Abstract

With the emergence of diffuse optical tomography (DOT) as a non-invasive imaging modality, there is a requirement to evaluate the performance of the developed DOT systems on clinically relevant tasks. One such important task is the detection of high-absorption signals in the tissue. To investigate signal detectability in DOT systems for system optimization, an appropriate approach is to use the Bayesian ideal observer, but this observer is computationally very intensive. It has been shown that the Fisher information can be used as a surrogate figure of merit (SFoM) that approximates the ideal observer performance. In this paper, we present a theoretical framework to use the Fisher information for investigating signal detectability in DOT systems. The usage of Fisher information requires evaluating the gradient of the photon distribution function with respect to the absorption coefficients. We derive the expressions to compute the gradient of the photon distribution function with respect to the scattering and absorption coefficients. We find that computing these gradients simply requires executing the radiative transport equation with a different source term. We then demonstrate the application of the SFoM to investigate signal detectability in DOT by performing various simulation studies, which help to validate the proposed framework and also present some insights on signal detectability in DOT.

© 2013 OSA

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2012 (4)

Y. Zhan, A. T. Eggebrecht, J. P. Culver, and H. Dehghani, “Image quality analysis of high-density diffuse optical tomography incorporating a subject-specific head model,” Front Neuroenergetics4, 6 (2012).
[CrossRef] [PubMed]

A. K. Jha, M. A. Kupinski, T. Masumura, E. Clarkson, A. A. Maslov, and H. H. Barrett, “Simulating photon-transport in uniform media using the radiative transfer equation: A study using the Neumann-series approach,” J. Opt. Soc. Amer. A29, 1741–1757 (2012).
[CrossRef]

V. C. Kavuri, Z. J. Lin, F. Tian, and H. Liu, “Sparsity enhanced spatial resolution and depth localization in diffuse optical tomography,” Biomed. Opt. Express3, 943–957 (2012).
[CrossRef] [PubMed]

A. K. Jha, M. A. Kupinski, H. H. Barrett, E. Clarkson, and J. H. Hartman, “Three-dimensional Neumann-series approach to model light transport in nonuniform media,” J. Opt. Soc. Am. A29, 1885–1899 (2012).
[CrossRef]

2011 (2)

D. Kang and M. A. Kupinski, “Signal detectability in diffusive media using phased arrays in conjunction with detector arrays,” Opt. Express19, 12261–12274 (2011).
[CrossRef] [PubMed]

N. Biswal, Y. Xu, and Q. Zhu, “Imaging tumor oxyhemoglobin and deoxyhemoglobin concentrations with ultrasound-guided diffuse optical tomography.” Tech. Cancer Res. Treatment10, 417 (2011).

2010 (2)

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]

E. Clarkson and F. Shen, “Fisher information and surrogate figures of merit for the task-based assessment of image quality,” J. Opt. Soc. Am. A27, 2313–2326 (2010).
[CrossRef]

2009 (7)

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]

M. Chu, K. Vishwanath, A. D. Klose, and H. Dehghani, “Light transport in biological tissue using three-dimensional frequency-domain simplified spherical harmonics equations,” Phys. Med. Biol.54, 2493–2509 (2009).
[CrossRef] [PubMed]

S. van de Ven, S. Elias, A. Wiethoff, M. van der Voort, A. Leproux, T. Nielsen, B. Brendel, L. Bakker, M. van der Mark, W. Mali, and P. Luijten, “Diffuse optical tomography of the breast: initial validation in benign cysts,” Mol. Imaging Biol.11, 64–70 (2009).
[CrossRef]

R. Ziegler, B. Brendel, A. Schipper, R. Harbers, M. v. Beek, H. Rinneberg, and T. Nielsen, “Investigation of detection limits for diffuse optical tomography systems: I. Theory and experiment,” Phys. Med. Biol.54, 399–412 (2009).
[CrossRef]

R. Ziegler, B. Brendel, H. Rinneberg, and T. Nielsen, “Investigation of detection limits for diffuse optical tomography systems: II. Analysis of slab and cup geometry for breast imaging,” Phys. Med. Biol.54, 413–431 (2009).
[CrossRef]

A. Gibson and H. Dehghani, “Diffuse optical imaging,” Phil. Tran. A. Math. Phys. Eng. Sci.367, 3055–3072 (2009).
[CrossRef]

H. Dehghani, S. Srinivasan, B. W. Pogue, and A. Gibson, “Numerical modelling and image reconstruction in diffuse optical tomography,” Phil. Trans. Royal Soc. A367, 3073–3093 (2009).
[CrossRef]

2008 (1)

2007 (1)

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. Nat. Acad. Sciences104, 12169–12174 (2007).
[CrossRef]

2006 (6)

S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “In vivo hemoglobin and water concentrations, oxygen saturation, and scattering estimates from near-infrared breast tomography using spectral reconstruction,” Acad. Radiol.13, 195–202 (2006).
[CrossRef] [PubMed]

T. Austin, A. P. Gibson, G. Branco, R. M. Yusof, S. R. Arridge, J. H. Meek, J. S. Wyatt, D. T. Delpy, and J. C. Hebden, “Three dimensional optical imaging of blood volume and oxygenation in the neonatal brain,” Neuroimage31, 1426–1433 (2006).
[CrossRef] [PubMed]

B. W. Pogue, S. C. Davis, X. Song, B. A. Brooksby, H. Dehghani, and K. D. Paulsen, “Image analysis methods for diffuse optical tomography,” J. Biomed. Opt.11, 33001 (2006).
[CrossRef] [PubMed]

G. Gulsen, O. Birgul, M. B. Unlu, R. Shafiiha, and O. Nalcioglu, “Combined diffuse optical tomography (DOT) and MRI system for cancer imaging in small animals,” Tech. Cancer Res. Treatment5, 351–363 (2006).

A. Klose and E. Larsen, “Light transport in biological tissue based on the simplified spherical harmonics equations,” J. Comput. Phys.220, 441–470 (2006).
[CrossRef]

F. Shen and E. Clarkson, “Using Fisher information to approximate ideal-observer performance on detection tasks for lumpy-background images,” J. Opt. Soc. Am. A23, 2406–2414 (2006).
[CrossRef]

2005 (3)

T. Tarvainen, M. Vauhkonen, V. Kolehmainen, and J. P. Kaipio, “Hybrid radiative-transfer-diffusion model for optical tomography,” Appl. Opt.44, 876–886 (2005).
[CrossRef] [PubMed]

A. H. Hielscher, “Optical tomographic imaging of small animals,” Curr. Opinion in Biotech.16, 79–88 (2005).
[CrossRef]

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol.50, 1–43 (2005).
[CrossRef]

2004 (3)

A. H. Hielscher, A. D. Klose, A. K. Scheel, B. Moa-Anderson, M. Backhaus, U. Netz, and J. Beuthan, “Sagittal laser optical tomography for imaging of rheumatoid finger joints,” Phys. Med. Biol.49, 1147–1163 (2004).
[CrossRef] [PubMed]

S. P. Morgan, “Detection performance of a diffusive wave phased array,” Appl. Opt.43, 2071–2078 (2004).
[CrossRef] [PubMed]

E. Aydin, C. de Oliveira, and A. Goddard, “A finite element-spherical harmonics radiation transport model for photon migration in turbid media,” J. Quant. Spectr. Rad. Trans.84, 247–260 (2004).
[CrossRef]

2003 (5)

2001 (2)

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, 57–75 (2001).
[CrossRef]

Y. Chen, C. Mu, X. Intes, and B. Chance, “Signal-to-noise analysis for detection sensitivity of small absorbing heterogeneity in turbid media with single-source and dual-interfering-source,” Opt. Express9, 212–224 (2001).
[CrossRef] [PubMed]

2000 (2)

1999 (1)

Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Weitz, and P. Sheng, “Wave transport in random media: the ballistic to diffusive transition,” Phys. Rev. E60, 4843–4850 (1999).
[CrossRef]

1998 (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]

1941 (1)

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J.93, 70–83 (1941).
[CrossRef]

Alcouffe, R. E.

A. H. Hielscher and R. E. Alcouffe, “Discrete-ordinate transport simulations of light propagation in highly forward scattering heterogeneous media,” in “Advances in Optical Imaging and Photon Migration,” (Optical Society of America, 1998), p. ATuC2.

Arridge, S. R.

T. Austin, A. P. Gibson, G. Branco, R. M. Yusof, S. R. Arridge, J. H. Meek, J. S. Wyatt, D. T. Delpy, and J. C. Hebden, “Three dimensional optical imaging of blood volume and oxygenation in the neonatal brain,” Neuroimage31, 1426–1433 (2006).
[CrossRef] [PubMed]

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol.50, 1–43 (2005).
[CrossRef]

S. R. Arridge and W. R. B. Lionheart, “Nonuniqueness in diffusion-based optical tomography,” Opt. Lett.23, 882–884 (1998).
[CrossRef]

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.

T. Austin, A. P. Gibson, G. Branco, R. M. Yusof, S. R. Arridge, J. H. Meek, J. S. Wyatt, D. T. Delpy, and J. C. Hebden, “Three dimensional optical imaging of blood volume and oxygenation in the neonatal brain,” Neuroimage31, 1426–1433 (2006).
[CrossRef] [PubMed]

Aydin, E.

E. Aydin, C. de Oliveira, and A. Goddard, “A finite element-spherical harmonics radiation transport model for photon migration in turbid media,” J. Quant. Spectr. Rad. Trans.84, 247–260 (2004).
[CrossRef]

Backhaus, M.

A. H. Hielscher, A. D. Klose, A. K. Scheel, B. Moa-Anderson, M. Backhaus, U. Netz, and J. Beuthan, “Sagittal laser optical tomography for imaging of rheumatoid finger joints,” Phys. Med. Biol.49, 1147–1163 (2004).
[CrossRef] [PubMed]

Bakker, L.

S. van de Ven, S. Elias, A. Wiethoff, M. van der Voort, A. Leproux, T. Nielsen, B. Brendel, L. Bakker, M. van der Mark, W. Mali, and P. Luijten, “Diffuse optical tomography of the breast: initial validation in benign cysts,” Mol. Imaging Biol.11, 64–70 (2009).
[CrossRef]

Barbieri, B.

Barrett, H. H.

A. K. Jha, M. A. Kupinski, T. Masumura, E. Clarkson, A. A. Maslov, and H. H. Barrett, “Simulating photon-transport in uniform media using the radiative transfer equation: A study using the Neumann-series approach,” J. Opt. Soc. Amer. A29, 1741–1757 (2012).
[CrossRef]

A. K. Jha, M. A. Kupinski, H. H. Barrett, E. Clarkson, and J. H. Hartman, “Three-dimensional Neumann-series approach to model light transport in nonuniform media,” J. Opt. Soc. Am. A29, 1885–1899 (2012).
[CrossRef]

H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley, 2004), 1st ed.

Beek, M. v.

R. Ziegler, B. Brendel, A. Schipper, R. Harbers, M. v. Beek, H. Rinneberg, and T. Nielsen, “Investigation of detection limits for diffuse optical tomography systems: I. Theory and experiment,” Phys. Med. Biol.54, 399–412 (2009).
[CrossRef]

Beuthan, J.

A. H. Hielscher, A. D. Klose, A. K. Scheel, B. Moa-Anderson, M. Backhaus, U. Netz, and J. Beuthan, “Sagittal laser optical tomography for imaging of rheumatoid finger joints,” Phys. Med. Biol.49, 1147–1163 (2004).
[CrossRef] [PubMed]

Birgul, O.

G. Gulsen, O. Birgul, M. B. Unlu, R. Shafiiha, and O. Nalcioglu, “Combined diffuse optical tomography (DOT) and MRI system for cancer imaging in small animals,” Tech. Cancer Res. Treatment5, 351–363 (2006).

Biswal, N.

N. Biswal, Y. Xu, and Q. Zhu, “Imaging tumor oxyhemoglobin and deoxyhemoglobin concentrations with ultrasound-guided diffuse optical tomography.” Tech. Cancer Res. Treatment10, 417 (2011).

Boas, D. A.

Branco, G.

T. Austin, A. P. Gibson, G. Branco, R. M. Yusof, S. R. Arridge, J. H. Meek, J. S. Wyatt, D. T. Delpy, and J. C. Hebden, “Three dimensional optical imaging of blood volume and oxygenation in the neonatal brain,” Neuroimage31, 1426–1433 (2006).
[CrossRef] [PubMed]

Brendel, B.

R. Ziegler, B. Brendel, H. Rinneberg, and T. Nielsen, “Investigation of detection limits for diffuse optical tomography systems: II. Analysis of slab and cup geometry for breast imaging,” Phys. Med. Biol.54, 413–431 (2009).
[CrossRef]

S. van de Ven, S. Elias, A. Wiethoff, M. van der Voort, A. Leproux, T. Nielsen, B. Brendel, L. Bakker, M. van der Mark, W. Mali, and P. Luijten, “Diffuse optical tomography of the breast: initial validation in benign cysts,” Mol. Imaging Biol.11, 64–70 (2009).
[CrossRef]

R. Ziegler, B. Brendel, A. Schipper, R. Harbers, M. v. Beek, H. Rinneberg, and T. Nielsen, “Investigation of detection limits for diffuse optical tomography systems: I. Theory and experiment,” Phys. Med. Biol.54, 399–412 (2009).
[CrossRef]

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, 57–75 (2001).
[CrossRef]

Brooksby, B. A.

B. W. Pogue, S. C. Davis, X. Song, B. A. Brooksby, H. Dehghani, and K. D. Paulsen, “Image analysis methods for diffuse optical tomography,” J. Biomed. Opt.11, 33001 (2006).
[CrossRef] [PubMed]

Brukilacchio, T. J.

Chance, B.

Y. Chen, C. Mu, X. Intes, and B. Chance, “Signal-to-noise analysis for detection sensitivity of small absorbing heterogeneity in turbid media with single-source and dual-interfering-source,” Opt. Express9, 212–224 (2001).
[CrossRef] [PubMed]

X. Intes, J. Yu, A. Yodh, and B. Chance, “Development and evaluation of a multi-wavelength multi-channel time resolved optical instrument for NIR/MRI mammography co-registration,” in “Proceedings of the IEEE 28th Annual Northeast Bioengineering Conference,” (2002), pp. 91–92.

Chaves, T.

Chen, Y.

Chorlton, M.

Chu, M.

M. Chu, K. Vishwanath, A. D. Klose, and H. Dehghani, “Light transport in biological tissue using three-dimensional frequency-domain simplified spherical harmonics equations,” Phys. Med. Biol.54, 2493–2509 (2009).
[CrossRef] [PubMed]

Clarkson, E.

A. K. Jha, M. A. Kupinski, H. H. Barrett, E. Clarkson, and J. H. Hartman, “Three-dimensional Neumann-series approach to model light transport in nonuniform media,” J. Opt. Soc. Am. A29, 1885–1899 (2012).
[CrossRef]

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A. K. Jha, M. A. Kupinski, H. H. Barrett, E. Clarkson, and J. H. Hartman, “Three-dimensional Neumann-series approach to model light transport in nonuniform media,” J. Opt. Soc. Am. A29, 1885–1899 (2012).
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S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “In vivo hemoglobin and water concentrations, oxygen saturation, and scattering estimates from near-infrared breast tomography using spectral reconstruction,” Acad. Radiol.13, 195–202 (2006).
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M. Chu, K. Vishwanath, A. D. Klose, and H. Dehghani, “Light transport in biological tissue using three-dimensional frequency-domain simplified spherical harmonics equations,” Phys. Med. Biol.54, 2493–2509 (2009).
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A. Klose and E. Larsen, “Light transport in biological tissue based on the simplified spherical harmonics equations,” J. Comput. Phys.220, 441–470 (2006).
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S. van de Ven, S. Elias, A. Wiethoff, M. van der Voort, A. Leproux, T. Nielsen, B. Brendel, L. Bakker, M. van der Mark, W. Mali, and P. Luijten, “Diffuse optical tomography of the breast: initial validation in benign cysts,” Mol. Imaging Biol.11, 64–70 (2009).
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Liu, H.

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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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S. van de Ven, S. Elias, A. Wiethoff, M. van der Voort, A. Leproux, T. Nielsen, B. Brendel, L. Bakker, M. van der Mark, W. Mali, and P. Luijten, “Diffuse optical tomography of the breast: initial validation in benign cysts,” Mol. Imaging Biol.11, 64–70 (2009).
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S. van de Ven, S. Elias, A. Wiethoff, M. van der Voort, A. Leproux, T. Nielsen, B. Brendel, L. Bakker, M. van der Mark, W. Mali, and P. Luijten, “Diffuse optical tomography of the breast: initial validation in benign cysts,” Mol. Imaging Biol.11, 64–70 (2009).
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A. K. Jha, M. A. Kupinski, T. Masumura, E. Clarkson, A. A. Maslov, and H. H. Barrett, “Simulating photon-transport in uniform media using the radiative transfer equation: A study using the Neumann-series approach,” J. Opt. Soc. Amer. A29, 1741–1757 (2012).
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A. K. Jha, M. A. Kupinski, T. Masumura, E. Clarkson, A. A. Maslov, and H. H. Barrett, “Simulating photon-transport in uniform media using the radiative transfer equation: A study using the Neumann-series approach,” J. Opt. Soc. Amer. A29, 1741–1757 (2012).
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T. Austin, A. P. Gibson, G. Branco, R. M. Yusof, S. R. Arridge, J. H. Meek, J. S. Wyatt, D. T. Delpy, and J. C. Hebden, “Three dimensional optical imaging of blood volume and oxygenation in the neonatal brain,” Neuroimage31, 1426–1433 (2006).
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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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A. H. Hielscher, A. D. Klose, A. K. Scheel, B. Moa-Anderson, M. Backhaus, U. Netz, and J. Beuthan, “Sagittal laser optical tomography for imaging of rheumatoid finger joints,” Phys. Med. Biol.49, 1147–1163 (2004).
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G. Gulsen, O. Birgul, M. B. Unlu, R. Shafiiha, and O. Nalcioglu, “Combined diffuse optical tomography (DOT) and MRI system for cancer imaging in small animals,” Tech. Cancer Res. Treatment5, 351–363 (2006).

Netz, U.

A. H. Hielscher, A. D. Klose, A. K. Scheel, B. Moa-Anderson, M. Backhaus, U. Netz, and J. Beuthan, “Sagittal laser optical tomography for imaging of rheumatoid finger joints,” Phys. Med. Biol.49, 1147–1163 (2004).
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S. van de Ven, S. Elias, A. Wiethoff, M. van der Voort, A. Leproux, T. Nielsen, B. Brendel, L. Bakker, M. van der Mark, W. Mali, and P. Luijten, “Diffuse optical tomography of the breast: initial validation in benign cysts,” Mol. Imaging Biol.11, 64–70 (2009).
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R. Ziegler, B. Brendel, A. Schipper, R. Harbers, M. v. Beek, H. Rinneberg, and T. Nielsen, “Investigation of detection limits for diffuse optical tomography systems: I. Theory and experiment,” Phys. Med. Biol.54, 399–412 (2009).
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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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Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Weitz, and P. Sheng, “Wave transport in random media: the ballistic to diffusive transition,” Phys. Rev. E60, 4843–4850 (1999).
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S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “In vivo hemoglobin and water concentrations, oxygen saturation, and scattering estimates from near-infrared breast tomography using spectral reconstruction,” Acad. Radiol.13, 195–202 (2006).
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B. W. Pogue, S. C. Davis, X. Song, B. A. Brooksby, H. Dehghani, and K. D. Paulsen, “Image analysis methods for diffuse optical tomography,” J. Biomed. Opt.11, 33001 (2006).
[CrossRef] [PubMed]

H. Dehghani, B. W. Pogue, S. P. Poplack, and K. D. Paulsen, “Multiwavelength three-dimensional near-infrared tomography of the breast: initial simulation, phantom, and clinical results,” App. Optics42, 135–146 (2003).
[CrossRef]

Pogue, B. W.

H. Dehghani, S. Srinivasan, B. W. Pogue, and A. Gibson, “Numerical modelling and image reconstruction in diffuse optical tomography,” Phil. Trans. Royal Soc. A367, 3073–3093 (2009).
[CrossRef]

B. W. Pogue, S. C. Davis, X. Song, B. A. Brooksby, H. Dehghani, and K. D. Paulsen, “Image analysis methods for diffuse optical tomography,” J. Biomed. Opt.11, 33001 (2006).
[CrossRef] [PubMed]

S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “In vivo hemoglobin and water concentrations, oxygen saturation, and scattering estimates from near-infrared breast tomography using spectral reconstruction,” Acad. Radiol.13, 195–202 (2006).
[CrossRef] [PubMed]

H. Dehghani, B. W. Pogue, S. P. Poplack, and K. D. Paulsen, “Multiwavelength three-dimensional near-infrared tomography of the breast: initial simulation, phantom, and clinical results,” App. Optics42, 135–146 (2003).
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S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “In vivo hemoglobin and water concentrations, oxygen saturation, and scattering estimates from near-infrared breast tomography using spectral reconstruction,” Acad. Radiol.13, 195–202 (2006).
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H. Dehghani, B. W. Pogue, S. P. Poplack, and K. D. Paulsen, “Multiwavelength three-dimensional near-infrared tomography of the breast: initial simulation, phantom, and clinical results,” App. Optics42, 135–146 (2003).
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R. Ziegler, B. Brendel, A. Schipper, R. Harbers, M. v. Beek, H. Rinneberg, and T. Nielsen, “Investigation of detection limits for diffuse optical tomography systems: I. Theory and experiment,” Phys. Med. Biol.54, 399–412 (2009).
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[CrossRef]

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A. H. Hielscher, A. D. Klose, A. K. Scheel, B. Moa-Anderson, M. Backhaus, U. Netz, and J. Beuthan, “Sagittal laser optical tomography for imaging of rheumatoid finger joints,” Phys. Med. Biol.49, 1147–1163 (2004).
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R. Ziegler, B. Brendel, A. Schipper, R. Harbers, M. v. Beek, H. Rinneberg, and T. Nielsen, “Investigation of detection limits for diffuse optical tomography systems: I. Theory and experiment,” Phys. Med. Biol.54, 399–412 (2009).
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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. Nat. Acad. Sciences104, 12169–12174 (2007).
[CrossRef]

Schriemer, H. P.

Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Weitz, and P. Sheng, “Wave transport in random media: the ballistic to diffusive transition,” Phys. Rev. E60, 4843–4850 (1999).
[CrossRef]

Schweiger, M.

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]

Shafiiha, R.

G. Gulsen, O. Birgul, M. B. Unlu, R. Shafiiha, and O. Nalcioglu, “Combined diffuse optical tomography (DOT) and MRI system for cancer imaging in small animals,” Tech. Cancer Res. Treatment5, 351–363 (2006).

Shen, F.

Sheng, P.

Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Weitz, and P. Sheng, “Wave transport in random media: the ballistic to diffusive transition,” Phys. Rev. E60, 4843–4850 (1999).
[CrossRef]

Siegel, A. M.

Soho, S.

S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “In vivo hemoglobin and water concentrations, oxygen saturation, and scattering estimates from near-infrared breast tomography using spectral reconstruction,” Acad. Radiol.13, 195–202 (2006).
[CrossRef] [PubMed]

Song, X.

H. Niu, P. Guo, X. Song, and T. Jiang, “Improving depth resolution of diffuse optical tomography with an exponential adjustment method based on maximum singular value of layered sensitivity,” Chin. Opt. Lett.6, 886–888 (2008).
[CrossRef]

B. W. Pogue, S. C. Davis, X. Song, B. A. Brooksby, H. Dehghani, and K. D. Paulsen, “Image analysis methods for diffuse optical tomography,” J. Biomed. Opt.11, 33001 (2006).
[CrossRef] [PubMed]

Spott, T.

Srinivasan, S.

H. Dehghani, S. Srinivasan, B. W. Pogue, and A. Gibson, “Numerical modelling and image reconstruction in diffuse optical tomography,” Phil. Trans. Royal Soc. A367, 3073–3093 (2009).
[CrossRef]

S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “In vivo hemoglobin and water concentrations, oxygen saturation, and scattering estimates from near-infrared breast tomography using spectral reconstruction,” Acad. Radiol.13, 195–202 (2006).
[CrossRef] [PubMed]

Stott, J.

Stott, J. J.

Svaasand, L. O.

Tarvainen, T.

Tian, F.

V. C. Kavuri, Z. J. Lin, F. Tian, and H. Liu, “Sparsity enhanced spatial resolution and depth localization in diffuse optical tomography,” Biomed. Opt. Express3, 943–957 (2012).
[CrossRef] [PubMed]

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]

Toronov, V.

Tosteson, T. D.

S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “In vivo hemoglobin and water concentrations, oxygen saturation, and scattering estimates from near-infrared breast tomography using spectral reconstruction,” Acad. Radiol.13, 195–202 (2006).
[CrossRef] [PubMed]

Unlu, M. B.

G. Gulsen, O. Birgul, M. B. Unlu, R. Shafiiha, and O. Nalcioglu, “Combined diffuse optical tomography (DOT) and MRI system for cancer imaging in small animals,” Tech. Cancer Res. Treatment5, 351–363 (2006).

van de Ven, S.

S. van de Ven, S. Elias, A. Wiethoff, M. van der Voort, A. Leproux, T. Nielsen, B. Brendel, L. Bakker, M. van der Mark, W. Mali, and P. Luijten, “Diffuse optical tomography of the breast: initial validation in benign cysts,” Mol. Imaging Biol.11, 64–70 (2009).
[CrossRef]

van der Mark, M.

S. van de Ven, S. Elias, A. Wiethoff, M. van der Voort, A. Leproux, T. Nielsen, B. Brendel, L. Bakker, M. van der Mark, W. Mali, and P. Luijten, “Diffuse optical tomography of the breast: initial validation in benign cysts,” Mol. Imaging Biol.11, 64–70 (2009).
[CrossRef]

van der Voort, M.

S. van de Ven, S. Elias, A. Wiethoff, M. van der Voort, A. Leproux, T. Nielsen, B. Brendel, L. Bakker, M. van der Mark, W. Mali, and P. Luijten, “Diffuse optical tomography of the breast: initial validation in benign cysts,” Mol. Imaging Biol.11, 64–70 (2009).
[CrossRef]

Vauhkonen, M.

Vishwanath, K.

M. Chu, K. Vishwanath, A. D. Klose, and H. Dehghani, “Light transport in biological tissue using three-dimensional frequency-domain simplified spherical harmonics equations,” Phys. Med. Biol.54, 2493–2509 (2009).
[CrossRef] [PubMed]

Webb, A.

Weitz, D. A.

Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Weitz, and P. Sheng, “Wave transport in random media: the ballistic to diffusive transition,” Phys. Rev. E60, 4843–4850 (1999).
[CrossRef]

White, B. R.

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. Nat. Acad. Sciences104, 12169–12174 (2007).
[CrossRef]

Wiethoff, A.

S. van de Ven, S. Elias, A. Wiethoff, M. van der Voort, A. Leproux, T. Nielsen, B. Brendel, L. Bakker, M. van der Mark, W. Mali, and P. Luijten, “Diffuse optical tomography of the breast: initial validation in benign cysts,” Mol. Imaging Biol.11, 64–70 (2009).
[CrossRef]

Wu, T.

Wyatt, J. S.

T. Austin, A. P. Gibson, G. Branco, R. M. Yusof, S. R. Arridge, J. H. Meek, J. S. Wyatt, D. T. Delpy, and J. C. Hebden, “Three dimensional optical imaging of blood volume and oxygenation in the neonatal brain,” Neuroimage31, 1426–1433 (2006).
[CrossRef] [PubMed]

Xu, Y.

N. Biswal, Y. Xu, and Q. Zhu, “Imaging tumor oxyhemoglobin and deoxyhemoglobin concentrations with ultrasound-guided diffuse optical tomography.” Tech. Cancer Res. Treatment10, 417 (2011).

Yodh, A.

X. Intes, J. Yu, A. Yodh, and B. Chance, “Development and evaluation of a multi-wavelength multi-channel time resolved optical instrument for NIR/MRI mammography co-registration,” in “Proceedings of the IEEE 28th Annual Northeast Bioengineering Conference,” (2002), pp. 91–92.

Yong, K.

Young, S.

S. Young, M. A. Kupinski, and A. K. Jha, “Estimating signal detectability in a model diffuse optical imaging system,” in “Biomedical Optics,” (Optical Society of America, 2010), p. BSuD26.

Yu, J.

X. Intes, J. Yu, A. Yodh, and B. Chance, “Development and evaluation of a multi-wavelength multi-channel time resolved optical instrument for NIR/MRI mammography co-registration,” in “Proceedings of the IEEE 28th Annual Northeast Bioengineering Conference,” (2002), pp. 91–92.

Yusof, R. M.

T. Austin, A. P. Gibson, G. Branco, R. M. Yusof, S. R. Arridge, J. H. Meek, J. S. Wyatt, D. T. Delpy, and J. C. Hebden, “Three dimensional optical imaging of blood volume and oxygenation in the neonatal brain,” Neuroimage31, 1426–1433 (2006).
[CrossRef] [PubMed]

Zeff, B. W.

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. Nat. Acad. Sciences104, 12169–12174 (2007).
[CrossRef]

Zhan, Y.

Y. Zhan, A. T. Eggebrecht, J. P. Culver, and H. Dehghani, “Image quality analysis of high-density diffuse optical tomography incorporating a subject-specific head model,” Front Neuroenergetics4, 6 (2012).
[CrossRef] [PubMed]

Zhang, Q.

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]

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, 57–75 (2001).
[CrossRef]

Zhang, Z. Q.

Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Weitz, and P. Sheng, “Wave transport in random media: the ballistic to diffusive transition,” Phys. Rev. E60, 4843–4850 (1999).
[CrossRef]

Zhu, Q.

N. Biswal, Y. Xu, and Q. Zhu, “Imaging tumor oxyhemoglobin and deoxyhemoglobin concentrations with ultrasound-guided diffuse optical tomography.” Tech. Cancer Res. Treatment10, 417 (2011).

Ziegler, R.

R. Ziegler, B. Brendel, A. Schipper, R. Harbers, M. v. Beek, H. Rinneberg, and T. Nielsen, “Investigation of detection limits for diffuse optical tomography systems: I. Theory and experiment,” Phys. Med. Biol.54, 399–412 (2009).
[CrossRef]

R. Ziegler, B. Brendel, H. Rinneberg, and T. Nielsen, “Investigation of detection limits for diffuse optical tomography systems: II. Analysis of slab and cup geometry for breast imaging,” Phys. Med. Biol.54, 413–431 (2009).
[CrossRef]

Acad. Radiol. (1)

S. Srinivasan, B. W. Pogue, S. Jiang, H. Dehghani, C. Kogel, S. Soho, J. J. Gibson, T. D. Tosteson, S. P. Poplack, and K. D. Paulsen, “In vivo hemoglobin and water concentrations, oxygen saturation, and scattering estimates from near-infrared breast tomography using spectral reconstruction,” Acad. Radiol.13, 195–202 (2006).
[CrossRef] [PubMed]

App. Optics (1)

H. Dehghani, B. W. Pogue, S. P. Poplack, and K. D. Paulsen, “Multiwavelength three-dimensional near-infrared tomography of the breast: initial simulation, phantom, and clinical results,” App. Optics42, 135–146 (2003).
[CrossRef]

Appl. Opt. (4)

Astrophys. J. (1)

L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J.93, 70–83 (1941).
[CrossRef]

Biomed. Opt. Express (1)

Chin. Opt. Lett. (1)

Curr. Opinion in Biotech. (1)

A. H. Hielscher, “Optical tomographic imaging of small animals,” Curr. Opinion in Biotech.16, 79–88 (2005).
[CrossRef]

Front Neuroenergetics (1)

Y. Zhan, A. T. Eggebrecht, J. P. Culver, and H. Dehghani, “Image quality analysis of high-density diffuse optical tomography incorporating a subject-specific head model,” Front Neuroenergetics4, 6 (2012).
[CrossRef] [PubMed]

IEEE Signal Process. Mag. (1)

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, 57–75 (2001).
[CrossRef]

J. Biomed Opt. (1)

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]

J. Biomed. Opt. (1)

B. W. Pogue, S. C. Davis, X. Song, B. A. Brooksby, H. Dehghani, and K. D. Paulsen, “Image analysis methods for diffuse optical tomography,” J. Biomed. Opt.11, 33001 (2006).
[CrossRef] [PubMed]

J. Comput. Phys. (1)

A. Klose and E. Larsen, “Light transport in biological tissue based on the simplified spherical harmonics equations,” J. Comput. Phys.220, 441–470 (2006).
[CrossRef]

J. Opt. Soc. Am. A (3)

J. Opt. Soc. Amer. A (1)

A. K. Jha, M. A. Kupinski, T. Masumura, E. Clarkson, A. A. Maslov, and H. H. Barrett, “Simulating photon-transport in uniform media using the radiative transfer equation: A study using the Neumann-series approach,” J. Opt. Soc. Amer. A29, 1741–1757 (2012).
[CrossRef]

J. Quant. Spectr. Rad. Trans. (1)

E. Aydin, C. de Oliveira, and A. Goddard, “A finite element-spherical harmonics radiation transport model for photon migration in turbid media,” J. Quant. Spectr. Rad. Trans.84, 247–260 (2004).
[CrossRef]

Med. Phys. (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]

Mol. Imaging Biol. (1)

S. van de Ven, S. Elias, A. Wiethoff, M. van der Voort, A. Leproux, T. Nielsen, B. Brendel, L. Bakker, M. van der Mark, W. Mali, and P. Luijten, “Diffuse optical tomography of the breast: initial validation in benign cysts,” Mol. Imaging Biol.11, 64–70 (2009).
[CrossRef]

Neuroimage (1)

T. Austin, A. P. Gibson, G. Branco, R. M. Yusof, S. R. Arridge, J. H. Meek, J. S. Wyatt, D. T. Delpy, and J. C. Hebden, “Three dimensional optical imaging of blood volume and oxygenation in the neonatal brain,” Neuroimage31, 1426–1433 (2006).
[CrossRef] [PubMed]

Opt. Express (5)

Opt. Lett. (2)

Phil. Tran. A. Math. Phys. Eng. Sci. (1)

A. Gibson and H. Dehghani, “Diffuse optical imaging,” Phil. Tran. A. Math. Phys. Eng. Sci.367, 3055–3072 (2009).
[CrossRef]

Phil. Trans. Royal Soc. A (1)

H. Dehghani, S. Srinivasan, B. W. Pogue, and A. Gibson, “Numerical modelling and image reconstruction in diffuse optical tomography,” Phil. Trans. Royal Soc. A367, 3073–3093 (2009).
[CrossRef]

Phys. Med. Biol. (5)

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol.50, 1–43 (2005).
[CrossRef]

A. H. Hielscher, A. D. Klose, A. K. Scheel, B. Moa-Anderson, M. Backhaus, U. Netz, and J. Beuthan, “Sagittal laser optical tomography for imaging of rheumatoid finger joints,” Phys. Med. Biol.49, 1147–1163 (2004).
[CrossRef] [PubMed]

R. Ziegler, B. Brendel, A. Schipper, R. Harbers, M. v. Beek, H. Rinneberg, and T. Nielsen, “Investigation of detection limits for diffuse optical tomography systems: I. Theory and experiment,” Phys. Med. Biol.54, 399–412 (2009).
[CrossRef]

R. Ziegler, B. Brendel, H. Rinneberg, and T. Nielsen, “Investigation of detection limits for diffuse optical tomography systems: II. Analysis of slab and cup geometry for breast imaging,” Phys. Med. Biol.54, 413–431 (2009).
[CrossRef]

M. Chu, K. Vishwanath, A. D. Klose, and H. Dehghani, “Light transport in biological tissue using three-dimensional frequency-domain simplified spherical harmonics equations,” Phys. Med. Biol.54, 2493–2509 (2009).
[CrossRef] [PubMed]

Phys. Rev. E (1)

Z. Q. Zhang, I. P. Jones, H. P. Schriemer, J. H. Page, D. A. Weitz, and P. Sheng, “Wave transport in random media: the ballistic to diffusive transition,” Phys. Rev. E60, 4843–4850 (1999).
[CrossRef]

Proc. Nat. Acad. Sciences (1)

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. Nat. Acad. Sciences104, 12169–12174 (2007).
[CrossRef]

Proc. SPIE Medical Imaging (1)

M. A. Kupinski, E. Clarkson, K. Gross, and J. W. Hoppin, “Optimizing imaging hardware for estimation tasks,” in “Proc. SPIE Medical Imaging,” (2003), pp. 309–313.
[CrossRef]

Tech. Cancer Res. Treatment (2)

G. Gulsen, O. Birgul, M. B. Unlu, R. Shafiiha, and O. Nalcioglu, “Combined diffuse optical tomography (DOT) and MRI system for cancer imaging in small animals,” Tech. Cancer Res. Treatment5, 351–363 (2006).

N. Biswal, Y. Xu, and Q. Zhu, “Imaging tumor oxyhemoglobin and deoxyhemoglobin concentrations with ultrasound-guided diffuse optical tomography.” Tech. Cancer Res. Treatment10, 417 (2011).

Other (6)

S. Young, M. A. Kupinski, and A. K. Jha, “Estimating signal detectability in a model diffuse optical imaging system,” in “Biomedical Optics,” (Optical Society of America, 2010), p. BSuD26.

X. Intes, J. Yu, A. Yodh, and B. Chance, “Development and evaluation of a multi-wavelength multi-channel time resolved optical instrument for NIR/MRI mammography co-registration,” in “Proceedings of the IEEE 28th Annual Northeast Bioengineering Conference,” (2002), pp. 91–92.

A. K. Jha, “Retrieving Information from Scattered Photons in Medical Imaging,” Ph.D. thesis, College of Optical Sciences, University of Arizona, Tucson, AZ, USA (2013).

B. W. Miller, “High-Resolution Gamma-Ray Imaging with Columnar Scintillators,” Ph.D. thesis, College of Optical Sciences, University of Arizona, Tucson, AZ, USA (2011).

A. H. Hielscher and R. E. Alcouffe, “Discrete-ordinate transport simulations of light propagation in highly forward scattering heterogeneous media,” in “Advances in Optical Imaging and Photon Migration,” (Optical Society of America, 1998), p. ATuC2.

H. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley, 2004), 1st ed.

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Figures (5)

Fig. 1
Fig. 1

The simulated DOT setup for our experiments.

Fig. 2
Fig. 2

The signal detectability as a function of the depth of the signal for on-axis and off-axis signal locations.

Fig. 3
Fig. 3

The signal detectability as a function of the scattering coefficient for different signal depths

Fig. 4
Fig. 4

The signal detectability as a function of the signal size for different signal depths

Fig. 5
Fig. 5

The signal detectability as a function of the signal contrast for different signal sizes

Equations (46)

Equations on this page are rendered with MathJax. Learn more.

t ( g ) = pr ( g | H 1 ) pr ( g | H 0 ) .
μ a ( r ) = n = 1 N μ a , n ϕ n ( r )
μ s ( r ) = n = 1 N μ s , n ϕ n ( r )
μ ( r ) = n = 1 N μ n ϕ n ( r ) ,
t μ ( g ) = pr ( g | μ ) pr ( g | μ 0 ) .
AUC ( μ ) = 1 2 [ 1 + erf ( d ( μ ) 2 ) ] ,
d 2 ( μ ) ( μ μ 0 ) F ( μ 0 ) ( μ μ 0 )
pr ( g | μ ) = m = 1 M [ g ¯ m ( μ ) ] g m exp [ g ¯ m ( μ ) ] g m !
( g ¯ | μ ) = log pr ( g | μ ) = m = 1 M g m log g ¯ m ( μ ) g ¯ m ( μ ) log ( g m ! ) .
F i j ( μ ) = 2 μ j μ i g | μ
μ i = m = 1 M ( g m g ¯ m ( μ ) 1 ) g ¯ m ( μ ) μ i .
2 μ j μ i = m = 1 M ( g m g ¯ m ( μ ) 2 g ¯ m ( μ ) μ j ) g ¯ m ( μ ) μ i + ( g m g ¯ m ( μ ) 1 ) 2 g ¯ m ( μ ) μ j μ i .
F i j ( μ ) = 2 μ j μ i g | μ = m = 1 M 1 g ¯ m ( μ ) g ¯ m ( μ ) μ j g ¯ m ( μ ) μ i ,
[ 𝒦 w ] ( r , s ^ ) = μ s ( r ) c m 4 π d Ω p ( s ^ , s ^ | r ) w ( r , s ^ ) ,
p ( s ^ , s ^ | r ) = 1 4 π { 1 α 2 [ 1 + α 2 2 α cos ( s ^ s ^ ) ] 3 / 2 }
[ 𝒳 w ] ( r , s ^ ) = 1 c m 0 d l w ( r s ^ l , s ^ ) exp [ 0 l d l μ tot ( r s ^ l ) ] ,
w = 𝒳 Ξ + 𝒳 𝒦 w
g ¯ m ( μ ) = d 3 r d Ω h m ( r , s ^ ) w ( r , s ^ ) ,
g ¯ m ( μ ) = ( h m , w ) .
g ¯ m ( μ ) μ n = ( h m , w μ n ) .
w μ n = 𝒳 μ n Ξ + 𝒳 μ n 𝒦 w + 𝒳 𝒦 μ n w + 𝒳 𝒦 w μ n .
[ 𝒳 Ξ ] ( r , s ^ ) = 1 c m 0 d l Ξ ( r s ^ l , s ^ ) exp [ 0 l d l μ tot ( r s ^ l ) ] .
μ tot ( r ) = μ s ( r ) + μ a ( r ) = n = 1 N ( μ s , n + μ a , n ) ϕ n ( r )
𝒳 μ n Ξ ( r , s ^ ) = 1 c m 0 d l Ξ ( r s ^ l , s ^ ) exp [ 0 l d l μ tot ( r s ^ l ) ] [ 0 l d l ϕ n ( r s ^ l ) ] .
step ( l ) = { 0 , l < 0 1 , otherwise
𝒳 μ n Ξ ( r , s ^ ) = 1 c m × 0 d l Ξ ( r s ^ l , s ^ ) exp [ 0 l d l μ tot ( r s ^ l ) ] [ 0 d l [ step ( l ) step ( l l ) ] ϕ n ( r s ^ l ) ] = 1 c m 0 d l 0 d l [ step ( l ) step ( l l ) ] ϕ n ( r s ^ l ) Ξ ( r s ^ l , s ^ ) exp [ 0 l d l μ tot ( r s ^ l ) ]
𝒳 μ n Ξ ( r , s ^ ) = 1 c m 0 d l ϕ n ( r s ^ l ) 0 d l [ step ( l ) step ( l l ) ] Ξ ( r s ^ l , s ^ ) exp [ 0 l d l μ tot ( r s ^ l ) ] = 1 c m 0 d l ϕ n ( r s ^ l ) l d l Ξ ( r s ^ l , s ^ ) exp [ 0 l d l μ tot ( r s ^ l ) ] ,
𝒳 μ n Ξ ( r , s ^ ) = 1 c m × 0 d l ϕ n ( r s ^ l ) exp [ 0 l d l μ tot ( r s ^ l ) ] l d l Ξ ( r s ^ l , s ^ ) exp [ l l d l μ tot ( r s ^ l ) d l ] .
l d l Ξ ( r s ^ l , s ^ ) exp [ l l d l μ tot ( r s ^ l ) ] = 0 d l ˜ Ξ ( r s ^ l s ^ l ˜ , s ^ ) exp [ l l + l ˜ d l μ tot ( r s ^ l ) ] = 0 d l ˜ Ξ ( r s ^ l s ^ l ˜ , s ^ ) exp [ 0 l ˜ d l ^ μ tot ( r s ^ l s ^ l ^ ) ] ,
0 d l ˜ Ξ ( r s ^ l s ^ l ˜ , s ^ ) exp [ 0 l ˜ d l ^ μ tot ( r s ^ l s ^ l ^ ) ] = c m [ 𝒳 Ξ ] ( r s ^ l , s ^ ) .
𝒳 μ n Ξ ( r , s ^ ) = 1 c m 0 d l ϕ n ( r s ^ l ) c m { [ 𝒳 Ξ ] ( r s ^ l , s ^ ) } exp [ 0 l d l μ tot ( r s ^ l ) ] ,
𝒳 μ n Ξ = c m 𝒳 ( ϕ n 𝒳 Ξ )
𝒳 μ n 𝒦 w = c m 𝒳 ( ϕ n 𝒳 𝒦 w )
𝒳 μ n Ξ + 𝒳 μ n 𝒦 w = c m 𝒳 [ ϕ n ( 𝒳 Ξ + 𝒳 𝒦 w ) ] = c m 𝒳 ( ϕ n w ) ,
𝒦 μ n w = μ n c m μ s ( r ) d Ω p ( s ^ , s ^ ) w ( r , s ^ ) = β c m ϕ n ( r ) d Ω p ( s ^ , s ^ ) w ( r , s ^ ) ,
[ 𝒦 1 w ] ( r , s ^ ) = c m d Ω p ( s ^ , s ^ ) w ( r , s ^ ) ,
𝒦 μ n w = β ϕ n 𝒦 1 w .
𝒳 𝒦 μ n w = 𝒳 ( β ϕ n 𝒦 1 w ) .
w μ n = 𝒳 ( c m ϕ n w + β ϕ n 𝒦 1 w ) + 𝒳 𝒦 w μ n .
s n = c m ϕ n w + β ϕ n 𝒦 1 w .
w μ n = 𝒳 s n + 𝒳 𝒦 𝒳 s n + 𝒳 𝒦 𝒳 s n +
W d = AS n + ADAS n + ADADAS n +
d 2 ( μ ) = Δ μ a 2 F a 1 , a 1 ,
F a 1 , a 1 = m = 1 M 1 g ¯ m ( g ¯ m μ a , 1 ) 2 .
g ¯ m μ a , 1 = ( h m , w μ a , 1 ) .
s n = c m ϕ 1 w .

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