Abstract

Polarization sensitive second harmonic generation (pSHG) microscopy has been used previously to characterize the structure of collagen fibers in corneal samples. Due to the typical organization of the corneal stroma, the information that pSHG provides may be misleading in points where two different collagen fiber bundles orient along different direction crossings. Here, a simulation that illustrates the problem is presented, along with a novel method that is capable of identifying these crossing points. These results can be used to improve the evaluation of corneal collagen structure, and it has been applied to analyze pSHG data acquired from healthy and keratoconic human corneal samples.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2015 (2)

2013 (1)

S. I. C. O. Santos, M. Mathew, O. E. Olarte, S. Psilodimitrakopoulos, and P. Loza-Alvarez, “Femtosecond laser axotomy in caenorhabditis elegans and collateral damage assessment using a combination of linear and nonlinear imaging techniques,” PLoS One 8(3), e58600 (2013).
[Crossref] [PubMed]

2012 (3)

2011 (1)

2010 (2)

S. Psilodimitrakopoulos, I. Amat-Roldan, P. Loza-Alvarez, and D. Artigas, “Estimating the helical pitch angle of amylopectin in starch using polarization second harmonic generation microscopy,” J. Opt. 12(8), 084007 (2010).
[Crossref]

I. Amat-Roldan, S. Psilodimitrakopoulos, P. Loza-Alvarez, and D. Artigas, “Fast image analysis in polarization SHG microscopy,” Opt. Express 18(16), 17209–17219 (2010).
[Crossref] [PubMed]

2009 (5)

S. Psilodimitrakopoulos, S. I. Santos, I. Amat-Roldan, A. K. Thayil, D. Artigas, and P. Loza-Alvarez, “In vivo, pixel-resolution mapping of thick filaments’ orientation in nonfibrilar muscle using polarization-sensitive second harmonic generation microscopy,” J. Biomed. Opt. 14(1), 014001 (2009).
[Crossref] [PubMed]

S. Psilodimitrakopoulos, D. Artigas, G. Soria, I. Amat-Roldan, A. M. Planas, and P. Loza-Alvarez, “Quantitative discrimination between endogenous SHG sources in mammalian tissue, based on their polarization response,” Opt. Express 17(12), 10168–10176 (2009).
[Crossref] [PubMed]

W.-L. Chen, T.-H. Li, P.-J. Su, C.-K. Chou, P. T. Fwu, and S.-J. Lin, “Second harmonic generation χ tensor microscopy for tissue imaging,” Appl. Phys. Lett. 94(18), 183902 (2009).
[Crossref]

S. Psilodimitrakopoulos, V. Petegnief, G. Soria, I. Amat-Roldan, D. Artigas, A. M. Planas, and P. Loza-Alvarez, “Estimation of the effective orientation of the SHG source in primary cortical neurons,” Opt. Express 17(16), 14418–14425 (2009).
[Crossref] [PubMed]

M. Mathew, S. I. C. O. Santos, D. Zalvidea, and P. Loza-Alvarez, “Multimodal optical workstation for simultaneous linear, nonlinear microscopy and nanomanipulation: Upgrading a commercial confocal inverted microscope,” Rev. Sci. Instrum. 80(7), 073701 (2009).
[Crossref] [PubMed]

2008 (2)

2007 (2)

F. Tiaho, G. Recher, and D. Rouède, “Estimation of helical angles of myosin and collagen by second harmonic generation imaging microscopy,” Opt. Express 15(19), 12286–12295 (2007).
[Crossref] [PubMed]

N. Morishige, A. J. Wahlert, M. C. Kenney, D. J. Brown, K. Kawamoto, T. Chikama, T. Nishida, and J. V. Jester, “Second-harmonic imaging microscopy of normal human and keratoconus cornea,” Invest. Ophthalmol. Vis. Sci. 48(3), 1087–1094 (2007).
[Crossref] [PubMed]

2006 (1)

H.-Y. Tan, Y. Sun, W. Lo, S.-J. Lin, C.-H. Hsiao, Y.-F. Chen, S. C.-M. Huang, W.-C. Lin, S.-H. Jee, H.-S. Yu, and C. Y. Dong, “Multiphoton fluorescence and second harmonic generation imaging of the structural alterations in keratoconus ex vivo,” Invest. Ophthalmol. Vis. Sci. 47(12), 5251–5259 (2006).
[Crossref] [PubMed]

2005 (1)

K. M. Meek, S. J. Tuft, Y. Huang, P. S. Gill, S. Hayes, R. H. Newton, and A. J. Bron, “Changes in collagen orientation and distribution in keratoconus corneas,” Invest. Ophthalmol. Vis. Sci. 46(6), 1948–1956 (2005).
[Crossref] [PubMed]

2004 (1)

S.-W. Chu, S.-Y. Chen, G.-W. Chern, T.-H. Tsai, Y.-C. Chen, B.-L. Lin, and C.-K. Sun, “Studies of χ(2)/χ(3) Tensors in Submicron-Scaled Bio-Tissues by Polarization Harmonics Optical Microscopy,” Biophys. J. 86(6), 3914–3922 (2004).
[Crossref] [PubMed]

2002 (1)

P. Stoller, K. M. Reiser, P. M. Celliers, and A. M. Rubenchik, “Polarization-modulated second harmonic generation in collagen,” Biophys. J. 82(6), 3330–3342 (2002).
[Crossref] [PubMed]

1998 (1)

W. Radner, M. Zehetmayer, C. Skorpik, and R. Mallinger, “Altered organization of collagen in the apex of keratoconus corneas,” Ophthalmic Res. 30(5), 327–332 (1998).
[Crossref] [PubMed]

1997 (1)

A. Daxer and P. Fratzl, “Collagen fibril orientation in the human corneal stroma and its implication in keratoconus,” Invest. Ophthalmol. Vis. Sci. 38(1), 121–129 (1997).
[PubMed]

1957 (1)

D. M. Maurice, “The structure and transparency of the cornea,” J. Physiol. 136(2), 263–286 (1957).
[Crossref] [PubMed]

Alkilani, A.

Amat-Roldan, I.

S. Psilodimitrakopoulos, I. Amat-Roldan, P. Loza-Alvarez, and D. Artigas, “Effect of molecular organization on the image histograms of polarization SHG microscopy,” Biomed. Opt. Express 3(10), 2681–2693 (2012).
[Crossref] [PubMed]

I. Amat-Roldan, S. Psilodimitrakopoulos, P. Loza-Alvarez, and D. Artigas, “Fast image analysis in polarization SHG microscopy,” Opt. Express 18(16), 17209–17219 (2010).
[Crossref] [PubMed]

S. Psilodimitrakopoulos, I. Amat-Roldan, P. Loza-Alvarez, and D. Artigas, “Estimating the helical pitch angle of amylopectin in starch using polarization second harmonic generation microscopy,” J. Opt. 12(8), 084007 (2010).
[Crossref]

S. Psilodimitrakopoulos, S. I. Santos, I. Amat-Roldan, A. K. Thayil, D. Artigas, and P. Loza-Alvarez, “In vivo, pixel-resolution mapping of thick filaments’ orientation in nonfibrilar muscle using polarization-sensitive second harmonic generation microscopy,” J. Biomed. Opt. 14(1), 014001 (2009).
[Crossref] [PubMed]

S. Psilodimitrakopoulos, V. Petegnief, G. Soria, I. Amat-Roldan, D. Artigas, A. M. Planas, and P. Loza-Alvarez, “Estimation of the effective orientation of the SHG source in primary cortical neurons,” Opt. Express 17(16), 14418–14425 (2009).
[Crossref] [PubMed]

S. Psilodimitrakopoulos, D. Artigas, G. Soria, I. Amat-Roldan, A. M. Planas, and P. Loza-Alvarez, “Quantitative discrimination between endogenous SHG sources in mammalian tissue, based on their polarization response,” Opt. Express 17(12), 10168–10176 (2009).
[Crossref] [PubMed]

Artigas, D.

S. Psilodimitrakopoulos, I. Amat-Roldan, P. Loza-Alvarez, and D. Artigas, “Effect of molecular organization on the image histograms of polarization SHG microscopy,” Biomed. Opt. Express 3(10), 2681–2693 (2012).
[Crossref] [PubMed]

I. Amat-Roldan, S. Psilodimitrakopoulos, P. Loza-Alvarez, and D. Artigas, “Fast image analysis in polarization SHG microscopy,” Opt. Express 18(16), 17209–17219 (2010).
[Crossref] [PubMed]

S. Psilodimitrakopoulos, I. Amat-Roldan, P. Loza-Alvarez, and D. Artigas, “Estimating the helical pitch angle of amylopectin in starch using polarization second harmonic generation microscopy,” J. Opt. 12(8), 084007 (2010).
[Crossref]

S. Psilodimitrakopoulos, D. Artigas, G. Soria, I. Amat-Roldan, A. M. Planas, and P. Loza-Alvarez, “Quantitative discrimination between endogenous SHG sources in mammalian tissue, based on their polarization response,” Opt. Express 17(12), 10168–10176 (2009).
[Crossref] [PubMed]

S. Psilodimitrakopoulos, V. Petegnief, G. Soria, I. Amat-Roldan, D. Artigas, A. M. Planas, and P. Loza-Alvarez, “Estimation of the effective orientation of the SHG source in primary cortical neurons,” Opt. Express 17(16), 14418–14425 (2009).
[Crossref] [PubMed]

S. Psilodimitrakopoulos, S. I. Santos, I. Amat-Roldan, A. K. Thayil, D. Artigas, and P. Loza-Alvarez, “In vivo, pixel-resolution mapping of thick filaments’ orientation in nonfibrilar muscle using polarization-sensitive second harmonic generation microscopy,” J. Biomed. Opt. 14(1), 014001 (2009).
[Crossref] [PubMed]

Boryskina, O. P.

Brasselet, S.

Bron, A. J.

K. M. Meek, S. J. Tuft, Y. Huang, P. S. Gill, S. Hayes, R. H. Newton, and A. J. Bron, “Changes in collagen orientation and distribution in keratoconus corneas,” Invest. Ophthalmol. Vis. Sci. 46(6), 1948–1956 (2005).
[Crossref] [PubMed]

Brown, D. J.

N. Morishige, A. J. Wahlert, M. C. Kenney, D. J. Brown, K. Kawamoto, T. Chikama, T. Nishida, and J. V. Jester, “Second-harmonic imaging microscopy of normal human and keratoconus cornea,” Invest. Ophthalmol. Vis. Sci. 48(3), 1087–1094 (2007).
[Crossref] [PubMed]

Brown, E. B.

Burke, R. M.

Celliers, P. M.

P. Stoller, K. M. Reiser, P. M. Celliers, and A. M. Rubenchik, “Polarization-modulated second harmonic generation in collagen,” Biophys. J. 82(6), 3330–3342 (2002).
[Crossref] [PubMed]

Chen, S.-J.

W. Lo, W.-L. Chen, C.-M. Hsueh, A. A. Ghazaryan, S.-J. Chen, D. H.-K. Ma, C.-Y. Dong, and H.-Y. Tan, “Fast Fourier Transform-Based Analysis of Second-Harmonic Generation Image in Keratoconic Cornea2D-FFT Analysis of Structural Change in Keratoconic Cornea,” Invest. Ophthalmol. Vis. Sci. 53(7), 3501–3507 (2012).
[Crossref] [PubMed]

Chen, S.-Y.

S.-W. Chu, S.-Y. Chen, G.-W. Chern, T.-H. Tsai, Y.-C. Chen, B.-L. Lin, and C.-K. Sun, “Studies of χ(2)/χ(3) Tensors in Submicron-Scaled Bio-Tissues by Polarization Harmonics Optical Microscopy,” Biophys. J. 86(6), 3914–3922 (2004).
[Crossref] [PubMed]

Chen, W.-L.

W. Lo, W.-L. Chen, C.-M. Hsueh, A. A. Ghazaryan, S.-J. Chen, D. H.-K. Ma, C.-Y. Dong, and H.-Y. Tan, “Fast Fourier Transform-Based Analysis of Second-Harmonic Generation Image in Keratoconic Cornea2D-FFT Analysis of Structural Change in Keratoconic Cornea,” Invest. Ophthalmol. Vis. Sci. 53(7), 3501–3507 (2012).
[Crossref] [PubMed]

W.-L. Chen, T.-H. Li, P.-J. Su, C.-K. Chou, P. T. Fwu, and S.-J. Lin, “Second harmonic generation χ tensor microscopy for tissue imaging,” Appl. Phys. Lett. 94(18), 183902 (2009).
[Crossref]

Chen, Y.-C.

S.-W. Chu, S.-Y. Chen, G.-W. Chern, T.-H. Tsai, Y.-C. Chen, B.-L. Lin, and C.-K. Sun, “Studies of χ(2)/χ(3) Tensors in Submicron-Scaled Bio-Tissues by Polarization Harmonics Optical Microscopy,” Biophys. J. 86(6), 3914–3922 (2004).
[Crossref] [PubMed]

Chen, Y.-F.

H.-Y. Tan, Y. Sun, W. Lo, S.-J. Lin, C.-H. Hsiao, Y.-F. Chen, S. C.-M. Huang, W.-C. Lin, S.-H. Jee, H.-S. Yu, and C. Y. Dong, “Multiphoton fluorescence and second harmonic generation imaging of the structural alterations in keratoconus ex vivo,” Invest. Ophthalmol. Vis. Sci. 47(12), 5251–5259 (2006).
[Crossref] [PubMed]

Chern, G.-W.

S.-W. Chu, S.-Y. Chen, G.-W. Chern, T.-H. Tsai, Y.-C. Chen, B.-L. Lin, and C.-K. Sun, “Studies of χ(2)/χ(3) Tensors in Submicron-Scaled Bio-Tissues by Polarization Harmonics Optical Microscopy,” Biophys. J. 86(6), 3914–3922 (2004).
[Crossref] [PubMed]

Chikama, T.

N. Morishige, A. J. Wahlert, M. C. Kenney, D. J. Brown, K. Kawamoto, T. Chikama, T. Nishida, and J. V. Jester, “Second-harmonic imaging microscopy of normal human and keratoconus cornea,” Invest. Ophthalmol. Vis. Sci. 48(3), 1087–1094 (2007).
[Crossref] [PubMed]

Chou, C.-K.

W.-L. Chen, T.-H. Li, P.-J. Su, C.-K. Chou, P. T. Fwu, and S.-J. Lin, “Second harmonic generation χ tensor microscopy for tissue imaging,” Appl. Phys. Lett. 94(18), 183902 (2009).
[Crossref]

Chu, S.-W.

S.-W. Chu, S.-Y. Chen, G.-W. Chern, T.-H. Tsai, Y.-C. Chen, B.-L. Lin, and C.-K. Sun, “Studies of χ(2)/χ(3) Tensors in Submicron-Scaled Bio-Tissues by Polarization Harmonics Optical Microscopy,” Biophys. J. 86(6), 3914–3922 (2004).
[Crossref] [PubMed]

Daxer, A.

A. Daxer and P. Fratzl, “Collagen fibril orientation in the human corneal stroma and its implication in keratoconus,” Invest. Ophthalmol. Vis. Sci. 38(1), 121–129 (1997).
[PubMed]

Dong, C. Y.

H.-Y. Tan, Y. Sun, W. Lo, S.-J. Lin, C.-H. Hsiao, Y.-F. Chen, S. C.-M. Huang, W.-C. Lin, S.-H. Jee, H.-S. Yu, and C. Y. Dong, “Multiphoton fluorescence and second harmonic generation imaging of the structural alterations in keratoconus ex vivo,” Invest. Ophthalmol. Vis. Sci. 47(12), 5251–5259 (2006).
[Crossref] [PubMed]

Dong, C.-Y.

W. Lo, W.-L. Chen, C.-M. Hsueh, A. A. Ghazaryan, S.-J. Chen, D. H.-K. Ma, C.-Y. Dong, and H.-Y. Tan, “Fast Fourier Transform-Based Analysis of Second-Harmonic Generation Image in Keratoconic Cornea2D-FFT Analysis of Structural Change in Keratoconic Cornea,” Invest. Ophthalmol. Vis. Sci. 53(7), 3501–3507 (2012).
[Crossref] [PubMed]

Fleury, V.

Fratzl, P.

A. Daxer and P. Fratzl, “Collagen fibril orientation in the human corneal stroma and its implication in keratoconus,” Invest. Ophthalmol. Vis. Sci. 38(1), 121–129 (1997).
[PubMed]

Fwu, P. T.

W.-L. Chen, T.-H. Li, P.-J. Su, C.-K. Chou, P. T. Fwu, and S.-J. Lin, “Second harmonic generation χ tensor microscopy for tissue imaging,” Appl. Phys. Lett. 94(18), 183902 (2009).
[Crossref]

Ghazaryan, A. A.

W. Lo, W.-L. Chen, C.-M. Hsueh, A. A. Ghazaryan, S.-J. Chen, D. H.-K. Ma, C.-Y. Dong, and H.-Y. Tan, “Fast Fourier Transform-Based Analysis of Second-Harmonic Generation Image in Keratoconic Cornea2D-FFT Analysis of Structural Change in Keratoconic Cornea,” Invest. Ophthalmol. Vis. Sci. 53(7), 3501–3507 (2012).
[Crossref] [PubMed]

Gill, P. S.

K. M. Meek, S. J. Tuft, Y. Huang, P. S. Gill, S. Hayes, R. H. Newton, and A. J. Bron, “Changes in collagen orientation and distribution in keratoconus corneas,” Invest. Ophthalmol. Vis. Sci. 46(6), 1948–1956 (2005).
[Crossref] [PubMed]

Guilbert, T.

Gusachenko, I.

Han, X.

Hayes, S.

K. M. Meek, S. J. Tuft, Y. Huang, P. S. Gill, S. Hayes, R. H. Newton, and A. J. Bron, “Changes in collagen orientation and distribution in keratoconus corneas,” Invest. Ophthalmol. Vis. Sci. 46(6), 1948–1956 (2005).
[Crossref] [PubMed]

Hsiao, C.-H.

H.-Y. Tan, Y. Sun, W. Lo, S.-J. Lin, C.-H. Hsiao, Y.-F. Chen, S. C.-M. Huang, W.-C. Lin, S.-H. Jee, H.-S. Yu, and C. Y. Dong, “Multiphoton fluorescence and second harmonic generation imaging of the structural alterations in keratoconus ex vivo,” Invest. Ophthalmol. Vis. Sci. 47(12), 5251–5259 (2006).
[Crossref] [PubMed]

Hsueh, C.-M.

W. Lo, W.-L. Chen, C.-M. Hsueh, A. A. Ghazaryan, S.-J. Chen, D. H.-K. Ma, C.-Y. Dong, and H.-Y. Tan, “Fast Fourier Transform-Based Analysis of Second-Harmonic Generation Image in Keratoconic Cornea2D-FFT Analysis of Structural Change in Keratoconic Cornea,” Invest. Ophthalmol. Vis. Sci. 53(7), 3501–3507 (2012).
[Crossref] [PubMed]

Huang, S. C.-M.

H.-Y. Tan, Y. Sun, W. Lo, S.-J. Lin, C.-H. Hsiao, Y.-F. Chen, S. C.-M. Huang, W.-C. Lin, S.-H. Jee, H.-S. Yu, and C. Y. Dong, “Multiphoton fluorescence and second harmonic generation imaging of the structural alterations in keratoconus ex vivo,” Invest. Ophthalmol. Vis. Sci. 47(12), 5251–5259 (2006).
[Crossref] [PubMed]

Huang, Y.

K. M. Meek, S. J. Tuft, Y. Huang, P. S. Gill, S. Hayes, R. H. Newton, and A. J. Bron, “Changes in collagen orientation and distribution in keratoconus corneas,” Invest. Ophthalmol. Vis. Sci. 46(6), 1948–1956 (2005).
[Crossref] [PubMed]

Jee, S.-H.

H.-Y. Tan, Y. Sun, W. Lo, S.-J. Lin, C.-H. Hsiao, Y.-F. Chen, S. C.-M. Huang, W.-C. Lin, S.-H. Jee, H.-S. Yu, and C. Y. Dong, “Multiphoton fluorescence and second harmonic generation imaging of the structural alterations in keratoconus ex vivo,” Invest. Ophthalmol. Vis. Sci. 47(12), 5251–5259 (2006).
[Crossref] [PubMed]

Jester, J. V.

N. Morishige, A. J. Wahlert, M. C. Kenney, D. J. Brown, K. Kawamoto, T. Chikama, T. Nishida, and J. V. Jester, “Second-harmonic imaging microscopy of normal human and keratoconus cornea,” Invest. Ophthalmol. Vis. Sci. 48(3), 1087–1094 (2007).
[Crossref] [PubMed]

Kawamoto, K.

N. Morishige, A. J. Wahlert, M. C. Kenney, D. J. Brown, K. Kawamoto, T. Chikama, T. Nishida, and J. V. Jester, “Second-harmonic imaging microscopy of normal human and keratoconus cornea,” Invest. Ophthalmol. Vis. Sci. 48(3), 1087–1094 (2007).
[Crossref] [PubMed]

Kenney, M. C.

N. Morishige, A. J. Wahlert, M. C. Kenney, D. J. Brown, K. Kawamoto, T. Chikama, T. Nishida, and J. V. Jester, “Second-harmonic imaging microscopy of normal human and keratoconus cornea,” Invest. Ophthalmol. Vis. Sci. 48(3), 1087–1094 (2007).
[Crossref] [PubMed]

Knupp, C.

K. M. Meek and C. Knupp, “Corneal structure and transparency,” Prog. Retin. Eye Res. 49, 1–16 (2015).
[Crossref] [PubMed]

Kowalczuk, L.

Lamarre, I.

Latour, G.

Le Grand, Y.

Li, T.-H.

W.-L. Chen, T.-H. Li, P.-J. Su, C.-K. Chou, P. T. Fwu, and S.-J. Lin, “Second harmonic generation χ tensor microscopy for tissue imaging,” Appl. Phys. Lett. 94(18), 183902 (2009).
[Crossref]

Lin, B.-L.

S.-W. Chu, S.-Y. Chen, G.-W. Chern, T.-H. Tsai, Y.-C. Chen, B.-L. Lin, and C.-K. Sun, “Studies of χ(2)/χ(3) Tensors in Submicron-Scaled Bio-Tissues by Polarization Harmonics Optical Microscopy,” Biophys. J. 86(6), 3914–3922 (2004).
[Crossref] [PubMed]

Lin, S.-J.

W.-L. Chen, T.-H. Li, P.-J. Su, C.-K. Chou, P. T. Fwu, and S.-J. Lin, “Second harmonic generation χ tensor microscopy for tissue imaging,” Appl. Phys. Lett. 94(18), 183902 (2009).
[Crossref]

H.-Y. Tan, Y. Sun, W. Lo, S.-J. Lin, C.-H. Hsiao, Y.-F. Chen, S. C.-M. Huang, W.-C. Lin, S.-H. Jee, H.-S. Yu, and C. Y. Dong, “Multiphoton fluorescence and second harmonic generation imaging of the structural alterations in keratoconus ex vivo,” Invest. Ophthalmol. Vis. Sci. 47(12), 5251–5259 (2006).
[Crossref] [PubMed]

Lin, W.-C.

H.-Y. Tan, Y. Sun, W. Lo, S.-J. Lin, C.-H. Hsiao, Y.-F. Chen, S. C.-M. Huang, W.-C. Lin, S.-H. Jee, H.-S. Yu, and C. Y. Dong, “Multiphoton fluorescence and second harmonic generation imaging of the structural alterations in keratoconus ex vivo,” Invest. Ophthalmol. Vis. Sci. 47(12), 5251–5259 (2006).
[Crossref] [PubMed]

Lo, W.

W. Lo, W.-L. Chen, C.-M. Hsueh, A. A. Ghazaryan, S.-J. Chen, D. H.-K. Ma, C.-Y. Dong, and H.-Y. Tan, “Fast Fourier Transform-Based Analysis of Second-Harmonic Generation Image in Keratoconic Cornea2D-FFT Analysis of Structural Change in Keratoconic Cornea,” Invest. Ophthalmol. Vis. Sci. 53(7), 3501–3507 (2012).
[Crossref] [PubMed]

H.-Y. Tan, Y. Sun, W. Lo, S.-J. Lin, C.-H. Hsiao, Y.-F. Chen, S. C.-M. Huang, W.-C. Lin, S.-H. Jee, H.-S. Yu, and C. Y. Dong, “Multiphoton fluorescence and second harmonic generation imaging of the structural alterations in keratoconus ex vivo,” Invest. Ophthalmol. Vis. Sci. 47(12), 5251–5259 (2006).
[Crossref] [PubMed]

Lombardo, G.

Lombardo, M.

Loza-Alvarez, P.

M. Lombardo, D. Merino, P. Loza-Alvarez, and G. Lombardo, “Translational label-free nonlinear imaging biomarkers to classify the human corneal microstructure,” Biomed. Opt. Express 6(8), 2803–2818 (2015).
[Crossref] [PubMed]

S. I. C. O. Santos, M. Mathew, O. E. Olarte, S. Psilodimitrakopoulos, and P. Loza-Alvarez, “Femtosecond laser axotomy in caenorhabditis elegans and collateral damage assessment using a combination of linear and nonlinear imaging techniques,” PLoS One 8(3), e58600 (2013).
[Crossref] [PubMed]

S. Psilodimitrakopoulos, I. Amat-Roldan, P. Loza-Alvarez, and D. Artigas, “Effect of molecular organization on the image histograms of polarization SHG microscopy,” Biomed. Opt. Express 3(10), 2681–2693 (2012).
[Crossref] [PubMed]

I. Amat-Roldan, S. Psilodimitrakopoulos, P. Loza-Alvarez, and D. Artigas, “Fast image analysis in polarization SHG microscopy,” Opt. Express 18(16), 17209–17219 (2010).
[Crossref] [PubMed]

S. Psilodimitrakopoulos, I. Amat-Roldan, P. Loza-Alvarez, and D. Artigas, “Estimating the helical pitch angle of amylopectin in starch using polarization second harmonic generation microscopy,” J. Opt. 12(8), 084007 (2010).
[Crossref]

M. Mathew, S. I. C. O. Santos, D. Zalvidea, and P. Loza-Alvarez, “Multimodal optical workstation for simultaneous linear, nonlinear microscopy and nanomanipulation: Upgrading a commercial confocal inverted microscope,” Rev. Sci. Instrum. 80(7), 073701 (2009).
[Crossref] [PubMed]

S. Psilodimitrakopoulos, S. I. Santos, I. Amat-Roldan, A. K. Thayil, D. Artigas, and P. Loza-Alvarez, “In vivo, pixel-resolution mapping of thick filaments’ orientation in nonfibrilar muscle using polarization-sensitive second harmonic generation microscopy,” J. Biomed. Opt. 14(1), 014001 (2009).
[Crossref] [PubMed]

S. Psilodimitrakopoulos, V. Petegnief, G. Soria, I. Amat-Roldan, D. Artigas, A. M. Planas, and P. Loza-Alvarez, “Estimation of the effective orientation of the SHG source in primary cortical neurons,” Opt. Express 17(16), 14418–14425 (2009).
[Crossref] [PubMed]

S. Psilodimitrakopoulos, D. Artigas, G. Soria, I. Amat-Roldan, A. M. Planas, and P. Loza-Alvarez, “Quantitative discrimination between endogenous SHG sources in mammalian tissue, based on their polarization response,” Opt. Express 17(12), 10168–10176 (2009).
[Crossref] [PubMed]

Ma, D. H.-K.

W. Lo, W.-L. Chen, C.-M. Hsueh, A. A. Ghazaryan, S.-J. Chen, D. H.-K. Ma, C.-Y. Dong, and H.-Y. Tan, “Fast Fourier Transform-Based Analysis of Second-Harmonic Generation Image in Keratoconic Cornea2D-FFT Analysis of Structural Change in Keratoconic Cornea,” Invest. Ophthalmol. Vis. Sci. 53(7), 3501–3507 (2012).
[Crossref] [PubMed]

Mallinger, R.

W. Radner, M. Zehetmayer, C. Skorpik, and R. Mallinger, “Altered organization of collagen in the apex of keratoconus corneas,” Ophthalmic Res. 30(5), 327–332 (1998).
[Crossref] [PubMed]

Mathew, M.

S. I. C. O. Santos, M. Mathew, O. E. Olarte, S. Psilodimitrakopoulos, and P. Loza-Alvarez, “Femtosecond laser axotomy in caenorhabditis elegans and collateral damage assessment using a combination of linear and nonlinear imaging techniques,” PLoS One 8(3), e58600 (2013).
[Crossref] [PubMed]

M. Mathew, S. I. C. O. Santos, D. Zalvidea, and P. Loza-Alvarez, “Multimodal optical workstation for simultaneous linear, nonlinear microscopy and nanomanipulation: Upgrading a commercial confocal inverted microscope,” Rev. Sci. Instrum. 80(7), 073701 (2009).
[Crossref] [PubMed]

Maurice, D. M.

D. M. Maurice, “The structure and transparency of the cornea,” J. Physiol. 136(2), 263–286 (1957).
[Crossref] [PubMed]

Meek, K. M.

K. M. Meek and C. Knupp, “Corneal structure and transparency,” Prog. Retin. Eye Res. 49, 1–16 (2015).
[Crossref] [PubMed]

K. M. Meek, S. J. Tuft, Y. Huang, P. S. Gill, S. Hayes, R. H. Newton, and A. J. Bron, “Changes in collagen orientation and distribution in keratoconus corneas,” Invest. Ophthalmol. Vis. Sci. 46(6), 1948–1956 (2005).
[Crossref] [PubMed]

Merino, D.

Morishige, N.

N. Morishige, A. J. Wahlert, M. C. Kenney, D. J. Brown, K. Kawamoto, T. Chikama, T. Nishida, and J. V. Jester, “Second-harmonic imaging microscopy of normal human and keratoconus cornea,” Invest. Ophthalmol. Vis. Sci. 48(3), 1087–1094 (2007).
[Crossref] [PubMed]

Newton, R. H.

K. M. Meek, S. J. Tuft, Y. Huang, P. S. Gill, S. Hayes, R. H. Newton, and A. J. Bron, “Changes in collagen orientation and distribution in keratoconus corneas,” Invest. Ophthalmol. Vis. Sci. 46(6), 1948–1956 (2005).
[Crossref] [PubMed]

Nishida, T.

N. Morishige, A. J. Wahlert, M. C. Kenney, D. J. Brown, K. Kawamoto, T. Chikama, T. Nishida, and J. V. Jester, “Second-harmonic imaging microscopy of normal human and keratoconus cornea,” Invest. Ophthalmol. Vis. Sci. 48(3), 1087–1094 (2007).
[Crossref] [PubMed]

Odin, C.

Olarte, O. E.

S. I. C. O. Santos, M. Mathew, O. E. Olarte, S. Psilodimitrakopoulos, and P. Loza-Alvarez, “Femtosecond laser axotomy in caenorhabditis elegans and collateral damage assessment using a combination of linear and nonlinear imaging techniques,” PLoS One 8(3), e58600 (2013).
[Crossref] [PubMed]

Petegnief, V.

Planas, A. M.

Psilodimitrakopoulos, S.

S. I. C. O. Santos, M. Mathew, O. E. Olarte, S. Psilodimitrakopoulos, and P. Loza-Alvarez, “Femtosecond laser axotomy in caenorhabditis elegans and collateral damage assessment using a combination of linear and nonlinear imaging techniques,” PLoS One 8(3), e58600 (2013).
[Crossref] [PubMed]

S. Psilodimitrakopoulos, I. Amat-Roldan, P. Loza-Alvarez, and D. Artigas, “Effect of molecular organization on the image histograms of polarization SHG microscopy,” Biomed. Opt. Express 3(10), 2681–2693 (2012).
[Crossref] [PubMed]

I. Amat-Roldan, S. Psilodimitrakopoulos, P. Loza-Alvarez, and D. Artigas, “Fast image analysis in polarization SHG microscopy,” Opt. Express 18(16), 17209–17219 (2010).
[Crossref] [PubMed]

S. Psilodimitrakopoulos, I. Amat-Roldan, P. Loza-Alvarez, and D. Artigas, “Estimating the helical pitch angle of amylopectin in starch using polarization second harmonic generation microscopy,” J. Opt. 12(8), 084007 (2010).
[Crossref]

S. Psilodimitrakopoulos, S. I. Santos, I. Amat-Roldan, A. K. Thayil, D. Artigas, and P. Loza-Alvarez, “In vivo, pixel-resolution mapping of thick filaments’ orientation in nonfibrilar muscle using polarization-sensitive second harmonic generation microscopy,” J. Biomed. Opt. 14(1), 014001 (2009).
[Crossref] [PubMed]

S. Psilodimitrakopoulos, V. Petegnief, G. Soria, I. Amat-Roldan, D. Artigas, A. M. Planas, and P. Loza-Alvarez, “Estimation of the effective orientation of the SHG source in primary cortical neurons,” Opt. Express 17(16), 14418–14425 (2009).
[Crossref] [PubMed]

S. Psilodimitrakopoulos, D. Artigas, G. Soria, I. Amat-Roldan, A. M. Planas, and P. Loza-Alvarez, “Quantitative discrimination between endogenous SHG sources in mammalian tissue, based on their polarization response,” Opt. Express 17(12), 10168–10176 (2009).
[Crossref] [PubMed]

Radner, W.

W. Radner, M. Zehetmayer, C. Skorpik, and R. Mallinger, “Altered organization of collagen in the apex of keratoconus corneas,” Ophthalmic Res. 30(5), 327–332 (1998).
[Crossref] [PubMed]

Recher, G.

Réfrégier, P.

Reiser, K. M.

P. Stoller, K. M. Reiser, P. M. Celliers, and A. M. Rubenchik, “Polarization-modulated second harmonic generation in collagen,” Biophys. J. 82(6), 3330–3342 (2002).
[Crossref] [PubMed]

Roche, M.

Rouède, D.

Rubenchik, A. M.

P. Stoller, K. M. Reiser, P. M. Celliers, and A. M. Rubenchik, “Polarization-modulated second harmonic generation in collagen,” Biophys. J. 82(6), 3330–3342 (2002).
[Crossref] [PubMed]

Santos, S. I.

S. Psilodimitrakopoulos, S. I. Santos, I. Amat-Roldan, A. K. Thayil, D. Artigas, and P. Loza-Alvarez, “In vivo, pixel-resolution mapping of thick filaments’ orientation in nonfibrilar muscle using polarization-sensitive second harmonic generation microscopy,” J. Biomed. Opt. 14(1), 014001 (2009).
[Crossref] [PubMed]

Santos, S. I. C. O.

S. I. C. O. Santos, M. Mathew, O. E. Olarte, S. Psilodimitrakopoulos, and P. Loza-Alvarez, “Femtosecond laser axotomy in caenorhabditis elegans and collateral damage assessment using a combination of linear and nonlinear imaging techniques,” PLoS One 8(3), e58600 (2013).
[Crossref] [PubMed]

M. Mathew, S. I. C. O. Santos, D. Zalvidea, and P. Loza-Alvarez, “Multimodal optical workstation for simultaneous linear, nonlinear microscopy and nanomanipulation: Upgrading a commercial confocal inverted microscope,” Rev. Sci. Instrum. 80(7), 073701 (2009).
[Crossref] [PubMed]

Schanne-Klein, M.-C.

Skorpik, C.

W. Radner, M. Zehetmayer, C. Skorpik, and R. Mallinger, “Altered organization of collagen in the apex of keratoconus corneas,” Ophthalmic Res. 30(5), 327–332 (1998).
[Crossref] [PubMed]

Soria, G.

Stoller, P.

P. Stoller, K. M. Reiser, P. M. Celliers, and A. M. Rubenchik, “Polarization-modulated second harmonic generation in collagen,” Biophys. J. 82(6), 3330–3342 (2002).
[Crossref] [PubMed]

Su, P.-J.

W.-L. Chen, T.-H. Li, P.-J. Su, C.-K. Chou, P. T. Fwu, and S.-J. Lin, “Second harmonic generation χ tensor microscopy for tissue imaging,” Appl. Phys. Lett. 94(18), 183902 (2009).
[Crossref]

Sun, C.-K.

S.-W. Chu, S.-Y. Chen, G.-W. Chern, T.-H. Tsai, Y.-C. Chen, B.-L. Lin, and C.-K. Sun, “Studies of χ(2)/χ(3) Tensors in Submicron-Scaled Bio-Tissues by Polarization Harmonics Optical Microscopy,” Biophys. J. 86(6), 3914–3922 (2004).
[Crossref] [PubMed]

Sun, Y.

H.-Y. Tan, Y. Sun, W. Lo, S.-J. Lin, C.-H. Hsiao, Y.-F. Chen, S. C.-M. Huang, W.-C. Lin, S.-H. Jee, H.-S. Yu, and C. Y. Dong, “Multiphoton fluorescence and second harmonic generation imaging of the structural alterations in keratoconus ex vivo,” Invest. Ophthalmol. Vis. Sci. 47(12), 5251–5259 (2006).
[Crossref] [PubMed]

Tan, H.-Y.

W. Lo, W.-L. Chen, C.-M. Hsueh, A. A. Ghazaryan, S.-J. Chen, D. H.-K. Ma, C.-Y. Dong, and H.-Y. Tan, “Fast Fourier Transform-Based Analysis of Second-Harmonic Generation Image in Keratoconic Cornea2D-FFT Analysis of Structural Change in Keratoconic Cornea,” Invest. Ophthalmol. Vis. Sci. 53(7), 3501–3507 (2012).
[Crossref] [PubMed]

H.-Y. Tan, Y. Sun, W. Lo, S.-J. Lin, C.-H. Hsiao, Y.-F. Chen, S. C.-M. Huang, W.-C. Lin, S.-H. Jee, H.-S. Yu, and C. Y. Dong, “Multiphoton fluorescence and second harmonic generation imaging of the structural alterations in keratoconus ex vivo,” Invest. Ophthalmol. Vis. Sci. 47(12), 5251–5259 (2006).
[Crossref] [PubMed]

Tang, P.

Thayil, A. K.

S. Psilodimitrakopoulos, S. I. Santos, I. Amat-Roldan, A. K. Thayil, D. Artigas, and P. Loza-Alvarez, “In vivo, pixel-resolution mapping of thick filaments’ orientation in nonfibrilar muscle using polarization-sensitive second harmonic generation microscopy,” J. Biomed. Opt. 14(1), 014001 (2009).
[Crossref] [PubMed]

Tiaho, F.

Tsai, T.-H.

S.-W. Chu, S.-Y. Chen, G.-W. Chern, T.-H. Tsai, Y.-C. Chen, B.-L. Lin, and C.-K. Sun, “Studies of χ(2)/χ(3) Tensors in Submicron-Scaled Bio-Tissues by Polarization Harmonics Optical Microscopy,” Biophys. J. 86(6), 3914–3922 (2004).
[Crossref] [PubMed]

Tuft, S. J.

K. M. Meek, S. J. Tuft, Y. Huang, P. S. Gill, S. Hayes, R. H. Newton, and A. J. Bron, “Changes in collagen orientation and distribution in keratoconus corneas,” Invest. Ophthalmol. Vis. Sci. 46(6), 1948–1956 (2005).
[Crossref] [PubMed]

Wahlert, A. J.

N. Morishige, A. J. Wahlert, M. C. Kenney, D. J. Brown, K. Kawamoto, T. Chikama, T. Nishida, and J. V. Jester, “Second-harmonic imaging microscopy of normal human and keratoconus cornea,” Invest. Ophthalmol. Vis. Sci. 48(3), 1087–1094 (2007).
[Crossref] [PubMed]

Yu, H.-S.

H.-Y. Tan, Y. Sun, W. Lo, S.-J. Lin, C.-H. Hsiao, Y.-F. Chen, S. C.-M. Huang, W.-C. Lin, S.-H. Jee, H.-S. Yu, and C. Y. Dong, “Multiphoton fluorescence and second harmonic generation imaging of the structural alterations in keratoconus ex vivo,” Invest. Ophthalmol. Vis. Sci. 47(12), 5251–5259 (2006).
[Crossref] [PubMed]

Zalvidea, D.

M. Mathew, S. I. C. O. Santos, D. Zalvidea, and P. Loza-Alvarez, “Multimodal optical workstation for simultaneous linear, nonlinear microscopy and nanomanipulation: Upgrading a commercial confocal inverted microscope,” Rev. Sci. Instrum. 80(7), 073701 (2009).
[Crossref] [PubMed]

Zehetmayer, M.

W. Radner, M. Zehetmayer, C. Skorpik, and R. Mallinger, “Altered organization of collagen in the apex of keratoconus corneas,” Ophthalmic Res. 30(5), 327–332 (1998).
[Crossref] [PubMed]

Zettel, M. L.

Appl. Phys. Lett. (1)

W.-L. Chen, T.-H. Li, P.-J. Su, C.-K. Chou, P. T. Fwu, and S.-J. Lin, “Second harmonic generation χ tensor microscopy for tissue imaging,” Appl. Phys. Lett. 94(18), 183902 (2009).
[Crossref]

Biomed. Opt. Express (3)

Biophys. J. (2)

P. Stoller, K. M. Reiser, P. M. Celliers, and A. M. Rubenchik, “Polarization-modulated second harmonic generation in collagen,” Biophys. J. 82(6), 3330–3342 (2002).
[Crossref] [PubMed]

S.-W. Chu, S.-Y. Chen, G.-W. Chern, T.-H. Tsai, Y.-C. Chen, B.-L. Lin, and C.-K. Sun, “Studies of χ(2)/χ(3) Tensors in Submicron-Scaled Bio-Tissues by Polarization Harmonics Optical Microscopy,” Biophys. J. 86(6), 3914–3922 (2004).
[Crossref] [PubMed]

Invest. Ophthalmol. Vis. Sci. (5)

H.-Y. Tan, Y. Sun, W. Lo, S.-J. Lin, C.-H. Hsiao, Y.-F. Chen, S. C.-M. Huang, W.-C. Lin, S.-H. Jee, H.-S. Yu, and C. Y. Dong, “Multiphoton fluorescence and second harmonic generation imaging of the structural alterations in keratoconus ex vivo,” Invest. Ophthalmol. Vis. Sci. 47(12), 5251–5259 (2006).
[Crossref] [PubMed]

N. Morishige, A. J. Wahlert, M. C. Kenney, D. J. Brown, K. Kawamoto, T. Chikama, T. Nishida, and J. V. Jester, “Second-harmonic imaging microscopy of normal human and keratoconus cornea,” Invest. Ophthalmol. Vis. Sci. 48(3), 1087–1094 (2007).
[Crossref] [PubMed]

W. Lo, W.-L. Chen, C.-M. Hsueh, A. A. Ghazaryan, S.-J. Chen, D. H.-K. Ma, C.-Y. Dong, and H.-Y. Tan, “Fast Fourier Transform-Based Analysis of Second-Harmonic Generation Image in Keratoconic Cornea2D-FFT Analysis of Structural Change in Keratoconic Cornea,” Invest. Ophthalmol. Vis. Sci. 53(7), 3501–3507 (2012).
[Crossref] [PubMed]

A. Daxer and P. Fratzl, “Collagen fibril orientation in the human corneal stroma and its implication in keratoconus,” Invest. Ophthalmol. Vis. Sci. 38(1), 121–129 (1997).
[PubMed]

K. M. Meek, S. J. Tuft, Y. Huang, P. S. Gill, S. Hayes, R. H. Newton, and A. J. Bron, “Changes in collagen orientation and distribution in keratoconus corneas,” Invest. Ophthalmol. Vis. Sci. 46(6), 1948–1956 (2005).
[Crossref] [PubMed]

J. Biomed. Opt. (1)

S. Psilodimitrakopoulos, S. I. Santos, I. Amat-Roldan, A. K. Thayil, D. Artigas, and P. Loza-Alvarez, “In vivo, pixel-resolution mapping of thick filaments’ orientation in nonfibrilar muscle using polarization-sensitive second harmonic generation microscopy,” J. Biomed. Opt. 14(1), 014001 (2009).
[Crossref] [PubMed]

J. Opt. (1)

S. Psilodimitrakopoulos, I. Amat-Roldan, P. Loza-Alvarez, and D. Artigas, “Estimating the helical pitch angle of amylopectin in starch using polarization second harmonic generation microscopy,” J. Opt. 12(8), 084007 (2010).
[Crossref]

J. Physiol. (1)

D. M. Maurice, “The structure and transparency of the cornea,” J. Physiol. 136(2), 263–286 (1957).
[Crossref] [PubMed]

Ophthalmic Res. (1)

W. Radner, M. Zehetmayer, C. Skorpik, and R. Mallinger, “Altered organization of collagen in the apex of keratoconus corneas,” Ophthalmic Res. 30(5), 327–332 (1998).
[Crossref] [PubMed]

Opt. Express (6)

Opt. Lett. (1)

PLoS One (1)

S. I. C. O. Santos, M. Mathew, O. E. Olarte, S. Psilodimitrakopoulos, and P. Loza-Alvarez, “Femtosecond laser axotomy in caenorhabditis elegans and collateral damage assessment using a combination of linear and nonlinear imaging techniques,” PLoS One 8(3), e58600 (2013).
[Crossref] [PubMed]

Prog. Retin. Eye Res. (1)

K. M. Meek and C. Knupp, “Corneal structure and transparency,” Prog. Retin. Eye Res. 49, 1–16 (2015).
[Crossref] [PubMed]

Rev. Sci. Instrum. (1)

M. Mathew, S. I. C. O. Santos, D. Zalvidea, and P. Loza-Alvarez, “Multimodal optical workstation for simultaneous linear, nonlinear microscopy and nanomanipulation: Upgrading a commercial confocal inverted microscope,” Rev. Sci. Instrum. 80(7), 073701 (2009).
[Crossref] [PubMed]

Other (2)

G. Latour, I. Gusachenko, L. Kowalczuk, I. Lamarre, and M.-C. Schanne-Klein, “In vivo multiphoton imaging of the cornea: polarization-resolved second harmonic generation from stromal collagen,” in A. Periasamy, K. König, and P. T. C. So, eds. Proceedings Volume 8226, Multiphoton Microscopy in the Biomedical Sciences XII (SPIE, 2012), p. 82262I.

R. Mercatelli, F. Ratto, F. Rossi, L. Menabuoni, A. Malandrini, F. Tatini, R. Nicoletti, R. Pini, F. S. Pavone, and R. Cicchi, “Three-dimensional mapping of corneal lamellar orientation by means of backward-scattered SHG microscopy,” in E. Beaurepaire, F. S. Pavone, and P. T. C. So, eds. Proceedings Volume 10414, Advances in Microscopic Imaging (SPIE, 2017), p. 1041407.

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Figures (9)

Fig. 1
Fig. 1 The values of the orientation data for collagen fibers, a) φ1 and b) φ2, used as input to generate the theoretical data set of two crossing collagen fibers.
Fig. 2
Fig. 2 The results of the pSHG model used with the theoretical data simulating two crossing collagen fibers generated as described. a) Values of φres and b) Values of θres.
Fig. 3
Fig. 3 The values of φres calculated by our simulation using a) Eq. (4), φres,2 = arg[a2exp(i2ϕ)]/2, and b) Eq. (5), φres,4 = arg[a4exp(i4ϕ)]/4. c) The values of φ’res,2, calculated by phase wrapping the values of φres,2 between ± 45°. d) The values of Δφres = (φ’res,2res,4).
Fig. 4
Fig. 4 a) Values of Δφres identical to those in Fig. 3(d). b) Histogram of the values in a). c) Values of Δφres only for the left-hand side (pixels with crossing fibers) of the data in a). d) Histogram of the values in c). e) Values of Δφres only for the right-hand side (pixels with parallel fibers) of the data in a). f) Histogram of the values in e).
Fig. 5
Fig. 5 a) Average intensity of the 9 pSHG images acquired from a starch granule. Scale bar is 10 μm. b) Values of φres,2 calculated from the image in a). c) The same image as in b) after applying a mask calculated using Δφres as explained in the text. d) The histogram of θe values.
Fig. 6
Fig. 6 pSHG data acquired from a normal cornea at a depth of 220 µm. a) Average intensity of the 9 pSHG images acquired. Scale bar is 10 µm. b) Collagen fiber orientations (φres,2) calculated using the pSHG model, Eq. (4). The circle highlights an area where the results of φres,2 do not correspond with apparent orientations of the fibers in a). In particular, in the white circle, a fiber in red (70°) appears to be aligned at around 30°-40°. Also, in the red circle, a fiber in green (0°) seems aligned along −70°. c) The same results as in b) after rejecting the pixels where crossing fibers have been identified, including these conflicting fibers.
Fig. 7
Fig. 7 Comparison of the pixels where pSHG signal has been identified as generated from a single collagen fiber using a) anisotropy parameter information, and b) our method based on Δφres information. c) Pixels that have been identified by only one of the methods. This image is used as a way to determine the differences between the two methods.
Fig. 8
Fig. 8 Helical pitch angle of collagen fibers in normal cornea, a) before and b) after detecting and rejecting pixels with crossing fibers. Scale bar in a) is 10 μm. c) Helical pitch angle distribution and the resulting fit based on a lognormal function for the data in figures a) and b).
Fig. 9
Fig. 9 pSHG data acquired from a keratoconic cornea at a depth of 45 µm. a) Average intensity of the 9 pSHG images acquired. Scale bar is 10 µm. b) Collagen fiber orientations (φres,2) calculated using the pSHG model. c) The same results as in b) after rejecting the pixels where crossing fibers have been identified.

Equations (9)

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I SHG (φ,α)= a 0 + a 2 cos2(φα)+ a 4 cos4(φα),
i(φ,Ω)= a 0 δ(0)+ a 2 exp(i2φ)δ(1Ω)+ a 4 exp(i4φ)δ(2Ω)+c.c,
tan 2 θ e = 2 a 0 + a 2 + a 4 a 0 a 2 + a 4 .
φ=arg[ a 2 exp(i2φ)]/2.
φ=arg[ a 4 exp(i4φ)]/4.
a 0 = A 2 8 [ 12 tan 4 ( θ e ) + 4 tan 2 ( θ e ) +7],
a 2 = A 2 2 [ 4 tan 4 ( θ e ) 1],
a 4 = A 2 8 [ 4 tan 4 ( θ e ) 4 tan 2 ( θ e ) 3].
I TOT (α)= I SHG,1 + I SHG,2 = I SHG (α, φ 1 )+ I SHG (α, φ 2 ),