Abstract

Image reconstruction in optical tomography is a nonlinear and generally ill- posed inverse problem. Noise in the measured surface data can give rise to substantial artifacts in the recovered volume images of optical coefficients. Apart from random shot noise caused by the limited number of photons detected at the measurement site, another class of systematic noise is associated with losses specific to individual source and detector locations. A common cause for such losses in data acquisition systems based on fiber-optic light delivery is the imperfect coupling between the fiber tips and the skin of the patient because of air gaps or surface moisture. Thus the term coupling errors was coined for this type of data noise. However, source and detector specific errors can also occur in noncontact measurement systems not using fiber-optic delivery, for example, owing to local skin pigmentation, hair and hair follicles, or instrumentation calibration errors. Often it is not possible to quantify coupling effects in a way that allows us to remove them from the data or incorporate them into the light transport model. We present an alternative method of eliminating coupling errors by regarding the complex-valued coupling factors for each source and detector as unknowns in the reconstruction process and recovering them simultaneously with the images of absorption and scattering. Our method takes into account the possibility that coupling effects have an influence on both the amplitude and the phase shift of the measurements. Reconstructions from simulated and experimental phantom data are presented, which show that including the coupling coefficients in the reconstruction greatly improves the recovery of absorption and scattering images.

© 2007 Optical Society of America

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2005 (5)

A. P. Gibson, J. Hebden, and S. R. Arridge, "Recent advances in diffuse optical tomography," Phys. Med. Biol. 50, R1-R43 (2005).
[CrossRef] [PubMed]

A. P. Gibson, T. Austin, N. L. Everdell, M. Schweiger, S. R. Arridge, J. H. Meek, J. S. Wyatt, D. T. Delpy, and J. C. Hebden, "Three-dimensional whole-head optical tomography of passive motor evoked responses in the neonate," Neuroimage 30, 521- 528 (2005).
[CrossRef] [PubMed]

M. Schweiger, S. R. Arridge, and I. Nissilä, "Gauss-Newton method for image reconstruction in diffuse optical tomography," Phys. Med. Biol. 50, 2365-2386 (2005).
[CrossRef] [PubMed]

I. Nissilä, T. Noponen, K. Kotilahti, T. Tarvainen, M. Schweiger, L. Lipiäinen, S. R. Arridge, and T. Katila, "Instrumentation and calibration methods for the multichannel measurement of phase and amplitude in optical tomography," Rev. Sci. Instrum. 76, 044302 (2005).
[CrossRef]

T. Tarvainen, V. Kolehmainen, M. Vauhkonen, A. Vanne, A. P. Gibson, M. Schweiger, S. R. Arridge, and J. P. Kaipio, "Computational calibration method for optical tomography," Appl. Opt. 44, 1879-1888 (2005).
[CrossRef] [PubMed]

2004 (1)

C. Li and H. Jiang, "A calibration method in diffuse optical tomography," J. Opt. A 6, 844-852 (2004).
[CrossRef]

2003 (2)

2002 (1)

2001 (3)

2000 (3)

C. H. Schmitz, H. L. Graber, H. Luo, I. Arif, J. Hira, Y. Pei, A. Bluestone, S. Zhong, R. Andronica, I. Soller, N. Ramirez, S.-L. S. Barbour, and R. L. Barbour, "Instrumentation and calibration protocol for imaging dynamic features in dense-scattering media by optical tomography," Appl. Opt. 39, 6466-6486 (2000).
[CrossRef]

B. W. Pogue, K. D. Paulsen, C. Abele, and H. Kaufman, "Calibration of near-infrared frequency-domain tissue spectroscopy for absolute absorption coefficient quantitation in neonatal head-simulating phantoms," J. Biomed. Opt. 5, 185-193 (2000).
[CrossRef] [PubMed]

D. Hawysz and E. M. Sevick-Muraca, "Developments towards diagnostic breast cancer imaging using near-infrared optical measurements and fluorescent contrast agents," Neoplasia 2, 388-417 (2000).
[CrossRef]

1999 (2)

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

A. D. Klose and A. H. Hielscher, "Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer," Med. Phys. 26, 1698-1707 (1999).
[CrossRef] [PubMed]

1998 (1)

O. Dorn, "A transport-backtransport method for optical tomography," Inverse Probl. 14, 1107-1130 (1998).
[CrossRef]

1997 (1)

1995 (2)

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

A. Yodh and B. Chance, "Spectroscopy and imaging with diffusing light," Phys. Today 48(3), 34-40 (1995).
[CrossRef]

1994 (1)

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, 299-309 (1993).
[CrossRef] [PubMed]

1989 (1)

1988 (1)

M. Cope and D. T. Delpy, "System for long term measurement of cerebral blood and tissue oxygenation on newborn infants by near-infrared transillumination," Med. Biol. Eng. Comput. 26, 289-294 (1988).
[CrossRef] [PubMed]

Abdoulaev, G.

Abele, C.

B. W. Pogue, K. D. Paulsen, C. Abele, and H. Kaufman, "Calibration of near-infrared frequency-domain tissue spectroscopy for absolute absorption coefficient quantitation in neonatal head-simulating phantoms," J. Biomed. Opt. 5, 185-193 (2000).
[CrossRef] [PubMed]

Andronica, R.

Arif, I.

Arridge, S. R.

A. P. Gibson, T. Austin, N. L. Everdell, M. Schweiger, S. R. Arridge, J. H. Meek, J. S. Wyatt, D. T. Delpy, and J. C. Hebden, "Three-dimensional whole-head optical tomography of passive motor evoked responses in the neonate," Neuroimage 30, 521- 528 (2005).
[CrossRef] [PubMed]

M. Schweiger, S. R. Arridge, and I. Nissilä, "Gauss-Newton method for image reconstruction in diffuse optical tomography," Phys. Med. Biol. 50, 2365-2386 (2005).
[CrossRef] [PubMed]

I. Nissilä, T. Noponen, K. Kotilahti, T. Tarvainen, M. Schweiger, L. Lipiäinen, S. R. Arridge, and T. Katila, "Instrumentation and calibration methods for the multichannel measurement of phase and amplitude in optical tomography," Rev. Sci. Instrum. 76, 044302 (2005).
[CrossRef]

A. P. Gibson, J. Hebden, and S. R. Arridge, "Recent advances in diffuse optical tomography," Phys. Med. Biol. 50, R1-R43 (2005).
[CrossRef] [PubMed]

T. Tarvainen, V. Kolehmainen, M. Vauhkonen, A. Vanne, A. P. Gibson, M. Schweiger, S. R. Arridge, and J. P. Kaipio, "Computational calibration method for optical tomography," Appl. Opt. 44, 1879-1888 (2005).
[CrossRef] [PubMed]

A. Gibson, R. M. Yusof, H. Deghani, J. Riley, N. Everdell, R. Richards, J. C. Hebden, M. Schweiger, S. R. Arridge, and D. T. Delpy, "Optical tomography of a realistic neonatal head phantom," Appl. Opt. 42, 3109-3116 (2003).
[CrossRef] [PubMed]

J. J. Stott, J. P. Culver, S. R. Arridge, and D. A. Boas, "Optode positional calibration in diffuse optical tomography," Appl. Opt. 42, 3154-3162 (2003).
[CrossRef] [PubMed]

D. A. Boas, T. J. Gaudette, and S. R. Arridge, "Simultaneous imaging and optode calibration with diffusive optical tomography," Opt. Express 8, 263-270 (2001).
[CrossRef] [PubMed]

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

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, "The finite element model for the propagation of light in scattering media: boundary and source conditions," Med. Phys. 22, 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, 299-309 (1993).
[CrossRef] [PubMed]

Austin, T.

A. P. Gibson, T. Austin, N. L. Everdell, M. Schweiger, S. R. Arridge, J. H. Meek, J. S. Wyatt, D. T. Delpy, and J. C. Hebden, "Three-dimensional whole-head optical tomography of passive motor evoked responses in the neonate," Neuroimage 30, 521- 528 (2005).
[CrossRef] [PubMed]

Barbour, R. L.

Barbour, S.-L. S.

Bluestone, A.

Bluestone, Y.

Boas, D. A.

Bouman, C. A.

Brooks, D. H.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, "Imaging the body with diffuse optical tomography," IEEE Signal Process Mag. 18(6), 57-75 (2001).
[CrossRef]

Chance, B.

Cope, M.

M. Cope and D. T. Delpy, "System for long term measurement of cerebral blood and tissue oxygenation on newborn infants by near-infrared transillumination," Med. Biol. Eng. Comput. 26, 289-294 (1988).
[CrossRef] [PubMed]

Culver, J. P.

Deghani, H.

Delpy, D. T.

A. P. Gibson, T. Austin, N. L. Everdell, M. Schweiger, S. R. Arridge, J. H. Meek, J. S. Wyatt, D. T. Delpy, and J. C. Hebden, "Three-dimensional whole-head optical tomography of passive motor evoked responses in the neonate," Neuroimage 30, 521- 528 (2005).
[CrossRef] [PubMed]

A. Gibson, R. M. Yusof, H. Deghani, J. Riley, N. Everdell, R. Richards, J. C. Hebden, M. Schweiger, S. R. Arridge, and D. T. Delpy, "Optical tomography of a realistic neonatal head phantom," Appl. Opt. 42, 3109-3116 (2003).
[CrossRef] [PubMed]

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, "The finite element model for the propagation of light in scattering media: boundary and source conditions," Med. Phys. 22, 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, 299-309 (1993).
[CrossRef] [PubMed]

M. Cope and D. T. Delpy, "System for long term measurement of cerebral blood and tissue oxygenation on newborn infants by near-infrared transillumination," Med. Biol. Eng. Comput. 26, 289-294 (1988).
[CrossRef] [PubMed]

DiMarzio, C. A.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, "Imaging the body with diffuse optical tomography," IEEE Signal Process Mag. 18(6), 57-75 (2001).
[CrossRef]

Dorn, O.

O. Dorn, "A transport-backtransport method for optical tomography," Inverse Probl. 14, 1107-1130 (1998).
[CrossRef]

Everdell, N.

Everdell, N. L.

A. P. Gibson, T. Austin, N. L. Everdell, M. Schweiger, S. R. Arridge, J. H. Meek, J. S. Wyatt, D. T. Delpy, and J. C. Hebden, "Three-dimensional whole-head optical tomography of passive motor evoked responses in the neonate," Neuroimage 30, 521- 528 (2005).
[CrossRef] [PubMed]

Feng, T.-C.

Gaudette, R. J.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, "Imaging the body with diffuse optical tomography," IEEE Signal Process Mag. 18(6), 57-75 (2001).
[CrossRef]

Gaudette, T. J.

Gibson, A.

Gibson, A. P.

T. Tarvainen, V. Kolehmainen, M. Vauhkonen, A. Vanne, A. P. Gibson, M. Schweiger, S. R. Arridge, and J. P. Kaipio, "Computational calibration method for optical tomography," Appl. Opt. 44, 1879-1888 (2005).
[CrossRef] [PubMed]

A. P. Gibson, J. Hebden, and S. R. Arridge, "Recent advances in diffuse optical tomography," Phys. Med. Biol. 50, R1-R43 (2005).
[CrossRef] [PubMed]

A. P. Gibson, T. Austin, N. L. Everdell, M. Schweiger, S. R. Arridge, J. H. Meek, J. S. Wyatt, D. T. Delpy, and J. C. Hebden, "Three-dimensional whole-head optical tomography of passive motor evoked responses in the neonate," Neuroimage 30, 521- 528 (2005).
[CrossRef] [PubMed]

Graber, H. L.

Grosenick, D.

Haskell, R. C.

Hawysz, D.

D. Hawysz and E. M. Sevick-Muraca, "Developments towards diagnostic breast cancer imaging using near-infrared optical measurements and fluorescent contrast agents," Neoplasia 2, 388-417 (2000).
[CrossRef]

Hebden, J.

A. P. Gibson, J. Hebden, and S. R. Arridge, "Recent advances in diffuse optical tomography," Phys. Med. Biol. 50, R1-R43 (2005).
[CrossRef] [PubMed]

Hebden, J. C.

A. P. Gibson, T. Austin, N. L. Everdell, M. Schweiger, S. R. Arridge, J. H. Meek, J. S. Wyatt, D. T. Delpy, and J. C. Hebden, "Three-dimensional whole-head optical tomography of passive motor evoked responses in the neonate," Neuroimage 30, 521- 528 (2005).
[CrossRef] [PubMed]

A. Gibson, R. M. Yusof, H. Deghani, J. Riley, N. Everdell, R. Richards, J. C. Hebden, M. Schweiger, S. R. Arridge, and D. T. Delpy, "Optical tomography of a realistic neonatal head phantom," Appl. Opt. 42, 3109-3116 (2003).
[CrossRef] [PubMed]

Hielscher, A. H.

Y. Bluestone, G. Abdoulaev, C. H. Schmitz, R. L. Barbour, and A. H. Hielscher, "Three-dimensional optical tomography of hemodynamics in the human head," Opt. Express 9, 272-286 (2001).
[CrossRef] [PubMed]

A. D. Klose and A. H. Hielscher, "Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer," Med. Phys. 26, 1698-1707 (1999).
[CrossRef] [PubMed]

Hira, J.

Hiraoka, M.

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, "The finite element model for the propagation of light in scattering media: boundary and source conditions," Med. Phys. 22, 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, 299-309 (1993).
[CrossRef] [PubMed]

Ishimaru, A.

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

Jiang, H.

C. Li and H. Jiang, "A calibration method in diffuse optical tomography," J. Opt. A 6, 844-852 (2004).
[CrossRef]

Kaipio, J. P.

Katila, T.

I. Nissilä, T. Noponen, K. Kotilahti, T. Tarvainen, M. Schweiger, L. Lipiäinen, S. R. Arridge, and T. Katila, "Instrumentation and calibration methods for the multichannel measurement of phase and amplitude in optical tomography," Rev. Sci. Instrum. 76, 044302 (2005).
[CrossRef]

Kaufman, H.

B. W. Pogue, K. D. Paulsen, C. Abele, and H. Kaufman, "Calibration of near-infrared frequency-domain tissue spectroscopy for absolute absorption coefficient quantitation in neonatal head-simulating phantoms," J. Biomed. Opt. 5, 185-193 (2000).
[CrossRef] [PubMed]

Kilmer, M.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, "Imaging the body with diffuse optical tomography," IEEE Signal Process Mag. 18(6), 57-75 (2001).
[CrossRef]

Klose, A. D.

A. D. Klose and A. H. Hielscher, "Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer," Med. Phys. 26, 1698-1707 (1999).
[CrossRef] [PubMed]

Kolehmainen, V.

Kotilahti, K.

I. Nissilä, T. Noponen, K. Kotilahti, T. Tarvainen, M. Schweiger, L. Lipiäinen, S. R. Arridge, and T. Katila, "Instrumentation and calibration methods for the multichannel measurement of phase and amplitude in optical tomography," Rev. Sci. Instrum. 76, 044302 (2005).
[CrossRef]

Li, C.

C. Li and H. Jiang, "A calibration method in diffuse optical tomography," J. Opt. A 6, 844-852 (2004).
[CrossRef]

Lipiäinen, L.

I. Nissilä, T. Noponen, K. Kotilahti, T. Tarvainen, M. Schweiger, L. Lipiäinen, S. R. Arridge, and T. Katila, "Instrumentation and calibration methods for the multichannel measurement of phase and amplitude in optical tomography," Rev. Sci. Instrum. 76, 044302 (2005).
[CrossRef]

Luo, H.

McAdams, M. S.

McBride, T. O.

T. O. McBride, B. W. Pogue, U. L. Österberg, and K. D. Paulsen, "Strategies for absolute calibration of near-infrared tomographic tissue imaging," in Oxygen Transport to Tissue XXIV, J. F. Dunn and H. M. Swartz, eds. (Kluwer/Plenum, 2003), pp. 85-99.

Meek, J. H.

A. P. Gibson, T. Austin, N. L. Everdell, M. Schweiger, S. R. Arridge, J. H. Meek, J. S. Wyatt, D. T. Delpy, and J. C. Hebden, "Three-dimensional whole-head optical tomography of passive motor evoked responses in the neonate," Neuroimage 30, 521- 528 (2005).
[CrossRef] [PubMed]

Millane, R. P.

Miller, E. L.

D. A. Boas, D. H. Brooks, E. L. Miller, C. A. DiMarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, "Imaging the body with diffuse optical tomography," IEEE Signal Process Mag. 18(6), 57-75 (2001).
[CrossRef]

Milstein, A. B.

Nissilä, I.

M. Schweiger, S. R. Arridge, and I. Nissilä, "Gauss-Newton method for image reconstruction in diffuse optical tomography," Phys. Med. Biol. 50, 2365-2386 (2005).
[CrossRef] [PubMed]

I. Nissilä, T. Noponen, K. Kotilahti, T. Tarvainen, M. Schweiger, L. Lipiäinen, S. R. Arridge, and T. Katila, "Instrumentation and calibration methods for the multichannel measurement of phase and amplitude in optical tomography," Rev. Sci. Instrum. 76, 044302 (2005).
[CrossRef]

Noponen, T.

I. Nissilä, T. Noponen, K. Kotilahti, T. Tarvainen, M. Schweiger, L. Lipiäinen, S. R. Arridge, and T. Katila, "Instrumentation and calibration methods for the multichannel measurement of phase and amplitude in optical tomography," Rev. Sci. Instrum. 76, 044302 (2005).
[CrossRef]

Oh, S.

Österberg, U. L.

T. O. McBride, B. W. Pogue, U. L. Österberg, and K. D. Paulsen, "Strategies for absolute calibration of near-infrared tomographic tissue imaging," in Oxygen Transport to Tissue XXIV, J. F. Dunn and H. M. Swartz, eds. (Kluwer/Plenum, 2003), pp. 85-99.

Patterson, M. S.

Paulsen, K. D.

B. W. Pogue, K. D. Paulsen, C. Abele, and H. Kaufman, "Calibration of near-infrared frequency-domain tissue spectroscopy for absolute absorption coefficient quantitation in neonatal head-simulating phantoms," J. Biomed. Opt. 5, 185-193 (2000).
[CrossRef] [PubMed]

T. O. McBride, B. W. Pogue, U. L. Österberg, and K. D. Paulsen, "Strategies for absolute calibration of near-infrared tomographic tissue imaging," in Oxygen Transport to Tissue XXIV, J. F. Dunn and H. M. Swartz, eds. (Kluwer/Plenum, 2003), pp. 85-99.

Pei, Y.

Pogue, B. W.

B. W. Pogue, K. D. Paulsen, C. Abele, and H. Kaufman, "Calibration of near-infrared frequency-domain tissue spectroscopy for absolute absorption coefficient quantitation in neonatal head-simulating phantoms," J. Biomed. Opt. 5, 185-193 (2000).
[CrossRef] [PubMed]

T. O. McBride, B. W. Pogue, U. L. Österberg, and K. D. Paulsen, "Strategies for absolute calibration of near-infrared tomographic tissue imaging," in Oxygen Transport to Tissue XXIV, J. F. Dunn and H. M. Swartz, eds. (Kluwer/Plenum, 2003), pp. 85-99.

Ramirez, N.

Richards, R.

Riley, J.

Rinneberg, H.

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Appl. Opt. (6)

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S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, "A finite element approach for modeling photon transport in tissue," Med. Phys. 20, 299-309 (1993).
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M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, "The finite element model for the propagation of light in scattering media: boundary and source conditions," Med. Phys. 22, 1779-1792 (1995).
[CrossRef] [PubMed]

Neoplasia (1)

D. Hawysz and E. M. Sevick-Muraca, "Developments towards diagnostic breast cancer imaging using near-infrared optical measurements and fluorescent contrast agents," Neoplasia 2, 388-417 (2000).
[CrossRef]

Neuroimage (1)

A. P. Gibson, T. Austin, N. L. Everdell, M. Schweiger, S. R. Arridge, J. H. Meek, J. S. Wyatt, D. T. Delpy, and J. C. Hebden, "Three-dimensional whole-head optical tomography of passive motor evoked responses in the neonate," Neuroimage 30, 521- 528 (2005).
[CrossRef] [PubMed]

Opt. Express (2)

Phys. Med. Biol. (2)

M. Schweiger, S. R. Arridge, and I. Nissilä, "Gauss-Newton method for image reconstruction in diffuse optical tomography," Phys. Med. Biol. 50, 2365-2386 (2005).
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A. Yodh and B. Chance, "Spectroscopy and imaging with diffusing light," Phys. Today 48(3), 34-40 (1995).
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Rev. Sci. Instrum. (1)

I. Nissilä, T. Noponen, K. Kotilahti, T. Tarvainen, M. Schweiger, L. Lipiäinen, S. R. Arridge, and T. Katila, "Instrumentation and calibration methods for the multichannel measurement of phase and amplitude in optical tomography," Rev. Sci. Instrum. 76, 044302 (2005).
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Figures (16)

Fig. 1
Fig. 1

Block structure of the augmented Jacobian including optical parameters, source and detector coupling blocks.

Fig. 2
Fig. 2

(Color online) Target and reconstructed images of absorption (top) and scattering distributions (bottom) for the 2D test case. Image columns from left to right: target, reconstruction from noise-free data, and reconstruction from data with 1% additive Gaussian noise.

Fig. 3
Fig. 3

Log amplitude and phase data at all detectors for a single source. Data without coupling contamination (solid curve) and with the three cases of coupling coefficients.

Fig. 4
Fig. 4

(Color online) Reconstructions from data in the presence of coupling errors. Shown are absorption images (left) and scattering images (right) for the three levels of coupling contamination, without additional random noise. In each image pair, the left image shows the conventional reconstruction without accounting for coupling noise, while the right image is obtained when coupling coefficients are included in the reconstruction.

Fig. 5
Fig. 5

(Color online) Reconstructions from data in the presence of coupling and random errors. The arrangement of images is the same as for Fig. 4.

Fig. 6
Fig. 6

(Color online) Reconstructed source and detector coupling coefficients for ln A and φ. The results for cases 1, 2, and 3 are scaled so that they can be compared to common target coefficients (solid curve).

Fig. 7
Fig. 7

L 2 error of reconstructed coupling coefficients as a function of iteration count for the three cases considered. Top, results from data without additional random noise. Bottom, results from data with additional 0.5% Gaussian-distributed random noise.

Fig. 8
Fig. 8

Objective functions for the three coupling cases as a function of iteration count. Top, no additional random data noise (N0). Bottom, 0.5% Gaussian-distributed random data noise.

Fig. 9
Fig. 9

(Color online) Effect of coupling coefficient reconstruction on the recovery of a superficial perturbation. (a) Target image with boundary absorption inclusion. (b) Cross sections of the target and reconstructed absorption images along the line indicated in (a).

Fig. 10
Fig. 10

(Color online) Reconstructed coupling coefficients of log amplitude (top) and phase (bottom) for the superficial inclusion in Fig. 9.

Fig. 11
Fig. 11

(Color online) Phantom setup for experimental data acquisition: location of embedded inclusions and arrangement of sources and detectors on the surface.

Fig. 12
Fig. 12

Subset of 100 measurements of the experimental data vectors of log amplitude (top) and phase (bottom). Shown are the data differences between inhomogeneous and homogeneous phantom for the two cases with and without hair under the optodes.

Fig. 13
Fig. 13

(Color online) Cross sections of reconstruction of absorption (top) and scattering distributions (bottom) from difference phantom data. Columns from left to right, reconstruction of uncontaminated data, reconstruction of hair data without recovery of coupling coefficients, reconstruction of hair data with coupling coefficients. Target locations are marked with a black circle.

Fig. 14
Fig. 14

(Color online) Cross sections of reconstructions of absorption (top) and scattering distributions (bottom) from absolute phantom data. Columns from left to right, reconstruction of uncontaminated data, reconstruction of hair data without recovery of coupling coefficients, reconstruction of hair data with coupling coefficients. Target locations are marked with a black circle.

Fig. 15
Fig. 15

Source and detector coupling coefficients ln A ( α , β ) (top) and φ ( α , β ) (bottom) recovered during absolute reconstruction of experimental phantom data.

Fig. 16
Fig. 16

(Color online) Effect of hair on the difference data for (a) log A, (b) phase. Dashed curve is the product of source and detector coupling coefficients obtained in the reconstruction; solid curve is the measured difference between clean measurement of the inhomogeneous phantom and a measurement where the hair is in place.

Tables (1)

Tables Icon

Table 1 Levels of Coupling and Random Error in the Simulated Data Sets Used for 2D Reconstructions

Equations (33)

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( κ ( r ) + μ a ( r ) + i ω c ) ϕ ( r , ω ) = q ( r , ω ) , r Ω ,
ϕ ( ξ , ω ) + 2 κ ( ξ ) ζ n ϕ ( ξ , ω ) = 0 , ξ Ω ,
Γ ( ξ ) = κ ( ξ ) n ϕ ( ξ , ω ) = 1 2 ζ ϕ ( ξ , ω ) , ξ Ω .
y i j = Ω w j ( ξ ) Γ i ( ξ ) d ξ = Ω w j ( ξ ) 2 ζ ϕ i ( ξ , ω ) d ξ ,
y = f ( x ) ,
x ( r ) k = 1 B x k b k ( r ) ,
ϕ ˜ i ( r , ω ) = α i ϕ i ( r , ω ) , i = 1 S ,
Γ ˜ i j ( ξ ) = β j 1 2 ζ ϕ ˜ i ( ξ , ω ) = α i β j Γ i j ( ξ ) , i = 1 S ,
j = 1 M ,
y ˜ = f ˜ ( x ˜ ) ,
f ˜ i j ( x ˜ ) = α i β j f i j ( x ) .
η = arg min η ¯ [ i ( η ¯ α i α i ( 0 ) ) 2 + j ( η ¯ 1 β j β j ( 0 ) ) 2 ] .
x k + 1 = x k + γ k [ J T ( x k ) J ( x k ) + τ ψ ( x k ) ] 1 [ J T ( x k ) ( y f ( x k ) ) τ ψ ( x k ) ]
J ˜ = [ f ˜ i j ( x ˜ ) x k f ˜ i j ( x ˜ ) α l f ˜ i j ( x ˜ ) β m ] .
f ˜ i j ( x ˜ ) x k = α i β j f i j ( x ) x k ,
f ˜ i j ( x ˜ ) α l = { β j f i j ( x ) if   i = l 0 otherwise ,
f ˜ i j ( x ˜ ) β m = { α i f i j ( x ) if   j = m 0 otherwise .
y i j = A i j exp   i φ i j { ln A i j = Re ( ln y i j ) φ i j = Im ( ln y i j ) .
α i = A i ( α ) exp i φ i ( α ) , β j = A j ( β ) exp i φ i ( β ) .
f ^ i j ( x ˜ ) := ln f ˜ i j ( x ˜ ) = ln ( A ˜ i j ) + i ( φ ˜ i j ) ,
A ˜ i j = A i j A i ( α ) A j ( β ) , φ ˜ i j = φ i j + φ i ( α ) + φ j ( β ) .  
J ^ = [ ln A ˜ i j μ a k ln A ˜ i j κ k ln A ˜ i j ln A k ( α ) ln A ˜ i j φ k ( α ) ln A ˜ i j ln A k ( β ) ln A ˜ i j φ k ( β ) φ ˜ i j μ a k φ ˜ i j κ k φ ˜ i j ln A k ( α ) φ ˜ i j φ k ( α ) φ ˜ i j ln A k ( β ) φ ˜ i j φ k ( β ) ] ,
ln A ˜ i j x k = 1 A i j A i j x k = ln A i j x k ,
φ ˜ i j x k = φ i j x k ,
ln A ˜ i j ln A k ( α ) = { 1 if   i = k 0 otherwise ,
φ ˜ i j ln A k ( α ) = 0 ,
ln A ˜ i j ln A k ( β ) = { 1 if   j = k 0 otherwise ,
φ ˜ i j ln A k ( β ) = 0 ,
ln A ˜ i j φ k ( α ) = 0 ,
φ ˜ i j φ k ( α ) = { 1 if   i = k 0 otherwise ,
ln A ˜ i j φ k ( β ) = 0 ,
φ ˜ i j φ k ( β ) = { 1 if   j = k 0 otherwise .
x ˜ T ( x ˜ ) = { ln μ a μ ¯ a ( 0 ) , ln κ κ ¯ ( 0 ) , ln α α ¯ ( 0 ) , ln β β ¯ ( 0 ) } ,

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