Abstract

Optical image reconstruction in a heterogeneous turbid medium with the use of frequency-domain measurements is investigated in detail. A finite-element reconstruction algorithm for optical data based on a diffusion equation approximation is presented and confirmed by a series of simulations and experiments using phantoms having optical properties in the range of those expected for tissues. Simultaneous reconstruction of absorption and scattering coefficients is achieved both theoretically and experimentally. Images with different target locations and contrast levels between target and background are also successfully recovered. All reconstructed images from both simulated and experimental data are derived directly from absolute optical data in which no differential measurement scheme is used. Results from the use of simulated and measured data suggest that quantitative images can be produced in terms of absorption and scattering coefficient values and location, size, and shape of heterogeneities within a circular background region over a range of contrast levels. Further, the effects of modulation frequency are found to be relatively modest, although boundary conditions appear to be important factors.

© 1996 Optical Society of America

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1995 (3)

1994 (5)

1993 (4)

1992 (4)

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]

R. Alfano, P. P. Ho, K. M. Yoo, “Photons for prompt tumor detection,” Phys. World 5, 37–40 (1992).

S. J. Madsen, M. S. Patterson, B. C. Wilson, “The use of india ink as an optical absorber in tissue-simulating phantoms,” Phys. Med. Biol. 37, 985–993 (1992).
[CrossRef] [PubMed]

T. Farrell, M. S. Patterson, B. C. Wilson, “A diffusion theory model of spatially resolved, steady-state, diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

1991 (1)

1990 (1)

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

1984 (1)

R. J. Bartrum, J. C. Crow, “Transillumination lightscanning to diagnose breast cancer: a feasibility study,” Am. J. Roentgenol. 142, 409–414 (1984).
[CrossRef]

1983 (1)

E. Gratton, M. Limkeman, “A continuously variable frequency cross-correlation phase fluorometer with picosecond resolution,” Biophys. J. 44, 315–324 (1983).
[CrossRef] [PubMed]

1973 (1)

1929 (1)

M. Cutler, “Transillumination as an aid to diagnosis of breast lesions,” Surg. Gynecol. Obstet. 48, 721–729 (1929).

Alfano, R.

R. Alfano, P. P. Ho, K. M. Yoo, “Photons for prompt tumor detection,” Phys. World 5, 37–40 (1992).

Alfano, R. R.

Aronson, R.

H. Graber, J. Chang, R. Aronson, R. L. Barbour, “A perturbation model for imaging in dense scattering media: derivation and evaluation of imaging operators,” in Ref. 6, pp. 121–143.

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 Ref. 6, pp. 87–120.

Arridge, S. R.

S. R. Arridge, M. Schweijerand, D. T. Delpy, “Iterative reconstruction of near infrared absorption images,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1767, 372–383 (1992).
[CrossRef]

S. R. Arridge, M. Schweiger, M. Hiraoka, D. T. Delpy, “Performance of an iterative reconstruction algorithm for near infrared absorption and scattering imaging,” in Photon Migration and Imaging in Random Media and Tissues, R. R. Alfano, B. Chance, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1888, 360–371 (1993).
[CrossRef]

S. R. Arridge, “Forward and inverse problems in time-resolved infrared imaging,” in Ref. 6, pp. 35–64.

Barbieri, B.

Barbour, R. L.

H. Graber, J. Chang, R. Aronson, R. L. Barbour, “A perturbation model for imaging in dense scattering media: derivation and evaluation of imaging operators,” in Ref. 6, pp. 121–143.

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 Ref. 6, pp. 87–120.

Bartrum, R. J.

R. J. Bartrum, J. C. Crow, “Transillumination lightscanning to diagnose breast cancer: a feasibility study,” Am. J. Roentgenol. 142, 409–414 (1984).
[CrossRef]

Boas, D. A.

Burch, C. L.

Chance, B.

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]

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]

Chang, J.

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 Ref. 6, pp. 87–120.

H. Graber, J. Chang, R. Aronson, R. L. Barbour, “A perturbation model for imaging in dense scattering media: derivation and evaluation of imaging operators,” in Ref. 6, pp. 121–143.

Crow, J. C.

R. J. Bartrum, J. C. Crow, “Transillumination lightscanning to diagnose breast cancer: a feasibility study,” Am. J. Roentgenol. 142, 409–414 (1984).
[CrossRef]

Cutler, M.

M. Cutler, “Transillumination as an aid to diagnosis of breast lesions,” Surg. Gynecol. Obstet. 48, 721–729 (1929).

Das, B. B.

Delpy, D. T.

J. C. Hebden, D. T. Delpy, “Enhanced time-resolved imaging with a diffusion model of photon transport,” Opt. Lett. 19, 311–313 (1994).
[CrossRef] [PubMed]

S. R. Arridge, M. Schweijerand, D. T. Delpy, “Iterative reconstruction of near infrared absorption images,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1767, 372–383 (1992).
[CrossRef]

S. R. Arridge, M. Schweiger, M. Hiraoka, D. T. Delpy, “Performance of an iterative reconstruction algorithm for near infrared absorption and scattering imaging,” in Photon Migration and Imaging in Random Media and Tissues, R. R. Alfano, B. Chance, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1888, 360–371 (1993).
[CrossRef]

Fantini, S.

Farrell, T.

T. Farrell, M. S. Patterson, B. C. Wilson, “A diffusion theory model of spatially resolved, steady-state, diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

Feng, T.

Fishkin, 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]

Fishkin, J. B.

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 tissues in the visible and near infrared,” Phys. Med. Biol. 35, 1317–1334 (1990).
[CrossRef] [PubMed]

Frisoli, J. J.

Graber, H.

H. Graber, J. Chang, R. Aronson, R. L. Barbour, “A perturbation model for imaging in dense scattering media: derivation and evaluation of imaging operators,” in Ref. 6, pp. 121–143.

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 Ref. 6, pp. 87–120.

Gratton, E.

Hale, G. M.

Haskell, R. C.

Hebden, J. C.

Hiraoka, M.

S. R. Arridge, M. Schweiger, M. Hiraoka, D. T. Delpy, “Performance of an iterative reconstruction algorithm for near infrared absorption and scattering imaging,” in Photon Migration and Imaging in Random Media and Tissues, R. R. Alfano, B. Chance, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1888, 360–371 (1993).
[CrossRef]

Ho, P. P.

R. Alfano, P. P. Ho, K. M. Yoo, “Photons for prompt tumor detection,” Phys. World 5, 37–40 (1992).

Ishimaru, A.

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

Jiang, H.

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]

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

B. W. Pogue, M. S. Patterson, H. Jiang, K. D. Paulsen, “Initial assessment of a simple system for frequency domain diffuse optical tomography,” Phys. Med. Biol. (to be published).

H. Jiang, K. D. Paulsen, U. L. Osterberg, “Optical image reconstruction using DC data: simulations and experiments,” Phys. Med. Biol. (to be published).

H. Jiang, K. D. Paulsen, U. L. Osterberg, “Indirect optical image reconstruction with a cw He–Ne laser for breast cancer detection,” in Optical Tomography: Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2389, 615–620 (1995).
[CrossRef]

Kaneko, M.

Y. Yamishita, M. Kaneko, “Infrared diaphanoscopy for medical diagnosis,” in Ref. 6, pp. 283–316.

Lakowicz, J. R.

Limkeman, M.

E. Gratton, M. Limkeman, “A continuously variable frequency cross-correlation phase fluorometer with picosecond resolution,” Biophys. J. 44, 315–324 (1983).
[CrossRef] [PubMed]

Madsen, S. J.

S. J. Madsen, M. S. Patterson, B. C. Wilson, “The use of india ink as an optical absorber in tissue-simulating phantoms,” Phys. Med. Biol. 37, 985–993 (1992).
[CrossRef] [PubMed]

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]

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.

Moes, C. J.

O’Leary, M. A.

Osterberg, U. L.

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

H. Jiang, K. D. Paulsen, U. L. Osterberg, “Indirect optical image reconstruction with a cw He–Ne laser for breast cancer detection,” in Optical Tomography: Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2389, 615–620 (1995).
[CrossRef]

H. Jiang, K. D. Paulsen, U. L. Osterberg, “Optical image reconstruction using DC data: simulations and experiments,” Phys. Med. Biol. (to be published).

Patterson, M. S.

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

B. W. Pogue, M. S. Patterson, “Frequency-domain optical absorption spectroscopy of finite tissue volumes using diffusion theory,” Phys. Med. Biol. 39, 1157–1180 (1994).
[CrossRef] [PubMed]

T. Farrell, M. S. Patterson, B. C. Wilson, “A diffusion theory model of spatially resolved, steady-state, diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[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]

S. J. Madsen, M. S. Patterson, B. C. Wilson, “The use of india ink as an optical absorber in tissue-simulating phantoms,” Phys. Med. Biol. 37, 985–993 (1992).
[CrossRef] [PubMed]

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

B. C. Wilson, M. S. Patterson, B. W. Pogue, “Instrumentation for in vivotissue spectroscopy and imaging,” in Medical Lasers and Systems II, D. M. Harris, C. M. Penney, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1892, 132–147 (1993).
[CrossRef]

B. W. Pogue, M. S. Patterson, H. Jiang, K. D. Paulsen, “Initial assessment of a simple system for frequency domain diffuse optical tomography,” Phys. Med. Biol. (to be published).

B. W. Pogue, M. S. Patterson, “Forward and inverse optical imaging using a multigrid finite difference calculation,” in Advances in Optical Imaging and Photon Migration, Vol. 21 of 1994 OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 176–180.

Paulsen, K. D.

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]

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

B. W. Pogue, M. S. Patterson, H. Jiang, K. D. Paulsen, “Initial assessment of a simple system for frequency domain diffuse optical tomography,” Phys. Med. Biol. (to be published).

H. Jiang, K. D. Paulsen, U. L. Osterberg, “Optical image reconstruction using DC data: simulations and experiments,” Phys. Med. Biol. (to be published).

H. Jiang, K. D. Paulsen, U. L. Osterberg, “Indirect optical image reconstruction with a cw He–Ne laser for breast cancer detection,” in Optical Tomography: Photon Migration, and Spectroscopy of Tissue and Model Media: Theory, Human Studies, and Instrumentation, B. Chance, R. R. Alfano, eds., Proc. Soc. Photo-Opt. Instrum. Eng.2389, 615–620 (1995).
[CrossRef]

Peters, V. G.

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

Pogue, B. W.

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

B. W. Pogue, M. S. Patterson, “Frequency-domain optical absorption spectroscopy of finite tissue volumes using diffusion theory,” Phys. Med. Biol. 39, 1157–1180 (1994).
[CrossRef] [PubMed]

B. W. Pogue, M. S. Patterson, H. Jiang, K. D. Paulsen, “Initial assessment of a simple system for frequency domain diffuse optical tomography,” Phys. Med. Biol. (to be published).

B. W. Pogue, M. S. Patterson, “Forward and inverse optical imaging using a multigrid finite difference calculation,” in Advances in Optical Imaging and Photon Migration, Vol. 21 of 1994 OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994), pp. 176–180.

B. C. Wilson, M. S. Patterson, B. W. Pogue, “Instrumentation for in vivotissue spectroscopy and imaging,” in Medical Lasers and Systems II, D. M. Harris, C. M. Penney, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1892, 132–147 (1993).
[CrossRef]

Prahl, A.

Querry, M. R.

Schweiger, M.

S. R. Arridge, M. Schweiger, M. Hiraoka, D. T. Delpy, “Performance of an iterative reconstruction algorithm for near infrared absorption and scattering imaging,” in Photon Migration and Imaging in Random Media and Tissues, R. R. Alfano, B. Chance, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1888, 360–371 (1993).
[CrossRef]

Schweijerand, M.

S. R. Arridge, M. Schweijerand, D. T. Delpy, “Iterative reconstruction of near infrared absorption images,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1767, 372–383 (1992).
[CrossRef]

Sevick, E. M.

E. M. Sevick, J. J. 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]

Svaasand, L. O.

Tromberg, B.

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]

van Germert, M. J.

van Marle, J.

van Staveren, H. J.

Wang, Y.

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 Ref. 6, pp. 87–120.

Wilson, B. C.

S. J. Madsen, M. S. Patterson, B. C. Wilson, “The use of india ink as an optical absorber in tissue-simulating phantoms,” Phys. Med. Biol. 37, 985–993 (1992).
[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]

T. Farrell, M. S. Patterson, B. C. Wilson, “A diffusion theory model of spatially resolved, steady-state, diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19, 879–888 (1992).
[CrossRef] [PubMed]

B. C. Wilson, M. S. Patterson, B. W. Pogue, “Instrumentation for in vivotissue spectroscopy and imaging,” in Medical Lasers and Systems II, D. M. Harris, C. M. Penney, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1892, 132–147 (1993).
[CrossRef]

Wong, K. S.

Wyman, D. R.

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Figures (16)

Fig. 1
Fig. 1

Experimental setup used for the frequency-domain measurements. Ref., reference; Sig., signal.

Fig. 2
Fig. 2

(a) Phantom geometry for the off-centered target case. The centered target case is identical except that the center of the internal heterogeneity is concentric with the background region. (b) Photograph of the phantom system used in this study. On the top of the phantom a target suspension system has been incorporated into a rotatable stage (scaled precisely with less than 0.5° error), which provided accurate manipulations during the data collection procedures.

Fig. 3
Fig. 3

Comparison between measured and computed data using type I and type III BC’s at the 16 detector positions around the phantom for one source excitation location (0° at the surface) in a homogeneous medium having μ s = 0.6 mm - 1 and μa = 0.006 mm−1: (a) normalized logarithmic AC amplitude, (b) phase shift between the detector and the source position. The horizontal axes express angle along the boundary surface (in degrees).

Fig. 4
Fig. 4

Simulated simultaneous reconstruction of both diffusion and absorption coefficients with 2:1 contrast for an eccentrically located target: (a) exact D image, (b) D reconstruction with no noise added, (c) D reconstruction with 1% random noise added, (d) D reconstruction with 5% random noise added, (e) exact μa image, (f) μa reconstruction with no noise added, (g) μa reconstruction with 1% random noise added, (h) μa reconstruction with 5% random noise added.

Fig. 5
Fig. 5

Comparison of exact and simulated reconstructions along transects AB and CD shown in Fig. 2(a) for an eccentrically located target with different noise levels: (a) D profiles along transect AB, (b) μa profiles along transect AB, (c) D profiles along transect CD, (d) μa profiles along transect CD. The horizontal axes indicate either transect AB or CD with millimeter units.

Fig. 6
Fig. 6

Reconstructed images from simulated data (no added noise) obtained from an eccentrically located target having different contrasts with the background medium: (a) D image with 2:1 contrast level, (b) D image with 5:1 contrast level, (c) D image with 10:1 contrast level, (d) μa image with 2:1 contrast level, (e) μa image with 5:1 contrast level, (f) μa image with 10:1 contrast level.

Fig. 7
Fig. 7

Comparison of exact and simulated reconstruction profiles along transects AB and CD shown in Fig. 2(a) for an eccentrically located target with different contrast levels: (a) D profiles along transect AB, (b) μa profiles along transect AB, (c) D profiles along transect CD, (d) μa profiles along transect CD. The horizontal axes indicate either transect AB or CD with millimeter units.

Fig. 8
Fig. 8

Simulated reconstructions (no added noise) for a centrally located target having 5:1 contrast excited at different modulation frequencies: (a) exact D image, (b) D image at f = 50 MHz, (c) D image at f = 200 MHz, (d) D image at f = 300 MHz, (e) exact μa image, (f) μa image at f = 50 MHz, (g) μa image at f = 200 MHz, (h) μa image at f = 300 MHz.

Fig. 9
Fig. 9

Simultaneous reconstruction of both diffusion and absorption profiles based on experimental data obtained from a centrally located target having 2:1 contrast with the background: (a) exact D image, (b) reconstructed D image, (c) exact μa image, (d) reconstructed μa image.

Fig. 10
Fig. 10

Same as Fig. 9, except that the eccentrically located target configuration is used.

Fig. 11
Fig. 11

Comparison of exact and reconstructed profiles along transects AB and CD shown in Fig. 2(a) based on experimental data obtained from eccentrically and centrally located targets with 2:1 contrast. (a) and (e): D profiles along transect AB for off-centered and centered, respectively; (b) and (f): μa profiles along transect AB for off-centered and centered, respectively; (c) and (g): D profiles along transect CD for off-centered and centered, respectively; (d) and (h): μa profiles along transect CD for off-centered and centered, respectively. The horizontal axes indicate either transect AB or CD with millimeter units.

Fig. 12
Fig. 12

Reconstructed images based on experimental data obtained from a centrally located target having different contrast levels relative to the background medium: (a) D image with 2:1 contrast level, (b) D image with 5:1 contrast level, (c) D image with 10:1 contrast level, (d) μa image with 2:1 contrast level, (e) μa image with 5:1 contrast level, (f) μa image with 10:1 contrast level.

Fig. 13
Fig. 13

Same as Fig. 12, except that the eccentrically located target configuration is used.

Fig. 14
Fig. 14

Same as Fig. 7, except that experimental data are used. Fig. 15. Same as Fig. 8, except that experimental data are used.

Fig. 15
Fig. 15

where frequencies from 50 MHz to 300 MHz for the centered target location configuration were used. Clearly they confirmed the results in the simulations. Table 8 provides a quantitative verification of this observation.

Fig. 16
Fig. 16

Reconstruction in which the centrally located target tube was filled with the same medium as that of the background: (a) D image, (b) μa image.

Tables (8)

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Table 1 Image Errors (Absolute Difference between the True and the Reconstructed Values) for Images from Simulated Data with Different Noise Levelsa

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Table 2 Geometric Information for Reconstructed Images from Simulated Data with Different Noise Levelsa

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Table 3 Reconstructed Contrast Levels between the Target and the Background and Optical Property Ratios of the Target between the Different Contrast Levels for Images from Simulated Data, Where the Target Is Located Off-centera

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Table 4 Image Errors (Absolute Difference between the True and Reconstructed Values) for Images from Simulated Data with a Centered Target and Different Modulation Frequencya

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Table 5 Image Errors (Absolute Difference between the True and Reconstructed Values) for Images from Experimental Data with Centered and Off-Centered Target Locationsa

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Table 6 Geometric Information for Reconstructed Images from Experimental Data with Centered and Off-Centered Target Locationsa

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Table 7 Reconstructed Contrast Levels between the Target and the Background and Optical Property Ratios of the Target between the Different Contrast Levels for Images Obtained from Experimental Data, Where the Target Is Located Off-Centera

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Table 8 Image Errors (Absolute Difference between the True and Reconstructed Values) for Images from Experimental Data with Centered Target and Different Modulation Frequencya

Equations (14)

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· D ( r ) Φ ( r , ω ) - [ μ a ( r ) - i ω c ] Φ ( r , ω ) = - S ( r , ω ) ,
D ( r ) = 1 3 [ μ a ( r ) + μ s ( r ) ] ,
[ A ] { Φ } = { b } .
b i = - S ψ i + ( α + i β ) j = 1 N Φ j ψ j ψ i d s Φ = { Φ 1 , Φ 2 , Φ N } T ,
Φ ( D ˜ , μ ˜ a ) = Φ ( D , μ a ) + Φ D Δ D + Φ μ a Δ μ a + ,
J Δ χ = Φ o - Φ c ,
Φ R ( D ˜ , μ ˜ a ) = Φ R ( D , μ a ) + Φ R D Δ D + Φ R μ a Δ μ a + ,
Φ I ( D ˜ , μ ˜ a ) = Φ I ( D , μ a ) + Φ I D Δ D + Φ I μ a Δ μ a + ,
J ˜ Δ χ = Φ ˜ o - Φ ˜ c ,
J ˜ = [ Φ R , 1 D 1 Φ R , 1 D 2 Φ R , 1 D K Φ R , 1 μ 1 Φ R , 1 μ 2 Φ R , 1 μ L Φ I , 1 D 1 Φ I , 1 D 2 Φ I , 1 D K Φ I , 1 μ 1 Φ I , 1 μ 2 Φ I , 1 μ L Φ R , M D 1 Φ R , M D 2 Φ R , M D K Φ R , M μ 1 Φ R , M μ 2 Φ R , M μ L Φ I , M D 1 Φ I , M D 2 Φ I , M D K Φ I , M μ 1 Φ I , M μ 2 Φ I , M μ L ] ,
Δ χ = ( Δ D 1 , Δ D 2 , , Δ D K , Δ μ 1 , Δ μ 2 , , Δ μ L ) T ,
Φ ˜ o = ( Φ R , 1 o , Φ I , 1 o , Φ R , 2 o , Φ I , 2 o , , Φ R , M o , Φ I , M o ) T ,
Φ ˜ c = ( Φ R , 1 c , Φ I , 1 c , Φ R , 2 c , Φ I , 2 c , , Φ R , M c , Φ I , M c ) T ,
χ new i = ( 1 - θ ) χ old i + θ N * j = 1 N * χ old j ,

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