Abstract

Estimation of polarimetric parameters has been a fundamental issue to assess biological tissues that have form birefringence or polarization scrambling in polarization-sensitive optical coherence tomography (PS-OCT). We present a mathematical framework to provide a maximum likelihood estimation of the target covariance matrix and its incoherent target decomposition to estimate a Jones matrix of a dominant scattering mechanism, called Cloude-Pottier decomposition, thereby deriving the phase retardation and the optic axis of the sample. In addition, we introduce entropy that shows the randomness of the polarization property. Underestimation of the entropy at a low sampling number is mitigated by asymptotic quasi maximum likelihood estimator. A bias of the entropy from random noises is corrected to show only the polarization property inherent in the sample. The theory is validated with experimental measurements of a glass plate and waveplates, and applied to the imaging of a healthy human eye anterior segment as an image filter.

© 2016 Optical Society of America

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Corrections

Masahiro Yamanari, Satoru Tsuda, Taiki Kokubun, Yukihiro Shiga, Kazuko Omodaka, Naoko Aizawa, Yu Yokoyama, Noriko Himori, Shiho Kunimatsu-Sanuki, Kazuichi Maruyama, Hiroshi Kunikata, and Toru Nakazawa, "Estimation of Jones matrix, birefringence and entropy using Cloude-Pottier decomposition in polarization-sensitive optical coherence tomography: erratum," Biomed. Opt. Express 7, 4636-4638 (2016)
https://www.osapublishing.org/boe/abstract.cfm?uri=boe-7-11-4636

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References

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2016 (2)

2015 (7)

N. Ortega-Quijano, F. Fanjul-Vélez, and J. L. Arce-Diego, “Physically meaningful depolarization metric based on the differential Mueller matrix,” Opt. Lett. 40(14), 3280–3283 (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(12), 4951–4974 (2015).
[Crossref] [PubMed]

N. Lippok, M. Villiger, and B. E. Bouma, “Degree of polarization (uniformity) and depolarization index: unambiguous depolarization contrast for optical coherence tomography,” Opt. Lett. 40(17), 3954–3957 (2015).
[Crossref] [PubMed]

M. J. A. Girard, M. Ang, C. W. Chung, M. Farook, N. Strouthidis, J. S. Mehta, and J. M. Mari, “Enhancement of Corneal Visibility in Optical Coherence Tomography Images Using Corneal Adaptive Compensation,” Transl. Vis. Sci. Technol. 4(3), 3 (2015).
[Crossref] [PubMed]

B. Baumann, J. Schirmer, S. Rauscher, S. Fialová, M. Glösmann, M. Augustin, M. Pircher, M. Gröger, and C. K. Hitzenberger, “Melanin Pigmentation in Rat Eyes: In Vivo Imaging by Polarization-Sensitive Optical Coherence Tomography and Comparison to Histology,” Invest. Ophthalmol. Vis. Sci. 56(12), 7462–7472 (2015).
[Crossref] [PubMed]

J. Li, F. Feroldi, J. de Lange, J. M. A. Daniels, K. Grünberg, and J. F. de Boer, “Polarization sensitive optical frequency domain imaging system for endobronchial imaging,” Opt. Express 23(3), 3390–3402 (2015).
[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(2), 369–389 (2015).
[Crossref] [PubMed]

2014 (10)

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(13), 16472–16492 (2014).
[Crossref] [PubMed]

Z. Wang, H.-C. Lee, O. O. Ahsen, B. Lee, W. Choi, B. Potsaid, J. Liu, V. Jayaraman, A. Cable, M. F. Kraus, K. Liang, J. Hornegger, and J. G. Fujimoto, “Depth-encoded all-fiber swept source polarization sensitive OCT,” Biomed. Opt. Express 5(9), 2931–2949 (2014).
[Crossref] [PubMed]

B. Braaf, K. A. Vermeer, M. de Groot, K. V. Vienola, and J. F. de Boer, “Fiber-based polarization-sensitive OCT of the human retina with correction of system polarization distortions,” Biomed. Opt. Express 5(8), 2736–2758 (2014).
[Crossref] [PubMed]

C. López-Martínez, A. Alonso-González, and X. Fabregas, “Perturbation Analysis of Eigenvector-Based Target Decomposition Theorems in Radar Polarimetry,” IEEE Trans. Geosci. Remote Sens. 52(4), 2081–2095 (2014).
[Crossref]

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(24), 6783–6786 (2014).
[Crossref] [PubMed]

K. A. Vermeer, J. Mo, J. J. A. Weda, H. G. Lemij, and J. F. de Boer, “Depth-resolved model-based reconstruction of attenuation coefficients in optical coherence tomography,” Biomed. Opt. Express 5(1), 322–337 (2014).
[Crossref] [PubMed]

Z. Lu, D. Kasaragod, and S. J. Matcher, “Conical scan polarization-sensitive optical coherence tomography,” Biomed. Opt. Express 5(3), 752–762 (2014).
[Crossref] [PubMed]

Y.-J. Hong, M. Miura, M. J. Ju, S. Makita, T. Iwasaki, and Y. Yasuno, “Simultaneous Investigation of Vascular and Retinal Pigment Epithelial Pathologies of Exudative Macular Diseases by Multifunctional Optical Coherence Tomography,” Invest. Ophthalmol. Vis. Sci. 55(8), 5016–5031 (2014).
[Crossref] [PubMed]

M. Villiger and B. E. Bouma, “Practical decomposition for physically admissible differential Mueller matrices,” Opt. Lett. 39(7), 1779–1782 (2014).
[Crossref] [PubMed]

J. J. Gil, “Review on Mueller matrix algebra for the analysis of polarimetric measurements,” J. Appl. Remote Sens. 8(1), 081599 (2014).
[Crossref]

2013 (4)

2012 (2)

S. Savenkov, A. Priezzhev, Y. Oberemok, P. Silfsten, T. Ervasti, J. Ketolainen, and K.-E. Peiponen, “Characterization of porous media by means of the depolarization metrics,” J. Quant. Spectrosc. Radiat. Transf. 113(18), 2503–2511 (2012).
[Crossref]

B. Baumann, S. O. Baumann, T. Konegger, M. Pircher, E. Götzinger, F. Schlanitz, C. Schütze, H. Sattmann, M. Litschauer, U. Schmidt-Erfurth, and C. K. Hitzenberger, “Polarization sensitive optical coherence tomography of melanin provides intrinsic contrast based on depolarization,” Biomed. Opt. Express 3(7), 1670–1683 (2012).
[Crossref] [PubMed]

2011 (5)

2010 (3)

2009 (2)

E. Götzinger, B. Baumann, M. Pircher, and C. K. Hitzenberger, “Polarization maintaining fiber based ultra-high resolution spectral domain polarization sensitive optical coherence tomography,” Opt. Express 17(25), 22704–22717 (2009).
[Crossref] [PubMed]

B. Baumann, E. Götzinger, M. Pircher, and C. K. Hitzenberger, “Measurements of depolarization distribution in the healthy human macula by polarization sensitive OCT,” J. Biophotonics 2(6-7), 426–434 (2009).
[Crossref] [PubMed]

2008 (5)

M. Miura, M. Yamanari, T. Iwasaki, A. E. Elsner, S. Makita, T. Yatagai, and Y. Yasuno, “Imaging Polarimetry in Age-Related Macular Degeneration,” Invest. Ophthalmol. Vis. Sci. 49(6), 2661–2667 (2008).
[Crossref] [PubMed]

J.-S. Lee, T. L. Ainsworth, J. P. Kelly, and C. López-Martínez, “Evaluation and Bias Removal of Multilook Effect on Entropy/Alpha/Anisotropy in Polarimetric SAR Decomposition,” IEEE Trans. Geosci. Remote Sens. 46(10), 3039–3052 (2008).
[Crossref]

J. C. Mansfield, C. P. Winlove, J. Moger, and S. J. Matcher, “Collagen fiber arrangement in normal and diseased cartilage studied by polarization sensitive nonlinear microscopy,” J. Biomed. Opt. 13(4), 044020 (2008).
[Crossref] [PubMed]

M. Yamanari, S. Makita, and Y. Yasuno, “Polarization-sensitive swept-source optical coherence tomography with continuous source polarization modulation,” Opt. Express 16(8), 5892–5906 (2008).
[Crossref] [PubMed]

E. Götzinger, M. Pircher, W. Geitzenauer, C. Ahlers, B. Baumann, S. Michels, U. Schmidt-Erfurth, and C. K. Hitzenberger, “Retinal pigment epithelium segmentation by polarization sensitive optical coherence tomography,” Opt. Express 16(21), 16410–16422 (2008).
[Crossref] [PubMed]

2007 (1)

F. Cao, W. Hong, Y. Wu, and E. Pottier, “An Unsupervised Segmentation With an Adaptive Number of Clusters Using the SPAN/H/α/A Space and the Complex Wishart Clustering for Fully Polarimetric SAR Data Analysis,” IEEE Trans. Geosci. Remote Sens. 45(11), 3454–3467 (2007).
[Crossref]

2005 (4)

A. Aiello and J. P. Woerdman, “Physical Bounds to the Entropy-Depolarization Relation in Random Light Scattering,” Phys. Rev. Lett. 94(9), 090406 (2005).
[Crossref] [PubMed]

C. López-Martínez, E. Pottier, and S. R. Cloude, “Statistical Assessment of Eigenvector-Based Target Decomposition Theorems in Radar Polarimetry,” IEEE Trans. Geosci. Remote Sens. 43(9), 2058–2074 (2005).
[Crossref]

B. H. Park, M. C. Pierce, B. Cense, and J. F. de Boer, “Optic axis determination accuracy for fiber-based polarization-sensitive optical coherence tomography,” Opt. Lett. 30(19), 2587–2589 (2005).
[Crossref] [PubMed]

B. Park, M. C. Pierce, B. Cense, S.-H. Yun, M. Mujat, G. Tearney, B. Bouma, and J. de Boer, “Real-time fiber-based multi-functional spectral-domain optical coherence tomography at 1.3 µm,” Opt. Express 13(11), 3931–3944 (2005).
[Crossref] [PubMed]

2004 (5)

2003 (1)

2002 (2)

2001 (2)

C. Hitzenberger, E. Goetzinger, M. Sticker, M. Pircher, and A. Fercher, “Measurement and imaging of birefringence and optic axis orientation by phase resolved polarization sensitive optical coherence tomography,” Opt. Express 9(13), 780–790 (2001).
[Crossref] [PubMed]

L. Ferro-Famil, E. Pottier, and J.-S. Lee, “Unsupervised classification of multifrequency and fully polarimetric SAR images based on the H/A/Alpha-Wishart classifier,” IEEE Trans. Geosci. Remote Sens. 39(11), 2332–2342 (2001).
[Crossref]

1999 (3)

1998 (2)

1997 (2)

J. F. de Boer, T. E. Milner, M. J. C. van Gemert, and J. S. Nelson, “Two-dimensional birefringence imaging in biological tissue by polarization-sensitive optical coherence tomography,” Opt. Lett. 22(12), 934–936 (1997).
[Crossref] [PubMed]

S. R. Cloude and E. Pottier, “An entropy based classification scheme for land applications of polarimetric SAR,” IEEE Trans. Geosci. Remote Sens. 35(1), 68–78 (1997).
[Crossref]

1996 (1)

S. R. Cloude and E. Pottier, “A review of target decomposition theorems in radar polarimetry,” IEEE Trans. Geosci. Remote Sens. 34(2), 498–518 (1996).
[Crossref]

1995 (1)

S. R. Cloude and E. Pottier, “Concept of polarization entropy in optical scattering,” Opt. Eng. 34(6), 1599–1610 (1995).
[Crossref]

1994 (2)

J.-S. Lee, K. W. Hoppel, S. A. Mango, and A. R. Miller, “Intensity and phase statistics of multilook polarimetric and interferometric SAR imagery,” IEEE Trans. Geosci. Remote Sens. 32(5), 1017–1028 (1994).
[Crossref]

J. S. Lee, M. R. Grunes, and R. Kwok, “Classification of multi-look polarimetric SAR imagery based on complex Wishart distribution,” Int. J. Remote Sens. 15(11), 2299–2311 (1994).
[Crossref]

1993 (1)

1992 (1)

1991 (1)

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

1986 (1)

S. R. Cloude, “Group theory and polarisation algebra,” Optik (Stuttg.) 75(1), 26–36 (1986).

1973 (2)

H. Takenaka, “A Unified Formalism for Polarization Optics by Using Group Theory I (Theory),” Jpn. J. Appl. Phys. 12(2), 226–231 (1973).
[Crossref]

H. Takenaka, “A unified formalism for polarization optics by using group theory,” Nouvelle Revue d’Optique 4(1), 37–41 (1973).
[Crossref]

1965 (1)

L. Mandel and E. Wolf, “Coherence Properties of Optical Fields,” Rev. Mod. Phys. 37(2), 231–287 (1965).
[Crossref]

1963 (1)

N. R. Goodman, “Statistical Analysis Based on a Certain Multivariate Complex Gaussian Distribution (An Introduction),” Ann. Math. Stat. 34(1), 152–177 (1963).
[Crossref]

1941 (1)

Ahlers, C.

Ahsen, O. O.

Aiello, A.

A. Aiello and J. P. Woerdman, “Physical Bounds to the Entropy-Depolarization Relation in Random Light Scattering,” Phys. Rev. Lett. 94(9), 090406 (2005).
[Crossref] [PubMed]

Ainsworth, T. L.

J.-S. Lee, T. L. Ainsworth, J. P. Kelly, and C. López-Martínez, “Evaluation and Bias Removal of Multilook Effect on Entropy/Alpha/Anisotropy in Polarimetric SAR Decomposition,” IEEE Trans. Geosci. Remote Sens. 46(10), 3039–3052 (2008).
[Crossref]

J.-S. Lee, M. R. Grunes, T. L. Ainsworth, L.-J. Du, D. L. Schuler, and S. R. Cloude, “Unsupervised classification using polarimetric decomposition and the complex Wishart classifier,” IEEE Trans. Geosci. Remote Sens. 37(5), 2249–2258 (1999).
[Crossref]

Alonso-González, A.

C. López-Martínez, A. Alonso-González, and X. Fabregas, “Perturbation Analysis of Eigenvector-Based Target Decomposition Theorems in Radar Polarimetry,” IEEE Trans. Geosci. Remote Sens. 52(4), 2081–2095 (2014).
[Crossref]

Alvarez-Perez, J. L.

J. L. Alvarez-Perez, “Coherence, Polarization, and Statistical Independence in Cloude-Pottier’s Radar Polarimetry,” IEEE Trans. Geosci. Remote Sens. 49(1), 426–441 (2011).
[Crossref]

Ang, M.

M. J. A. Girard, M. Ang, C. W. Chung, M. Farook, N. Strouthidis, J. S. Mehta, and J. M. Mari, “Enhancement of Corneal Visibility in Optical Coherence Tomography Images Using Corneal Adaptive Compensation,” Transl. Vis. Sci. Technol. 4(3), 3 (2015).
[Crossref] [PubMed]

Arce-Diego, J. L.

Augustin, M.

S. Fialová, M. Augustin, M. Glösmann, T. Himmel, S. Rauscher, M. Gröger, M. Pircher, C. K. Hitzenberger, and B. Baumann, “Polarization properties of single layers in the posterior eyes of mice and rats investigated using high resolution polarization sensitive optical coherence tomography,” Biomed. Opt. Express 7(4), 1479–1495 (2016).
[Crossref] [PubMed]

B. Baumann, J. Schirmer, S. Rauscher, S. Fialová, M. Glösmann, M. Augustin, M. Pircher, M. Gröger, and C. K. Hitzenberger, “Melanin Pigmentation in Rat Eyes: In Vivo Imaging by Polarization-Sensitive Optical Coherence Tomography and Comparison to Histology,” Invest. Ophthalmol. Vis. Sci. 56(12), 7462–7472 (2015).
[Crossref] [PubMed]

Barakat, R.

Baumann, B.

S. Fialová, M. Augustin, M. Glösmann, T. Himmel, S. Rauscher, M. Gröger, M. Pircher, C. K. Hitzenberger, and B. Baumann, “Polarization properties of single layers in the posterior eyes of mice and rats investigated using high resolution polarization sensitive optical coherence tomography,” Biomed. Opt. Express 7(4), 1479–1495 (2016).
[Crossref] [PubMed]

B. Baumann, J. Schirmer, S. Rauscher, S. Fialová, M. Glösmann, M. Augustin, M. Pircher, M. Gröger, and C. K. Hitzenberger, “Melanin Pigmentation in Rat Eyes: In Vivo Imaging by Polarization-Sensitive Optical Coherence Tomography and Comparison to Histology,” Invest. Ophthalmol. Vis. Sci. 56(12), 7462–7472 (2015).
[Crossref] [PubMed]

B. Baumann, S. O. Baumann, T. Konegger, M. Pircher, E. Götzinger, F. Schlanitz, C. Schütze, H. Sattmann, M. Litschauer, U. Schmidt-Erfurth, and C. K. Hitzenberger, “Polarization sensitive optical coherence tomography of melanin provides intrinsic contrast based on depolarization,” Biomed. Opt. Express 3(7), 1670–1683 (2012).
[Crossref] [PubMed]

E. Götzinger, M. Pircher, B. Baumann, T. Schmoll, H. Sattmann, R. A. Leitgeb, and C. K. Hitzenberger, “Speckle noise reduction in high speed polarization sensitive spectral domain optical coherence tomography,” Opt. Express 19(15), 14568–14585 (2011).
[Crossref] [PubMed]

E. Götzinger, B. Baumann, M. Pircher, and C. K. Hitzenberger, “Polarization maintaining fiber based ultra-high resolution spectral domain polarization sensitive optical coherence tomography,” Opt. Express 17(25), 22704–22717 (2009).
[Crossref] [PubMed]

B. Baumann, E. Götzinger, M. Pircher, and C. K. Hitzenberger, “Measurements of depolarization distribution in the healthy human macula by polarization sensitive OCT,” J. Biophotonics 2(6-7), 426–434 (2009).
[Crossref] [PubMed]

E. Götzinger, M. Pircher, W. Geitzenauer, C. Ahlers, B. Baumann, S. Michels, U. Schmidt-Erfurth, and C. K. Hitzenberger, “Retinal pigment epithelium segmentation by polarization sensitive optical coherence tomography,” Opt. Express 16(21), 16410–16422 (2008).
[Crossref] [PubMed]

Baumann, S. O.

Beheregaray, S.

Bizheva, K.

Boerner, W. M.

R. Touzi, W. M. Boerner, J. S. Lee, and E. Lueneburg, “A review of polarimetry in the context of synthetic aperture radar: concepts and information extraction,” Can. J. Rem. Sens. 30(3), 380–407 (2004).
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Bouma, B.

Bouma, B. E.

Braaf, B.

Brosseau, C.

Cable, A.

Cao, F.

F. Cao, W. Hong, Y. Wu, and E. Pottier, “An Unsupervised Segmentation With an Adaptive Number of Clusters Using the SPAN/H/α/A Space and the Complex Wishart Clustering for Fully Polarimetric SAR Data Analysis,” IEEE Trans. Geosci. Remote Sens. 45(11), 3454–3467 (2007).
[Crossref]

Cense, B.

Chang, W.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Choi, W.

Chung, C. W.

M. J. A. Girard, M. Ang, C. W. Chung, M. Farook, N. Strouthidis, J. S. Mehta, and J. M. Mari, “Enhancement of Corneal Visibility in Optical Coherence Tomography Images Using Corneal Adaptive Compensation,” Transl. Vis. Sci. Technol. 4(3), 3 (2015).
[Crossref] [PubMed]

Cloude, S. R.

C. López-Martínez, E. Pottier, and S. R. Cloude, “Statistical Assessment of Eigenvector-Based Target Decomposition Theorems in Radar Polarimetry,” IEEE Trans. Geosci. Remote Sens. 43(9), 2058–2074 (2005).
[Crossref]

J.-S. Lee, M. R. Grunes, T. L. Ainsworth, L.-J. Du, D. L. Schuler, and S. R. Cloude, “Unsupervised classification using polarimetric decomposition and the complex Wishart classifier,” IEEE Trans. Geosci. Remote Sens. 37(5), 2249–2258 (1999).
[Crossref]

S. R. Cloude and E. Pottier, “An entropy based classification scheme for land applications of polarimetric SAR,” IEEE Trans. Geosci. Remote Sens. 35(1), 68–78 (1997).
[Crossref]

S. R. Cloude and E. Pottier, “A review of target decomposition theorems in radar polarimetry,” IEEE Trans. Geosci. Remote Sens. 34(2), 498–518 (1996).
[Crossref]

S. R. Cloude and E. Pottier, “Concept of polarization entropy in optical scattering,” Opt. Eng. 34(6), 1599–1610 (1995).
[Crossref]

S. R. Cloude, “Group theory and polarisation algebra,” Optik (Stuttg.) 75(1), 26–36 (1986).

Collet, C.

Colston, B. W.

Da Silva, L. B.

Daniels, J. M. A.

de Boer, J.

de Boer, J. F.

J. Li, F. Feroldi, J. de Lange, J. M. A. Daniels, K. Grünberg, and J. F. de Boer, “Polarization sensitive optical frequency domain imaging system for endobronchial imaging,” Opt. Express 23(3), 3390–3402 (2015).
[Crossref] [PubMed]

B. Braaf, K. A. Vermeer, M. de Groot, K. V. Vienola, and J. F. de Boer, “Fiber-based polarization-sensitive OCT of the human retina with correction of system polarization distortions,” Biomed. Opt. Express 5(8), 2736–2758 (2014).
[Crossref] [PubMed]

K. A. Vermeer, J. Mo, J. J. A. Weda, H. G. Lemij, and J. F. de Boer, “Depth-resolved model-based reconstruction of attenuation coefficients in optical coherence tomography,” Biomed. Opt. Express 5(1), 322–337 (2014).
[Crossref] [PubMed]

B. H. Park, M. C. Pierce, B. Cense, and J. F. de Boer, “Optic axis determination accuracy for fiber-based polarization-sensitive optical coherence tomography,” Opt. Lett. 30(19), 2587–2589 (2005).
[Crossref] [PubMed]

B. H. Park, M. C. Pierce, B. Cense, and J. F. de Boer, “Jones matrix analysis for a polarization-sensitive optical coherence tomography system using fiber-optic components,” Opt. Lett. 29(21), 2512–2514 (2004).
[Crossref] [PubMed]

J. F. de Boer, T. E. Milner, and J. S. Nelson, “Determination of the depth-resolved Stokes parameters of light backscattered from turbid media by use of polarization-sensitive optical coherence tomography,” Opt. Lett. 24(5), 300–302 (1999).
[Crossref] [PubMed]

J. F. de Boer, T. E. Milner, M. J. C. van Gemert, and J. S. Nelson, “Two-dimensional birefringence imaging in biological tissue by polarization-sensitive optical coherence tomography,” Opt. Lett. 22(12), 934–936 (1997).
[Crossref] [PubMed]

de Groot, M.

de Lange, J.

Du, L.-J.

J.-S. Lee, M. R. Grunes, T. L. Ainsworth, L.-J. Du, D. L. Schuler, and S. R. Cloude, “Unsupervised classification using polarimetric decomposition and the complex Wishart classifier,” IEEE Trans. Geosci. Remote Sens. 37(5), 2249–2258 (1999).
[Crossref]

Duan, L.

Ellerbee Bowden, A. K.

Elsner, A. E.

M. Miura, M. Yamanari, T. Iwasaki, A. E. Elsner, S. Makita, T. Yatagai, and Y. Yasuno, “Imaging Polarimetry in Age-Related Macular Degeneration,” Invest. Ophthalmol. Vis. Sci. 49(6), 2661–2667 (2008).
[Crossref] [PubMed]

Ervasti, T.

S. Savenkov, A. Priezzhev, Y. Oberemok, P. Silfsten, T. Ervasti, J. Ketolainen, and K.-E. Peiponen, “Characterization of porous media by means of the depolarization metrics,” J. Quant. Spectrosc. Radiat. Transf. 113(18), 2503–2511 (2012).
[Crossref]

Everett, M. J.

Fabregas, X.

C. López-Martínez, A. Alonso-González, and X. Fabregas, “Perturbation Analysis of Eigenvector-Based Target Decomposition Theorems in Radar Polarimetry,” IEEE Trans. Geosci. Remote Sens. 52(4), 2081–2095 (2014).
[Crossref]

Fan, C.

Fanjul-Vélez, F.

Farook, M.

M. J. A. Girard, M. Ang, C. W. Chung, M. Farook, N. Strouthidis, J. S. Mehta, and J. M. Mari, “Enhancement of Corneal Visibility in Optical Coherence Tomography Images Using Corneal Adaptive Compensation,” Transl. Vis. Sci. Technol. 4(3), 3 (2015).
[Crossref] [PubMed]

Fercher, A.

Feroldi, F.

Ferro-Famil, L.

J.-S. Lee, M. R. Grunes, E. Pottier, and L. Ferro-Famil, “Unsupervised terrain classification preserving polarimetric scattering characteristics,” IEEE Trans. Geosci. Remote Sens. 42(4), 722–731 (2004).
[Crossref]

L. Ferro-Famil, E. Pottier, and J.-S. Lee, “Unsupervised classification of multifrequency and fully polarimetric SAR images based on the H/A/Alpha-Wishart classifier,” IEEE Trans. Geosci. Remote Sens. 39(11), 2332–2342 (2001).
[Crossref]

Fialová, S.

S. Fialová, M. Augustin, M. Glösmann, T. Himmel, S. Rauscher, M. Gröger, M. Pircher, C. K. Hitzenberger, and B. Baumann, “Polarization properties of single layers in the posterior eyes of mice and rats investigated using high resolution polarization sensitive optical coherence tomography,” Biomed. Opt. Express 7(4), 1479–1495 (2016).
[Crossref] [PubMed]

B. Baumann, J. Schirmer, S. Rauscher, S. Fialová, M. Glösmann, M. Augustin, M. Pircher, M. Gröger, and C. K. Hitzenberger, “Melanin Pigmentation in Rat Eyes: In Vivo Imaging by Polarization-Sensitive Optical Coherence Tomography and Comparison to Histology,” Invest. Ophthalmol. Vis. Sci. 56(12), 7462–7472 (2015).
[Crossref] [PubMed]

Flotte, T.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Fujimoto, J.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Fujimoto, J. G.

Fukuda, S.

Geissbuehler, M.

Geitzenauer, W.

Gil, J. J.

J. J. Gil, “Review on Mueller matrix algebra for the analysis of polarimetric measurements,” J. Appl. Remote Sens. 8(1), 081599 (2014).
[Crossref]

Girard, M. J. A.

M. J. A. Girard, M. Ang, C. W. Chung, M. Farook, N. Strouthidis, J. S. Mehta, and J. M. Mari, “Enhancement of Corneal Visibility in Optical Coherence Tomography Images Using Corneal Adaptive Compensation,” Transl. Vis. Sci. Technol. 4(3), 3 (2015).
[Crossref] [PubMed]

J. M. Mari, N. G. Strouthidis, S. C. Park, and M. J. A. Girard, “Enhancement of Lamina Cribrosa Visibility in Optical Coherence Tomography Images Using Adaptive Compensation,” Invest. Ophthalmol. Vis. Sci. 54(3), 2238–2247 (2013).
[Crossref] [PubMed]

Glösmann, M.

S. Fialová, M. Augustin, M. Glösmann, T. Himmel, S. Rauscher, M. Gröger, M. Pircher, C. K. Hitzenberger, and B. Baumann, “Polarization properties of single layers in the posterior eyes of mice and rats investigated using high resolution polarization sensitive optical coherence tomography,” Biomed. Opt. Express 7(4), 1479–1495 (2016).
[Crossref] [PubMed]

B. Baumann, J. Schirmer, S. Rauscher, S. Fialová, M. Glösmann, M. Augustin, M. Pircher, M. Gröger, and C. K. Hitzenberger, “Melanin Pigmentation in Rat Eyes: In Vivo Imaging by Polarization-Sensitive Optical Coherence Tomography and Comparison to Histology,” Invest. Ophthalmol. Vis. Sci. 56(12), 7462–7472 (2015).
[Crossref] [PubMed]

Goetzinger, E.

Goodman, N. R.

N. R. Goodman, “Statistical Analysis Based on a Certain Multivariate Complex Gaussian Distribution (An Introduction),” Ann. Math. Stat. 34(1), 152–177 (1963).
[Crossref]

Götzinger, E.

Gregory, K.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
[Crossref] [PubMed]

Gröger, M.

S. Fialová, M. Augustin, M. Glösmann, T. Himmel, S. Rauscher, M. Gröger, M. Pircher, C. K. Hitzenberger, and B. Baumann, “Polarization properties of single layers in the posterior eyes of mice and rats investigated using high resolution polarization sensitive optical coherence tomography,” Biomed. Opt. Express 7(4), 1479–1495 (2016).
[Crossref] [PubMed]

B. Baumann, J. Schirmer, S. Rauscher, S. Fialová, M. Glösmann, M. Augustin, M. Pircher, M. Gröger, and C. K. Hitzenberger, “Melanin Pigmentation in Rat Eyes: In Vivo Imaging by Polarization-Sensitive Optical Coherence Tomography and Comparison to Histology,” Invest. Ophthalmol. Vis. Sci. 56(12), 7462–7472 (2015).
[Crossref] [PubMed]

Grünberg, K.

Grunes, M. R.

J.-S. Lee, M. R. Grunes, E. Pottier, and L. Ferro-Famil, “Unsupervised terrain classification preserving polarimetric scattering characteristics,” IEEE Trans. Geosci. Remote Sens. 42(4), 722–731 (2004).
[Crossref]

J.-S. Lee, M. R. Grunes, T. L. Ainsworth, L.-J. Du, D. L. Schuler, and S. R. Cloude, “Unsupervised classification using polarimetric decomposition and the complex Wishart classifier,” IEEE Trans. Geosci. Remote Sens. 37(5), 2249–2258 (1999).
[Crossref]

J. S. Lee, M. R. Grunes, and R. Kwok, “Classification of multi-look polarimetric SAR imagery based on complex Wishart distribution,” Int. J. Remote Sens. 15(11), 2299–2311 (1994).
[Crossref]

Hee, M. R.

M. R. Hee, D. Huang, E. A. Swanson, and J. G. Fujimoto, “Polarization-sensitive low-coherence reflectometer for birefringence characterization and ranging,” J. Opt. Soc. Am. B 9(6), 903–908 (1992).
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B. Baumann, J. Schirmer, S. Rauscher, S. Fialová, M. Glösmann, M. Augustin, M. Pircher, M. Gröger, and C. K. Hitzenberger, “Melanin Pigmentation in Rat Eyes: In Vivo Imaging by Polarization-Sensitive Optical Coherence Tomography and Comparison to Histology,” Invest. Ophthalmol. Vis. Sci. 56(12), 7462–7472 (2015).
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E. Götzinger, M. Pircher, B. Baumann, T. Schmoll, H. Sattmann, R. A. Leitgeb, and C. K. Hitzenberger, “Speckle noise reduction in high speed polarization sensitive spectral domain optical coherence tomography,” Opt. Express 19(15), 14568–14585 (2011).
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E. Götzinger, B. Baumann, M. Pircher, and C. K. Hitzenberger, “Polarization maintaining fiber based ultra-high resolution spectral domain polarization sensitive optical coherence tomography,” Opt. Express 17(25), 22704–22717 (2009).
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B. Baumann, E. Götzinger, M. Pircher, and C. K. Hitzenberger, “Measurements of depolarization distribution in the healthy human macula by polarization sensitive OCT,” J. Biophotonics 2(6-7), 426–434 (2009).
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E. Götzinger, M. Pircher, W. Geitzenauer, C. Ahlers, B. Baumann, S. Michels, U. Schmidt-Erfurth, and C. K. Hitzenberger, “Retinal pigment epithelium segmentation by polarization sensitive optical coherence tomography,” Opt. Express 16(21), 16410–16422 (2008).
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F. Cao, W. Hong, Y. Wu, and E. Pottier, “An Unsupervised Segmentation With an Adaptive Number of Clusters Using the SPAN/H/α/A Space and the Complex Wishart Clustering for Fully Polarimetric SAR Data Analysis,” IEEE Trans. Geosci. Remote Sens. 45(11), 3454–3467 (2007).
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J.-S. Lee, T. L. Ainsworth, J. P. Kelly, and C. López-Martínez, “Evaluation and Bias Removal of Multilook Effect on Entropy/Alpha/Anisotropy in Polarimetric SAR Decomposition,” IEEE Trans. Geosci. Remote Sens. 46(10), 3039–3052 (2008).
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J.-S. Lee, M. R. Grunes, T. L. Ainsworth, L.-J. Du, D. L. Schuler, and S. R. Cloude, “Unsupervised classification using polarimetric decomposition and the complex Wishart classifier,” IEEE Trans. Geosci. Remote Sens. 37(5), 2249–2258 (1999).
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Z. Lu, D. Kasaragod, and S. J. Matcher, “Conical scan polarization-sensitive optical coherence tomography,” Biomed. Opt. Express 5(3), 752–762 (2014).
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Z. Lu, D. K. Kasaragod, and S. J. Matcher, “Method to calibrate phase fluctuation in polarization-sensitive swept-source optical coherence tomography,” J. Biomed. Opt. 16(7), 070502 (2011).
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J.-S. Lee, K. W. Hoppel, S. A. Mango, and A. R. Miller, “Intensity and phase statistics of multilook polarimetric and interferometric SAR imagery,” IEEE Trans. Geosci. Remote Sens. 32(5), 1017–1028 (1994).
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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(12), 4951–4974 (2015).
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Y.-J. Hong, M. Miura, M. J. Ju, S. Makita, T. Iwasaki, and Y. Yasuno, “Simultaneous Investigation of Vascular and Retinal Pigment Epithelial Pathologies of Exudative Macular Diseases by Multifunctional Optical Coherence Tomography,” Invest. Ophthalmol. Vis. Sci. 55(8), 5016–5031 (2014).
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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(24), 6783–6786 (2014).
[Crossref] [PubMed]

M. Miura, M. Yamanari, T. Iwasaki, A. E. Elsner, S. Makita, T. Yatagai, and Y. Yasuno, “Imaging Polarimetry in Age-Related Macular Degeneration,” Invest. Ophthalmol. Vis. Sci. 49(6), 2661–2667 (2008).
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Moayed, A. A.

Moger, J.

J. C. Mansfield, C. P. Winlove, J. Moger, and S. J. Matcher, “Collagen fiber arrangement in normal and diseased cartilage studied by polarization sensitive nonlinear microscopy,” J. Biomed. Opt. 13(4), 044020 (2008).
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R. H. Newton and K. M. Meek, “The Integration of the Corneal and Limbal Fibrils in the Human Eye,” Biophys. J. 75(5), 2508–2512 (1998).
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S. Savenkov, A. Priezzhev, Y. Oberemok, P. Silfsten, T. Ervasti, J. Ketolainen, and K.-E. Peiponen, “Characterization of porous media by means of the depolarization metrics,” J. Quant. Spectrosc. Radiat. Transf. 113(18), 2503–2511 (2012).
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Park, S. C.

J. M. Mari, N. G. Strouthidis, S. C. Park, and M. J. A. Girard, “Enhancement of Lamina Cribrosa Visibility in Optical Coherence Tomography Images Using Adaptive Compensation,” Invest. Ophthalmol. Vis. Sci. 54(3), 2238–2247 (2013).
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S. Savenkov, A. Priezzhev, Y. Oberemok, P. Silfsten, T. Ervasti, J. Ketolainen, and K.-E. Peiponen, “Characterization of porous media by means of the depolarization metrics,” J. Quant. Spectrosc. Radiat. Transf. 113(18), 2503–2511 (2012).
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Pircher, M.

S. Fialová, M. Augustin, M. Glösmann, T. Himmel, S. Rauscher, M. Gröger, M. Pircher, C. K. Hitzenberger, and B. Baumann, “Polarization properties of single layers in the posterior eyes of mice and rats investigated using high resolution polarization sensitive optical coherence tomography,” Biomed. Opt. Express 7(4), 1479–1495 (2016).
[Crossref] [PubMed]

B. Baumann, J. Schirmer, S. Rauscher, S. Fialová, M. Glösmann, M. Augustin, M. Pircher, M. Gröger, and C. K. Hitzenberger, “Melanin Pigmentation in Rat Eyes: In Vivo Imaging by Polarization-Sensitive Optical Coherence Tomography and Comparison to Histology,” Invest. Ophthalmol. Vis. Sci. 56(12), 7462–7472 (2015).
[Crossref] [PubMed]

B. Baumann, S. O. Baumann, T. Konegger, M. Pircher, E. Götzinger, F. Schlanitz, C. Schütze, H. Sattmann, M. Litschauer, U. Schmidt-Erfurth, and C. K. Hitzenberger, “Polarization sensitive optical coherence tomography of melanin provides intrinsic contrast based on depolarization,” Biomed. Opt. Express 3(7), 1670–1683 (2012).
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E. Götzinger, M. Pircher, B. Baumann, T. Schmoll, H. Sattmann, R. A. Leitgeb, and C. K. Hitzenberger, “Speckle noise reduction in high speed polarization sensitive spectral domain optical coherence tomography,” Opt. Express 19(15), 14568–14585 (2011).
[Crossref] [PubMed]

E. Götzinger, B. Baumann, M. Pircher, and C. K. Hitzenberger, “Polarization maintaining fiber based ultra-high resolution spectral domain polarization sensitive optical coherence tomography,” Opt. Express 17(25), 22704–22717 (2009).
[Crossref] [PubMed]

B. Baumann, E. Götzinger, M. Pircher, and C. K. Hitzenberger, “Measurements of depolarization distribution in the healthy human macula by polarization sensitive OCT,” J. Biophotonics 2(6-7), 426–434 (2009).
[Crossref] [PubMed]

E. Götzinger, M. Pircher, W. Geitzenauer, C. Ahlers, B. Baumann, S. Michels, U. Schmidt-Erfurth, and C. K. Hitzenberger, “Retinal pigment epithelium segmentation by polarization sensitive optical coherence tomography,” Opt. Express 16(21), 16410–16422 (2008).
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Pottier, E.

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[Crossref]

C. López-Martínez, E. Pottier, and S. R. Cloude, “Statistical Assessment of Eigenvector-Based Target Decomposition Theorems in Radar Polarimetry,” IEEE Trans. Geosci. Remote Sens. 43(9), 2058–2074 (2005).
[Crossref]

J.-S. Lee, M. R. Grunes, E. Pottier, and L. Ferro-Famil, “Unsupervised terrain classification preserving polarimetric scattering characteristics,” IEEE Trans. Geosci. Remote Sens. 42(4), 722–731 (2004).
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L. Ferro-Famil, E. Pottier, and J.-S. Lee, “Unsupervised classification of multifrequency and fully polarimetric SAR images based on the H/A/Alpha-Wishart classifier,” IEEE Trans. Geosci. Remote Sens. 39(11), 2332–2342 (2001).
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S. Savenkov, A. Priezzhev, Y. Oberemok, P. Silfsten, T. Ervasti, J. Ketolainen, and K.-E. Peiponen, “Characterization of porous media by means of the depolarization metrics,” J. Quant. Spectrosc. Radiat. Transf. 113(18), 2503–2511 (2012).
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D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
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S. Fialová, M. Augustin, M. Glösmann, T. Himmel, S. Rauscher, M. Gröger, M. Pircher, C. K. Hitzenberger, and B. Baumann, “Polarization properties of single layers in the posterior eyes of mice and rats investigated using high resolution polarization sensitive optical coherence tomography,” Biomed. Opt. Express 7(4), 1479–1495 (2016).
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B. Baumann, J. Schirmer, S. Rauscher, S. Fialová, M. Glösmann, M. Augustin, M. Pircher, M. Gröger, and C. K. Hitzenberger, “Melanin Pigmentation in Rat Eyes: In Vivo Imaging by Polarization-Sensitive Optical Coherence Tomography and Comparison to Histology,” Invest. Ophthalmol. Vis. Sci. 56(12), 7462–7472 (2015).
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Sattmann, H.

Savenkov, S.

S. Savenkov, A. Priezzhev, Y. Oberemok, P. Silfsten, T. Ervasti, J. Ketolainen, and K.-E. Peiponen, “Characterization of porous media by means of the depolarization metrics,” J. Quant. Spectrosc. Radiat. Transf. 113(18), 2503–2511 (2012).
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Schirmer, J.

B. Baumann, J. Schirmer, S. Rauscher, S. Fialová, M. Glösmann, M. Augustin, M. Pircher, M. Gröger, and C. K. Hitzenberger, “Melanin Pigmentation in Rat Eyes: In Vivo Imaging by Polarization-Sensitive Optical Coherence Tomography and Comparison to Histology,” Invest. Ophthalmol. Vis. Sci. 56(12), 7462–7472 (2015).
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J.-S. Lee, M. R. Grunes, T. L. Ainsworth, L.-J. Du, D. L. Schuler, and S. R. Cloude, “Unsupervised classification using polarimetric decomposition and the complex Wishart classifier,” IEEE Trans. Geosci. Remote Sens. 37(5), 2249–2258 (1999).
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D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. Fujimoto, “Optical coherence tomography,” Science 254(5035), 1178–1181 (1991).
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Figures (7)

Fig. 1
Fig. 1

Plots of phase retardations of the glass plate (a), the EWP (b) and the QWP (c). The horizontal axes show the index of A-scan. For the plots of mean, E(R) and R1, all A-scans from 0 to n-th indices are used for the estimation. Since it is clear that the true values of (a) and (c) are on the sides of the measurable range, true values are not shown for them.

Fig. 2
Fig. 2

Box plots of the optic axis estimated by Cloude-Pottier decomposition. The sample was the EWP. Different number of A-scans were used for the ensemble average of (T) as indicated by the figure legend, and the distributions of the estimated optic axes were shown. The plots of the different number of A-scans were displaced horizontally from the true set orientation only for the purpose of visualization. An interquartile range, a median and 100% fraction range of the distribution are indicated by the box, the bar in the box and the whiskers, respectively.

Fig. 3
Fig. 3

Measured entropy without (solid line) or with (dashed line) AQ-MLE. The horizontal axis indicates the number of A-scans n used for the ensemble average of (T). The raw OCT data were measured at different signal intensities, and the results were plotted with different colors. The samples and the measurement settings are shown in Table 1.

Fig. 4
Fig. 4

Estimation results of Hnoise at a single depth. The settings (a)-(f) in the figure legend correspond to Table 1. The black solid line shows ideal estimation where the measured entropy equals Hnoise.

Fig. 5
Fig. 5

Estimation results of Hnoise for the local Jones matrix. The samples and settings of (a)-(f) correspond to Table 1. The black solid line shows ideal estimation where the measured entropy equals Hnoise.

Fig. 6
Fig. 6

Images of the anterior eye segment around the angle at the superior region. Polarization-diverse signal intensity (a), the attenuation coefficient (b), the local retardation R1 (c), the entropy without the correction methods (e), the entropy with AQ-MLE and the bias correction of Hnoise (g). Figures (d), (f) and (h) are the composite images of (c), (e) and (g) with (b), respectively. The scale bars indicate 1 mm in air. Ametrine color map [66] was used for (c)-(h).

Fig. 7
Fig. 7

Local retardation images of the anterior eye segment processed by the coherent Gaussian spatial filter (a) and Cloude-Pottier decomposition (b). The image (b) is exactly same as Fig. 6(c). The scale bars indicate 1 mm in air.

Tables (1)

Tables Icon

Table 1 Relative signal intensities between the Jones-matrix elements used for the test target analysis

Equations (46)

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J=[ E 1 ¯ E 2 ¯ ]=[ J 1H J 2H J 1V J 2V ],
κ ¯ = 1 2 Tr( JΨ ),
{ Ψ L }={ 2[ 1 0 0 0 ] 2[ 0 1 0 0 ] 2[ 0 0 1 0 ] 2[ 0 0 0 1 ] }.
κ L ¯ = [ J 1H J 2H J 1V J 2V ] T ,
p( κ ¯ )= 1 π m det(C) exp( κ ¯ C 1 κ ¯ ),
Z= 1 n i=1 n κ ¯ i κ ¯ i ,
p(Z)= n mn [det(Z)] nm [det(C)] n Γ ˜ m (n) exp[ nTr( C 1 Z ) ],
Γ ˜ m (n)= π m(m1)/2 i=1 m Γ(ni+1) ,
T κ L ¯ κ L ¯ ¯ =[ | J 1H | 2 ¯ J 1H J 2H * ¯ J 1H J 1V * ¯ J 1H J 2V * ¯ J 2H J 1H * ¯ | J 2H | 2 ¯ J 2H J 1V * ¯ J 2H J 2V * ¯ J 1V J 1H * ¯ J 1V J 2H * ¯ | J 1V | 2 ¯ J 1V J 2V * ¯ J 2V J 1H * ¯ J 2V J 2H * ¯ J 2V J 1V * ¯ | J 2V | 2 ¯ ],
T=UA U , A=diag[ λ 1 , λ 2 , λ 3 , λ 4 ], λ 1 λ 2 λ 3 λ 4 0, U=[ e 1 ¯ e 2 ¯ e 3 ¯ e 4 ¯ ],
λ i = λ i j 4 λ j ,i=1,2,3,4,
e i ¯ := [ a i b i c i d i ] T , L i =[ a i b i c i d i ].
E sample (z δz 2 ) 2 E sample (z+ δz 2 ) 2 E sample 1 (z δz 2 ) E sample (z+ δz 2 ) 2 E sample 1 (z δz 2 ) E sample (z+ δz 2 ),
R i =arg( ξ 1 (i) ξ 2 (i)* ),
E(R)= i λ i R i .
H= i λ i log 4 λ i ,
l i = λ i λ i n j=1,ji m λ j λ i λ j O( n 1 ),i=1,2,,m ,
H ^ = i=1 4 l i log 4 l i , l i = l i i=1 4 l i ,
H measured = H subject + H noise ,
H noise ( E 1 ¯ , E 2 ¯ ) H noise ( E 1 ¯ )+ H noise ( E 2 ¯ ),
H noise ( E 1 ¯ , E 2 ¯ )= H noise ( E 1 ¯ )+ H noise ( E 2 ¯ ).
H noise ( E i ¯ )= j=1 2 ζ j (i) log( ζ j (i) ) ,
ζ j (i) = 1± P (i) 2 ,
J measured =[ E 1 ¯ + n 1 ¯ E 2 ¯ + n 2 ¯ ]=[ E 1H + n 1H E 2H + n 2H E 1V + n 1V E 2V + n 2V ]=[ g 1H g 2H g 1V g 2V ].
[ s 0 (1) s 1 (1) s 2 (1) s 3 (1) ]=[ | g 1H | 2 + | g 1V | 2 | g 1H | 2 | g 1V | 2 2Re[ g 1H g 1V * ] 2Im[ g 1H g 1V * ] ],
[ s 0 (1) s 1 (1) s 2 (1) s 3 (1) ]:=[ | E 1H | 2 + | E 1V | 2 | E 1H | 2 | E 1V | 2 2Re[ E 1H E 1V * ] 2Im[ E 1H E 1V * ] ]=[ | E 1H | 2 + | E 1V | 2 | E 1H | 2 | E 1V | 2 2| E 1H || E 1V |cosδ 2| E 1H || E 1V |sinδ ],
[ s 0 (1) s 1 (1) s 2 (1) s 3 (1) ]:=[ | E 1H | 2 ¯ + | E 1V | 2 ¯ | E 1H | 2 ¯ | E 1V | 2 ¯ 2 | E 1H | ¯ | E 1V | ¯ cosδ 2 | E 1H | ¯ | E 1V | ¯ sinδ ]=[ | g 1H | 2 ¯ + | g 1V | 2 ¯ ( | n 1H | 2 ¯ + | n 1V | 2 ¯ ) | g 1H | 2 ¯ | g 1V | 2 ¯ ( | n 1H | 2 ¯ | n 1V | 2 ¯ ) 2 | g 1H | 2 ¯ | n 1H | 2 ¯ | g 1V | 2 ¯ | n 1V | 2 ¯ cosδ 2 | g 1H | 2 ¯ | n 1H | 2 ¯ | g 1V | 2 ¯ | n 1V | 2 ¯ sinδ ],
P (1) = { s 1 (1) } 2 + { s 2 (1) } 2 + { s 3 (1) } 2 s 0 (1) = [ | g 1H | 2 ¯ | g 1V | 2 ¯ ( | n 1H | 2 ¯ | n 1V | 2 ¯ ) ] 2 +4( | g 1H | 2 ¯ | n 1H | 2 ¯ )( | g 1V | 2 ¯ | n 1V | 2 ¯ ) | g 1H | 2 ¯ + | g 1V | 2 ¯ ( | n 1H | 2 ¯ + | n 1V | 2 ¯ ) .
E sample (z+ δz 2 )=[ a b c d ], E sample (z δz 2 )=[ e f g h ].
E sample 1 (z δz 2 ) E sample (z+ δz 2 )= [ det( E sample (z δz 2 ) ) ] 1 [ hafc hbfd ga+ec gb+ed ].
det( E sample 1 (z δz 2 ) ) E sample 1 (z δz 2 ) E sample (z+ δz 2 )=[ hafc hbfd ga+ec gb+ed ] [ ε 1 ¯ ε 2 ¯ ]=[ ε 1H ε 2H ε 1V ε 2V ].
H noise ( ε 1 ¯ , ε 2 ¯ )= H noise ( ε 1 ¯ )+ H noise ( ε 2 ¯ ) H noise ( ε 1 ¯ : ε 2 ¯ ),
H noise ( ε 1 ¯ : ε 2 ¯ )= H noise ( ε 1 ¯ ) H noise ( ε 1 ¯ | ε 2 ¯ ) = H noise ( ε 2 ¯ ) H noise ( ε 2 ¯ | ε 1 ¯ ),
H noise ( ε 1 ¯ , ε 2 ¯ )= H noise ( ε 2 ¯ )+ H noise ( ε 1 ¯ | ε 2 ¯ ) = H noise ( ε 1 ¯ )+ H noise ( ε 2 ¯ | ε 1 ¯ ).
Var( ε 1H )=Var(hafc)=Var(h)Var(a)+Var(h) { E(a) } 2 + { E(h) } 2 Var(a) +Var(f)Var(c)+Var(f) { E(c) } 2 + { E(f) } 2 Var(c),
Var( ε 1V )=Var(ga+ec)=Var(g)Var(a)+Var(g) { E(a) } 2 + { E(g) } 2 Var(a) +Var(e)Var(c)+Var(e) { E(c) } 2 + { E(e) } 2 Var(c),
Var( ε 2H )=Var(hbfd)=Var(h)Var(b)+Var(h) { E(b) } 2 + { E(h) } 2 Var(b) +Var(f)Var(d)+Var(f) { E(d) } 2 + { E(f) } 2 Var(d),
Var( ε 2V )=Var(gb+ed)=Var(g)Var(b)+Var(g) { E(b) } 2 + { E(g) } 2 Var(b) +Var(e)Var(d)+Var(e) { E(d) } 2 + { E(e) } 2 Var(d),
Var( ε 1H | ε 2 ¯ )= { E(h) } 2 Var(a)+ { E(f) } 2 Var(c),
Var( ε 1V | ε 2 ¯ )= { E(g) } 2 Var(a)+ { E(e) } 2 Var(c),
Var( ε 2H | ε 1 ¯ )= { E(h) } 2 Var(b)+ { E(f) } 2 Var(d),
Var( ε 2V | ε 1 ¯ )= { E(g) } 2 Var(b)+ { E(e) } 2 Var(d).
P ( ε 1 ¯ ) = [ | ε 1H | 2 ¯ | ε 1V | 2 ¯ ( Var( ε 1H )Var( ε 1V ) ) ] 2 +4( | ε 1H | 2 ¯ Var( ε 1H ) )( | ε 1V | 2 ¯ Var( ε 1V ) ) | ε 1H | 2 ¯ + | ε 1V | 2 ¯ ( Var( ε 1H )+Var( ε 1V ) ) ,
P ( ε 2 ¯ ) = [ | ε 2H | 2 ¯ | ε 2V | 2 ¯ ( Var( ε 2H )Var( ε 2V ) ) ] 2 +4( | ε 2H | 2 ¯ Var( ε 2H ) )( | ε 2V | 2 ¯ Var( ε 2V ) ) | ε 2H | 2 ¯ + | ε 2V | 2 ¯ ( Var( ε 2H )+Var( ε 2V ) ) ,
P ( ε 1 ¯ | ε 2 ¯ ) = [ | ε 1H | 2 ¯ | ε 1V | 2 ¯ ( Var( ε 1H | ε 2 ¯ )Var( ε 1V | ε 2 ¯ ) ) ] 2 +4( | ε 1H | 2 ¯ Var( ε 1H | ε 2 ¯ ) )( | ε 1V | 2 ¯ Var( ε 1V | ε 2 ¯ ) ) | ε 1H | 2 ¯ + | ε 1V | 2 ¯ ( Var( ε 1H | ε 2 ¯ )+Var( ε 1V | ε 2 ¯ ) ) ,
P ( ε 2 ¯ | ε 1 ¯ ) = [ | ε 2H | 2 ¯ | ε 2V | 2 ¯ ( Var( ε 2H | ε 1 ¯ )Var( ε 2V | ε 1 ¯ ) ) ] 2 +4( | ε 2H | 2 ¯ Var( ε 2H | ε 1 ¯ ) )( | ε 2V | 2 ¯ Var( ε 2V | ε 1 ¯ ) ) | ε 2H | 2 ¯ + | ε 2V | 2 ¯ ( Var( ε 2H | ε 1 ¯ )+Var( ε 2V | ε 1 ¯ ) ) .

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