Abstract

Optical diffusion tomography is a method for reconstructing three-dimensional optical properties from light that passes through a highly scattering medium. Computing reconstructions from such data requires the solution of a nonlinear inverse problem. The situation is further complicated by the fact that while reconstruction algorithms typically assume exact knowledge of the optical source and detector coupling coefficients, these coupling coefficients are generally not available in practical measurement systems. A new method for estimating these unknown coupling coefficients in the three-dimensional reconstruction process is described. The joint problem of coefficient estimation and three-dimensional reconstruction is formulated in a Bayesian framework, and the resulting estimates are computed by using a variation of iterative coordinate descent optimization that is adapted for this problem. Simulations show that this approach is an accurate and efficient method for simultaneous reconstruction of absorption and diffusion coefficients as well as the coupling coefficients. A simple experimental result validates the approach.

© 2002 Optical Society of America

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2002

T. O. McBride, B. W. Pogue, S. Poplack, S. Soho, W. A. Wells, S. Jiang, U. Osterberg, K. D. Paulsen, “Multispectral near-infrared tomography: a case study in compensating for water and lipid content in hemoglobin imaging of the breast,” J. Biomed. Opt. 7, 72–79 (2002).
[CrossRef] [PubMed]

A. B. Milstein, S. Oh, J. S. Reynolds, K. J. Webb, C. A. Bouman, R. P. Millane, “Three-dimensional Bayesian optical diffusion tomography using experimental data,” Opt. Lett. 27, 95–97 (2002).
[CrossRef]

2001

B. W. Pogue, S. P. Poplack, T. O. McBride, W. A. Wells, K. S. Osterman, U. L. Osterberg, K. D. Paulsen, “Quantitative hemoglobin tomography with diffuse near-infrared spectroscopy: pilot results in the breast,” Radiology 218, 261–266 (2001).
[CrossRef] [PubMed]

J. C. Ye, C. A. Bouman, K. J. Webb, R. P. Millane, “Nonlinear multigrid algorithms for Bayesian optical diffusion tomography,” IEEE Trans. Image Process. 10, 909–922 (2001).
[CrossRef]

D. Boas, T. Gaudette, S. Arridge, “Simultaneous imaging and optode calibration with diffuse optical tomography,” Opt. Express 8, 263–270 (2001), www.opticsexpress.org .
[CrossRef] [PubMed]

2000

B. W. Pogue, C. Willscher, T. O. McBride, U. L. Osterberg, K. D. Paulsen, “Contrast-detail analysis for detection and characterization with near-infrared diffuse tomography,” Med. Phys. 27, 2693–2700 (2000).
[CrossRef]

N. Iftimia, H. Jiang, “Quantitative optical image reconstructions of turbid media by use of direct-current measurements,” Appl. Opt. 39, 5256–5261 (2000).
[CrossRef]

1999

1998

S. R. Arridge, M. Schweiger, “A gradient-based optimisation scheme for optical tomography,” Opt. Express 2, 213–226 (1998), www.opticsexpress.org .
[CrossRef] [PubMed]

H. Jiang, K. Paulsen, U. Osterberg, M. Patterson, “Improved continuous light diffusion imaging in single- and multi-target tissue-like phantoms,” Phys. Med. Biol. 43, 675–693 (1998).
[CrossRef] [PubMed]

S. S. Saquib, C. A. Bouman, K. Sauer, “ML parameter estimation for Markov random fields with applications to Bayesian tomography,” IEEE Trans. Image Process. 7, 1029–1044 (1998).
[CrossRef]

1996

H. Jiang, K. Paulsen, U. Osterberg, “Optical image reconstruction using dc data: simulations and experiments,” Phys. Med. Biol. 41, 1483–1498 (1996).
[CrossRef] [PubMed]

J. S. Reynolds, A. Przadka, S. Yeung, K. J. Webb, “Optical diffusion imaging: a comparative numerical and experimental study,” Appl. Opt. 35, 3671–3679 (1996).
[CrossRef] [PubMed]

1995

1993

C. A. Bouman, K. Sauer, “A generalized Gaussian image model for edge-preserving MAP estimation,” IEEE Trans. Image Process. 2, 296–310 (1993).
[CrossRef] [PubMed]

1987

S. Geman, D. McClure, “Statistical methods for tomographic image reconstruction,” Bull. Int. Stat. Inst. LII-4, 5–21 (1987).

1966

L. E. Baum, T. Petrie, “Statistical inference for probabilistic functions of finite state Markov chains,” Ann. Math. Stat. 37, 1554–1563 (1966).
[CrossRef]

Arridge, S.

Arridge, S. R.

Baum, L. E.

L. E. Baum, T. Petrie, “Statistical inference for probabilistic functions of finite state Markov chains,” Ann. Math. Stat. 37, 1554–1563 (1966).
[CrossRef]

Boas, D.

Bouman, C. A.

A. B. Milstein, S. Oh, J. S. Reynolds, K. J. Webb, C. A. Bouman, R. P. Millane, “Three-dimensional Bayesian optical diffusion tomography using experimental data,” Opt. Lett. 27, 95–97 (2002).
[CrossRef]

J. C. Ye, C. A. Bouman, K. J. Webb, R. P. Millane, “Nonlinear multigrid algorithms for Bayesian optical diffusion tomography,” IEEE Trans. Image Process. 10, 909–922 (2001).
[CrossRef]

J. C. Ye, K. J. Webb, C. A. Bouman, R. P. Millane, “Optical diffusion tomography using iterative coordinate descent optimization in a Bayesian framework,” J. Opt. Soc. Am. A 16, 2400–2412 (1999).
[CrossRef]

S. S. Saquib, C. A. Bouman, K. Sauer, “ML parameter estimation for Markov random fields with applications to Bayesian tomography,” IEEE Trans. Image Process. 7, 1029–1044 (1998).
[CrossRef]

C. A. Bouman, K. Sauer, “A generalized Gaussian image model for edge-preserving MAP estimation,” IEEE Trans. Image Process. 2, 296–310 (1993).
[CrossRef] [PubMed]

Cunningham, G. S.

S. S. Saquib, K. M. Hanson, G. S. Cunningham, “Model-based image reconstruction from time-resolved diffusion data,” in Medical Imaging 1997: Image Processing, K. M. Hanson, ed., Proc. SPIE3034, 369–380 (1997).
[CrossRef]

Downar, T. J.

Duderstadt, J. J.

J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976).

Gaudette, T.

Geman, S.

S. Geman, D. McClure, “Statistical methods for tomographic image reconstruction,” Bull. Int. Stat. Inst. LII-4, 5–21 (1987).

Hamilton, L. J.

J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976).

Hanson, K. M.

A. H. Hielscher, A. D. Klose, K. M. Hanson, “Gradient-based iterative image reconstruction scheme for time-resolved optical tomography,” IEEE Trans. Med. Imaging 18, 262–271 (1999).
[CrossRef] [PubMed]

S. S. Saquib, K. M. Hanson, G. S. Cunningham, “Model-based image reconstruction from time-resolved diffusion data,” in Medical Imaging 1997: Image Processing, K. M. Hanson, ed., Proc. SPIE3034, 369–380 (1997).
[CrossRef]

Hielscher, A. H.

A. H. Hielscher, A. D. Klose, K. M. Hanson, “Gradient-based iterative image reconstruction scheme for time-resolved optical tomography,” IEEE Trans. Med. Imaging 18, 262–271 (1999).
[CrossRef] [PubMed]

Iftimia, N.

Ishimaru, A.

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

Jiang, H.

N. Iftimia, H. Jiang, “Quantitative optical image reconstructions of turbid media by use of direct-current measurements,” Appl. Opt. 39, 5256–5261 (2000).
[CrossRef]

H. Jiang, K. Paulsen, U. Osterberg, M. Patterson, “Improved continuous light diffusion imaging in single- and multi-target tissue-like phantoms,” Phys. Med. Biol. 43, 675–693 (1998).
[CrossRef] [PubMed]

H. Jiang, K. Paulsen, U. Osterberg, “Optical image reconstruction using dc data: simulations and experiments,” Phys. Med. Biol. 41, 1483–1498 (1996).
[CrossRef] [PubMed]

Jiang, S.

T. O. McBride, B. W. Pogue, S. Poplack, S. Soho, W. A. Wells, S. Jiang, U. Osterberg, K. D. Paulsen, “Multispectral near-infrared tomography: a case study in compensating for water and lipid content in hemoglobin imaging of the breast,” J. Biomed. Opt. 7, 72–79 (2002).
[CrossRef] [PubMed]

Klose, A. D.

A. H. Hielscher, A. D. Klose, K. M. Hanson, “Gradient-based iterative image reconstruction scheme for time-resolved optical tomography,” IEEE Trans. Med. Imaging 18, 262–271 (1999).
[CrossRef] [PubMed]

Lange, K.

K. Lange, “An overview of Bayesian methods in image reconstruction,” in Digital Image Synthesis and Inverse Optics, A. F. Gmitro, P. S. Idell, I. J. LaHaie, eds., Proc. SPIE1351, 270–287 (1990).
[CrossRef]

McBride, T. O.

T. O. McBride, B. W. Pogue, S. Poplack, S. Soho, W. A. Wells, S. Jiang, U. Osterberg, K. D. Paulsen, “Multispectral near-infrared tomography: a case study in compensating for water and lipid content in hemoglobin imaging of the breast,” J. Biomed. Opt. 7, 72–79 (2002).
[CrossRef] [PubMed]

B. W. Pogue, S. P. Poplack, T. O. McBride, W. A. Wells, K. S. Osterman, U. L. Osterberg, K. D. Paulsen, “Quantitative hemoglobin tomography with diffuse near-infrared spectroscopy: pilot results in the breast,” Radiology 218, 261–266 (2001).
[CrossRef] [PubMed]

B. W. Pogue, C. Willscher, T. O. McBride, U. L. Osterberg, K. D. Paulsen, “Contrast-detail analysis for detection and characterization with near-infrared diffuse tomography,” Med. Phys. 27, 2693–2700 (2000).
[CrossRef]

McClure, D.

S. Geman, D. McClure, “Statistical methods for tomographic image reconstruction,” Bull. Int. Stat. Inst. LII-4, 5–21 (1987).

Millane, R. P.

Milstein, A. B.

Mohammad-Djafari, A.

A. Mohammad-Djafari, “On the estimation of hyperparameters in Bayesian approach of solving inverse problems,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 495–498.

A. Mohammad-Djafari, “Joint estimation of parameters and hyperparameters in a Bayesian approach of solving inverse problems,” in Proceedings of IEEE International Conference on Image Processing (Institute of Electrical and Electronics Engineers, New York, 1996), Vol. II, pp. 473–476.

Oh, S.

Osterberg, U.

T. O. McBride, B. W. Pogue, S. Poplack, S. Soho, W. A. Wells, S. Jiang, U. Osterberg, K. D. Paulsen, “Multispectral near-infrared tomography: a case study in compensating for water and lipid content in hemoglobin imaging of the breast,” J. Biomed. Opt. 7, 72–79 (2002).
[CrossRef] [PubMed]

H. Jiang, K. Paulsen, U. Osterberg, M. Patterson, “Improved continuous light diffusion imaging in single- and multi-target tissue-like phantoms,” Phys. Med. Biol. 43, 675–693 (1998).
[CrossRef] [PubMed]

H. Jiang, K. Paulsen, U. Osterberg, “Optical image reconstruction using dc data: simulations and experiments,” Phys. Med. Biol. 41, 1483–1498 (1996).
[CrossRef] [PubMed]

Osterberg, U. L.

B. W. Pogue, S. P. Poplack, T. O. McBride, W. A. Wells, K. S. Osterman, U. L. Osterberg, K. D. Paulsen, “Quantitative hemoglobin tomography with diffuse near-infrared spectroscopy: pilot results in the breast,” Radiology 218, 261–266 (2001).
[CrossRef] [PubMed]

B. W. Pogue, C. Willscher, T. O. McBride, U. L. Osterberg, K. D. Paulsen, “Contrast-detail analysis for detection and characterization with near-infrared diffuse tomography,” Med. Phys. 27, 2693–2700 (2000).
[CrossRef]

Osterman, K. S.

B. W. Pogue, S. P. Poplack, T. O. McBride, W. A. Wells, K. S. Osterman, U. L. Osterberg, K. D. Paulsen, “Quantitative hemoglobin tomography with diffuse near-infrared spectroscopy: pilot results in the breast,” Radiology 218, 261–266 (2001).
[CrossRef] [PubMed]

Patterson, M.

H. Jiang, K. Paulsen, U. Osterberg, M. Patterson, “Improved continuous light diffusion imaging in single- and multi-target tissue-like phantoms,” Phys. Med. Biol. 43, 675–693 (1998).
[CrossRef] [PubMed]

Paulsen, K.

H. Jiang, K. Paulsen, U. Osterberg, M. Patterson, “Improved continuous light diffusion imaging in single- and multi-target tissue-like phantoms,” Phys. Med. Biol. 43, 675–693 (1998).
[CrossRef] [PubMed]

H. Jiang, K. Paulsen, U. Osterberg, “Optical image reconstruction using dc data: simulations and experiments,” Phys. Med. Biol. 41, 1483–1498 (1996).
[CrossRef] [PubMed]

Paulsen, K. D.

T. O. McBride, B. W. Pogue, S. Poplack, S. Soho, W. A. Wells, S. Jiang, U. Osterberg, K. D. Paulsen, “Multispectral near-infrared tomography: a case study in compensating for water and lipid content in hemoglobin imaging of the breast,” J. Biomed. Opt. 7, 72–79 (2002).
[CrossRef] [PubMed]

B. W. Pogue, S. P. Poplack, T. O. McBride, W. A. Wells, K. S. Osterman, U. L. Osterberg, K. D. Paulsen, “Quantitative hemoglobin tomography with diffuse near-infrared spectroscopy: pilot results in the breast,” Radiology 218, 261–266 (2001).
[CrossRef] [PubMed]

B. W. Pogue, C. Willscher, T. O. McBride, U. L. Osterberg, K. D. Paulsen, “Contrast-detail analysis for detection and characterization with near-infrared diffuse tomography,” Med. Phys. 27, 2693–2700 (2000).
[CrossRef]

Petrie, T.

L. E. Baum, T. Petrie, “Statistical inference for probabilistic functions of finite state Markov chains,” Ann. Math. Stat. 37, 1554–1563 (1966).
[CrossRef]

Pogue, B. W.

T. O. McBride, B. W. Pogue, S. Poplack, S. Soho, W. A. Wells, S. Jiang, U. Osterberg, K. D. Paulsen, “Multispectral near-infrared tomography: a case study in compensating for water and lipid content in hemoglobin imaging of the breast,” J. Biomed. Opt. 7, 72–79 (2002).
[CrossRef] [PubMed]

B. W. Pogue, S. P. Poplack, T. O. McBride, W. A. Wells, K. S. Osterman, U. L. Osterberg, K. D. Paulsen, “Quantitative hemoglobin tomography with diffuse near-infrared spectroscopy: pilot results in the breast,” Radiology 218, 261–266 (2001).
[CrossRef] [PubMed]

B. W. Pogue, C. Willscher, T. O. McBride, U. L. Osterberg, K. D. Paulsen, “Contrast-detail analysis for detection and characterization with near-infrared diffuse tomography,” Med. Phys. 27, 2693–2700 (2000).
[CrossRef]

Poplack, S.

T. O. McBride, B. W. Pogue, S. Poplack, S. Soho, W. A. Wells, S. Jiang, U. Osterberg, K. D. Paulsen, “Multispectral near-infrared tomography: a case study in compensating for water and lipid content in hemoglobin imaging of the breast,” J. Biomed. Opt. 7, 72–79 (2002).
[CrossRef] [PubMed]

Poplack, S. P.

B. W. Pogue, S. P. Poplack, T. O. McBride, W. A. Wells, K. S. Osterman, U. L. Osterberg, K. D. Paulsen, “Quantitative hemoglobin tomography with diffuse near-infrared spectroscopy: pilot results in the breast,” Radiology 218, 261–266 (2001).
[CrossRef] [PubMed]

Przadka, A.

Reynolds, J. S.

Saquib, S. S.

S. S. Saquib, C. A. Bouman, K. Sauer, “ML parameter estimation for Markov random fields with applications to Bayesian tomography,” IEEE Trans. Image Process. 7, 1029–1044 (1998).
[CrossRef]

S. S. Saquib, K. M. Hanson, G. S. Cunningham, “Model-based image reconstruction from time-resolved diffusion data,” in Medical Imaging 1997: Image Processing, K. M. Hanson, ed., Proc. SPIE3034, 369–380 (1997).
[CrossRef]

Sauer, K.

S. S. Saquib, C. A. Bouman, K. Sauer, “ML parameter estimation for Markov random fields with applications to Bayesian tomography,” IEEE Trans. Image Process. 7, 1029–1044 (1998).
[CrossRef]

C. A. Bouman, K. Sauer, “A generalized Gaussian image model for edge-preserving MAP estimation,” IEEE Trans. Image Process. 2, 296–310 (1993).
[CrossRef] [PubMed]

Schweiger, M.

Soho, S.

T. O. McBride, B. W. Pogue, S. Poplack, S. Soho, W. A. Wells, S. Jiang, U. Osterberg, K. D. Paulsen, “Multispectral near-infrared tomography: a case study in compensating for water and lipid content in hemoglobin imaging of the breast,” J. Biomed. Opt. 7, 72–79 (2002).
[CrossRef] [PubMed]

Webb, K. J.

Wells, W. A.

T. O. McBride, B. W. Pogue, S. Poplack, S. Soho, W. A. Wells, S. Jiang, U. Osterberg, K. D. Paulsen, “Multispectral near-infrared tomography: a case study in compensating for water and lipid content in hemoglobin imaging of the breast,” J. Biomed. Opt. 7, 72–79 (2002).
[CrossRef] [PubMed]

B. W. Pogue, S. P. Poplack, T. O. McBride, W. A. Wells, K. S. Osterman, U. L. Osterberg, K. D. Paulsen, “Quantitative hemoglobin tomography with diffuse near-infrared spectroscopy: pilot results in the breast,” Radiology 218, 261–266 (2001).
[CrossRef] [PubMed]

Willscher, C.

B. W. Pogue, C. Willscher, T. O. McBride, U. L. Osterberg, K. D. Paulsen, “Contrast-detail analysis for detection and characterization with near-infrared diffuse tomography,” Med. Phys. 27, 2693–2700 (2000).
[CrossRef]

Ye, J. C.

Yeung, S.

Ann. Math. Stat.

L. E. Baum, T. Petrie, “Statistical inference for probabilistic functions of finite state Markov chains,” Ann. Math. Stat. 37, 1554–1563 (1966).
[CrossRef]

Appl. Opt.

Bull. Int. Stat. Inst.

S. Geman, D. McClure, “Statistical methods for tomographic image reconstruction,” Bull. Int. Stat. Inst. LII-4, 5–21 (1987).

IEEE Trans. Image Process.

S. S. Saquib, C. A. Bouman, K. Sauer, “ML parameter estimation for Markov random fields with applications to Bayesian tomography,” IEEE Trans. Image Process. 7, 1029–1044 (1998).
[CrossRef]

C. A. Bouman, K. Sauer, “A generalized Gaussian image model for edge-preserving MAP estimation,” IEEE Trans. Image Process. 2, 296–310 (1993).
[CrossRef] [PubMed]

J. C. Ye, C. A. Bouman, K. J. Webb, R. P. Millane, “Nonlinear multigrid algorithms for Bayesian optical diffusion tomography,” IEEE Trans. Image Process. 10, 909–922 (2001).
[CrossRef]

IEEE Trans. Med. Imaging

A. H. Hielscher, A. D. Klose, K. M. Hanson, “Gradient-based iterative image reconstruction scheme for time-resolved optical tomography,” IEEE Trans. Med. Imaging 18, 262–271 (1999).
[CrossRef] [PubMed]

Inverse Probl.

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

J. Biomed. Opt.

T. O. McBride, B. W. Pogue, S. Poplack, S. Soho, W. A. Wells, S. Jiang, U. Osterberg, K. D. Paulsen, “Multispectral near-infrared tomography: a case study in compensating for water and lipid content in hemoglobin imaging of the breast,” J. Biomed. Opt. 7, 72–79 (2002).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A

Med. Phys.

B. W. Pogue, C. Willscher, T. O. McBride, U. L. Osterberg, K. D. Paulsen, “Contrast-detail analysis for detection and characterization with near-infrared diffuse tomography,” Med. Phys. 27, 2693–2700 (2000).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Med. Biol.

H. Jiang, K. Paulsen, U. Osterberg, “Optical image reconstruction using dc data: simulations and experiments,” Phys. Med. Biol. 41, 1483–1498 (1996).
[CrossRef] [PubMed]

H. Jiang, K. Paulsen, U. Osterberg, M. Patterson, “Improved continuous light diffusion imaging in single- and multi-target tissue-like phantoms,” Phys. Med. Biol. 43, 675–693 (1998).
[CrossRef] [PubMed]

Radiology

B. W. Pogue, S. P. Poplack, T. O. McBride, W. A. Wells, K. S. Osterman, U. L. Osterberg, K. D. Paulsen, “Quantitative hemoglobin tomography with diffuse near-infrared spectroscopy: pilot results in the breast,” Radiology 218, 261–266 (2001).
[CrossRef] [PubMed]

Other

S. S. Saquib, K. M. Hanson, G. S. Cunningham, “Model-based image reconstruction from time-resolved diffusion data,” in Medical Imaging 1997: Image Processing, K. M. Hanson, ed., Proc. SPIE3034, 369–380 (1997).
[CrossRef]

J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976).

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

A. Mohammad-Djafari, “On the estimation of hyperparameters in Bayesian approach of solving inverse problems,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 495–498.

A. Mohammad-Djafari, “Joint estimation of parameters and hyperparameters in a Bayesian approach of solving inverse problems,” in Proceedings of IEEE International Conference on Image Processing (Institute of Electrical and Electronics Engineers, New York, 1996), Vol. II, pp. 473–476.

K. Lange, “An overview of Bayesian methods in image reconstruction,” in Digital Image Synthesis and Inverse Optics, A. F. Gmitro, P. S. Idell, I. J. LaHaie, eds., Proc. SPIE1351, 270–287 (1990).
[CrossRef]

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Fig. 1
Fig. 1

Pseudocode specification for (a) the overall optimization procedure and (b) the image update by one ICD scan.

Fig. 2
Fig. 2

Isosurface plots at 0.04 cm-1 and 0.02 cm, respectively, for μa (left column) and D (right column) for Phantom A: (a), (b) original tissue phantom; (c), (d) reconstructions with source–detector calibration; (e), (f) reconstructions using the correct weights; (g), (h) reconstructions without calibration.

Fig. 3
Fig. 3

Cross sections through the centers of the inhomogeneities at z=0.5 cm and z=1.5 cm, respectively, for μa (left column) and D (right column) of Phantom A: (a), (b) original tissue phantom; (c), (d) reconstructions with source–detector calibration; (e), (f) reconstructions using the correct weights, (g), (h) reconstructions without calibration.

Fig. 4
Fig. 4

Isosurface plots at 0.04 cm-1 and 0.02 cm, respectively, for μa (left column) and D (right column) for Phantom B: (a), (b) original tissue phantom; (c), (d) reconstructions with source–detector calibration; (e), (f) reconstructions using the correct weights; (g), (h) reconstructions without calibration.

Fig. 5
Fig. 5

Cross sections through the centers of the inhomogeneities at z=0.0 cm and z=0.25 cm, respectively, for μa (left column) and D (right column) of Phantom B: (a), (b) original tissue phantom; (c), (d) reconstructions with source–detector calibration; (e), (f) reconstructions using the correct weights; (g), (h) reconstructions without calibration.

Fig. 6
Fig. 6

(a) Locations of sources and detectors, (b) several levels of boundaries: from outer boundary, zero-flux boundary, physical boundary, source–detector boundary, and imaging boundary.

Fig. 7
Fig. 7

(a) Source–detector coupling coefficients used in the simulations. Estimation error of coupling coefficients for (b) Phantom A and (c) Phantom B after 30 iterations. Note that the scale of (b) and (c) is 10 times of that of (a).

Fig. 8
Fig. 8

NRMSE between the phantom and the reconstructed images for (a) Phantom A and (b) Phantom B.

Fig. 9
Fig. 9

(a) RMS error in the estimated coupling coefficients versus iteration, (b) convergence of coupling coefficients for Group 1 (&sline;) and Group 2 (---) for Phantom B.

Fig. 10
Fig. 10

Image NRMSE comparison between the reconstruction with coupling coefficient calibration and the reconstruction with coupling coefficients fixed to 1+j0, for various standard deviations of coupling coefficients. Images were obtained after 30 iterations.

Fig. 11
Fig. 11

Cross sections of the reconstructed images of Phantom A without calibration through the centers of the inhomogeneities at z=0.5 cm for μa and z=1.5 cm for D for σcoeff=0.02 for (a) μa and (b) D and for σcoeff=0.04 for (c) μa and (d) D.

Fig. 12
Fig. 12

(a) Culture flask with the absorbing cylinder embedded in a scattering Intralipid solution, (b) schematic diagram of the apparatus used to collect data.

Fig. 13
Fig. 13

Cross sections for reconstructed images of an absorbing cylinder with (a) two complex-valued calibration coefficients, (b) a single complex calibration coefficient, (c) a single real calibration coefficient, and (d) all calibration coefficients assumed to be 1.

Equations (34)

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·[D(r)ϕk(r)]+[-μa(r)-jω/c]ϕk(r)
=-δ(r-ak),
x=[μa(r1),, μa(rN), D(r1),, D(rN)]T.
E[Ykm|x, sk, dm]=skdmϕk(bm; x),
y=[y11,, y1M, y21,, y2M,, yKM]T.
E[Y|x, s, d]=diag(sd)Φ(x),
Φ(x)=[ϕ1(b1; x), ϕ1(b2; x),, ϕ1(bM; x),ϕ2(b1; x),, ϕK(bM; x)]T.
f(x, s, d)=diag(sd)Φ(x).
p(y|x, s, d, α)=1(πα)P|Λ|-1exp-y-f(x, s, d)Λ2α,
(xˆMAP, sˆ, dˆ, αˆ)=arg max(x0, s, d, α){log p(x|y, s, d, α)}=arg max(x0, s, d, α){log p(y|x, s, d, α)+log p(x)},
p(x)=p([μa(r1), μa(r2),, μa(rN)]T)p([D(r1), D(r2),, D(rN)]T)
=1σ0Nz(p0)exp-1p0σ0p0{i, j}Nb0,i-j|xi-xj|p0
×1σ1Nz(p1)exp-1p1σ1p1{i, j}Nb1,i-j|xN+i-xN+j|p1
=u=011σuNz(pu)exp-1puσupu×{i, j}Nbu,i-j|xuN+i-xuN+j|pu,
c(x, s, d, α)=1αy-f(x, s, d)Λ2+P log α+u=011puσupu{i, j}Nbu,i-j|xuN+i-xuN+j|pu.
(xˆMAP, sˆ, dˆ, αˆ)=arg min(x0, s, d, α)c(x, s, d, α).
αˆargminα c(xˆ, sˆ, dˆ, α),
sˆarg mins c(xˆ, s, dˆ, αˆ),
dˆarg mind c(xˆ, sˆ, d, αˆ),
xˆICD_updatex{c(x, sˆ, dˆ, αˆ), xˆ},
αˆ1Py-f(xˆ, sˆ, dˆ)Λ2,
sˆk[diag(dˆ)Φk(s)(xˆ)]HΛk(s)y|diag(dˆ)Φk(s)(xˆ)|Λk(s)2,
k=1, 2,, K,
dˆm[diag(sˆ)Φm(d)(xˆ)]HΛm(d)y|diag(sˆ)Φm(d)(xˆ)|Λm(d)2,
m=1, 2,, M,
y-f(x, sˆ, dˆ)Λ2y-f(xˆ, sˆ, dˆ)-f(xˆ, sˆ, dˆ)ΔxΛ2,
c(x, sˆ, dˆ, αˆ)
1αˆ|z-f(xˆ, sˆ, dˆ)x|Λ2+u=011puσupu{i, j}Nbu,i-j|xuN+i-xuN+j|pu,
z=y-f(xˆ, sˆ, dˆ)+f(xˆ, sˆ, dˆ)xˆ.
xˆuN+iarg minxuN+i01αˆy-f(xˆ, sˆ, dˆ)-[f(xˆ, sˆ, dˆ)]*(uN+i)(xuN+i-xˆuN+i)Λ2+1puσupujNibu,i-j|xuN+i-xˆuN+j|pu,
f(xˆ, sˆ, dˆ)=f11(xˆ, sˆ1, dˆ1)μa(r1)f11(xˆ, sˆ1, dˆ1)μa(rN)f11(xˆ, sˆ1, dˆ1)D(r1)f11(xˆ, sˆ1, dˆ1)D(rN)f12(xˆ, sˆ1, dˆ2)μa(r1)f12(xˆ, sˆ1, dˆ2)μa(rN)f12(xˆ, sˆ1, dˆ2)D(r1)f12(xˆ, sˆ1, dˆ2)D(rN)f1M(xˆ, sˆ1, dˆM)μa(r1)f1M(xˆ, sˆ1, dˆM)μa(rN)f1M(xˆ, sˆ1, dˆM)D(r1)f1M(xˆ, sˆ1, dˆM)D(rN)f21(xˆ, sˆ2, dˆ1)μa(r1)f21(xˆ, sˆ2, dˆ1)μa(rN)f21(xˆ, sˆ2, dˆ1)D(r1)f21(xˆ, sˆ2, dˆ1)D(rN)fKM(xˆ, sˆK, dˆM)μa(r1)fKM(xˆ, sˆK, dˆM)μa(rN)fKM(xˆ, sˆK, dˆM)D(r1)fKM(xˆ, sˆK, dˆM)D(rN),
fkm(xˆ, sˆk, dˆm)μa(ri)=-sˆkdˆmg(bm, ri; xˆ)ϕk(ri; xˆ)A,
fkm(xˆ, sˆk, dˆm)D(ri)=-sˆkdˆmg(bm, ri; xˆ)·ϕk(ri; xˆ)A,
NRMSE=12u=01riR|xˆuN+i-xuN+i|2riR|xuN+i|21/2,

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