Abstract

This paper presents a noise-stochastic corrected maximum a posteriori estimator for birefringence imaging using Jones matrix optical coherence tomography. The estimator described in this paper is based on the relationship between probability distribution functions of the measured birefringence and the effective signal to noise ratio (ESNR) as well as the true birefringence and the true ESNR. The Monte Carlo method is used to numerically describe this relationship and adaptive 2D kernel density estimation provides the likelihood for a posteriori estimation of the true birefringence. Improved estimation is shown for the new estimator with stochastic model of ESNR in comparison to the old estimator, both based on the Jones matrix noise model. A comparison with the mean estimator is also done. Numerical simulation validates the superiority of the new estimator. The superior performance of the new estimator was also shown by in vivo measurement of optic nerve head.

© 2017 Optical Society of America

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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]

2016 (2)

M. Villiger, D. Lorenser, R. A. McLaughlin, B. C. Quirk, R. W. Kirk, B. E. Bouma, and D. D. Sampson, “Deep tissue volume imaging of birefringence through fibre-optic needle probes for the delineation of breast tumour,” Sci. Rep. 6, 28771 (2016).
[Crossref] [PubMed]

S. Makita, K. Kurokawa, Y.-J. Hong, M. Miura, and Y. Yasuno, “Noise-immune complex correlation for optical coherence angiography based on standard and Jones matrix optical coherence tomography,” Biomed. Opt. Express 7, 1525 (2016).
[Crossref] [PubMed]

2015 (3)

2014 (4)

2013 (4)

2012 (2)

2011 (4)

2010 (2)

Z. I. Botev, J. F. Grotowski, and D. P. Kroese, “Kernel density estimation via diffusion,” The Annals of Statistics 38, 2916–2957 (2010).
[Crossref]

S. Makita, M. Yamanari, and Y. Yasuno, “Generalized Jones matrix optical coherence tomography: performance and local birefringence imaging,” Opt. Express 18, 854–876 (2010).
[Crossref] [PubMed]

2008 (2)

M. Yamanari, M. Miura, S. Makita, T. Yatagai, and Y. Yasuno, “Phase retardation measurement of retinal nerve fiber layer by polarization-sensitive spectral-domain optical coherence tomography and scanning laser polarimetry,” J. Biomed. Opt. 13, 014013 (2008).
[Crossref] [PubMed]

R. F. Spaide, H. Koizumi, and M. C. Pozonni, “Enhanced depth imaging spectral-domain optical coherence tomography,” Am. J. Ophthalmol. 146, 496–500 (2008).
[Crossref] [PubMed]

2007 (1)

S. K. Nadkarni, M. C. Pierce, B. H. Park, J. F. de Boer, P. Whittaker, B. E. Bouma, J. E. Bressner, E. Halpern, S. L. Houser, and G. J. Tearney, “Measurement of Collagen and Smooth Muscle Cell Content in Atherosclerotic Plaques Using Polarization-Sensitive Optical Coherence Tomography,” J. Am. College Cardiol. 49, 1474–1481 (2007).
[Crossref]

2006 (1)

2004 (1)

2002 (2)

1992 (1)

1991 (1)

Y. Komai and T. Ushiki, “The three-dimensional organization of collagen fibrils in the human cornea and sclera,” Investigative Ophthalmol. Vis. Sci. 32, 2244–2258 (1991).

1990 (1)

A. R. Elkington, C. B. Inman, P. V. Steart, and R. O. Weller, “The structure of the lamina cribrosa of the human eye: an immunocytochemical and electron microscopical study,” Eye 4, 42–57 (1990).
[Crossref] [PubMed]

Ahrens, G.

Baumann, B.

Beheregaray, S.

D. Kasaragod, S. Makita, S. Fukuda, S. Beheregaray, T. Oshika, and Y. Yasuno, “Bayesian maximum likelihood estimator of phase retardation for quantitative polarization-sensitive optical coherence tomography,” Opt. Express 22, 16472–16492 (2014).
[Crossref] [PubMed]

S. Fukuda, S. Beheregaray, D. Kasaragod, S. Hoshi, G. Kishino, K. Ishii, Y. Yasuno, and T. Oshika, “Noninvasive Evaluation of Phase Retardation in Blebs After Glaucoma Surgery Using Anterior Segment Polarization-Sensitive Optical Coherence Tomography,” Investigative Ophthalmol. Vis. Sci. 55, 5200–5206 (2014).
[Crossref]

S. Fukuda, M. Yamanari, Y. Lim, S. Hoshi, S. Beheregaray, T. Oshika, and Y. Yasuno, “Keratoconus diagnosis using anterior segment polarization-sensitive optical coherence tomography,” Investigative Ophthalmol. Vis. Sci. 54, 1384–1391 (2013).
[Crossref]

Botev, Z. I.

Z. I. Botev, J. F. Grotowski, and D. P. Kroese, “Kernel density estimation via diffusion,” The Annals of Statistics 38, 2916–2957 (2010).
[Crossref]

Bouma, B. E.

M. Villiger, D. Lorenser, R. A. McLaughlin, B. C. Quirk, R. W. Kirk, B. E. Bouma, and D. D. Sampson, “Deep tissue volume imaging of birefringence through fibre-optic needle probes for the delineation of breast tumour,” Sci. Rep. 6, 28771 (2016).
[Crossref] [PubMed]

M. Villiger and B. E. Bouma, “Practical decomposition for physically admissible differential Mueller matrices,” Opt. Lett. 39, 1779 (2014).
[Crossref] [PubMed]

M. Villiger, E. Z. Zhang, S. K. Nadkarni, W.-Y. Oh, B. J. Vakoc, and B. E. Bouma, “Spectral binning for mitigation of polarization mode dispersion artifacts in catheter-based optical frequency domain imaging,” Opt. Express 21, 16353 (2013).
[Crossref] [PubMed]

S. K. Nadkarni, M. C. Pierce, B. H. Park, J. F. de Boer, P. Whittaker, B. E. Bouma, J. E. Bressner, E. Halpern, S. L. Houser, and G. J. Tearney, “Measurement of Collagen and Smooth Muscle Cell Content in Atherosclerotic Plaques Using Polarization-Sensitive Optical Coherence Tomography,” J. Am. College Cardiol. 49, 1474–1481 (2007).
[Crossref]

Box, G. E. P.

G. E. P. Box and G. C. Tiao, Bayesian Inference in Statistical Analysis (John Wiley & Sons, 1992).
[Crossref]

Bressner, J. E.

S. K. Nadkarni, M. C. Pierce, B. H. Park, J. F. de Boer, P. Whittaker, B. E. Bouma, J. E. Bressner, E. Halpern, S. L. Houser, and G. J. Tearney, “Measurement of Collagen and Smooth Muscle Cell Content in Atherosclerotic Plaques Using Polarization-Sensitive Optical Coherence Tomography,” J. Am. College Cardiol. 49, 1474–1481 (2007).
[Crossref]

Cense, B.

Chen, T. C.

Chen, Z.

Choi, W.

Chung, W. K.

de Boer, J. F.

S. K. Nadkarni, M. C. Pierce, B. H. Park, J. F. de Boer, P. Whittaker, B. E. Bouma, J. E. Bressner, E. Halpern, S. L. Houser, and G. J. Tearney, “Measurement of Collagen and Smooth Muscle Cell Content in Atherosclerotic Plaques Using Polarization-Sensitive Optical Coherence Tomography,” J. Am. College Cardiol. 49, 1474–1481 (2007).
[Crossref]

J. F. De Boer and T. E. Milner, “Review of polarization sensitive optical coherence tomography and Stokes vector determination,” J. Biomed. Opt. 7, 359–371 (2002).
[Crossref] [PubMed]

B. Cense, T. C. Chen, B. H. Park, M. C. Pierce, and J. F. de Boer, “In vivo depth-resolved birefringence measurements of the human retinal nerve fiber layer by polarization-sensitive optical coherence tomography,” Opt. Lett. 27, 1610 (2002).
[Crossref]

Duan, L.

Duker, J. S.

Elkington, A. R.

A. R. Elkington, C. B. Inman, P. V. Steart, and R. O. Weller, “The structure of the lamina cribrosa of the human eye: an immunocytochemical and electron microscopical study,” Eye 4, 42–57 (1990).
[Crossref] [PubMed]

Elsner, A. E.

Engelke, R.

Fujimoto, J. G.

Fukuda, S.

D. Kasaragod, S. Makita, S. Fukuda, S. Beheregaray, T. Oshika, and Y. Yasuno, “Bayesian maximum likelihood estimator of phase retardation for quantitative polarization-sensitive optical coherence tomography,” Opt. Express 22, 16472–16492 (2014).
[Crossref] [PubMed]

S. Fukuda, S. Beheregaray, D. Kasaragod, S. Hoshi, G. Kishino, K. Ishii, Y. Yasuno, and T. Oshika, “Noninvasive Evaluation of Phase Retardation in Blebs After Glaucoma Surgery Using Anterior Segment Polarization-Sensitive Optical Coherence Tomography,” Investigative Ophthalmol. Vis. Sci. 55, 5200–5206 (2014).
[Crossref]

S. Fukuda, M. Yamanari, Y. Lim, S. Hoshi, S. Beheregaray, T. Oshika, and Y. Yasuno, “Keratoconus diagnosis using anterior segment polarization-sensitive optical coherence tomography,” Investigative Ophthalmol. Vis. Sci. 54, 1384–1391 (2013).
[Crossref]

Y. Lim, M. Yamanari, S. Fukuda, Y. Kaji, T. Kiuchi, M. Miura, T. Oshika, and Y. Yasuno, “Birefringence measurement of cornea and anterior segment by office-based polarization-sensitive optical coherence tomography,” Biomed. Opt. Express 2, 2392–2402 (2011).
[Crossref] [PubMed]

Goetzinger, E.

Grotowski, J. F.

Z. I. Botev, J. F. Grotowski, and D. P. Kroese, “Kernel density estimation via diffusion,” The Annals of Statistics 38, 2916–2957 (2010).
[Crossref]

Gruetzner, G.

Guo, S.

Halpern, E.

S. K. Nadkarni, M. C. Pierce, B. H. Park, J. F. de Boer, P. Whittaker, B. E. Bouma, J. E. Bressner, E. Halpern, S. L. Houser, and G. J. Tearney, “Measurement of Collagen and Smooth Muscle Cell Content in Atherosclerotic Plaques Using Polarization-Sensitive Optical Coherence Tomography,” J. Am. College Cardiol. 49, 1474–1481 (2007).
[Crossref]

Hee, M. R.

Himori, N.

Hitzenberger, C. K.

Hong, Y. J.

Y. Yasuno, M.-J. Ju, Y. J. Hong, S. Makita, Y. Lim, and M. Yamanari, “Jones Matrix Based Polarization Sensitive Optical Coherence Tomography,” in Optical Coherence Tomography: Technology and Applications2nd ed. (Springer, 2015) pp. 1137–1162.
[Crossref]

Hong, Y.-J.

Hoshi, S.

S. Fukuda, S. Beheregaray, D. Kasaragod, S. Hoshi, G. Kishino, K. Ishii, Y. Yasuno, and T. Oshika, “Noninvasive Evaluation of Phase Retardation in Blebs After Glaucoma Surgery Using Anterior Segment Polarization-Sensitive Optical Coherence Tomography,” Investigative Ophthalmol. Vis. Sci. 55, 5200–5206 (2014).
[Crossref]

S. Fukuda, M. Yamanari, Y. Lim, S. Hoshi, S. Beheregaray, T. Oshika, and Y. Yasuno, “Keratoconus diagnosis using anterior segment polarization-sensitive optical coherence tomography,” Investigative Ophthalmol. Vis. Sci. 54, 1384–1391 (2013).
[Crossref]

Houser, S. L.

S. K. Nadkarni, M. C. Pierce, B. H. Park, J. F. de Boer, P. Whittaker, B. E. Bouma, J. E. Bressner, E. Halpern, S. L. Houser, and G. J. Tearney, “Measurement of Collagen and Smooth Muscle Cell Content in Atherosclerotic Plaques Using Polarization-Sensitive Optical Coherence Tomography,” J. Am. College Cardiol. 49, 1474–1481 (2007).
[Crossref]

Huang, D.

Ikuno, Y.

Inman, C. B.

A. R. Elkington, C. B. Inman, P. V. Steart, and R. O. Weller, “The structure of the lamina cribrosa of the human eye: an immunocytochemical and electron microscopical study,” Eye 4, 42–57 (1990).
[Crossref] [PubMed]

Ishii, K.

S. Fukuda, S. Beheregaray, D. Kasaragod, S. Hoshi, G. Kishino, K. Ishii, Y. Yasuno, and T. Oshika, “Noninvasive Evaluation of Phase Retardation in Blebs After Glaucoma Surgery Using Anterior Segment Polarization-Sensitive Optical Coherence Tomography,” Investigative Ophthalmol. Vis. Sci. 55, 5200–5206 (2014).
[Crossref]

Ju, M. J.

Ju, M.-J.

Y. Yasuno, M.-J. Ju, Y. J. Hong, S. Makita, Y. Lim, and M. Yamanari, “Jones Matrix Based Polarization Sensitive Optical Coherence Tomography,” in Optical Coherence Tomography: Technology and Applications2nd ed. (Springer, 2015) pp. 1137–1162.
[Crossref]

Kaji, Y.

Kasaragod, D.

Kim, B.

Kim, K. H.

Kirk, R. W.

M. Villiger, D. Lorenser, R. A. McLaughlin, B. C. Quirk, R. W. Kirk, B. E. Bouma, and D. D. Sampson, “Deep tissue volume imaging of birefringence through fibre-optic needle probes for the delineation of breast tumour,” Sci. Rep. 6, 28771 (2016).
[Crossref] [PubMed]

Kishino, G.

S. Fukuda, S. Beheregaray, D. Kasaragod, S. Hoshi, G. Kishino, K. Ishii, Y. Yasuno, and T. Oshika, “Noninvasive Evaluation of Phase Retardation in Blebs After Glaucoma Surgery Using Anterior Segment Polarization-Sensitive Optical Coherence Tomography,” Investigative Ophthalmol. Vis. Sci. 55, 5200–5206 (2014).
[Crossref]

Kiuchi, T.

Koizumi, H.

R. F. Spaide, H. Koizumi, and M. C. Pozonni, “Enhanced depth imaging spectral-domain optical coherence tomography,” Am. J. Ophthalmol. 146, 496–500 (2008).
[Crossref] [PubMed]

Kokubun, T.

Komai, Y.

Y. Komai and T. Ushiki, “The three-dimensional organization of collagen fibrils in the human cornea and sclera,” Investigative Ophthalmol. Vis. Sci. 32, 2244–2258 (1991).

Kroese, D. P.

Z. I. Botev, J. F. Grotowski, and D. P. Kroese, “Kernel density estimation via diffusion,” The Annals of Statistics 38, 2916–2957 (2010).
[Crossref]

Kunikata, H.

Kunimatsu-Sanuki, S.

Kurokawa, K.

Le, V.-H.

Lee, S.

Lim, Y.

M. J. Ju, Y.-J. Hong, S. Makita, Y. Lim, K. Kurokawa, L. Duan, M. Miura, S. Tang, and Y. Yasuno, “Advanced multi-contrast Jones matrix optical coherence tomography for Doppler and polarization sensitive imaging,” Opt. Express 21, 19412–19436 (2013).
[Crossref] [PubMed]

S. Fukuda, M. Yamanari, Y. Lim, S. Hoshi, S. Beheregaray, T. Oshika, and Y. Yasuno, “Keratoconus diagnosis using anterior segment polarization-sensitive optical coherence tomography,” Investigative Ophthalmol. Vis. Sci. 54, 1384–1391 (2013).
[Crossref]

Y. Lim, Y.-J. Hong, L. Duan, M. Yamanari, and Y. Yasuno, “Passive component based multifunctional Jones matrix swept source optical coherence tomography for Doppler and polarization imaging,” Opt. Lett. 37, 1958–1960 (2012).
[Crossref] [PubMed]

S. Sakai, M. Yamanari, Y. Lim, N. Nakagawa, and Y. Yasuno, “In vivo evaluation of human skin anisotropy by polarization-sensitive optical coherence tomography,” Biomed. Opt. Express 2, 2623–2631 (2011).
[Crossref] [PubMed]

Y. Lim, M. Yamanari, S. Fukuda, Y. Kaji, T. Kiuchi, M. Miura, T. Oshika, and Y. Yasuno, “Birefringence measurement of cornea and anterior segment by office-based polarization-sensitive optical coherence tomography,” Biomed. Opt. Express 2, 2392–2402 (2011).
[Crossref] [PubMed]

L. Duan, S. Makita, M. Yamanari, Y. Lim, and Y. Yasuno, “Monte-Carlo-based phase retardation estimator for polarization sensitive optical coherence tomography,” Opt. Express 19, 16330–16345 (2011).
[Crossref] [PubMed]

Y. Yasuno, M.-J. Ju, Y. J. Hong, S. Makita, Y. Lim, and M. Yamanari, “Jones Matrix Based Polarization Sensitive Optical Coherence Tomography,” in Optical Coherence Tomography: Technology and Applications2nd ed. (Springer, 2015) pp. 1137–1162.
[Crossref]

Lorenser, D.

M. Villiger, D. Lorenser, R. A. McLaughlin, B. C. Quirk, R. W. Kirk, B. E. Bouma, and D. D. Sampson, “Deep tissue volume imaging of birefringence through fibre-optic needle probes for the delineation of breast tumour,” Sci. Rep. 6, 28771 (2016).
[Crossref] [PubMed]

Makita, S.

S. Makita, K. Kurokawa, Y.-J. Hong, M. Miura, and Y. Yasuno, “Noise-immune complex correlation for optical coherence angiography based on standard and Jones matrix optical coherence tomography,” Biomed. Opt. Express 7, 1525 (2016).
[Crossref] [PubMed]

S. Sugiyama, Y.-J. Hong, D. Kasaragod, S. Makita, S. Uematsu, Y. Ikuno, M. Miura, and Y. Yasuno, “Birefringence imaging of posterior eye by multi-functional Jones matrix optical coherence tomography,” Biomed. Opt. Express 6, 4951 (2015).
[Crossref] [PubMed]

S. Makita, Y.-J. Hong, M. Miura, and Y. Yasuno, “Degree of polarization uniformity with high noise immunity using polarization-sensitive optical coherence tomography,” Opt. Lett. 39, 6783–6786 (2014).
[Crossref] [PubMed]

D. Kasaragod, S. Makita, S. Fukuda, S. Beheregaray, T. Oshika, and Y. Yasuno, “Bayesian maximum likelihood estimator of phase retardation for quantitative polarization-sensitive optical coherence tomography,” Opt. Express 22, 16472–16492 (2014).
[Crossref] [PubMed]

M. J. Ju, Y.-J. Hong, S. Makita, Y. Lim, K. Kurokawa, L. Duan, M. Miura, S. Tang, and Y. Yasuno, “Advanced multi-contrast Jones matrix optical coherence tomography for Doppler and polarization sensitive imaging,” Opt. Express 21, 19412–19436 (2013).
[Crossref] [PubMed]

L. Duan, S. Makita, M. Yamanari, Y. Lim, and Y. Yasuno, “Monte-Carlo-based phase retardation estimator for polarization sensitive optical coherence tomography,” Opt. Express 19, 16330–16345 (2011).
[Crossref] [PubMed]

S. Makita, M. Yamanari, and Y. Yasuno, “Generalized Jones matrix optical coherence tomography: performance and local birefringence imaging,” Opt. Express 18, 854–876 (2010).
[Crossref] [PubMed]

M. Yamanari, M. Miura, S. Makita, T. Yatagai, and Y. Yasuno, “Phase retardation measurement of retinal nerve fiber layer by polarization-sensitive spectral-domain optical coherence tomography and scanning laser polarimetry,” J. Biomed. Opt. 13, 014013 (2008).
[Crossref] [PubMed]

Y. Yasuno, M.-J. Ju, Y. J. Hong, S. Makita, Y. Lim, and M. Yamanari, “Jones Matrix Based Polarization Sensitive Optical Coherence Tomography,” in Optical Coherence Tomography: Technology and Applications2nd ed. (Springer, 2015) pp. 1137–1162.
[Crossref]

Maruyama, K.

McLaughlin, R. A.

M. Villiger, D. Lorenser, R. A. McLaughlin, B. C. Quirk, R. W. Kirk, B. E. Bouma, and D. D. Sampson, “Deep tissue volume imaging of birefringence through fibre-optic needle probes for the delineation of breast tumour,” Sci. Rep. 6, 28771 (2016).
[Crossref] [PubMed]

Miller, D. T.

Milner, T. E.

J. F. De Boer and T. E. Milner, “Review of polarization sensitive optical coherence tomography and Stokes vector determination,” J. Biomed. Opt. 7, 359–371 (2002).
[Crossref] [PubMed]

Miura, M.

S. Makita, K. Kurokawa, Y.-J. Hong, M. Miura, and Y. Yasuno, “Noise-immune complex correlation for optical coherence angiography based on standard and Jones matrix optical coherence tomography,” Biomed. Opt. Express 7, 1525 (2016).
[Crossref] [PubMed]

S. Sugiyama, Y.-J. Hong, D. Kasaragod, S. Makita, S. Uematsu, Y. Ikuno, M. Miura, and Y. Yasuno, “Birefringence imaging of posterior eye by multi-functional Jones matrix optical coherence tomography,” Biomed. Opt. Express 6, 4951 (2015).
[Crossref] [PubMed]

S. Makita, Y.-J. Hong, M. Miura, and Y. Yasuno, “Degree of polarization uniformity with high noise immunity using polarization-sensitive optical coherence tomography,” Opt. Lett. 39, 6783–6786 (2014).
[Crossref] [PubMed]

M. J. Ju, Y.-J. Hong, S. Makita, Y. Lim, K. Kurokawa, L. Duan, M. Miura, S. Tang, and Y. Yasuno, “Advanced multi-contrast Jones matrix optical coherence tomography for Doppler and polarization sensitive imaging,” Opt. Express 21, 19412–19436 (2013).
[Crossref] [PubMed]

Y. Lim, M. Yamanari, S. Fukuda, Y. Kaji, T. Kiuchi, M. Miura, T. Oshika, and Y. Yasuno, “Birefringence measurement of cornea and anterior segment by office-based polarization-sensitive optical coherence tomography,” Biomed. Opt. Express 2, 2392–2402 (2011).
[Crossref] [PubMed]

M. Yamanari, M. Miura, S. Makita, T. Yatagai, and Y. Yasuno, “Phase retardation measurement of retinal nerve fiber layer by polarization-sensitive spectral-domain optical coherence tomography and scanning laser polarimetry,” J. Biomed. Opt. 13, 014013 (2008).
[Crossref] [PubMed]

Nadkarni, S. K.

M. Villiger, E. Z. Zhang, S. K. Nadkarni, W.-Y. Oh, B. J. Vakoc, and B. E. Bouma, “Spectral binning for mitigation of polarization mode dispersion artifacts in catheter-based optical frequency domain imaging,” Opt. Express 21, 16353 (2013).
[Crossref] [PubMed]

S. K. Nadkarni, M. C. Pierce, B. H. Park, J. F. de Boer, P. Whittaker, B. E. Bouma, J. E. Bressner, E. Halpern, S. L. Houser, and G. J. Tearney, “Measurement of Collagen and Smooth Muscle Cell Content in Atherosclerotic Plaques Using Polarization-Sensitive Optical Coherence Tomography,” J. Am. College Cardiol. 49, 1474–1481 (2007).
[Crossref]

Nakagawa, N.

Nakazawa, T.

Nelson, J. S.

Oh, W.-Y.

Omodaka, K.

Oshika, T.

D. Kasaragod, S. Makita, S. Fukuda, S. Beheregaray, T. Oshika, and Y. Yasuno, “Bayesian maximum likelihood estimator of phase retardation for quantitative polarization-sensitive optical coherence tomography,” Opt. Express 22, 16472–16492 (2014).
[Crossref] [PubMed]

S. Fukuda, S. Beheregaray, D. Kasaragod, S. Hoshi, G. Kishino, K. Ishii, Y. Yasuno, and T. Oshika, “Noninvasive Evaluation of Phase Retardation in Blebs After Glaucoma Surgery Using Anterior Segment Polarization-Sensitive Optical Coherence Tomography,” Investigative Ophthalmol. Vis. Sci. 55, 5200–5206 (2014).
[Crossref]

S. Fukuda, M. Yamanari, Y. Lim, S. Hoshi, S. Beheregaray, T. Oshika, and Y. Yasuno, “Keratoconus diagnosis using anterior segment polarization-sensitive optical coherence tomography,” Investigative Ophthalmol. Vis. Sci. 54, 1384–1391 (2013).
[Crossref]

Y. Lim, M. Yamanari, S. Fukuda, Y. Kaji, T. Kiuchi, M. Miura, T. Oshika, and Y. Yasuno, “Birefringence measurement of cornea and anterior segment by office-based polarization-sensitive optical coherence tomography,” Biomed. Opt. Express 2, 2392–2402 (2011).
[Crossref] [PubMed]

Park, B. H.

S. K. Nadkarni, M. C. Pierce, B. H. Park, J. F. de Boer, P. Whittaker, B. E. Bouma, J. E. Bressner, E. Halpern, S. L. Houser, and G. J. Tearney, “Measurement of Collagen and Smooth Muscle Cell Content in Atherosclerotic Plaques Using Polarization-Sensitive Optical Coherence Tomography,” J. Am. College Cardiol. 49, 1474–1481 (2007).
[Crossref]

B. Cense, T. C. Chen, B. H. Park, M. C. Pierce, and J. F. de Boer, “In vivo depth-resolved birefringence measurements of the human retinal nerve fiber layer by polarization-sensitive optical coherence tomography,” Opt. Lett. 27, 1610 (2002).
[Crossref]

Pierce, M. C.

S. K. Nadkarni, M. C. Pierce, B. H. Park, J. F. de Boer, P. Whittaker, B. E. Bouma, J. E. Bressner, E. Halpern, S. L. Houser, and G. J. Tearney, “Measurement of Collagen and Smooth Muscle Cell Content in Atherosclerotic Plaques Using Polarization-Sensitive Optical Coherence Tomography,” J. Am. College Cardiol. 49, 1474–1481 (2007).
[Crossref]

B. Cense, T. C. Chen, B. H. Park, M. C. Pierce, and J. F. de Boer, “In vivo depth-resolved birefringence measurements of the human retinal nerve fiber layer by polarization-sensitive optical coherence tomography,” Opt. Lett. 27, 1610 (2002).
[Crossref]

Pircher, M.

Potsaid, B.

Pozonni, M. C.

R. F. Spaide, H. Koizumi, and M. C. Pozonni, “Enhanced depth imaging spectral-domain optical coherence tomography,” Am. J. Ophthalmol. 146, 496–500 (2008).
[Crossref] [PubMed]

Quirk, B. C.

M. Villiger, D. Lorenser, R. A. McLaughlin, B. C. Quirk, R. W. Kirk, B. E. Bouma, and D. D. Sampson, “Deep tissue volume imaging of birefringence through fibre-optic needle probes for the delineation of breast tumour,” Sci. Rep. 6, 28771 (2016).
[Crossref] [PubMed]

Ryu, M.

Sakai, S.

Sampson, D. D.

M. Villiger, D. Lorenser, R. A. McLaughlin, B. C. Quirk, R. W. Kirk, B. E. Bouma, and D. D. Sampson, “Deep tissue volume imaging of birefringence through fibre-optic needle probes for the delineation of breast tumour,” Sci. Rep. 6, 28771 (2016).
[Crossref] [PubMed]

Schmidt-Erfurth, U.

M. Pircher, C. K. Hitzenberger, and U. Schmidt-Erfurth, “Polarization sensitive optical coherence tomography in the human eye,” Progress in Retinal and Eye Research 30, 431–451 (2011).
[Crossref] [PubMed]

Shiga, Y.

Spaide, R. F.

R. F. Spaide, H. Koizumi, and M. C. Pozonni, “Enhanced depth imaging spectral-domain optical coherence tomography,” Am. J. Ophthalmol. 146, 496–500 (2008).
[Crossref] [PubMed]

Steart, P. V.

A. R. Elkington, C. B. Inman, P. V. Steart, and R. O. Weller, “The structure of the lamina cribrosa of the human eye: an immunocytochemical and electron microscopical study,” Eye 4, 42–57 (1990).
[Crossref] [PubMed]

Stifter, D.

Sugiyama, S.

Swanson, E. A.

Takahashi, H.

Tang, S.

Tearney, G. J.

S. K. Nadkarni, M. C. Pierce, B. H. Park, J. F. de Boer, P. Whittaker, B. E. Bouma, J. E. Bressner, E. Halpern, S. L. Houser, and G. J. Tearney, “Measurement of Collagen and Smooth Muscle Cell Content in Atherosclerotic Plaques Using Polarization-Sensitive Optical Coherence Tomography,” J. Am. College Cardiol. 49, 1474–1481 (2007).
[Crossref]

Tiao, G. C.

G. E. P. Box and G. C. Tiao, Bayesian Inference in Statistical Analysis (John Wiley & Sons, 1992).
[Crossref]

Tsuda, S.

Uematsu, S.

Ushiki, T.

Y. Komai and T. Ushiki, “The three-dimensional organization of collagen fibrils in the human cornea and sclera,” Investigative Ophthalmol. Vis. Sci. 32, 2244–2258 (1991).

Vakoc, B. J.

Villiger, M.

Wang, L.

Wang, Q.

Weller, R. O.

A. R. Elkington, C. B. Inman, P. V. Steart, and R. O. Weller, “The structure of the lamina cribrosa of the human eye: an immunocytochemical and electron microscopical study,” Eye 4, 42–57 (1990).
[Crossref] [PubMed]

Whittaker, P.

S. K. Nadkarni, M. C. Pierce, B. H. Park, J. F. de Boer, P. Whittaker, B. E. Bouma, J. E. Bressner, E. Halpern, S. L. Houser, and G. J. Tearney, “Measurement of Collagen and Smooth Muscle Cell Content in Atherosclerotic Plaques Using Polarization-Sensitive Optical Coherence Tomography,” J. Am. College Cardiol. 49, 1474–1481 (2007).
[Crossref]

Wiesauer, K.

Yamanari, M.

M. Yamanari, S. Tsuda, T. Kokubun, Y. Shiga, K. Omodaka, Y. Yokoyama, N. Himori, M. Ryu, S. Kunimatsu-Sanuki, H. Takahashi, K. Maruyama, H. Kunikata, and T. Nakazawa, “Fiber-based polarization-sensitive OCT for birefringence imaging of the anterior eye segment,” Biomed. Opt. Express 6, 369 (2015).
[Crossref] [PubMed]

S. Fukuda, M. Yamanari, Y. Lim, S. Hoshi, S. Beheregaray, T. Oshika, and Y. Yasuno, “Keratoconus diagnosis using anterior segment polarization-sensitive optical coherence tomography,” Investigative Ophthalmol. Vis. Sci. 54, 1384–1391 (2013).
[Crossref]

Y. Lim, Y.-J. Hong, L. Duan, M. Yamanari, and Y. Yasuno, “Passive component based multifunctional Jones matrix swept source optical coherence tomography for Doppler and polarization imaging,” Opt. Lett. 37, 1958–1960 (2012).
[Crossref] [PubMed]

S. Sakai, M. Yamanari, Y. Lim, N. Nakagawa, and Y. Yasuno, “In vivo evaluation of human skin anisotropy by polarization-sensitive optical coherence tomography,” Biomed. Opt. Express 2, 2623–2631 (2011).
[Crossref] [PubMed]

L. Duan, S. Makita, M. Yamanari, Y. Lim, and Y. Yasuno, “Monte-Carlo-based phase retardation estimator for polarization sensitive optical coherence tomography,” Opt. Express 19, 16330–16345 (2011).
[Crossref] [PubMed]

Y. Lim, M. Yamanari, S. Fukuda, Y. Kaji, T. Kiuchi, M. Miura, T. Oshika, and Y. Yasuno, “Birefringence measurement of cornea and anterior segment by office-based polarization-sensitive optical coherence tomography,” Biomed. Opt. Express 2, 2392–2402 (2011).
[Crossref] [PubMed]

S. Makita, M. Yamanari, and Y. Yasuno, “Generalized Jones matrix optical coherence tomography: performance and local birefringence imaging,” Opt. Express 18, 854–876 (2010).
[Crossref] [PubMed]

M. Yamanari, M. Miura, S. Makita, T. Yatagai, and Y. Yasuno, “Phase retardation measurement of retinal nerve fiber layer by polarization-sensitive spectral-domain optical coherence tomography and scanning laser polarimetry,” J. Biomed. Opt. 13, 014013 (2008).
[Crossref] [PubMed]

Y. Yasuno, M.-J. Ju, Y. J. Hong, S. Makita, Y. Lim, and M. Yamanari, “Jones Matrix Based Polarization Sensitive Optical Coherence Tomography,” in Optical Coherence Tomography: Technology and Applications2nd ed. (Springer, 2015) pp. 1137–1162.
[Crossref]

Yasuno, Y.

S. Makita, K. Kurokawa, Y.-J. Hong, M. Miura, and Y. Yasuno, “Noise-immune complex correlation for optical coherence angiography based on standard and Jones matrix optical coherence tomography,” Biomed. Opt. Express 7, 1525 (2016).
[Crossref] [PubMed]

S. Sugiyama, Y.-J. Hong, D. Kasaragod, S. Makita, S. Uematsu, Y. Ikuno, M. Miura, and Y. Yasuno, “Birefringence imaging of posterior eye by multi-functional Jones matrix optical coherence tomography,” Biomed. Opt. Express 6, 4951 (2015).
[Crossref] [PubMed]

S. Fukuda, S. Beheregaray, D. Kasaragod, S. Hoshi, G. Kishino, K. Ishii, Y. Yasuno, and T. Oshika, “Noninvasive Evaluation of Phase Retardation in Blebs After Glaucoma Surgery Using Anterior Segment Polarization-Sensitive Optical Coherence Tomography,” Investigative Ophthalmol. Vis. Sci. 55, 5200–5206 (2014).
[Crossref]

D. Kasaragod, S. Makita, S. Fukuda, S. Beheregaray, T. Oshika, and Y. Yasuno, “Bayesian maximum likelihood estimator of phase retardation for quantitative polarization-sensitive optical coherence tomography,” Opt. Express 22, 16472–16492 (2014).
[Crossref] [PubMed]

S. Makita, Y.-J. Hong, M. Miura, and Y. Yasuno, “Degree of polarization uniformity with high noise immunity using polarization-sensitive optical coherence tomography,” Opt. Lett. 39, 6783–6786 (2014).
[Crossref] [PubMed]

M. J. Ju, Y.-J. Hong, S. Makita, Y. Lim, K. Kurokawa, L. Duan, M. Miura, S. Tang, and Y. Yasuno, “Advanced multi-contrast Jones matrix optical coherence tomography for Doppler and polarization sensitive imaging,” Opt. Express 21, 19412–19436 (2013).
[Crossref] [PubMed]

S. Fukuda, M. Yamanari, Y. Lim, S. Hoshi, S. Beheregaray, T. Oshika, and Y. Yasuno, “Keratoconus diagnosis using anterior segment polarization-sensitive optical coherence tomography,” Investigative Ophthalmol. Vis. Sci. 54, 1384–1391 (2013).
[Crossref]

Y. Lim, Y.-J. Hong, L. Duan, M. Yamanari, and Y. Yasuno, “Passive component based multifunctional Jones matrix swept source optical coherence tomography for Doppler and polarization imaging,” Opt. Lett. 37, 1958–1960 (2012).
[Crossref] [PubMed]

Y. Lim, M. Yamanari, S. Fukuda, Y. Kaji, T. Kiuchi, M. Miura, T. Oshika, and Y. Yasuno, “Birefringence measurement of cornea and anterior segment by office-based polarization-sensitive optical coherence tomography,” Biomed. Opt. Express 2, 2392–2402 (2011).
[Crossref] [PubMed]

L. Duan, S. Makita, M. Yamanari, Y. Lim, and Y. Yasuno, “Monte-Carlo-based phase retardation estimator for polarization sensitive optical coherence tomography,” Opt. Express 19, 16330–16345 (2011).
[Crossref] [PubMed]

S. Sakai, M. Yamanari, Y. Lim, N. Nakagawa, and Y. Yasuno, “In vivo evaluation of human skin anisotropy by polarization-sensitive optical coherence tomography,” Biomed. Opt. Express 2, 2623–2631 (2011).
[Crossref] [PubMed]

S. Makita, M. Yamanari, and Y. Yasuno, “Generalized Jones matrix optical coherence tomography: performance and local birefringence imaging,” Opt. Express 18, 854–876 (2010).
[Crossref] [PubMed]

M. Yamanari, M. Miura, S. Makita, T. Yatagai, and Y. Yasuno, “Phase retardation measurement of retinal nerve fiber layer by polarization-sensitive spectral-domain optical coherence tomography and scanning laser polarimetry,” J. Biomed. Opt. 13, 014013 (2008).
[Crossref] [PubMed]

Y. Yasuno, M.-J. Ju, Y. J. Hong, S. Makita, Y. Lim, and M. Yamanari, “Jones Matrix Based Polarization Sensitive Optical Coherence Tomography,” in Optical Coherence Tomography: Technology and Applications2nd ed. (Springer, 2015) pp. 1137–1162.
[Crossref]

Yatagai, T.

M. Yamanari, M. Miura, S. Makita, T. Yatagai, and Y. Yasuno, “Phase retardation measurement of retinal nerve fiber layer by polarization-sensitive spectral-domain optical coherence tomography and scanning laser polarimetry,” J. Biomed. Opt. 13, 014013 (2008).
[Crossref] [PubMed]

Yokoyama, Y.

Yoon, C. J.

Yoon, Y.

Zhang, E. Z.

Zhang, J.

Zhao, L.

Am. J. Ophthalmol. (1)

R. F. Spaide, H. Koizumi, and M. C. Pozonni, “Enhanced depth imaging spectral-domain optical coherence tomography,” Am. J. Ophthalmol. 146, 496–500 (2008).
[Crossref] [PubMed]

Biomed. Opt. Express (7)

Y. Lim, M. Yamanari, S. Fukuda, Y. Kaji, T. Kiuchi, M. Miura, T. Oshika, and Y. Yasuno, “Birefringence measurement of cornea and anterior segment by office-based polarization-sensitive optical coherence tomography,” Biomed. Opt. Express 2, 2392–2402 (2011).
[Crossref] [PubMed]

S. Sakai, M. Yamanari, Y. Lim, N. Nakagawa, and Y. Yasuno, “In vivo evaluation of human skin anisotropy by polarization-sensitive optical coherence tomography,” Biomed. Opt. Express 2, 2623–2631 (2011).
[Crossref] [PubMed]

B. Cense, Q. Wang, S. Lee, L. Zhao, A. E. Elsner, C. K. Hitzenberger, and D. T. Miller, “Henle fiber layer phase retardation measured with polarization-sensitive optical coherence tomography,” Biomed. Opt. Express 4, 2296 (2013).
[Crossref] [PubMed]

M. Yamanari, S. Tsuda, T. Kokubun, Y. Shiga, K. Omodaka, Y. Yokoyama, N. Himori, M. Ryu, S. Kunimatsu-Sanuki, H. Takahashi, K. Maruyama, H. Kunikata, and T. Nakazawa, “Fiber-based polarization-sensitive OCT for birefringence imaging of the anterior eye segment,” Biomed. Opt. Express 6, 369 (2015).
[Crossref] [PubMed]

V.-H. Le, S. Lee, B. Kim, Y. Yoon, C. J. Yoon, W. K. Chung, and K. H. Kim, “Correlation between polarization sensitive optical coherence tomography and second harmonic generation microscopy in skin,” Biomed. Opt. Express 6, 2542–2551 (2015).
[Crossref] [PubMed]

S. Sugiyama, Y.-J. Hong, D. Kasaragod, S. Makita, S. Uematsu, Y. Ikuno, M. Miura, and Y. Yasuno, “Birefringence imaging of posterior eye by multi-functional Jones matrix optical coherence tomography,” Biomed. Opt. Express 6, 4951 (2015).
[Crossref] [PubMed]

S. Makita, K. Kurokawa, Y.-J. Hong, M. Miura, and Y. Yasuno, “Noise-immune complex correlation for optical coherence angiography based on standard and Jones matrix optical coherence tomography,” Biomed. Opt. Express 7, 1525 (2016).
[Crossref] [PubMed]

Eye (1)

A. R. Elkington, C. B. Inman, P. V. Steart, and R. O. Weller, “The structure of the lamina cribrosa of the human eye: an immunocytochemical and electron microscopical study,” Eye 4, 42–57 (1990).
[Crossref] [PubMed]

Investigative Ophthalmol. Vis. Sci. (3)

S. Fukuda, M. Yamanari, Y. Lim, S. Hoshi, S. Beheregaray, T. Oshika, and Y. Yasuno, “Keratoconus diagnosis using anterior segment polarization-sensitive optical coherence tomography,” Investigative Ophthalmol. Vis. Sci. 54, 1384–1391 (2013).
[Crossref]

S. Fukuda, S. Beheregaray, D. Kasaragod, S. Hoshi, G. Kishino, K. Ishii, Y. Yasuno, and T. Oshika, “Noninvasive Evaluation of Phase Retardation in Blebs After Glaucoma Surgery Using Anterior Segment Polarization-Sensitive Optical Coherence Tomography,” Investigative Ophthalmol. Vis. Sci. 55, 5200–5206 (2014).
[Crossref]

Y. Komai and T. Ushiki, “The three-dimensional organization of collagen fibrils in the human cornea and sclera,” Investigative Ophthalmol. Vis. Sci. 32, 2244–2258 (1991).

J. Am. College Cardiol. (1)

S. K. Nadkarni, M. C. Pierce, B. H. Park, J. F. de Boer, P. Whittaker, B. E. Bouma, J. E. Bressner, E. Halpern, S. L. Houser, and G. J. Tearney, “Measurement of Collagen and Smooth Muscle Cell Content in Atherosclerotic Plaques Using Polarization-Sensitive Optical Coherence Tomography,” J. Am. College Cardiol. 49, 1474–1481 (2007).
[Crossref]

J. Biomed. Opt. (2)

M. Yamanari, M. Miura, S. Makita, T. Yatagai, and Y. Yasuno, “Phase retardation measurement of retinal nerve fiber layer by polarization-sensitive spectral-domain optical coherence tomography and scanning laser polarimetry,” J. Biomed. Opt. 13, 014013 (2008).
[Crossref] [PubMed]

J. F. De Boer and T. E. Milner, “Review of polarization sensitive optical coherence tomography and Stokes vector determination,” J. Biomed. Opt. 7, 359–371 (2002).
[Crossref] [PubMed]

J. Opt. Soc. Am. B (1)

Opt. Express (7)

L. Duan, S. Makita, M. Yamanari, Y. Lim, and Y. Yasuno, “Monte-Carlo-based phase retardation estimator for polarization sensitive optical coherence tomography,” Opt. Express 19, 16330–16345 (2011).
[Crossref] [PubMed]

K. Wiesauer, M. Pircher, E. Goetzinger, C. K. Hitzenberger, R. Engelke, G. Ahrens, G. Gruetzner, and D. Stifter, “Transversal ultrahigh-resolution polarization sensitive optical coherence tomography for strain mapping in materials,” Opt. Express 14, 5945–5953 (2006).
[Crossref] [PubMed]

S. Makita, M. Yamanari, and Y. Yasuno, “Generalized Jones matrix optical coherence tomography: performance and local birefringence imaging,” Opt. Express 18, 854–876 (2010).
[Crossref] [PubMed]

D. Kasaragod, S. Makita, S. Fukuda, S. Beheregaray, T. Oshika, and Y. Yasuno, “Bayesian maximum likelihood estimator of phase retardation for quantitative polarization-sensitive optical coherence tomography,” Opt. Express 22, 16472–16492 (2014).
[Crossref] [PubMed]

B. Baumann, W. Choi, B. Potsaid, D. Huang, J. S. Duker, and J. G. Fujimoto, “Swept source / Fourier domain polarization sensitive optical coherence tomography with a passive polarization delay unit,” Opt. Express 20, 10218–10230 (2012).
[Crossref]

M. Villiger, E. Z. Zhang, S. K. Nadkarni, W.-Y. Oh, B. J. Vakoc, and B. E. Bouma, “Spectral binning for mitigation of polarization mode dispersion artifacts in catheter-based optical frequency domain imaging,” Opt. Express 21, 16353 (2013).
[Crossref] [PubMed]

M. J. Ju, Y.-J. Hong, S. Makita, Y. Lim, K. Kurokawa, L. Duan, M. Miura, S. Tang, and Y. Yasuno, “Advanced multi-contrast Jones matrix optical coherence tomography for Doppler and polarization sensitive imaging,” Opt. Express 21, 19412–19436 (2013).
[Crossref] [PubMed]

Opt. Lett. (5)

Progress in Retinal and Eye Research (1)

M. Pircher, C. K. Hitzenberger, and U. Schmidt-Erfurth, “Polarization sensitive optical coherence tomography in the human eye,” Progress in Retinal and Eye Research 30, 431–451 (2011).
[Crossref] [PubMed]

Sci. Rep. (1)

M. Villiger, D. Lorenser, R. A. McLaughlin, B. C. Quirk, R. W. Kirk, B. E. Bouma, and D. D. Sampson, “Deep tissue volume imaging of birefringence through fibre-optic needle probes for the delineation of breast tumour,” Sci. Rep. 6, 28771 (2016).
[Crossref] [PubMed]

The Annals of Statistics (1)

Z. I. Botev, J. F. Grotowski, and D. P. Kroese, “Kernel density estimation via diffusion,” The Annals of Statistics 38, 2916–2957 (2010).
[Crossref]

Other (2)

Y. Yasuno, M.-J. Ju, Y. J. Hong, S. Makita, Y. Lim, and M. Yamanari, “Jones Matrix Based Polarization Sensitive Optical Coherence Tomography,” in Optical Coherence Tomography: Technology and Applications2nd ed. (Springer, 2015) pp. 1137–1162.
[Crossref]

G. E. P. Box and G. C. Tiao, Bayesian Inference in Statistical Analysis (John Wiley & Sons, 1992).
[Crossref]

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Figures (5)

Fig. 1
Fig. 1

The process diagram of birefringence estimation. Green boxes and yellow rounded boxes indicate data and processes, respectively. The preparation process (upper dashed box) is performed only once, while the measurement and estimation process (lower dashed box) is performed for each measurement.

Fig. 2
Fig. 2

(a) Estimated phase retardation from 1,024 sets of the numerically simulated measured phase retardation and ESNR. The red, blue, and black curves represent the estimation results of the new MAP, old MAP, and mean estimator. The dashed horizontal lines represent 10 true phase retardation values. (b) Root mean square error of the estimations. The mean errors were computed over the 10 true phase retardations. The type of estimator is indicated by the same color in (a).

Fig. 3
Fig. 3

Comparison of the mean, old MAP, and new MAP estimators for in vivo posterior eye imaging. The right and left column represent the birefringence estimation without and with adaptive Jones matrix averaging (AJA), respectively. Each row represents the mean, old MAP, and new MAP estimations, from top to bottom.

Fig. 4
Fig. 4

The histograms of Fig. 3. The horizontal axis represents phase retardation. The subfigure labels correspond to those of Fig. 3. The dotted line represents the 2π/3, to which the mean estimation erroneously converges as the ESNR decreases.

Fig. 5
Fig. 5

Comparison of mean (a)–(c) and new MAP (d)–(f) estimators for 4, 16, and 64 repeated measurements. The first and second rows show pseudo-color birefringence images obtained using the mean and new MAP estimators, respectively. The third and fourth rows show the line profiles of birefringence along A1–A2 (third row) and B1–B2 (fourth row) lines. The black and red curves represent the mean and new MAP estimators, respectively. The positions of the profiles are indicated in (d).

Equations (19)

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J l ( z ; Z d ) J s ( z 2 ) J s ( z 1 ) 1 ,
b ( z ) = δ ( z ) / 2 k 0 Z d ,
J l ( z ; Z d ) = [ J s ( z 2 ) + N ( z 2 ) ] + [ J s ( z 1 ) + N ( z 1 ) ] 1 ,
ESNR = [ 1 4 j = 1 2 i = 1 2 ( I h ( j ) ( z i ) + I v ( j ) z i σ n 2 ( z i ) ) 1 ] 1 ,
p ( β | b , g ) p ( b , g | β ) π ( β ) ,
p ( b , g | β ) = + p ( b , g | β , γ ) p ( γ | β ) d γ ,
p ( b , g | β ) = + p ( b , g | β , γ ) p ( γ ) d γ ,
f ( β ; b , g ) p ( b , g | β ) + p ( b , g | β , γ ) p ( γ ) d γ .
p ( γ | g ) p ( g | γ ) π ( γ ) ,
p ( γ | g ) π ( γ ) + [ + p ( b , g | β , γ ) d b ] π ( β ) d β ,
f ( β ; b , g ) + p ( b , g | β , γ ) [ π ( γ ) p ( b , g | β , γ ) π ( β ) d b d β ] d γ .
p 1 ( β ) p ( β | b = b 1 , g = g 1 ) f ( β ; b 1 , g 1 ) ,
p 2 ( β ) f ( β ; b 2 , g 2 ) p 1 ( β ) = f ( β ; b 2 , g 2 ) f ( β ; b 1 , g 1 ) .
p N ( β ) i = 1 N f ( β ; b i , g i ) .
β ^ = arg max β p N ( β ) = arg max β i = 1 N f ( β ; b i , g i ) .
( β ^ ; Δ β ) = β ^ Δ β / 2 β ^ + Δ β / 2 p N ( β ) d β .
S = 2 Sigmoid ( a ) 1 ,
p ( γ ) p ( γ | g ) p ( g | γ ) π ( γ ) π ( γ ) p ( g | β , γ ) p ( β | γ ) d β π ( γ ) p ( g | β , γ ) p ( β ) d β .
p ( γ | g ) π ( γ ) + [ + p ( b , g | β , γ ) d b ] p ( β ) d β .

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