Abstract

We present a tumor localization method for diffuse optical tomography using linearly constrained minimum variance (LCMV) beam-forming. Beamforming is a spatial filtering technique where signals from certain directions can be enhanced while noise and interference from other directions are suppressed. In our method, we tessellate the domain into small voxels and regard each voxel as a possible position of abnormality (e.g., tumor). We then design a spatial filter based on the linearly constrained minimum variance criterion and apply it to each voxel in the domain. The abnormality is localized by observing the peak in the filter output signals. We test our method using simulated 3D examples. We assume a cubic transmission geometry and consider different cases where the abnormality is an absorber, a scatterer, and both. We also give examples showing the resolution of our method and its performance under different perturbation levels and noise levels. Simulation results show that LCMV beamforming can localize the abnormality well with good computational efficiency. It can be used alone for tumor localization and also as an effective preprocessing tool for improving the image reconstruction performances of other inverse methods.

© 2007 Optical Society of America

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2005 (1)

M. Guven, B. Yazici, X. Intes, and B. Chance, "Diffuse optical tomography with a priori anatomical information," Phys. Med. Biol. 50, 2837-2858 (2005).
[CrossRef] [PubMed]

2004 (1)

D. Grosenick, H. Wabnitz, K. T. Moesta, J. Mucke, M. M¨oller, C. Stroszczynski, J. St¨obel, B. Wassermann, P. M. Schlag, and H. Rinneberg, "Concentration and oxygen saturation of haemoglobin of 50 breast tumours determined by time-domain optical mammography," Phys. Med. Biol. 49, 1165-1181, (2004).
[CrossRef] [PubMed]

2003 (2)

2002 (1)

G. Strangman, D. Boas, and J. Sutton, "Non-invasive neuroimaging using near-infrared light," Biol. Psychiatry 52, 679-693 (2002).
[CrossRef] [PubMed]

2001 (2)

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

K. Sekihara, S. S. Nagarajan, D. Poeppel, A. Marantz, and Y. Miyashita, "Reconstructing spatio-temporal activities of neural sources using an MEG vector beamforming technique," IEEE Trans. Biomed. Eng. 48, 760-771 (2001).
[CrossRef] [PubMed]

2000 (2)

R. J. Gaudette, D. H. Brooks, C. A. DiMarzio, M. E. Kilmer, E. L. Miller, T. Gaudette, and D. A. Boas, "A comparison study of linear reconstruction techniques for diffuse optical tomographic imaging of absorption coefficient," Phys. Med. Biol. 45, 1051-1070 (2000).
[CrossRef] [PubMed]

M. J. Holboke, B. J. Tromberg, X. Li, N. Shah, J. Fishkin, D. Kidney, J. Butler, B. Chance, and A.G. Yodh, "Three-dimensional diffuse optical mammography with ultrasound localization in a human subject," J. Biomed. Opt. 5, 237-247 (2000).
[CrossRef] [PubMed]

1999 (4)

1997 (3)

B. D. Van Veen, W. van Drongelen, M. Yuchtman, and A. Suzuki, "Localization of brain electrial activity via linearly constrained minimum variance spatial filtering," IEEE Trans. Biomed. Eng. 44, 867-880 (1997).
[CrossRef] [PubMed]

D. A. Boas, "A fundamental limitation of linearized algorithms for diffuse optical tomography," Opt. Express 1, 404-413 (1997).
[CrossRef] [PubMed]

A. Villringer and B. Chance, "Non-invasive optical spectroscopy and imaging of human brain function," Trends Neurosci. 20, 435-442 (1997).
[CrossRef] [PubMed]

1996 (1)

M. Papazoglou and J. L. Krolik, "High resolution adaptive beamforming for three-dimensional acoustic imaging of zooplankton," J. Acoust. Soc. Am. 100, 3621-3630 (1996).
[CrossRef]

1995 (1)

1994 (1)

1989 (1)

M. S. Patterson, B. Chance, and B. C. Wilson, "Time resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties," J. Appl. Opt. 28, 2331-2336 (1989).
[CrossRef]

1988 (1)

B. D. Van Veen and K. M. Buckley, "Beamforming: A versatile approach to spatial filtering," IEEE ASSP. Magazine 5, 4-24 (1988).
[CrossRef]

1944 (1)

S. O. Rice, "Mathematical analysis of random noise," Bell Syst. Tech. J. 23, 282-332 (1944).

Aronson, R.

Arridge, S. R.

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

Boas, D.

G. Strangman, D. Boas, and J. Sutton, "Non-invasive neuroimaging using near-infrared light," Biol. Psychiatry 52, 679-693 (2002).
[CrossRef] [PubMed]

Boas, D. A.

R. J. Gaudette, D. H. Brooks, C. A. DiMarzio, M. E. Kilmer, E. L. Miller, T. Gaudette, and D. A. Boas, "A comparison study of linear reconstruction techniques for diffuse optical tomographic imaging of absorption coefficient," Phys. Med. Biol. 45, 1051-1070 (2000).
[CrossRef] [PubMed]

D. A. Boas, "A fundamental limitation of linearized algorithms for diffuse optical tomography," Opt. Express 1, 404-413 (1997).
[CrossRef] [PubMed]

Bouman, C. A.

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

Brooks, D. H.

R. J. Gaudette, D. H. Brooks, C. A. DiMarzio, M. E. Kilmer, E. L. Miller, T. Gaudette, and D. A. Boas, "A comparison study of linear reconstruction techniques for diffuse optical tomographic imaging of absorption coefficient," Phys. Med. Biol. 45, 1051-1070 (2000).
[CrossRef] [PubMed]

Buckley, K. M.

B. D. Van Veen and K. M. Buckley, "Beamforming: A versatile approach to spatial filtering," IEEE ASSP. Magazine 5, 4-24 (1988).
[CrossRef]

Butler, J.

M. J. Holboke, B. J. Tromberg, X. Li, N. Shah, J. Fishkin, D. Kidney, J. Butler, B. Chance, and A.G. Yodh, "Three-dimensional diffuse optical mammography with ultrasound localization in a human subject," J. Biomed. Opt. 5, 237-247 (2000).
[CrossRef] [PubMed]

Chance, B.

M. Guven, B. Yazici, X. Intes, and B. Chance, "Diffuse optical tomography with a priori anatomical information," Phys. Med. Biol. 50, 2837-2858 (2005).
[CrossRef] [PubMed]

X. Intes, J. Ripoll, Y. Chen, S. Nioka, A. Yodh, and B. Chance, "In vivo continuous-wave optical breast imaging enhanced with Indocyanine Green," Med. Phys. 30, 1039-1047 (2003).
[CrossRef] [PubMed]

M. J. Holboke, B. J. Tromberg, X. Li, N. Shah, J. Fishkin, D. Kidney, J. Butler, B. Chance, and A.G. Yodh, "Three-dimensional diffuse optical mammography with ultrasound localization in a human subject," J. Biomed. Opt. 5, 237-247 (2000).
[CrossRef] [PubMed]

V. Ntziachristos, B. Chance, and A. G. Yodh, "Differential diffuse optical tomography," Opt. Express 5, 230-242 (1999).
[CrossRef] [PubMed]

A. Villringer and B. Chance, "Non-invasive optical spectroscopy and imaging of human brain function," Trends Neurosci. 20, 435-442 (1997).
[CrossRef] [PubMed]

M. S. Patterson, B. Chance, and B. C. Wilson, "Time resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties," J. Appl. Opt. 28, 2331-2336 (1989).
[CrossRef]

Chen, Y.

X. Intes, J. Ripoll, Y. Chen, S. Nioka, A. Yodh, and B. Chance, "In vivo continuous-wave optical breast imaging enhanced with Indocyanine Green," Med. Phys. 30, 1039-1047 (2003).
[CrossRef] [PubMed]

DiMarzio, C. A.

R. J. Gaudette, D. H. Brooks, C. A. DiMarzio, M. E. Kilmer, E. L. Miller, T. Gaudette, and D. A. Boas, "A comparison study of linear reconstruction techniques for diffuse optical tomographic imaging of absorption coefficient," Phys. Med. Biol. 45, 1051-1070 (2000).
[CrossRef] [PubMed]

Feng, T. C.

Fishkin, J.

M. J. Holboke, B. J. Tromberg, X. Li, N. Shah, J. Fishkin, D. Kidney, J. Butler, B. Chance, and A.G. Yodh, "Three-dimensional diffuse optical mammography with ultrasound localization in a human subject," J. Biomed. Opt. 5, 237-247 (2000).
[CrossRef] [PubMed]

Gaudette, R. J.

R. J. Gaudette, D. H. Brooks, C. A. DiMarzio, M. E. Kilmer, E. L. Miller, T. Gaudette, and D. A. Boas, "A comparison study of linear reconstruction techniques for diffuse optical tomographic imaging of absorption coefficient," Phys. Med. Biol. 45, 1051-1070 (2000).
[CrossRef] [PubMed]

Gaudette, T.

R. J. Gaudette, D. H. Brooks, C. A. DiMarzio, M. E. Kilmer, E. L. Miller, T. Gaudette, and D. A. Boas, "A comparison study of linear reconstruction techniques for diffuse optical tomographic imaging of absorption coefficient," Phys. Med. Biol. 45, 1051-1070 (2000).
[CrossRef] [PubMed]

Grosenick, D.

Guven, M.

M. Guven, B. Yazici, X. Intes, and B. Chance, "Diffuse optical tomography with a priori anatomical information," Phys. Med. Biol. 50, 2837-2858 (2005).
[CrossRef] [PubMed]

Haskell, R. C.

Holboke, M. J.

M. J. Holboke, B. J. Tromberg, X. Li, N. Shah, J. Fishkin, D. Kidney, J. Butler, B. Chance, and A.G. Yodh, "Three-dimensional diffuse optical mammography with ultrasound localization in a human subject," J. Biomed. Opt. 5, 237-247 (2000).
[CrossRef] [PubMed]

Intes, X.

M. Guven, B. Yazici, X. Intes, and B. Chance, "Diffuse optical tomography with a priori anatomical information," Phys. Med. Biol. 50, 2837-2858 (2005).
[CrossRef] [PubMed]

X. Intes, J. Ripoll, Y. Chen, S. Nioka, A. Yodh, and B. Chance, "In vivo continuous-wave optical breast imaging enhanced with Indocyanine Green," Med. Phys. 30, 1039-1047 (2003).
[CrossRef] [PubMed]

Kidney, D.

M. J. Holboke, B. J. Tromberg, X. Li, N. Shah, J. Fishkin, D. Kidney, J. Butler, B. Chance, and A.G. Yodh, "Three-dimensional diffuse optical mammography with ultrasound localization in a human subject," J. Biomed. Opt. 5, 237-247 (2000).
[CrossRef] [PubMed]

Kilmer, M. E.

R. J. Gaudette, D. H. Brooks, C. A. DiMarzio, M. E. Kilmer, E. L. Miller, T. Gaudette, and D. A. Boas, "A comparison study of linear reconstruction techniques for diffuse optical tomographic imaging of absorption coefficient," Phys. Med. Biol. 45, 1051-1070 (2000).
[CrossRef] [PubMed]

Krolik, J. L.

M. Papazoglou and J. L. Krolik, "High resolution adaptive beamforming for three-dimensional acoustic imaging of zooplankton," J. Acoust. Soc. Am. 100, 3621-3630 (1996).
[CrossRef]

Li, X.

M. J. Holboke, B. J. Tromberg, X. Li, N. Shah, J. Fishkin, D. Kidney, J. Butler, B. Chance, and A.G. Yodh, "Three-dimensional diffuse optical mammography with ultrasound localization in a human subject," J. Biomed. Opt. 5, 237-247 (2000).
[CrossRef] [PubMed]

Macdonald, R.

Marantz, A.

K. Sekihara, S. S. Nagarajan, D. Poeppel, A. Marantz, and Y. Miyashita, "Reconstructing spatio-temporal activities of neural sources using an MEG vector beamforming technique," IEEE Trans. Biomed. Eng. 48, 760-771 (2001).
[CrossRef] [PubMed]

McAdams, M. S.

McBride, T.

Millane, R. P.

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

Miller, E. L.

R. J. Gaudette, D. H. Brooks, C. A. DiMarzio, M. E. Kilmer, E. L. Miller, T. Gaudette, and D. A. Boas, "A comparison study of linear reconstruction techniques for diffuse optical tomographic imaging of absorption coefficient," Phys. Med. Biol. 45, 1051-1070 (2000).
[CrossRef] [PubMed]

Miyashita, Y.

K. Sekihara, S. S. Nagarajan, D. Poeppel, A. Marantz, and Y. Miyashita, "Reconstructing spatio-temporal activities of neural sources using an MEG vector beamforming technique," IEEE Trans. Biomed. Eng. 48, 760-771 (2001).
[CrossRef] [PubMed]

Moesta, K. T.

D. Grosenick, H. Wabnitz, K. T. Moesta, J. Mucke, M. M¨oller, C. Stroszczynski, J. St¨obel, B. Wassermann, P. M. Schlag, and H. Rinneberg, "Concentration and oxygen saturation of haemoglobin of 50 breast tumours determined by time-domain optical mammography," Phys. Med. Biol. 49, 1165-1181, (2004).
[CrossRef] [PubMed]

D. Grosenick, H. Wabnitz, H. Rinneberg, K. T. Moesta, and P. M. Schlag, "Development of a time-domain optical mammograph and first in vivo applications," Appl. Opt. 38, 2927-2943 (1999).
[CrossRef]

Moesta, T.

Mucke, J.

D. Grosenick, H. Wabnitz, K. T. Moesta, J. Mucke, M. M¨oller, C. Stroszczynski, J. St¨obel, B. Wassermann, P. M. Schlag, and H. Rinneberg, "Concentration and oxygen saturation of haemoglobin of 50 breast tumours determined by time-domain optical mammography," Phys. Med. Biol. 49, 1165-1181, (2004).
[CrossRef] [PubMed]

D. Grosenick, T. Moesta, H. Wabnitz, J. Mucke, C. Stroszcynski, R. Macdonald, P. Schlag, and H. Rinnerberg, "Time-domain optical mammography: Initial clinial results on detection and characterization of breast tumors," Appl. Opt. 42, 3170-3186 (2003).
[CrossRef] [PubMed]

Nagarajan, S. S.

K. Sekihara, S. S. Nagarajan, D. Poeppel, A. Marantz, and Y. Miyashita, "Reconstructing spatio-temporal activities of neural sources using an MEG vector beamforming technique," IEEE Trans. Biomed. Eng. 48, 760-771 (2001).
[CrossRef] [PubMed]

Nioka, S.

X. Intes, J. Ripoll, Y. Chen, S. Nioka, A. Yodh, and B. Chance, "In vivo continuous-wave optical breast imaging enhanced with Indocyanine Green," Med. Phys. 30, 1039-1047 (2003).
[CrossRef] [PubMed]

Ntziachristos, V.

Osterberg, U.

Papazoglou, M.

M. Papazoglou and J. L. Krolik, "High resolution adaptive beamforming for three-dimensional acoustic imaging of zooplankton," J. Acoust. Soc. Am. 100, 3621-3630 (1996).
[CrossRef]

Patterson, M. S.

M. S. Patterson, B. Chance, and B. C. Wilson, "Time resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties," J. Appl. Opt. 28, 2331-2336 (1989).
[CrossRef]

Paulsen, K.

Poeppel, D.

K. Sekihara, S. S. Nagarajan, D. Poeppel, A. Marantz, and Y. Miyashita, "Reconstructing spatio-temporal activities of neural sources using an MEG vector beamforming technique," IEEE Trans. Biomed. Eng. 48, 760-771 (2001).
[CrossRef] [PubMed]

Pogue, B.

Prewitt, J.

Rice, S. O.

S. O. Rice, "Mathematical analysis of random noise," Bell Syst. Tech. J. 23, 282-332 (1944).

Rinneberg, H.

Rinnerberg, H.

Ripoll, J.

X. Intes, J. Ripoll, Y. Chen, S. Nioka, A. Yodh, and B. Chance, "In vivo continuous-wave optical breast imaging enhanced with Indocyanine Green," Med. Phys. 30, 1039-1047 (2003).
[CrossRef] [PubMed]

Schlag, P.

Schlag, P. M.

Sekihara, K.

K. Sekihara, S. S. Nagarajan, D. Poeppel, A. Marantz, and Y. Miyashita, "Reconstructing spatio-temporal activities of neural sources using an MEG vector beamforming technique," IEEE Trans. Biomed. Eng. 48, 760-771 (2001).
[CrossRef] [PubMed]

Shah, N.

M. J. Holboke, B. J. Tromberg, X. Li, N. Shah, J. Fishkin, D. Kidney, J. Butler, B. Chance, and A.G. Yodh, "Three-dimensional diffuse optical mammography with ultrasound localization in a human subject," J. Biomed. Opt. 5, 237-247 (2000).
[CrossRef] [PubMed]

Strangman, G.

G. Strangman, D. Boas, and J. Sutton, "Non-invasive neuroimaging using near-infrared light," Biol. Psychiatry 52, 679-693 (2002).
[CrossRef] [PubMed]

Stroszcynski, C.

Sutton, J.

G. Strangman, D. Boas, and J. Sutton, "Non-invasive neuroimaging using near-infrared light," Biol. Psychiatry 52, 679-693 (2002).
[CrossRef] [PubMed]

Suzuki, A.

B. D. Van Veen, W. van Drongelen, M. Yuchtman, and A. Suzuki, "Localization of brain electrial activity via linearly constrained minimum variance spatial filtering," IEEE Trans. Biomed. Eng. 44, 867-880 (1997).
[CrossRef] [PubMed]

Svaasand, L. O.

Tromberg, B. J.

M. J. Holboke, B. J. Tromberg, X. Li, N. Shah, J. Fishkin, D. Kidney, J. Butler, B. Chance, and A.G. Yodh, "Three-dimensional diffuse optical mammography with ultrasound localization in a human subject," J. Biomed. Opt. 5, 237-247 (2000).
[CrossRef] [PubMed]

R. C. Haskell, L. O. Svaasand, T. 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]

Tsay, T. T.

van Drongelen, W.

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J. C. Ye, C. A. Bouman, K. J. Webb, and R. P. Millane, "Nonlinear multigrid algorithms for Bayesian optical diffuse tomography," IEEE Trans. Image Process. 10, 909-922 (2001).
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M. Guven, B. Yazici, X. Intes, and B. Chance, "Diffuse optical tomography with a priori anatomical information," Phys. Med. Biol. 50, 2837-2858 (2005).
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Ye, J. C.

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

Yodh, A.

X. Intes, J. Ripoll, Y. Chen, S. Nioka, A. Yodh, and B. Chance, "In vivo continuous-wave optical breast imaging enhanced with Indocyanine Green," Med. Phys. 30, 1039-1047 (2003).
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Yodh, A. G.

Yodh, A.G.

M. J. Holboke, B. J. Tromberg, X. Li, N. Shah, J. Fishkin, D. Kidney, J. Butler, B. Chance, and A.G. Yodh, "Three-dimensional diffuse optical mammography with ultrasound localization in a human subject," J. Biomed. Opt. 5, 237-247 (2000).
[CrossRef] [PubMed]

Yuchtman, M.

B. D. Van Veen, W. van Drongelen, M. Yuchtman, and A. Suzuki, "Localization of brain electrial activity via linearly constrained minimum variance spatial filtering," IEEE Trans. Biomed. Eng. 44, 867-880 (1997).
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Appl. Opt. (3)

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IEEE ASSP. Magazine (1)

B. D. Van Veen and K. M. Buckley, "Beamforming: A versatile approach to spatial filtering," IEEE ASSP. Magazine 5, 4-24 (1988).
[CrossRef]

IEEE Trans. Biomed. Eng. (2)

K. Sekihara, S. S. Nagarajan, D. Poeppel, A. Marantz, and Y. Miyashita, "Reconstructing spatio-temporal activities of neural sources using an MEG vector beamforming technique," IEEE Trans. Biomed. Eng. 48, 760-771 (2001).
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IEEE Trans. Image Process. (1)

J. C. Ye, C. A. Bouman, K. J. Webb, and R. P. Millane, "Nonlinear multigrid algorithms for Bayesian optical diffuse tomography," IEEE Trans. Image Process. 10, 909-922 (2001).
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M. S. Patterson, B. Chance, and B. C. Wilson, "Time resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties," J. Appl. Opt. 28, 2331-2336 (1989).
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J. Biomed. Opt. (1)

M. J. Holboke, B. J. Tromberg, X. Li, N. Shah, J. Fishkin, D. Kidney, J. Butler, B. Chance, and A.G. Yodh, "Three-dimensional diffuse optical mammography with ultrasound localization in a human subject," J. Biomed. Opt. 5, 237-247 (2000).
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X. Intes, J. Ripoll, Y. Chen, S. Nioka, A. Yodh, and B. Chance, "In vivo continuous-wave optical breast imaging enhanced with Indocyanine Green," Med. Phys. 30, 1039-1047 (2003).
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Figures (10)

Fig. 1.
Fig. 1.

Simulation setup. The domain is composed of a 8×8×6 cm3 cube, and two possible spherical abnormalities with radius 1 cm are inserted. Twenty-five sources (circles) are placed on the bottom surface (z = 0 cm) and twenty-five detectors (black dots) on the top (z = 6 cm). The range along the x and y axes is [-4,4] cm.

Fig. 2.
Fig. 2.

Original (a) and reconstructed (b) δμ a distributions in cm-1, assuming only one absorbing abnormality. Small images show the cross-section layers at different z values with a 0.5 cm distance.

Fig. 3.
Fig. 3.

Original (a) and reconstructed (b) δμ ś distributions in cm-1, assuming only one scattering abnormality. Small images show the cross-section layers at different z values with a 0.5 cm distance.

Fig. 4.
Fig. 4.

Original (a) and reconstructed (b) δμ a and δμ ś distributions for one abnormality that is both absorptive and scattering. Small images show the cross-section layers of each parameter at different z values at 0.5 cm intervals.

Fig. 5.
Fig. 5.

Original (a) and reconstructed (b) δμ a and δμ ś distributions, assuming two abnormalities, one of which is absorbing and the other scattering. Small images show the cross-section layers at different z values at 0.5 cm intervals.

Fig. 6.
Fig. 6.

Reconstructed δμ a and δμ ś distributions after filtering out the effect of δμ ś in the recovered δμ a images. Small images show the cross-section layers of each parameter at different z values at 0.5 cm intervals.

Fig. 7.
Fig. 7.

Reconstructed μ a distributions using SIRT with (a) and without (b) the prior information provided by the LCMV solutions. The true abnormality location is shown in Fig. 2(a). Small images show the cross-section layers of each parameter at different z values at 0.5 cm intervals.

Fig. 8.
Fig. 8.

Reconstructed δμ a distributions in cm-1, assuming only one absorbing abnormality as shown in Fig. 2(a). (a) For true δμ a = 0.05 cm-1; (b) For true δμ a = 0.1 cm-1. Small images show the cross-section layers at different z values with a 0.5 cm distance.

Fig. 9.
Fig. 9.

Reconstructed δμ a distributions in cm-1 with different noise levels, assuming only one absorbing abnormality as shown in Fig. 2(a). (a) For σ = 0.1; (b) For σ = 1. Small images show the cross-section layers at different z values with a 0.5 cm distance.

Fig. 10.
Fig. 10.

Reconstructed δμ a distributions in cm-1, assuming two spherical absorbing abnormalities with R = 0.75 cm and δμ a = 0.2 cm-1. The centers of the two spheres are at (a) [- 1.5, 0.8, 2] cm and [1.5, -0.8, 4] cm; and (b) [-1, 0.5, 2.5] cm and [1, -0.5, 3.5] cm. Small images show the cross-section layers at different z values with a 0.5 cm distance.

Equations (27)

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. D ( r ) U r ω + c μ a ( r ) U r ω + jωU r ω = c q 0 r ω ,
μ a ( r ) = μ a 0 + δ μ a ( r ) ,
D ( r ) = D 0 + δD ( r ) ,
U r r s = U 0 r r s e ϕ sc r r s ,
ϕ sc c = A c Δ ,
ϕ sc c = [ ϕ sc r s 1 r d 1 , . . . , ϕ sc r s 1 r d k , . . . , ϕ sc r s N s r d N d ] T ,
Δ = [ δ μ a ( r 1 ) , . . . , δ μ a ( r j ) , . . . , δ μ a ( r N ) Δ a , δ D ( r 1 ) , . . . , δD ( r j ) , . . . , δ D ( r N ) Δ s ] T ,
A c = [ A 111 ca A 11 N ca A 111 cs A 11 N cs A ik 1 ca A ikN ca A ik 1 cs A ikN cs A N s N d 1 ca A N s N d N ca A N s N d 1 cs A N s N d N cs ] = [ A ca A cs ] .
A ikj ca = U 0 r s i r j G ( r j r d k ) h 3 D 0 U 0 r s i r d k ,
A ikj cs = U 0 r s i r j . G ( r j r d k ) h 3 D 0 U 0 r s i r d k
ϕ sc = A Δ ,
y = ϕ sc + e ,
= σ 2 [ 0 0 0 0 ] ,
0 = [ ϕ 11 c 0 0 ϕ N d N s c ] ,
y = l = 1 L H ind { l } δ μ a ( r ind { l } ) + e ,
ϕ ̂ i = W i T y , i = 1,2 , . . . , N tot
W i T H i = { 1 , i ind { } 0 , otherwise
min W i tr C ( ϕ ̂ ) subject to W i T H i = 1 ,
W i T = [ H i T C 1 ( y ) H i ] 1 H i T C 1 ( y ) .
C ̂ ( y ) = 1 M 1 m = 1 M ( y m y - ) ( y m y - ) T ,
L W D = tr { W T CW + 2 ( W T H I ) D } .
L W D = tr { W T CW + ( W T H I ) D + D T ( H T W I ) } .
L W D = tr { ( W T + D T H T C 1 ) C ( W + C 1 HL ) L L T L T H T C 1 HL } .
W = C 1 HD .
D T H T C 1 H = I ,
D T = ( H T C 1 H ) 1 .
W T = [ H T C 1 H ] 1 H T C 1 .

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