Abstract

High-density diffuse optical tomography (HD-DOT) is an emerging technique for visualizing the internal state of biological tissues. The large number of overlapping measurement channels due to the use of high-density probe arrays permits the reconstruction of the internal optical properties, even with a reflectance-only measurement. However, accurate three-dimensional reconstruction is still a challenging problem. First, the exponentially decaying sensitivity causes a systematic depth-localization error. Second, the nature of diffusive light makes the image blurred. In this paper, we propose a three-dimensional reconstruction method that overcomes these two problems by introducing sensitivity-normalized regularization and sparsity into the hierarchical Bayesian method. Phantom experiments were performed to validate the proposed method under three conditions of probe interval: 26 mm, 18.4 mm, and 13 mm. We found that two absorbers with distances shorter than the probe interval could be discriminated under the high-density conditions of 18.4-mm and 13-mm intervals. This discrimination ability was possible even if the depths of the two absorbers were different from each other. These results show the high spatial resolution of the proposed method in both depth and horizontal directions.

© 2012 OSA

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2012

F. Lucka, S. Pursiainen, M. Burger, and C. H. Wolters, “Hierarchical Bayesian inference for the EEG inverse problem using realistic FE head models: depth localization and source separation for focal primary currents,” Neuroimage61, 1364–1382 (2012).
[CrossRef] [PubMed]

T. Aihara, Y. Takeda, K. Takeda, W. Yasuda, T. Sato, Y. Otaka, T. Hanakawa, M. Honda, M. Liu, M. Kawato, M. Sato, and R. Osu, “Cortical current source estimation from electroencephalography in combination with near-infrared spectroscopy as a hierarchical prior,” NeuroImage59, 4006–4021 (2012).
[CrossRef]

2011

Q. Fang, J. Selb, S. A. Carp, G. Boverman, E. L. Miller, D. H. Brooks, R. H. Moore, D. B. Kopans, and D. A. Boas, “Combined optical and X-ray tomosynthesis breast imaging,” Radiology258, 89–97 (2011).
[CrossRef]

C. Habermehl, S. Holtze, J. Steinbrink, S. P. Koch, H. Obrig, J. Mehnert, and C. H. Schmitz, “Somatosensory activation of two fingers can be discriminated with ultrahigh-density diffuse optical tomography,” NeuroImage59, 3201–3211 (2011).
[CrossRef] [PubMed]

S. Okawa, Y. Hoshi, and Y. Yamada, “Improvement of image quality of time-domain diffuse optical tomography with lp sparsity regularization,” Biomed. Opt. Express2, 3334–3348 (2011).
[CrossRef] [PubMed]

2010

2009

2008

D. Wipf and S. Nagarajan, “A new view of automatic relevance determination,” Adv. Neural Inf. Process. Syst.20, 1625–1632 (2008).

H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using NIRFAST: Algorithm for numerical model and image reconstruction,” Commun. Numer. Methods Eng.25, 711–732 (2008).
[CrossRef] [PubMed]

2007

2005

D. A. Boas and A. M. Dale, “Simulation study of magnetic resonance imaging-guided cortically constrained diffuse optical tomography of human brain function,” Appl. Opt.44, 1957–1968 (2005).
[CrossRef] [PubMed]

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol.50, R1–R43 (2005).
[CrossRef] [PubMed]

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

M. Sato, T. Yoshioka, S. Kajihara, K. Toyama, N. Goda, K. Doya, and M. Kawato, “Hierarchical Bayesian estimation for MEG inverse problem,” NeuroImage23, 806–826 (2004).
[CrossRef] [PubMed]

2003

J. P. Culver, T. Durduran, D. Furuya, C. Cheung, J. H. Greenberg, and A. G. Yodh, “Diffuse optical tomography of cerebral blood flow, oxygenation, and metabolism in rat during focal ischemia,” J. Cereb. Blood Flow Metab.23, 911–924 (2003).
[CrossRef] [PubMed]

J. P. Culver, R. Choe, M. J. Holboke, L. Zubkov, T. Durduran, A. Slemp, V. Ntziachristos, B. Chance, and A. G. Yodh, “Three-dimensional diffuse optical tomography in the parallel plane transmission geometry: evaluation of a hybrid frequency domain/continuous wave clinical system for breast imaging,” Med. Phys.30, 235–247 (2003).
[CrossRef] [PubMed]

A. Li, E. L. Miller, M. E. Kilmer, T. J. Brukilacchio, T. Chaves, J. Stott, Q. Zhang, T. Wu, M. Chorlton, R. H. Moore, D. B. Kopans, and D. A. Boas, “Tomographic optical breast imaging guided by three-dimensional mammography,” Appl. Opt.42, 5181–5190 (2003).
[CrossRef] [PubMed]

2002

A. C. Faul and M. E. Tipping, “Analysis of sparse Bayesian learning,” Adv. Neural Inf. Process. Syst.14, 383–389 (2002).

F. Gao, H. Zhao, and Y Yamada, “Improvement of image quality in diffuse optical tomography by use of full time-resolved data,” Appl. Opt.41, 778–791 (2002).
[CrossRef] [PubMed]

2001

M. Sato, “Online model selection based on the variational Bayes,” Neural Comput.13, 1649–1681 (2001).
[CrossRef]

1999

1997

1995

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

A. Maki, Y. Yamashita, Y. Ito, E. Watanabe, Y. Mayanagi, and H. Koizumi, “Spatial and temporal analysis of human motor activity using noninvasive NIR topography,” Med. Phys.22, 1997–2005 (1995).
[CrossRef] [PubMed]

1994

1993

A. Villringer, J. Planck, C. Hock, L. Schleinkofer, and U. Dirnagl, “Near infrared spectroscopy (NIRS): a new tool to study hemodynamic changes during activation of brain function in human adults,” Neurosci. Lett.154, 101–104 (1993).
[CrossRef] [PubMed]

Y. Hoshi and M. Tamura, “Detection of dynamic changes in cerebral oxygenation coupled to neuronal function during mental work in man,” Neurosci. Lett.150, 5–8 (1993).
[CrossRef] [PubMed]

Abdelnour, F.

Adibi, A.

Aihara, T.

T. Aihara, Y. Takeda, K. Takeda, W. Yasuda, T. Sato, Y. Otaka, T. Hanakawa, M. Honda, M. Liu, M. Kawato, M. Sato, and R. Osu, “Cortical current source estimation from electroencephalography in combination with near-infrared spectroscopy as a hierarchical prior,” NeuroImage59, 4006–4021 (2012).
[CrossRef]

Akaike, H.

H. Akaike, “Likelihood and the Bayes procedure,” in Bayesian Statistics, J. M. Bernardo, M. H. De Groot, D. V. Lindley, and A. F. M. Smith, eds. (Univ. Press, Valencia, 1980), 143–166.

Arridge, S. R.

S. R. Arridge and J. C. Schotland, “Optical tomography: forward and inverse problems,” Inverse Probl.25, 123010 (2009).
[CrossRef]

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol.50, R1–R43 (2005).
[CrossRef] [PubMed]

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

E. Okada, M. Firbank, M. Schweiger, S. R. Arridge, M. Cope, and D. T. Delpy, “Theoretical and experimental investigation of near-infrared light propagation in a model of the adult head,” Appl. Opt.36, 21–31 (1997).
[CrossRef] [PubMed]

Attias, H.

H. Attias, “Inferring parameters and structure of latent variable models by variational Bayes,” Proc. 15th Conf. on Uncertainty in Artificial Intelligence, Morgan Kaufmann, 21–30 (1999).

Bishop, C. M.

C. M. Bishop, Pattern Recognition and Machine Learning (Springer, New York, 2006).

Boas, D. A.

Boverman, G.

Q. Fang, J. Selb, S. A. Carp, G. Boverman, E. L. Miller, D. H. Brooks, R. H. Moore, D. B. Kopans, and D. A. Boas, “Combined optical and X-ray tomosynthesis breast imaging,” Radiology258, 89–97 (2011).
[CrossRef]

Brooks, D. H.

Q. Fang, J. Selb, S. A. Carp, G. Boverman, E. L. Miller, D. H. Brooks, R. H. Moore, D. B. Kopans, and D. A. Boas, “Combined optical and X-ray tomosynthesis breast imaging,” Radiology258, 89–97 (2011).
[CrossRef]

Brukilacchio, T. J.

Burger, M.

F. Lucka, S. Pursiainen, M. Burger, and C. H. Wolters, “Hierarchical Bayesian inference for the EEG inverse problem using realistic FE head models: depth localization and source separation for focal primary currents,” Neuroimage61, 1364–1382 (2012).
[CrossRef] [PubMed]

Cao, N.

Carp, S. A.

Q. Fang, J. Selb, S. A. Carp, G. Boverman, E. L. Miller, D. H. Brooks, R. H. Moore, D. B. Kopans, and D. A. Boas, “Combined optical and X-ray tomosynthesis breast imaging,” Radiology258, 89–97 (2011).
[CrossRef]

Carpenter, C. M.

H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using NIRFAST: Algorithm for numerical model and image reconstruction,” Commun. Numer. Methods Eng.25, 711–732 (2008).
[CrossRef] [PubMed]

Chan, T. F.

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]

J. P. Culver, R. Choe, M. J. Holboke, L. Zubkov, T. Durduran, A. Slemp, V. Ntziachristos, B. Chance, and A. G. Yodh, “Three-dimensional diffuse optical tomography in the parallel plane transmission geometry: evaluation of a hybrid frequency domain/continuous wave clinical system for breast imaging,” Med. Phys.30, 235–247 (2003).
[CrossRef] [PubMed]

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

Chatziioannou, A. F.

Chaves, T.

Cheung, C.

J. P. Culver, T. Durduran, D. Furuya, C. Cheung, J. H. Greenberg, and A. G. Yodh, “Diffuse optical tomography of cerebral blood flow, oxygenation, and metabolism in rat during focal ischemia,” J. Cereb. Blood Flow Metab.23, 911–924 (2003).
[CrossRef] [PubMed]

Choe, R.

J. P. Culver, R. Choe, M. J. Holboke, L. Zubkov, T. Durduran, A. Slemp, V. Ntziachristos, B. Chance, and A. G. Yodh, “Three-dimensional diffuse optical tomography in the parallel plane transmission geometry: evaluation of a hybrid frequency domain/continuous wave clinical system for breast imaging,” Med. Phys.30, 235–247 (2003).
[CrossRef] [PubMed]

Chorlton, M.

Cope, M.

Culver, J. P.

H. Dehghani, B. R. White, B. W. Zeff, A. Tizzard, and J. P. Culver, “Depth sensitivity and image reconstruction analysis of dense imaging arrays for mapping brain function with diffuse optical tomography,” Appl. Opt.48, D137–D143 (2009).
[CrossRef] [PubMed]

B. W. Zeff, B. R. White, H. Dehghani, B. L. Schlaggar, and J. P. Culver, “Retinotopic mapping of adult human visual cortex with high-density diffuse optical tomography,” Proc. Natl. Acad. Sci. U.S.A.104, 12169–12174 (2007).
[CrossRef] [PubMed]

J. P. Culver, R. Choe, M. J. Holboke, L. Zubkov, T. Durduran, A. Slemp, V. Ntziachristos, B. Chance, and A. G. Yodh, “Three-dimensional diffuse optical tomography in the parallel plane transmission geometry: evaluation of a hybrid frequency domain/continuous wave clinical system for breast imaging,” Med. Phys.30, 235–247 (2003).
[CrossRef] [PubMed]

J. P. Culver, T. Durduran, D. Furuya, C. Cheung, J. H. Greenberg, and A. G. Yodh, “Diffuse optical tomography of cerebral blood flow, oxygenation, and metabolism in rat during focal ischemia,” J. Cereb. Blood Flow Metab.23, 911–924 (2003).
[CrossRef] [PubMed]

Dale, A. M.

Davis, S. C.

H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using NIRFAST: Algorithm for numerical model and image reconstruction,” Commun. Numer. Methods Eng.25, 711–732 (2008).
[CrossRef] [PubMed]

Dehghani, H.

H. Dehghani, B. R. White, B. W. Zeff, A. Tizzard, and J. P. Culver, “Depth sensitivity and image reconstruction analysis of dense imaging arrays for mapping brain function with diffuse optical tomography,” Appl. Opt.48, D137–D143 (2009).
[CrossRef] [PubMed]

H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using NIRFAST: Algorithm for numerical model and image reconstruction,” Commun. Numer. Methods Eng.25, 711–732 (2008).
[CrossRef] [PubMed]

B. W. Zeff, B. R. White, H. Dehghani, B. L. Schlaggar, and J. P. Culver, “Retinotopic mapping of adult human visual cortex with high-density diffuse optical tomography,” Proc. Natl. Acad. Sci. U.S.A.104, 12169–12174 (2007).
[CrossRef] [PubMed]

Delpy, D. T.

Dhamne, S.

H. Niu, Z. J. Lin, F. Tian, S. Dhamne, and H. Liu, “Comprehensive investigation of three-dimensional diffuse optical tomography with depth compensation algorithm,” J. Biomed. Opt.15, 046005 (2010).
[CrossRef] [PubMed]

Dirnagl, U.

A. Villringer, J. Planck, C. Hock, L. Schleinkofer, and U. Dirnagl, “Near infrared spectroscopy (NIRS): a new tool to study hemodynamic changes during activation of brain function in human adults,” Neurosci. Lett.154, 101–104 (1993).
[CrossRef] [PubMed]

Douraghy, A.

Doya, K.

M. Sato, T. Yoshioka, S. Kajihara, K. Toyama, N. Goda, K. Doya, and M. Kawato, “Hierarchical Bayesian estimation for MEG inverse problem,” NeuroImage23, 806–826 (2004).
[CrossRef] [PubMed]

Durduran, T.

J. P. Culver, R. Choe, M. J. Holboke, L. Zubkov, T. Durduran, A. Slemp, V. Ntziachristos, B. Chance, and A. G. Yodh, “Three-dimensional diffuse optical tomography in the parallel plane transmission geometry: evaluation of a hybrid frequency domain/continuous wave clinical system for breast imaging,” Med. Phys.30, 235–247 (2003).
[CrossRef] [PubMed]

J. P. Culver, T. Durduran, D. Furuya, C. Cheung, J. H. Greenberg, and A. G. Yodh, “Diffuse optical tomography of cerebral blood flow, oxygenation, and metabolism in rat during focal ischemia,” J. Cereb. Blood Flow Metab.23, 911–924 (2003).
[CrossRef] [PubMed]

Eames, M. E.

H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using NIRFAST: Algorithm for numerical model and image reconstruction,” Commun. Numer. Methods Eng.25, 711–732 (2008).
[CrossRef] [PubMed]

Eftekhar, A. A.

Fang, Q.

Q. Fang, J. Selb, S. A. Carp, G. Boverman, E. L. Miller, D. H. Brooks, R. H. Moore, D. B. Kopans, and D. A. Boas, “Combined optical and X-ray tomosynthesis breast imaging,” Radiology258, 89–97 (2011).
[CrossRef]

Q. Fang and D. A. Boas, “Monte Carlo simulation of photon migration in 3D turbid media accelerated by graphics processing units,” Opt. Express17, 20178–20190 (2009).
[CrossRef] [PubMed]

Faul, A. C.

A. C. Faul and M. E. Tipping, “Analysis of sparse Bayesian learning,” Adv. Neural Inf. Process. Syst.14, 383–389 (2002).

Feng, T. C.

Firbank, M.

Furuya, D.

J. P. Culver, T. Durduran, D. Furuya, C. Cheung, J. H. Greenberg, and A. G. Yodh, “Diffuse optical tomography of cerebral blood flow, oxygenation, and metabolism in rat during focal ischemia,” J. Cereb. Blood Flow Metab.23, 911–924 (2003).
[CrossRef] [PubMed]

Gao, F.

Genovese, C.

Gibson, A. P.

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol.50, R1–R43 (2005).
[CrossRef] [PubMed]

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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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C. Habermehl, S. Holtze, J. Steinbrink, S. P. Koch, H. Obrig, J. Mehnert, and C. H. Schmitz, “Somatosensory activation of two fingers can be discriminated with ultrahigh-density diffuse optical tomography,” NeuroImage59, 3201–3211 (2011).
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C. Habermehl, S. Holtze, J. Steinbrink, S. P. Koch, H. Obrig, J. Mehnert, and C. H. Schmitz, “Somatosensory activation of two fingers can be discriminated with ultrahigh-density diffuse optical tomography,” NeuroImage59, 3201–3211 (2011).
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A. Li, E. L. Miller, M. E. Kilmer, T. J. Brukilacchio, T. Chaves, J. Stott, Q. Zhang, T. Wu, M. Chorlton, R. H. Moore, D. B. Kopans, and D. A. Boas, “Tomographic optical breast imaging guided by three-dimensional mammography,” Appl. Opt.42, 5181–5190 (2003).
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T. Aihara, Y. Takeda, K. Takeda, W. Yasuda, T. Sato, Y. Otaka, T. Hanakawa, M. Honda, M. Liu, M. Kawato, M. Sato, and R. Osu, “Cortical current source estimation from electroencephalography in combination with near-infrared spectroscopy as a hierarchical prior,” NeuroImage59, 4006–4021 (2012).
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B. W. Zeff, B. R. White, H. Dehghani, B. L. Schlaggar, and J. P. Culver, “Retinotopic mapping of adult human visual cortex with high-density diffuse optical tomography,” Proc. Natl. Acad. Sci. U.S.A.104, 12169–12174 (2007).
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C. Habermehl, S. Holtze, J. Steinbrink, S. P. Koch, H. Obrig, J. Mehnert, and C. H. Schmitz, “Somatosensory activation of two fingers can be discriminated with ultrahigh-density diffuse optical tomography,” NeuroImage59, 3201–3211 (2011).
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C. Habermehl, S. Holtze, J. Steinbrink, S. P. Koch, H. Obrig, J. Mehnert, and C. H. Schmitz, “Somatosensory activation of two fingers can be discriminated with ultrahigh-density diffuse optical tomography,” NeuroImage59, 3201–3211 (2011).
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T. Aihara, Y. Takeda, K. Takeda, W. Yasuda, T. Sato, Y. Otaka, T. Hanakawa, M. Honda, M. Liu, M. Kawato, M. Sato, and R. Osu, “Cortical current source estimation from electroencephalography in combination with near-infrared spectroscopy as a hierarchical prior,” NeuroImage59, 4006–4021 (2012).
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Y. Hoshi and M. Tamura, “Detection of dynamic changes in cerebral oxygenation coupled to neuronal function during mental work in man,” Neurosci. Lett.150, 5–8 (1993).
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H. Niu, Z. J. Lin, F. Tian, S. Dhamne, and H. Liu, “Comprehensive investigation of three-dimensional diffuse optical tomography with depth compensation algorithm,” J. Biomed. Opt.15, 046005 (2010).
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M. Sato, T. Yoshioka, S. Kajihara, K. Toyama, N. Goda, K. Doya, and M. Kawato, “Hierarchical Bayesian estimation for MEG inverse problem,” NeuroImage23, 806–826 (2004).
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Tsay, T. T.

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Figures (5)

Fig. 1
Fig. 1

Setup of liquid phantom experiment. (a) Schematic of the phantom. (b) Experimental device. (c) Source and detector probes are on the bottom of the tank, z = 0. (d) The xy positions of the absorber in measurement: (i) center (0, 0), (ii) midpoint ( l 2, 0), (iii) source ( l 2, l 2), (iv) detector ( l 2, + l 2), (v) second absorber position ( l 2 + d , 0). Here, l denotes the probe interval and d denotes the horizontal distance between the centers of the two absorbers. The red circle represents the source and the blue square represents the detector.

Fig. 2
Fig. 2

Example of 3D reconstruction image. The three-dimensional image (tomography) is represented in gray scale and the xy maximum intensity projection (topography) is represented in green scale on the z = 0 surface. (a) True distribution of the absorption change. The positions of the centers of the two absorbers are (−9.2, 0, 10) [mm] and (3.3, 0, 15) [mm]. The red cylinder represents the source and the blue square prism represents the detector. (b) Solution of the Tikhonov regularization with spatially uniform regularization. (c) Solution of the Tikhonov regularization with sensitivity-normalized regularization. (d) Solution of the hierarchical Bayesian estimation method using the result of (c) as an initial value.

Fig. 3
Fig. 3

Estimation results of one-absorber experiment. (a) Estimation results under various depth and probe-interval conditions. From left, results under the depth conditions of the absorber: z = 7.5 mm, 10 mm, 12.5 mm, 15 mm, 17.5 mm, 20 mm, 22.5 mm. From top, results under three conditions of the probe interval: l = 13 mm, 18.4 mm, 26 mm. The white circle represents the true position of the absorption change. The estimation result is surrounded by a bold frame if the positional error was within one voxel (2.5 mm) in each x, y, z direction and the maximum estimation value was greater than 0.025mm−1. (b) The projection region is the central 7.5-mm-thick layer including the true distribution of the absorption change. The estimation images of (a) are obtained by projecting the maximum estimation value within the projection region in the y direction onto the xz space. (c) Estimated depth as a function of the true depth.

Fig. 4
Fig. 4

Estimation results for the 18.4-mm probe-interval condition. From top, the results under four conditions of horizontal distance: d = 17.5 mm, 15 mm, 12.5 mm, 10 mm. From left, the results under four conditions of depth: z = 12.5 mm, 15 mm, 17.5 mm, 20 mm. The white circles represent the true positions of the absorption change. The estimation image is surrounded by a bold frame if the positional errors of both absorbers were within one voxel (2.5 mm) in each x, y, z direction and both maximum estimation values were greater than 0.025mm−1.

Fig. 5
Fig. 5

Estimation results for the 18.4-mm probe-interval condition. From top, the results under four conditions of horizontal distance: d = 17.5 mm, 15 mm, 12.5 mm, 10 mm. From left, the results under four conditions of depth: z = 7.5&12.5 mm, 10&15 mm, 12.5&17.5 mm, 15&20 mm. The white circles represent the true positions of the absorption change. The estimation image is surrounded by a bold frame if the positional errors of both absorbers were within one voxel (2.5 mm) in each x, y, z direction and both maximum estimation values were greater than 0.025mm−1.

Tables (3)

Tables Icon

Table 1 Depth limit of spherical absorber’s position where accurate estimation is possible.

Tables Icon

Table 2 Depth limit where the two absorbers are accurately estimated.

Tables Icon

Table 3 Depth limit where two absorbers at different depths are accurately estimated.

Equations (28)

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ϕ pert ( r s , r d ) = d r Φ 0 ( r s , r ) Φ 0 ( r , r d ) Φ 0 ( r s , r d ) δ μ a ( r ) ,
C ( x t ; λ ) = Σ y 1 2 ( y t A x t ) 2 2 + λ x t 2 2 ,
C D ( x t ; λ ) = Σ y 1 2 ( y t A x t ) 2 2 + λ D 1 2 x t 2 2 ,
D = diag ( ρ + β 1 N ) ,
ρ i = ( A T Σ y 1 A ) i i ,
x ^ t , D = ( A T Σ y 1 A + λ D ) 1 A T Σ y 1 y t .
P ( Y | X , σ ) = t = 1 T N ( y t ; A x t , σ 1 Σ y ) .
P ( σ ) σ 1 .
P ( Z | λ ) = t = 1 T N ( z t ; 0 , Λ 1 ) ,
P ( X | λ ) = t = 1 T N ( x t ; 0 , W Λ 1 W T ) .
P ( λ ) = i = 1 N Γ ( λ i ; λ ¯ 0 i , γ 0 i ) ,
P ( X , λ , σ | Y ) = P ( Y , X , λ , σ ) P ( Y ) = P ( Y | X , σ ) P ( X | λ ) P ( λ ) P ( σ ) d X d λ d σ P ( Y | X , σ ) P ( X | λ ) P ( λ ) P ( σ ) .
F ( Q ) = d X d λ d σ Q ( X , λ , σ ) log P ( Y , X , λ , σ ) Q ( X , λ , σ ) , = log P ( Y ) KL [ Q ( X , λ , σ ) | | P ( X , λ , σ | Y ) ] ,
Q ( X , λ , σ ) = Q X ( X ) Q λ ( λ , σ ) .
log Q X ( X ) = d λ d σ Q λ ( λ , σ ) log P ( Y , X , λ , σ ) = t = 1 T log N ( x t ; x ¯ t , Σ x ) ,
log Q λ ( λ , σ ) = d X Q X ( X ) log P ( Y , X , λ , σ ) = i = 1 N log Γ ( λ i ; λ ¯ i , γ i ) + log Γ ( σ ; σ ¯ , γ σ ) ,
F ( Q ) = t = 1 T F t ( x ¯ t , λ ¯ , σ ¯ ) + f ( λ ¯ , σ ¯ , Σ x ) ,
F t ( x ¯ t , λ ¯ , σ ¯ ) = 1 2 [ σ ¯ Σ y 1 2 ( y t A x ¯ t ) 2 2 + Λ ¯ 1 2 W 1 x ¯ t 2 2 ] .
F ( Q ) = 1 2 t = 1 T [ σ ¯ Σ y 1 2 ( y t A x ¯ t ) 2 2 + Λ ¯ 1 2 W 1 x ¯ t 2 2 ] + f ( λ ¯ , σ ¯ , Σ x ) ,
f ( λ ¯ , σ ¯ , Σ x ) = i = 1 N γ 0 i [ log ( λ ¯ i / λ 0 i ) ( λ ¯ i / λ 0 i ) + 1 ] + T 2 [ log | Σ x | + log | Λ ¯ | + log | σ ¯ Σ y 1 | ] T 2 [ tr ( Σ x ( σ ¯ A T Σ y 1 A + W 1 T Λ ¯ W 1 ) ) ] + const .
Σ : = σ ¯ 1 + Σ y + AW Λ ¯ 1 W T A T ,
h : = diag [ Λ ¯ 1 W T A T Σ 1 Y Y T Σ 1 AW + T ( I N Λ ¯ 1 W T A T Σ 1 AW ) ] ,
S : = tr [ σ ¯ 1 Σ y Σ 1 Y Y T Σ 1 + T Λ ¯ 1 W T A T Σ 1 AW ] .
γ i : = γ 0 i + T 2 ,
γ i λ ¯ i 1 : = γ 0 i λ ¯ 0 i 1 + 1 2 h i λ ¯ i 1 ,
γ σ : = 1 2 M T ,
γ σ σ ¯ 1 : = 1 2 S σ ¯ 1 .
X ^ B = W Λ ¯ 1 W T A T Σ 1 Y .

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