridge, M. Schweijer, M. Hiraoka, D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med Phys. 20, 299–308 (1993).
[CrossRef] [PubMed]

Hutchinson, C.

E. Sevick-Muraca, G. Lopez, T. Troy, J. Reynolds, C. Hutchinson, “Fluorescence and absorption contrast mechanisms for biomedical optical imaging using frequency-domain techniques,” Photochem. Photobiol. 66, 55–64 (1997).
[CrossRef] [PubMed]

Hutchinson, C. L.

T. L. Troy, L. Nelson-Larry, C. L. Hutchinson, E. M. Sevick-Muraca, “Investigation of exogenous contrast agents for biomedical optical imaging,” in Biomedical Optical Spectroscopy and Diagnostics, E. Sevick-Muraca, D. Benaron, eds., Vol. 3 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 104–106.

Imai, D.

S. Takahashi, D. Imai, Y. Yamada, “Fundamental 3D FEM analysis of light propagation in head model toward 3D optical tomography,” in Optical Tomography and Spectroscopy of Tissue: Theory, Instrumentation, Model and Human Studies II, B. Chance, R. Alfano, eds., Proc. SPIE2979, 130–138 (1997).
[CrossRef]

Itzkan, I.

Jailkumar, S.

Jarlman, O.

R. Berg, S. Anderson-Engels, O. Jarlman, S. Svanberg, “Tumor detection using time-resolved light transillumination,” in Future Trends in Biomedical Applications of Lasers, L. O. Svaasand, ed., Proc. SPIE1525, 59–67 (1991).
[CrossRef]

Jiang, H.

Johansson, J.

E. S. Andersson, J. Johansson, K. Svanberg, S. Svanberg, “Fluorescence imaging and point measurements of tissue: applications to the determination of malignant tumors and atherosclerotic lesions from normal tissue,” Photochem. Photobiol. 53, 807–814 (1991).

Kaneko, M.

Y. Yamishita, M. Kaneko, “Infrared diaphanoscopy for medical diagnosis,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Muller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. Van der Zee, eds. (SPIE Press, Bellingham, Wash., 1993), pp. 283–316.

Kramer, J.

R. Richards-Kortum, R. P. Rava, M. Fitzmaurice, L. L. Tong, N. Ratliff, J. Kramer, M. S. Feld, “A one layer model of laser-induced fluorescence for diagnosis of disease in human tissue: applications to atherosclerosis,” IEEE Trans. Biomed. Eng. 36, 1222–1232 (1989).
[CrossRef] [PubMed]

Lakowicz, J. R.

Lam, W.

R. R. Alfano, G. C. Tang, A. Pradhan, W. Lam, D. S. Choy, E. Opher, “Fluorescence spectra from cancerous and normal human breast and lung tissues,” IEEE J. Quantum Electron. QE-23, 1806–1811 (1987).
[CrossRef]

Li, X.

Li, X. D.

Lopez, G.

E. Sevick-Muraca, G. Lopez, T. Troy, J. Reynolds, C. Hutchinson, “Fluorescence and absorption contrast mechanisms for biomedical optical imaging using frequency-domain techniques,” Photochem. Photobiol. 66, 55–64 (1997).
[CrossRef] [PubMed]

Madsen, S.

B. J. Tromberg, S. Madsen, C. Chapman, L. Svaasand, R. Haskell, “Fluorescence energy transfer studies on the macrophage scavenger receptor,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of 1994 OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), p. 93.

Maier, J. S.

Mantulin, W.

E. Gratton, W. Mantulin, M. J. van de Ven, J. Fishkin, M. Maris, B. Chance, “A novel approach to laser tomography,” Bioimaging 1, 40–46 (1993).
[CrossRef]

Mantulin, W. W.

Maris, M.

E. Gratton, W. Mantulin, M. J. van de Ven, J. Fishkin, M. Maris, B. Chance, “A novel approach to laser tomography,” Bioimaging 1, 40–46 (1993).
[CrossRef]

McAdams, M. S.

Nelson-Larry, L.

T. L. Troy, L. Nelson-Larry, C. L. Hutchinson, E. M. Sevick-Muraca, “Investigation of exogenous contrast agents for biomedical optical imaging,” in Biomedical Optical Spectroscopy and Diagnostics, E. Sevick-Muraca, D. Benaron, eds., Vol. 3 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 104–106.

Nioka, S.

S. Zhao, Y. Yang, S. Nioka, B. Chance, “Human breast tumor detection using contrast agent,” in Integration of Medical Optical Imaging and Spectroscopy and Magnetic Resonance Imaging Symposium Abstracts (University of Pennsylvania, Philadelphia, Pa., 1994).

O’Leary, M.

O’Leary, M. A.

Opher, E.

R. R. Alfano, G. C. Tang, A. Pradhan, W. Lam, D. S. Choy, E. Opher, “Fluorescence spectra from cancerous and normal human breast and lung tissues,” IEEE J. Quantum Electron. QE-23, 1806–1811 (1987).
[CrossRef]

Osterberg, U. L.

Page, D.

T. Troy, D. Page, E. Sevick-Muraca, “Optical properties of normal and diseased breast tissues: prognosis for optical mammography,” J. Biomed. Opt. 1, 342–355 (1996).
[CrossRef] [PubMed]

Paithankar, D. Y.

Patterson, M. S.

H. Jiang, K. D. Paulsen, U. L. Osterberg, M. S. Patterson, “Frequency-domain optical image reconstruction for breast imaging: initial evaluation in multitarget tissuelike phantoms,” Med. Phys. 25, 183–193 (1998).
[CrossRef] [PubMed]

H. Jiang, K. D. Paulsen, U. L. Osterberg, M. S. Patterson, “Frequency-domain optical image reconstruction in heterogeneous media: an experimental study of single-target detectability,” Appl. Opt. 36, 52–63 (1997).
[CrossRef] [PubMed]

D. Y. Paithankar, A. U. Chen, B. W. Pogue, M. S. Patterson, E. M. Sevick-Muraca, “Imaging of fluorescent lifetime and yield from multiple scattered light reemitted from tissues and other random media,” Appl. Opt. 36, 2260–2272 (1997).
[CrossRef] [PubMed]

H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, M. S. Patterson, “Optical image reconstruction using frequency-domain data: simulations and experiments,” J. Opt. Soc. Am. A 13, 253–266 (1996).
[CrossRef]

H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue, M. S. Patterson, “Simultaneous reconstruction of absorption and scattering maps in turbid media from near-infrared frequency-domain data,” Opt. Lett. 20, 2128–2130 (1995).
[CrossRef] [PubMed]

M. S. Patterson, B. W. Pogue, “Mathematical model for time-resolved and frequency-domain fluorescence spectroscopy in biological tissues,” Appl. Opt. 33, 1963–1974 (1994).
[CrossRef] [PubMed]

B. C. Wilson, E. M. Sevick, M. S. Patterson, B. Chance, “Time-dependent optical spectroscopy and imaging for biomedical applications,” Proc. IEEE 80, 918–930 (1992).
[CrossRef]

V. G. Peters, D. R. Wyman, M. S. Patterson, G. L. Frank, “Optical properties of normal and diseased human breast tissue in the visible and near infrared,” Phys. Med. Biol. 35, 1317–1334 (1990).
[CrossRef] [PubMed]

Paulsen, K. D.

Perelman, L.

Peters, V. G.

V. G. Peters, D. R. Wyman, M. S. Patterson, G. L. Frank, “Optical properties of normal and diseased human breast tissue in the visible and near infrared,” Phys. Med. Biol. 35, 1317–1334 (1990).
[CrossRef] [PubMed]

Pham, D.

B. J. Tromberg, O. Coquoz, J. Fishkin, T. Pham, E. Anderson, J. Butler, M. Cahn, J. Gross, V. Venugopalan, D. Pham, “Noninvasive measurements of breast tissue optical properties using frequency-domain photon migration,” Philos. Trans. R. Soc. London B 352, 661–668 (1997).
[CrossRef]

Pham, T.

B. J. Tromberg, O. Coquoz, J. Fishkin, T. Pham, E. Anderson, J. Butler, M. Cahn, J. Gross, V. Venugopalan, D. Pham, “Noninvasive measurements of breast tissue optical properties using frequency-domain photon migration,” Philos. Trans. R. Soc. London B 352, 661–668 (1997).
[CrossRef]

Pogue, B. W.

Pradhan, A.

R. R. Alfano, G. C. Tang, A. Pradhan, W. Lam, D. S. Choy, E. Opher, “Fluorescence spectra from cancerous and normal human breast and lung tissues,” IEEE J. Quantum Electron. QE-23, 1806–1811 (1987).
[CrossRef]

Ratliff, N.

R. Richards-Kortum, R. P. Rava, M. Fitzmaurice, L. L. Tong, N. Ratliff, J. Kramer, M. S. Feld, “A one layer model of laser-induced fluorescence for diagnosis of disease in human tissue: applications to atherosclerosis,” IEEE Trans. Biomed. Eng. 36, 1222–1232 (1989).
[CrossRef] [PubMed]

Rava, R. P.

J. Wu, M. S. Feld, R. P. Rava, “Analytical model for extracting intrinsic fluorescence in turbid media,” Appl. Opt. 32, 3585–3595 (1993).
[CrossRef] [PubMed]

R. Richards-Kortum, R. P. Rava, M. Fitzmaurice, L. L. Tong, N. Ratliff, J. Kramer, M. S. Feld, “A one layer model of laser-induced fluorescence for diagnosis of disease in human tissue: applications to atherosclerosis,” IEEE Trans. Biomed. Eng. 36, 1222–1232 (1989).
[CrossRef] [PubMed]

Reynolds, J.

E. Sevick-Muraca, G. Lopez, T. Troy, J. Reynolds, C. Hutchinson, “Fluorescence and absorption contrast mechanisms for biomedical optical imaging using frequency-domain techniques,” Photochem. Photobiol. 66, 55–64 (1997).
[CrossRef] [PubMed]

Richards-Kortum, R.

A. J. Durkin, S. Jailkumar, R. Richards-Kortum, “Optically dilute, absorbing, and turbid phantoms for fluorescence spectroscopy of homogeneous and inhomogeneous samples,” Appl. Spectrosc. 47, 2114–2121 (1993).
[CrossRef]

R. Richards-Kortum, R. P. Rava, M. Fitzmaurice, L. L. Tong, N. Ratliff, J. Kramer, M. S. Feld, “A one layer model of laser-induced fluorescence for diagnosis of disease in human tissue: applications to atherosclerosis,” IEEE Trans. Biomed. Eng. 36, 1222–1232 (1989).
[CrossRef] [PubMed]

Schweijer, M.

M. Schweijer, S. R. Arridge, “Direct calculation with a finite-element method of the Laplace transform of the distribution of photon time of flight in tissue,” Appl. Opt. 36, 9042–9049 (1997).
[CrossRef]

S. R. Arridge, M. Schweijer, M. Hiraoka, D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med Phys. 20, 299–308 (1993).
[CrossRef] [PubMed]

Sevick, E. M.

E. M. Sevick, J. K. Frisoli, C. L. Burch, J. R. Lakowicz, “Localization of absorbers in scattering media by use of frequency domain measurements of time-dependent photon migration,” Appl. Opt. 33, 3562–3570 (1994).
[CrossRef] [PubMed]

B. C. Wilson, E. M. Sevick, M. S. Patterson, B. Chance, “Time-dependent optical spectroscopy and imaging for biomedical applications,” Proc. IEEE 80, 918–930 (1992).
[CrossRef]

Sevick-Muraca, E.

E. Sevick-Muraca, G. Lopez, T. Troy, J. Reynolds, C. Hutchinson, “Fluorescence and absorption contrast mechanisms for biomedical optical imaging using frequency-domain techniques,” Photochem. Photobiol. 66, 55–64 (1997).
[CrossRef] [PubMed]

T. Troy, D. Page, E. Sevick-Muraca, “Optical properties of normal and diseased breast tissues: prognosis for optical mammography,” J. Biomed. Opt. 1, 342–355 (1996).
[CrossRef] [PubMed]

Sevick-Muraca, E. M.

D. Y. Paithankar, A. U. Chen, B. W. Pogue, M. S. Patterson, E. M. Sevick-Muraca, “Imaging of fluorescent lifetime and yield from multiple scattered light reemitted from tissues and other random media,” Appl. Opt. 36, 2260–2272 (1997).
[CrossRef] [PubMed]

E. M. Sevick-Muraca, C. L. Burch, “Origin of phosphorescence signals reemitted from tissues,” Opt. Lett. 19, 1928–1930 (1994).
[CrossRef] [PubMed]

T. L. Troy, L. Nelson-Larry, C. L. Hutchinson, E. M. Sevick-Muraca, “Investigation of exogenous contrast agents for biomedical optical imaging,” in Biomedical Optical Spectroscopy and Diagnostics, E. Sevick-Muraca, D. Benaron, eds., Vol. 3 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 104–106.

Svaasand, L.

B. J. Tromberg, S. Madsen, C. Chapman, L. Svaasand, R. Haskell, “Fluorescence energy transfer studies on the macrophage scavenger receptor,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of 1994 OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), p. 93.

Svaasand, L. O.

Svanberg, K.

E. S. Andersson, J. Johansson, K. Svanberg, S. Svanberg, “Fluorescence imaging and point measurements of tissue: applications to the determination of malignant tumors and atherosclerotic lesions from normal tissue,” Photochem. Photobiol. 53, 807–814 (1991).

Svanberg, S.

E. S. Andersson, J. Johansson, K. Svanberg, S. Svanberg, “Fluorescence imaging and point measurements of tissue: applications to the determination of malignant tumors and atherosclerotic lesions from normal tissue,” Photochem. Photobiol. 53, 807–814 (1991).

R. Berg, S. Anderson-Engels, O. Jarlman, S. Svanberg, “Tumor detection using time-resolved light transillumination,” in Future Trends in Biomedical Applications of Lasers, L. O. Svaasand, ed., Proc. SPIE1525, 59–67 (1991).
[CrossRef]

Takahashi, S.

S. Takahashi, D. Imai, Y. Yamada, “Fundamental 3D FEM analysis of light propagation in head model toward 3D optical tomography,” in Optical Tomography and Spectroscopy of Tissue: Theory, Instrumentation, Model and Human Studies II, B. Chance, R. Alfano, eds., Proc. SPIE2979, 130–138 (1997).
[CrossRef]

Tang, G. C.

R. R. Alfano, G. C. Tang, A. Pradhan, W. Lam, D. S. Choy, E. Opher, “Fluorescence spectra from cancerous and normal human breast and lung tissues,” IEEE J. Quantum Electron. QE-23, 1806–1811 (1987).
[CrossRef]

Tong, L. L.

R. Richards-Kortum, R. P. Rava, M. Fitzmaurice, L. L. Tong, N. Ratliff, J. Kramer, M. S. Feld, “A one layer model of laser-induced fluorescence for diagnosis of disease in human tissue: applications to atherosclerosis,” IEEE Trans. Biomed. Eng. 36, 1222–1232 (1989).
[CrossRef] [PubMed]

Tromberg, B. J.

B. J. Tromberg, O. Coquoz, J. Fishkin, T. Pham, E. Anderson, J. Butler, M. Cahn, J. Gross, V. Venugopalan, D. Pham, “Noninvasive measurements of breast tissue optical properties using frequency-domain photon migration,” Philos. Trans. R. Soc. London B 352, 661–668 (1997).
[CrossRef]

R. C. Haskell, L. O. Svaasand, T. Tsay, T. C. Feng, M. S. McAdams, B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11, 2727–2741 (1994).
[CrossRef]

B. J. Tromberg, S. Madsen, C. Chapman, L. Svaasand, R. Haskell, “Fluorescence energy transfer studies on the macrophage scavenger receptor,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of 1994 OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), p. 93.

Troy, T.

E. Sevick-Muraca, G. Lopez, T. Troy, J. Reynolds, C. Hutchinson, “Fluorescence and absorption contrast mechanisms for biomedical optical imaging using frequency-domain techniques,” Photochem. Photobiol. 66, 55–64 (1997).
[CrossRef] [PubMed]

T. Troy, D. Page, E. Sevick-Muraca, “Optical properties of normal and diseased breast tissues: prognosis for optical mammography,” J. Biomed. Opt. 1, 342–355 (1996).
[CrossRef] [PubMed]

Troy, T. L.

T. L. Troy, L. Nelson-Larry, C. L. Hutchinson, E. M. Sevick-Muraca, “Investigation of exogenous contrast agents for biomedical optical imaging,” in Biomedical Optical Spectroscopy and Diagnostics, E. Sevick-Muraca, D. Benaron, eds., Vol. 3 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 104–106.

Tsay, T.

van de Ven, M. J.

E. Gratton, W. Mantulin, M. J. van de Ven, J. Fishkin, M. Maris, B. Chance, “A novel approach to laser tomography,” Bioimaging 1, 40–46 (1993).
[CrossRef]

Venugopalan, V.

B. J. Tromberg, O. Coquoz, J. Fishkin, T. Pham, E. Anderson, J. Butler, M. Cahn, J. Gross, V. Venugopalan, D. Pham, “Noninvasive measurements of breast tissue optical properties using frequency-domain photon migration,” Philos. Trans. R. Soc. London B 352, 661–668 (1997).
[CrossRef]

Wang, Y.

J. Wu, Y. Wang, L. Perelman, I. Itzkan, R. R. Dasari, M. S. Feld, “Time-resolved multichannel imaging of fluorescent objects embedded in turbid media,” Opt. Lett. 20, 489–491 (1995).
[CrossRef] [PubMed]

R. L. Barbour, H. Graber, Y. Wang, J. Chang, R. Aronson, “Perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Muller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. Van der Zee, eds. (SPIE Press, Bellingham, Wash., 1993), pp. 87–120.

Wilson, B. C.

B. C. Wilson, E. M. Sevick, M. S. Patterson, B. Chance, “Time-dependent optical spectroscopy and imaging for biomedical applications,” Proc. IEEE 80, 918–930 (1992).
[CrossRef]

Wong, K. S.

Wu, J.

Wyman, D. R.

V. G. Peters, D. R. Wyman, M. S. Patterson, G. L. Frank, “Optical properties of normal and diseased human breast tissue in the visible and near infrared,” Phys. Med. Biol. 35, 1317–1334 (1990).
[CrossRef] [PubMed]

Yamada, Y.

S. Takahashi, D. Imai, Y. Yamada, “Fundamental 3D FEM analysis of light propagation in head model toward 3D optical tomography,” in Optical Tomography and Spectroscopy of Tissue: Theory, Instrumentation, Model and Human Studies II, B. Chance, R. Alfano, eds., Proc. SPIE2979, 130–138 (1997).
[CrossRef]

Y. Yamada, Y. Hasegawa, “Time-dependent FEM analysis of photon migration in random media,” in Optical Tomography and Spectroscopy of Tissue: Theory, Instrumentation, Model and Human Studies, B. Chance, R. Alfano, eds., Proc. SPIE1888, 167–178 (1993).

Yamishita, Y.

Y. Yamishita, M. Kaneko, “Infrared diaphanoscopy for medical diagnosis,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Muller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. Van der Zee, eds. (SPIE Press, Bellingham, Wash., 1993), pp. 283–316.

Yang, Y.

Y. Yang, “Fluorescence spectroscopy as a photonic pathology method for detecting colon cancer,” Lasers Life Sci. 6, 259–276 (1995).

S. Zhao, Y. Yang, S. Nioka, B. Chance, “Human breast tumor detection using contrast agent,” in Integration of Medical Optical Imaging and Spectroscopy and Magnetic Resonance Imaging Symposium Abstracts (University of Pennsylvania, Philadelphia, Pa., 1994).

Yodh, A.

Yodh, A. G.

Yoo, K. M.

Zhao, S.

S. Zhao, Y. Yang, S. Nioka, B. Chance, “Human breast tumor detection using contrast agent,” in Integration of Medical Optical Imaging and Spectroscopy and Magnetic Resonance Imaging Symposium Abstracts (University of Pennsylvania, Philadelphia, Pa., 1994).

Appl. Opt.

J. C. Hebden, K. S. Wong, “Time-resolved optical tomography,” Appl. Opt. 32, 372–380 (1993).
[CrossRef] [PubMed]

E. M. Sevick, J. K. Frisoli, C. L. Burch, J. R. Lakowicz, “Localization of absorbers in scattering media by use of frequency domain measurements of time-dependent photon migration,” Appl. Opt. 33, 3562–3570 (1994).
[CrossRef] [PubMed]

X. Li, M. O’Leary, D. Boas, B. Chance, A. Yodh, “Fluorescent diffuse photon density waves in homogeneous and heterogeneous turbid media: analytic solutions and applications,” Appl. Opt. 35, 3746–3758 (1996).
[CrossRef] [PubMed]

A. E. Cerussi, J. S. Maier, S. Fantini, M. A. Franceschini, W. W. Mantulin, E. Gratton, “Experimental verification of a theory for the time-resolved fluorescence spectroscopy of thick tissues,” Appl. Opt. 36, 116–124 (1997).
[CrossRef] [PubMed]

M. S. Patterson, B. W. Pogue, “Mathematical model for time-resolved and frequency-domain fluorescence spectroscopy in biological tissues,” Appl. Opt. 33, 1963–1974 (1994).
[CrossRef] [PubMed]

J. Wu, M. S. Feld, R. P. Rava, “Analytical model for extracting intrinsic fluorescence in turbid media,” Appl. Opt. 32, 3585–3595 (1993).
[CrossRef] [PubMed]

D. Y. Paithankar, A. U. Chen, B. W. Pogue, M. S. Patterson, E. M. Sevick-Muraca, “Imaging of fluorescent lifetime and yield from multiple scattered light reemitted from tissues and other random media,” Appl. Opt. 36, 2260–2272 (1997).
[CrossRef] [PubMed]

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

H. Jiang, K. D. Paulsen, U. L. Osterberg, M. S. Patterson, “Frequency-domain optical image reconstruction in heterogeneous media: an experimental study of single-target detectability,” Appl. Opt. 36, 52–63 (1997).
[CrossRef] [PubMed]

M. Schweijer, S. R. Arridge, “Direct calculation with a finite-element method of the Laplace transform of the distribution of photon time of flight in tissue,” Appl. Opt. 36, 9042–9049 (1997).
[CrossRef]

Appl. Spectrosc.

Bioimaging

E. Gratton, W. Mantulin, M. J. van de Ven, J. Fishkin, M. Maris, B. Chance, “A novel approach to laser tomography,” Bioimaging 1, 40–46 (1993).
[CrossRef]

IEEE J. Quantum Electron.

R. R. Alfano, G. C. Tang, A. Pradhan, W. Lam, D. S. Choy, E. Opher, “Fluorescence spectra from cancerous and normal human breast and lung tissues,” IEEE J. Quantum Electron. QE-23, 1806–1811 (1987).
[CrossRef]

IEEE Trans. Biomed. Eng.

R. Richards-Kortum, R. P. Rava, M. Fitzmaurice, L. L. Tong, N. Ratliff, J. Kramer, M. S. Feld, “A one layer model of laser-induced fluorescence for diagnosis of disease in human tissue: applications to atherosclerosis,” IEEE Trans. Biomed. Eng. 36, 1222–1232 (1989).
[CrossRef] [PubMed]

J. Chang, H. Graber, R. L. Barbour, “Imaging of fluorescence in highly scattering media,” IEEE Trans. Biomed. Eng. 44, 810–822 (1997).
[CrossRef] [PubMed]

J. Biomed. Opt.

T. Troy, D. Page, E. Sevick-Muraca, “Optical properties of normal and diseased breast tissues: prognosis for optical mammography,” J. Biomed. Opt. 1, 342–355 (1996).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A

Lasers Life Sci.

Y. Yang, “Fluorescence spectroscopy as a photonic pathology method for detecting colon cancer,” Lasers Life Sci. 6, 259–276 (1995).

Med Phys.

S. R. Arridge, M. Schweijer, M. Hiraoka, D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med Phys. 20, 299–308 (1993).
[CrossRef] [PubMed]

Med. Phys.

H. Jiang, K. D. Paulsen, U. L. Osterberg, M. S. Patterson, “Frequency-domain optical image reconstruction for breast imaging: initial evaluation in multitarget tissuelike phantoms,” Med. Phys. 25, 183–193 (1998).
[CrossRef] [PubMed]

K. D. Paulsen, H. Jiang, “Spatially-varying optical property reconstruction using a finite element diffusion equation approximation,” Med. Phys. 22, 691–702 (1995).
[CrossRef] [PubMed]

Opt. Lett.

Philos. Trans. R. Soc. London B

B. J. Tromberg, O. Coquoz, J. Fishkin, T. Pham, E. Anderson, J. Butler, M. Cahn, J. Gross, V. Venugopalan, D. Pham, “Noninvasive measurements of breast tissue optical properties using frequency-domain photon migration,” Philos. Trans. R. Soc. London B 352, 661–668 (1997).
[CrossRef]

Photochem. Photobiol.

E. S. Andersson, J. Johansson, K. Svanberg, S. Svanberg, “Fluorescence imaging and point measurements of tissue: applications to the determination of malignant tumors and atherosclerotic lesions from normal tissue,” Photochem. Photobiol. 53, 807–814 (1991).

E. Sevick-Muraca, G. Lopez, T. Troy, J. Reynolds, C. Hutchinson, “Fluorescence and absorption contrast mechanisms for biomedical optical imaging using frequency-domain techniques,” Photochem. Photobiol. 66, 55–64 (1997).
[CrossRef] [PubMed]

Phys. Med. Biol.

V. G. Peters, D. R. Wyman, M. S. Patterson, G. L. Frank, “Optical properties of normal and diseased human breast tissue in the visible and near infrared,” Phys. Med. Biol. 35, 1317–1334 (1990).
[CrossRef] [PubMed]

Proc. IEEE

B. C. Wilson, E. M. Sevick, M. S. Patterson, B. Chance, “Time-dependent optical spectroscopy and imaging for biomedical applications,” Proc. IEEE 80, 918–930 (1992).
[CrossRef]

Other

Y. Yamishita, M. Kaneko, “Infrared diaphanoscopy for medical diagnosis,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Muller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. Van der Zee, eds. (SPIE Press, Bellingham, Wash., 1993), pp. 283–316.

R. L. Barbour, H. Graber, Y. Wang, J. Chang, R. Aronson, “Perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Muller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. Van der Zee, eds. (SPIE Press, Bellingham, Wash., 1993), pp. 87–120.

R. Berg, S. Anderson-Engels, O. Jarlman, S. Svanberg, “Tumor detection using time-resolved light transillumination,” in Future Trends in Biomedical Applications of Lasers, L. O. Svaasand, ed., Proc. SPIE1525, 59–67 (1991).
[CrossRef]

S. Zhao, Y. Yang, S. Nioka, B. Chance, “Human breast tumor detection using contrast agent,” in Integration of Medical Optical Imaging and Spectroscopy and Magnetic Resonance Imaging Symposium Abstracts (University of Pennsylvania, Philadelphia, Pa., 1994).

J. R. Lakowicz, Principles of Fluorescence Spectroscopy (Plenum, New York, 1983).
[CrossRef]

T. L. Troy, L. Nelson-Larry, C. L. Hutchinson, E. M. Sevick-Muraca, “Investigation of exogenous contrast agents for biomedical optical imaging,” in Biomedical Optical Spectroscopy and Diagnostics, E. Sevick-Muraca, D. Benaron, eds., Vol. 3 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 104–106.

B. J. Tromberg, S. Madsen, C. Chapman, L. Svaasand, R. Haskell, “Fluorescence energy transfer studies on the macrophage scavenger receptor,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, ed., Vol. 21 of 1994 OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), p. 93.

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Figures (3)

Fig. 1
Fig. 1

Geometry of the test case under study. Transects (AB, CD, EF, GH) used to quantify imaging performance are also shown.

Fig. 2
Fig. 2

Simulated simultaneous reconstruction of both fluorescent lifetime (τ, nanoseconds) and yield (η, dimensionless) in different noise conditions: (a) τ, reconstruction with no noise added; (b) τ, reconstruction with 5% random noise added; (c) η, reconstruction with no noise added; (d) η, reconstruction with 5% random noise added.

Fig. 3
Fig. 3

Comparison of exact and simulated reconstructions along transect AB shown in Fig. 1 with different noise levels: (a) τ profiles, (b) η profiles. The horizontal axes indicate transect AB with millimeter units.

Tables (2)

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Table 1 Geometric Information Derived from the Reconstructed Images in Different Noise Conditionsa

Tables Icon

Table 2 Image rms Errorsa for Reconstructed Fluorescent Properties in Different Noise Conditionsb

Equations (20)

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· D x r Φ x r ,   ω - μ a x r - i ω c Φ x r ,   ω = - S r ,   ω ,
· D m r Φ m r ,   ω - μ a m r - i ω c Φ m r ,   ω = - η r μ a x m Φ x r ,   ω 1 + i ω τ r 1 + ω 2 τ r 2 ,
D x , m r = 1 3 μ a x , m r + μ s x , m r ,
A x Φ x = b x ,
A m Φ m = b m .
a x ij = - D x ψ j · ψ i - μ a x - i ω c ψ j ψ i ,
a m ij = - D m ψ j · ψ i - μ a m - i ω c ψ j ψ i ,
b x i = - S ψ i + α   j = 1 M Φ x j     ψ j ψ i d s ,
b m i = - k = 1 K   η k ψ k μ a x m j = 1 N Φ x j ψ j ψ i 1 - i ω   l = 1 L   τ l ψ l 1 + ω 2 l = 1 L   τ l ψ l 2 + α   j = 1 M Φ m j     ψ j ψ i d s ,
Φ x , m = Φ x , m 1 ,   Φ x , m 2 ,     , Φ x , m N T ,
Φ m R τ ˜ ,   η ˜ = Φ m R τ ,   η + Φ m R τ   Δ τ + Φ m R η   Δ η + ,
Φ m I τ ˜ ,   η ˜ = Φ m I τ ,   η + Φ m I τ   Δ τ + Φ m I η   Δ η + ,
J Δ χ = Φ m o - Φ m c ,
J = Φ m , 1 R τ 1 Φ m , 1 R τ 2 Φ m , 1 R τ L Φ m , 1 R η 1 Φ m , 1 R η 2 Φ m , 1 R η K Φ m , 1 I τ 1 Φ m , 1 I τ 2 Φ m , 1 I τ L Φ m , 1 I η 1 Φ m , 1 I η 2 Φ m , 1 I η K Φ m , M R τ 1 Φ m , M R τ 2 Φ m , M R τ L Φ m , M R η 1 Φ m , M R η 2 Φ m , M R η K Φ m , M I τ 1 Φ m , M I τ 2 Φ m , M I τ L Φ m , M I η 1 Φ m , M I η 2 Φ m , M I η K ,
Δ χ = Δ τ 1 ,   Δ τ 2 ,     Δ τ L ,   Δ η 1 ,   Δ η 2 ,     Δ η K T ,
Φ m o = Φ m R 1 o ,   Φ m I 1 o ,   Φ m R 2 o ,   Φ m I 2 o     Φ m R M o ,   Φ m I M o T ,
Φ m c = Φ m R 1 c ,   Φ m I 1 c ,   Φ m R 2 c ,   Φ m I 2 c     Φ m R M c ,   Φ m I M c T ,
J T J + λ I Δ χ = J T Φ m o - Φ m c ,
χ new i = 1 - θ χ old i + θ N * j = 1 N *   χ old j ,
1 N ¯ i = 1 N ¯ χ exact - χ reconstructed χ exact 2 1 / 2 ,

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