Abstract

The polarization states of a skylight can be used for navigation by a lot of insects as well as by human beings. However, the polarization patterns of skylight are greatly influenced by the atmospheric conditions. This paper studied the polarization patterns of skylight under different sky conditions by polarized imaging measurements, in which the AOT (Aerosol Optical Thickness) and clouds were taken into account. The results showed that both the aerosol and cloud disturbed the polarization patterns of the skylight, but the patterns of AOP (Angle of Polarization) showed great robustness. The symmetry of the AOP images was found to be quite steady under most of the sky conditions. We proposed a navigation method by finding out the solar meridian according to the symmetry of an AOP map. The results showed that the solar meridian can be identified accurately under different situations. The calculation errors slightly fluctuated along with the aerosol and cloud.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref] [PubMed]
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2017 (1)

2016 (2)

2015 (2)

H. Lu, K. Zhao, Z. You, and K. Huang, “Angle algorithm based on Hough transform for imaging polarization navigation sensor,” Opt. Express 23(6), 7248–7262 (2015).
[Crossref] [PubMed]

T. Ma, X. Hu, L. Zhang, J. Lian, X. He, Y. Wang, and Z. Xian, “An evaluation of skylight polarization patterns for navigation,” Sensors (Basel) 15(3), 5895–5913 (2015).
[Crossref] [PubMed]

2014 (3)

Y. Zhang, H. Zhao, P. Song, S. Shi, W. Xu, and X. Liang, “Ground-based full-sky imaging polarimeter based on liquid crystal variable retarders,” Opt. Express 22(7), 8749–8764 (2014).
[Crossref] [PubMed]

Q. Liu, J. K. Chu, J. Wang, and L. Guan, “Research and simulation Analysis of Atmospheric polarization properties under water cloud condition,” Acta Opt. Sin. 34(3), 0301004 (2014).
[Crossref]

L. Hao, Z. Kaichun, M. Qiang, and Y. Zheng, “Design and implementation of detection system for skylight polarized pattern using continuously spiniling polarization analyzer,” J. Astronautics (Chinese) 35(9), 1087–1094 (2014).

2012 (2)

J. K. Chu, W. Wang, Y. Cui, W. Zhi, and Q. S. Gao, “Measurement for influence of aerosols on polarized sky radiance,” Optics and Precision Engineering 20(3), 520–526 (2012).
[Crossref]

J. Chahl and A. Mizutani, “Biomimetic attitude and orientation sensors,” IEEE Sens. J. 12(2), 289–297 (2012).
[Crossref]

2011 (4)

P. Kraft, C. Evangelista, M. Dacke, T. Labhart, and M. V. Srinivasan, “Honeybee navigation: following routes using polarized-light cues,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 703–708 (2011).
[Crossref] [PubMed]

M. Dacke, M. J. Byrne, E. Baird, C. H. Scholtz, and E. J. Warrant, “How dim is dim? Precision of the celestial compass in moonlight and sunlight,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 697–702 (2011).
[Crossref] [PubMed]

G. Horváth, A. Barta, I. Pomozi, B. Suhai, R. Hegedüs, S. Åkesson, B. Meyer-Rochow, and R. Wehner, “On the trail of Vikings with polarized skylight: experimental study of the atmospheric optical prerequisites allowing polarimetric navigation by Viking seafarers,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 772–782 (2011).
[Crossref] [PubMed]

N. J. Pust and J. A. Shaw, “Comparison of skylight polarization measurements and MODTRAN-P calculations,” J. Appl. Remote Sens. 5(1), 053529 (2011).
[Crossref]

2010 (1)

M. Sarkar, D. S. S. Bello, C. V. Hoot, and A. Theuwissen, “Biologically inspired autonomous agent navigation using an integrated polarization analyzing CMOS image sensor,” Procedia Eng. 5(3), 673–676 (2010).
[Crossref]

2008 (3)

J. K. Chu, K. C. Zhao, Q. Zhang, and T. C. Wang, “Construction and performance test of a novel polarization sensor for navigation,” Sensor. and Actuat. A. 148(1), 75–82 (2008).

Z. Fan, J. Gao, J. Wei, W. Zhang, and Y. Hu, “Investigation of atmospheric polarization measurement method imitating POL-neurons of cataglyphis,” Yiqi Yibiao Xuebao 29(4), 745–749 (2008).

N. J. Pust and J. A. Shaw, “Digital all-sky polarization imaging of partly cloudy skies,” Appl. Opt. 47(34), H190–H198 (2008).
[Crossref] [PubMed]

2007 (4)

R. Hegedüs, S. Åkesson, and G. Horváth, “Polarization patterns of thick clouds: overcast skies have distribution of the angle of polarization similar to that of clear skies,” J. Opt. Soc. Am. A 24(8), 2347–2356 (2007).
[Crossref] [PubMed]

M. J. Henze and T. Labhart, “Haze, clouds and limited sky visibility: polarotactic orientation of crickets under difficult stimulus conditions,” J. Exp. Biol. 210(18), 3266–3276 (2007).
[Crossref] [PubMed]

R. Hegedüs, S. Åkesson, R. Wehner, and G. Horváth, “Could Vikings have navigated under foggy and cloudy conditions by skylight polarization? On the atmospheric optical prerequisites of polarimetric Viking navigation under foggy and cloudy skies,” Proc. R. Soc. A 463(2080), 1081–1095 (2007).
[Crossref]

R. Hegedüs, A. Barta, B. Bernáth, V. B. Meyer-Rochow, and G. Horváth, “Imaging polarimetry of forest canopies: how the azimuth direction of the sun, occluded by vegetation, can be assessed from the polarization pattern of the sunlit foliage,” Appl. Opt. 46(23), 6019–6032 (2007).
[Crossref] [PubMed]

2006 (1)

2004 (2)

B. Suhai and G. Horváth, “How well does the Rayleigh model describe the E-vector distribution of skylight in clear and cloudy conditions? A full-sky polarimetric study,” J. Opt. Soc. Am. A 21(9), 1669–1676 (2004).
[Crossref] [PubMed]

A. Barta and G. Horváth, “Why is it advantageous for animals to detect celestial polarization in the ultraviolet? Skylight polarization under clouds and canopies is strongest in the UV,” J. Theor. Biol. 226(4), 429–437 (2004).
[Crossref] [PubMed]

2003 (1)

M. Dacke, D. E. Nilsson, C. H. Scholtz, M. Byrne, and E. J. Warrant, “Animal behaviour: Insect orientation to polarized moonlight,” Nature 424(6944), 33 (2003).
[Crossref] [PubMed]

2002 (1)

2001 (1)

I. Pomozi, G. Horváth, and R. Wehner, “How the clear-sky angle of polarization pattern continues underneath clouds: full-sky measurements and implications for animal orientation,” J. Exp. Biol. 204(Pt 17), 2933–2942 (2001).
[PubMed]

2000 (1)

D. Lambrinos, R. Möller, T. Labhart, R. Pfeifer, and R. Wehner, “A mobile robot employing insect strategies for navigation,” Robot. Auton. Syst. 30(1), 39–64 (2000).
[Crossref]

1997 (1)

D. Lambrinos, H. Kobayashi, R. Pfeifer, M. Maris, T. Labhart, and R. Wehner, “An autonomous agent navigating with a polarized light compass,” Adapt. Behav. 6(1), 131–161 (1997).
[Crossref]

1996 (1)

T. Labhart, “How polarization-sensitive interneurones of crickets perform at low degrees of polarization,” J. Exp. Biol. 199(Pt 7), 1467–1475 (1996).
[PubMed]

1991 (1)

K. F. Evans and G. L. Stephens, “A new polarized atmospheric radiative transfer model,” J. Quant. Spectrosc. Radiat. Transf. 46(5), 413–423 (1991).
[Crossref]

1986 (1)

K. Fent, “Polarized skylight orientation in the desert ant Cataglyphis,” J. Comp. Physiol. A Neuroethol. Sens. Neural Behav. Physiol. 158(2), 145–150 (1986).
[Crossref]

1984 (2)

S. Rossel and R. Wehner, “How bees analyse the polarization patterns in the sky,” J. Comp. Physiol. A Neuroethol. Sens. Neural Behav. Physiol. 154(5), 607–615 (1984).
[Crossref]

T. Labhart, B. Hodel, and I. Valenzuela, “The physiology of the cricket’s compound eye with particular reference to the anatomically specialized dorsal rim area,” J. Comp. Physiol. A Neuroethol. Sens. Neural Behav. Physiol. 155(3), 289–296 (1984).
[Crossref]

Åkesson, S.

G. Horváth, A. Barta, I. Pomozi, B. Suhai, R. Hegedüs, S. Åkesson, B. Meyer-Rochow, and R. Wehner, “On the trail of Vikings with polarized skylight: experimental study of the atmospheric optical prerequisites allowing polarimetric navigation by Viking seafarers,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 772–782 (2011).
[Crossref] [PubMed]

R. Hegedüs, S. Åkesson, R. Wehner, and G. Horváth, “Could Vikings have navigated under foggy and cloudy conditions by skylight polarization? On the atmospheric optical prerequisites of polarimetric Viking navigation under foggy and cloudy skies,” Proc. R. Soc. A 463(2080), 1081–1095 (2007).
[Crossref]

R. Hegedüs, S. Åkesson, and G. Horváth, “Polarization patterns of thick clouds: overcast skies have distribution of the angle of polarization similar to that of clear skies,” J. Opt. Soc. Am. A 24(8), 2347–2356 (2007).
[Crossref] [PubMed]

Baird, E.

M. Dacke, M. J. Byrne, E. Baird, C. H. Scholtz, and E. J. Warrant, “How dim is dim? Precision of the celestial compass in moonlight and sunlight,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 697–702 (2011).
[Crossref] [PubMed]

Barta, A.

G. Horváth, A. Barta, I. Pomozi, B. Suhai, R. Hegedüs, S. Åkesson, B. Meyer-Rochow, and R. Wehner, “On the trail of Vikings with polarized skylight: experimental study of the atmospheric optical prerequisites allowing polarimetric navigation by Viking seafarers,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 772–782 (2011).
[Crossref] [PubMed]

R. Hegedüs, A. Barta, B. Bernáth, V. B. Meyer-Rochow, and G. Horváth, “Imaging polarimetry of forest canopies: how the azimuth direction of the sun, occluded by vegetation, can be assessed from the polarization pattern of the sunlit foliage,” Appl. Opt. 46(23), 6019–6032 (2007).
[Crossref] [PubMed]

A. Barta and G. Horváth, “Why is it advantageous for animals to detect celestial polarization in the ultraviolet? Skylight polarization under clouds and canopies is strongest in the UV,” J. Theor. Biol. 226(4), 429–437 (2004).
[Crossref] [PubMed]

G. Horváth, A. Barta, J. Gál, B. Suhai, and O. Haiman, “Ground-based full-sky imaging polarimetry of rapidly changing skies and its use for polarimetric cloud detection,” Appl. Opt. 41(3), 543–559 (2002).
[Crossref] [PubMed]

Bello, D. S. S.

M. Sarkar, D. S. S. Bello, C. V. Hoot, and A. Theuwissen, “Biologically inspired autonomous agent navigation using an integrated polarization analyzing CMOS image sensor,” Procedia Eng. 5(3), 673–676 (2010).
[Crossref]

Bernáth, B.

Byrne, M.

M. Dacke, D. E. Nilsson, C. H. Scholtz, M. Byrne, and E. J. Warrant, “Animal behaviour: Insect orientation to polarized moonlight,” Nature 424(6944), 33 (2003).
[Crossref] [PubMed]

Byrne, M. J.

M. Dacke, M. J. Byrne, E. Baird, C. H. Scholtz, and E. J. Warrant, “How dim is dim? Precision of the celestial compass in moonlight and sunlight,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 697–702 (2011).
[Crossref] [PubMed]

Cao, Y.

Carey, N.

N. Carey and W. Sturzl, “An insect-inspired omnidirectional vision system including uv-sensitivity and polarization,” in Proceedings of IEEE International Conference on Computer Vision Workshops (IEEE, 2011), pp. 312–319.

Chahl, J.

J. Chahl and A. Mizutani, “Biomimetic attitude and orientation sensors,” IEEE Sens. J. 12(2), 289–297 (2012).
[Crossref]

Chu, J. K.

Q. Liu, J. K. Chu, J. Wang, and L. Guan, “Research and simulation Analysis of Atmospheric polarization properties under water cloud condition,” Acta Opt. Sin. 34(3), 0301004 (2014).
[Crossref]

J. K. Chu, W. Wang, Y. Cui, W. Zhi, and Q. S. Gao, “Measurement for influence of aerosols on polarized sky radiance,” Optics and Precision Engineering 20(3), 520–526 (2012).
[Crossref]

J. K. Chu, K. C. Zhao, Q. Zhang, and T. C. Wang, “Construction and performance test of a novel polarization sensor for navigation,” Sensor. and Actuat. A. 148(1), 75–82 (2008).

Cui, Y.

J. K. Chu, W. Wang, Y. Cui, W. Zhi, and Q. S. Gao, “Measurement for influence of aerosols on polarized sky radiance,” Optics and Precision Engineering 20(3), 520–526 (2012).
[Crossref]

Dacke, M.

M. Dacke, M. J. Byrne, E. Baird, C. H. Scholtz, and E. J. Warrant, “How dim is dim? Precision of the celestial compass in moonlight and sunlight,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 697–702 (2011).
[Crossref] [PubMed]

P. Kraft, C. Evangelista, M. Dacke, T. Labhart, and M. V. Srinivasan, “Honeybee navigation: following routes using polarized-light cues,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 703–708 (2011).
[Crossref] [PubMed]

M. Dacke, D. E. Nilsson, C. H. Scholtz, M. Byrne, and E. J. Warrant, “Animal behaviour: Insect orientation to polarized moonlight,” Nature 424(6944), 33 (2003).
[Crossref] [PubMed]

Evangelista, C.

P. Kraft, C. Evangelista, M. Dacke, T. Labhart, and M. V. Srinivasan, “Honeybee navigation: following routes using polarized-light cues,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 703–708 (2011).
[Crossref] [PubMed]

Evans, K. F.

K. F. Evans and G. L. Stephens, “A new polarized atmospheric radiative transfer model,” J. Quant. Spectrosc. Radiat. Transf. 46(5), 413–423 (1991).
[Crossref]

Fan, Z.

Z. Fan, J. Gao, J. Wei, W. Zhang, and Y. Hu, “Investigation of atmospheric polarization measurement method imitating POL-neurons of cataglyphis,” Yiqi Yibiao Xuebao 29(4), 745–749 (2008).

Fent, K.

K. Fent, “Polarized skylight orientation in the desert ant Cataglyphis,” J. Comp. Physiol. A Neuroethol. Sens. Neural Behav. Physiol. 158(2), 145–150 (1986).
[Crossref]

Gál, J.

Gao, J.

Z. Fan, J. Gao, J. Wei, W. Zhang, and Y. Hu, “Investigation of atmospheric polarization measurement method imitating POL-neurons of cataglyphis,” Yiqi Yibiao Xuebao 29(4), 745–749 (2008).

Gao, Q. S.

J. K. Chu, W. Wang, Y. Cui, W. Zhi, and Q. S. Gao, “Measurement for influence of aerosols on polarized sky radiance,” Optics and Precision Engineering 20(3), 520–526 (2012).
[Crossref]

Guan, L.

Q. Liu, J. K. Chu, J. Wang, and L. Guan, “Research and simulation Analysis of Atmospheric polarization properties under water cloud condition,” Acta Opt. Sin. 34(3), 0301004 (2014).
[Crossref]

Haiman, O.

Hao, L.

L. Hao, Z. Kaichun, M. Qiang, and Y. Zheng, “Design and implementation of detection system for skylight polarized pattern using continuously spiniling polarization analyzer,” J. Astronautics (Chinese) 35(9), 1087–1094 (2014).

He, X.

T. Ma, X. Hu, L. Zhang, J. Lian, X. He, Y. Wang, and Z. Xian, “An evaluation of skylight polarization patterns for navigation,” Sensors (Basel) 15(3), 5895–5913 (2015).
[Crossref] [PubMed]

Hegedüs, R.

G. Horváth, A. Barta, I. Pomozi, B. Suhai, R. Hegedüs, S. Åkesson, B. Meyer-Rochow, and R. Wehner, “On the trail of Vikings with polarized skylight: experimental study of the atmospheric optical prerequisites allowing polarimetric navigation by Viking seafarers,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 772–782 (2011).
[Crossref] [PubMed]

R. Hegedüs, S. Åkesson, R. Wehner, and G. Horváth, “Could Vikings have navigated under foggy and cloudy conditions by skylight polarization? On the atmospheric optical prerequisites of polarimetric Viking navigation under foggy and cloudy skies,” Proc. R. Soc. A 463(2080), 1081–1095 (2007).
[Crossref]

R. Hegedüs, S. Åkesson, and G. Horváth, “Polarization patterns of thick clouds: overcast skies have distribution of the angle of polarization similar to that of clear skies,” J. Opt. Soc. Am. A 24(8), 2347–2356 (2007).
[Crossref] [PubMed]

R. Hegedüs, A. Barta, B. Bernáth, V. B. Meyer-Rochow, and G. Horváth, “Imaging polarimetry of forest canopies: how the azimuth direction of the sun, occluded by vegetation, can be assessed from the polarization pattern of the sunlit foliage,” Appl. Opt. 46(23), 6019–6032 (2007).
[Crossref] [PubMed]

Henze, M. J.

M. J. Henze and T. Labhart, “Haze, clouds and limited sky visibility: polarotactic orientation of crickets under difficult stimulus conditions,” J. Exp. Biol. 210(18), 3266–3276 (2007).
[Crossref] [PubMed]

Hodel, B.

T. Labhart, B. Hodel, and I. Valenzuela, “The physiology of the cricket’s compound eye with particular reference to the anatomically specialized dorsal rim area,” J. Comp. Physiol. A Neuroethol. Sens. Neural Behav. Physiol. 155(3), 289–296 (1984).
[Crossref]

Hoot, C. V.

M. Sarkar, D. S. S. Bello, C. V. Hoot, and A. Theuwissen, “Biologically inspired autonomous agent navigation using an integrated polarization analyzing CMOS image sensor,” Procedia Eng. 5(3), 673–676 (2010).
[Crossref]

Horváth, G.

G. Horváth, A. Barta, I. Pomozi, B. Suhai, R. Hegedüs, S. Åkesson, B. Meyer-Rochow, and R. Wehner, “On the trail of Vikings with polarized skylight: experimental study of the atmospheric optical prerequisites allowing polarimetric navigation by Viking seafarers,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 772–782 (2011).
[Crossref] [PubMed]

R. Hegedüs, S. Åkesson, R. Wehner, and G. Horváth, “Could Vikings have navigated under foggy and cloudy conditions by skylight polarization? On the atmospheric optical prerequisites of polarimetric Viking navigation under foggy and cloudy skies,” Proc. R. Soc. A 463(2080), 1081–1095 (2007).
[Crossref]

R. Hegedüs, S. Åkesson, and G. Horváth, “Polarization patterns of thick clouds: overcast skies have distribution of the angle of polarization similar to that of clear skies,” J. Opt. Soc. Am. A 24(8), 2347–2356 (2007).
[Crossref] [PubMed]

R. Hegedüs, A. Barta, B. Bernáth, V. B. Meyer-Rochow, and G. Horváth, “Imaging polarimetry of forest canopies: how the azimuth direction of the sun, occluded by vegetation, can be assessed from the polarization pattern of the sunlit foliage,” Appl. Opt. 46(23), 6019–6032 (2007).
[Crossref] [PubMed]

B. Suhai and G. Horváth, “How well does the Rayleigh model describe the E-vector distribution of skylight in clear and cloudy conditions? A full-sky polarimetric study,” J. Opt. Soc. Am. A 21(9), 1669–1676 (2004).
[Crossref] [PubMed]

A. Barta and G. Horváth, “Why is it advantageous for animals to detect celestial polarization in the ultraviolet? Skylight polarization under clouds and canopies is strongest in the UV,” J. Theor. Biol. 226(4), 429–437 (2004).
[Crossref] [PubMed]

G. Horváth, A. Barta, J. Gál, B. Suhai, and O. Haiman, “Ground-based full-sky imaging polarimetry of rapidly changing skies and its use for polarimetric cloud detection,” Appl. Opt. 41(3), 543–559 (2002).
[Crossref] [PubMed]

I. Pomozi, G. Horváth, and R. Wehner, “How the clear-sky angle of polarization pattern continues underneath clouds: full-sky measurements and implications for animal orientation,” J. Exp. Biol. 204(Pt 17), 2933–2942 (2001).
[PubMed]

Hu, X.

T. Ma, X. Hu, L. Zhang, J. Lian, X. He, Y. Wang, and Z. Xian, “An evaluation of skylight polarization patterns for navigation,” Sensors (Basel) 15(3), 5895–5913 (2015).
[Crossref] [PubMed]

Hu, Y.

Z. Fan, J. Gao, J. Wei, W. Zhang, and Y. Hu, “Investigation of atmospheric polarization measurement method imitating POL-neurons of cataglyphis,” Yiqi Yibiao Xuebao 29(4), 745–749 (2008).

Huang, K.

Kaichun, Z.

L. Hao, Z. Kaichun, M. Qiang, and Y. Zheng, “Design and implementation of detection system for skylight polarized pattern using continuously spiniling polarization analyzer,” J. Astronautics (Chinese) 35(9), 1087–1094 (2014).

Klemm, F. J.

Kobayashi, H.

D. Lambrinos, H. Kobayashi, R. Pfeifer, M. Maris, T. Labhart, and R. Wehner, “An autonomous agent navigating with a polarized light compass,” Adapt. Behav. 6(1), 131–161 (1997).
[Crossref]

Kotchenova, S. Y.

Kraft, P.

P. Kraft, C. Evangelista, M. Dacke, T. Labhart, and M. V. Srinivasan, “Honeybee navigation: following routes using polarized-light cues,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 703–708 (2011).
[Crossref] [PubMed]

Labhart, T.

P. Kraft, C. Evangelista, M. Dacke, T. Labhart, and M. V. Srinivasan, “Honeybee navigation: following routes using polarized-light cues,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 703–708 (2011).
[Crossref] [PubMed]

M. J. Henze and T. Labhart, “Haze, clouds and limited sky visibility: polarotactic orientation of crickets under difficult stimulus conditions,” J. Exp. Biol. 210(18), 3266–3276 (2007).
[Crossref] [PubMed]

D. Lambrinos, R. Möller, T. Labhart, R. Pfeifer, and R. Wehner, “A mobile robot employing insect strategies for navigation,” Robot. Auton. Syst. 30(1), 39–64 (2000).
[Crossref]

D. Lambrinos, H. Kobayashi, R. Pfeifer, M. Maris, T. Labhart, and R. Wehner, “An autonomous agent navigating with a polarized light compass,” Adapt. Behav. 6(1), 131–161 (1997).
[Crossref]

T. Labhart, “How polarization-sensitive interneurones of crickets perform at low degrees of polarization,” J. Exp. Biol. 199(Pt 7), 1467–1475 (1996).
[PubMed]

T. Labhart, B. Hodel, and I. Valenzuela, “The physiology of the cricket’s compound eye with particular reference to the anatomically specialized dorsal rim area,” J. Comp. Physiol. A Neuroethol. Sens. Neural Behav. Physiol. 155(3), 289–296 (1984).
[Crossref]

Lambrinos, D.

D. Lambrinos, R. Möller, T. Labhart, R. Pfeifer, and R. Wehner, “A mobile robot employing insect strategies for navigation,” Robot. Auton. Syst. 30(1), 39–64 (2000).
[Crossref]

D. Lambrinos, H. Kobayashi, R. Pfeifer, M. Maris, T. Labhart, and R. Wehner, “An autonomous agent navigating with a polarized light compass,” Adapt. Behav. 6(1), 131–161 (1997).
[Crossref]

Li, D.

Lian, J.

T. Ma, X. Hu, L. Zhang, J. Lian, X. He, Y. Wang, and Z. Xian, “An evaluation of skylight polarization patterns for navigation,” Sensors (Basel) 15(3), 5895–5913 (2015).
[Crossref] [PubMed]

Liang, H.

S. Zhang, H. Liang, H. Zhu, D. Wang, and B. Yu, “A camera-based real-time polarization sensor and its application to mobile robot navigation,” in Proceedings of the 2014 IEEE International Conference on Robotics and Biomimetics (IEEE, 2014), pp. 271–276.
[Crossref]

Liang, X.

Liu, J.

Liu, Q.

Q. Liu, J. K. Chu, J. Wang, and L. Guan, “Research and simulation Analysis of Atmospheric polarization properties under water cloud condition,” Acta Opt. Sin. 34(3), 0301004 (2014).
[Crossref]

Lu, H.

Ma, T.

T. Ma, X. Hu, L. Zhang, J. Lian, X. He, Y. Wang, and Z. Xian, “An evaluation of skylight polarization patterns for navigation,” Sensors (Basel) 15(3), 5895–5913 (2015).
[Crossref] [PubMed]

Maris, M.

D. Lambrinos, H. Kobayashi, R. Pfeifer, M. Maris, T. Labhart, and R. Wehner, “An autonomous agent navigating with a polarized light compass,” Adapt. Behav. 6(1), 131–161 (1997).
[Crossref]

Matarrese, R.

Meyer-Rochow, B.

G. Horváth, A. Barta, I. Pomozi, B. Suhai, R. Hegedüs, S. Åkesson, B. Meyer-Rochow, and R. Wehner, “On the trail of Vikings with polarized skylight: experimental study of the atmospheric optical prerequisites allowing polarimetric navigation by Viking seafarers,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 772–782 (2011).
[Crossref] [PubMed]

Meyer-Rochow, V. B.

Mizutani, A.

J. Chahl and A. Mizutani, “Biomimetic attitude and orientation sensors,” IEEE Sens. J. 12(2), 289–297 (2012).
[Crossref]

Möller, R.

D. Lambrinos, R. Möller, T. Labhart, R. Pfeifer, and R. Wehner, “A mobile robot employing insect strategies for navigation,” Robot. Auton. Syst. 30(1), 39–64 (2000).
[Crossref]

Nilsson, D. E.

M. Dacke, D. E. Nilsson, C. H. Scholtz, M. Byrne, and E. J. Warrant, “Animal behaviour: Insect orientation to polarized moonlight,” Nature 424(6944), 33 (2003).
[Crossref] [PubMed]

Ning, Y.

Pfeifer, R.

D. Lambrinos, R. Möller, T. Labhart, R. Pfeifer, and R. Wehner, “A mobile robot employing insect strategies for navigation,” Robot. Auton. Syst. 30(1), 39–64 (2000).
[Crossref]

D. Lambrinos, H. Kobayashi, R. Pfeifer, M. Maris, T. Labhart, and R. Wehner, “An autonomous agent navigating with a polarized light compass,” Adapt. Behav. 6(1), 131–161 (1997).
[Crossref]

Pomozi, I.

G. Horváth, A. Barta, I. Pomozi, B. Suhai, R. Hegedüs, S. Åkesson, B. Meyer-Rochow, and R. Wehner, “On the trail of Vikings with polarized skylight: experimental study of the atmospheric optical prerequisites allowing polarimetric navigation by Viking seafarers,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 772–782 (2011).
[Crossref] [PubMed]

I. Pomozi, G. Horváth, and R. Wehner, “How the clear-sky angle of polarization pattern continues underneath clouds: full-sky measurements and implications for animal orientation,” J. Exp. Biol. 204(Pt 17), 2933–2942 (2001).
[PubMed]

Pust, N. J.

N. J. Pust and J. A. Shaw, “Comparison of skylight polarization measurements and MODTRAN-P calculations,” J. Appl. Remote Sens. 5(1), 053529 (2011).
[Crossref]

N. J. Pust and J. A. Shaw, “Digital all-sky polarization imaging of partly cloudy skies,” Appl. Opt. 47(34), H190–H198 (2008).
[Crossref] [PubMed]

Qiang, M.

L. Hao, Z. Kaichun, M. Qiang, and Y. Zheng, “Design and implementation of detection system for skylight polarized pattern using continuously spiniling polarization analyzer,” J. Astronautics (Chinese) 35(9), 1087–1094 (2014).

Ren, J.

Rossel, S.

S. Rossel and R. Wehner, “How bees analyse the polarization patterns in the sky,” J. Comp. Physiol. A Neuroethol. Sens. Neural Behav. Physiol. 154(5), 607–615 (1984).
[Crossref]

Sarkar, M.

M. Sarkar, D. S. S. Bello, C. V. Hoot, and A. Theuwissen, “Biologically inspired autonomous agent navigation using an integrated polarization analyzing CMOS image sensor,” Procedia Eng. 5(3), 673–676 (2010).
[Crossref]

Scholtz, C. H.

M. Dacke, M. J. Byrne, E. Baird, C. H. Scholtz, and E. J. Warrant, “How dim is dim? Precision of the celestial compass in moonlight and sunlight,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 697–702 (2011).
[Crossref] [PubMed]

M. Dacke, D. E. Nilsson, C. H. Scholtz, M. Byrne, and E. J. Warrant, “Animal behaviour: Insect orientation to polarized moonlight,” Nature 424(6944), 33 (2003).
[Crossref] [PubMed]

Shaw, J. A.

N. J. Pust and J. A. Shaw, “Comparison of skylight polarization measurements and MODTRAN-P calculations,” J. Appl. Remote Sens. 5(1), 053529 (2011).
[Crossref]

N. J. Pust and J. A. Shaw, “Digital all-sky polarization imaging of partly cloudy skies,” Appl. Opt. 47(34), H190–H198 (2008).
[Crossref] [PubMed]

Shen, C.

Shi, S.

Song, P.

Srinivasan, M. V.

P. Kraft, C. Evangelista, M. Dacke, T. Labhart, and M. V. Srinivasan, “Honeybee navigation: following routes using polarized-light cues,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 703–708 (2011).
[Crossref] [PubMed]

Stephens, G. L.

K. F. Evans and G. L. Stephens, “A new polarized atmospheric radiative transfer model,” J. Quant. Spectrosc. Radiat. Transf. 46(5), 413–423 (1991).
[Crossref]

Sturzl, W.

N. Carey and W. Sturzl, “An insect-inspired omnidirectional vision system including uv-sensitivity and polarization,” in Proceedings of IEEE International Conference on Computer Vision Workshops (IEEE, 2011), pp. 312–319.

Suhai, B.

G. Horváth, A. Barta, I. Pomozi, B. Suhai, R. Hegedüs, S. Åkesson, B. Meyer-Rochow, and R. Wehner, “On the trail of Vikings with polarized skylight: experimental study of the atmospheric optical prerequisites allowing polarimetric navigation by Viking seafarers,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 772–782 (2011).
[Crossref] [PubMed]

B. Suhai and G. Horváth, “How well does the Rayleigh model describe the E-vector distribution of skylight in clear and cloudy conditions? A full-sky polarimetric study,” J. Opt. Soc. Am. A 21(9), 1669–1676 (2004).
[Crossref] [PubMed]

G. Horváth, A. Barta, J. Gál, B. Suhai, and O. Haiman, “Ground-based full-sky imaging polarimetry of rapidly changing skies and its use for polarimetric cloud detection,” Appl. Opt. 41(3), 543–559 (2002).
[Crossref] [PubMed]

Tang, J.

Theuwissen, A.

M. Sarkar, D. S. S. Bello, C. V. Hoot, and A. Theuwissen, “Biologically inspired autonomous agent navigation using an integrated polarization analyzing CMOS image sensor,” Procedia Eng. 5(3), 673–676 (2010).
[Crossref]

Valenzuela, I.

T. Labhart, B. Hodel, and I. Valenzuela, “The physiology of the cricket’s compound eye with particular reference to the anatomically specialized dorsal rim area,” J. Comp. Physiol. A Neuroethol. Sens. Neural Behav. Physiol. 155(3), 289–296 (1984).
[Crossref]

Vermote, E. F.

Wang, C.

Wang, D.

S. Zhang, H. Liang, H. Zhu, D. Wang, and B. Yu, “A camera-based real-time polarization sensor and its application to mobile robot navigation,” in Proceedings of the 2014 IEEE International Conference on Robotics and Biomimetics (IEEE, 2014), pp. 271–276.
[Crossref]

Wang, F.

Wang, J.

Q. Liu, J. K. Chu, J. Wang, and L. Guan, “Research and simulation Analysis of Atmospheric polarization properties under water cloud condition,” Acta Opt. Sin. 34(3), 0301004 (2014).
[Crossref]

Wang, T. C.

J. K. Chu, K. C. Zhao, Q. Zhang, and T. C. Wang, “Construction and performance test of a novel polarization sensor for navigation,” Sensor. and Actuat. A. 148(1), 75–82 (2008).

Wang, W.

J. K. Chu, W. Wang, Y. Cui, W. Zhi, and Q. S. Gao, “Measurement for influence of aerosols on polarized sky radiance,” Optics and Precision Engineering 20(3), 520–526 (2012).
[Crossref]

Wang, Y.

T. Ma, X. Hu, L. Zhang, J. Lian, X. He, Y. Wang, and Z. Xian, “An evaluation of skylight polarization patterns for navigation,” Sensors (Basel) 15(3), 5895–5913 (2015).
[Crossref] [PubMed]

Warrant, E. J.

M. Dacke, M. J. Byrne, E. Baird, C. H. Scholtz, and E. J. Warrant, “How dim is dim? Precision of the celestial compass in moonlight and sunlight,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 697–702 (2011).
[Crossref] [PubMed]

M. Dacke, D. E. Nilsson, C. H. Scholtz, M. Byrne, and E. J. Warrant, “Animal behaviour: Insect orientation to polarized moonlight,” Nature 424(6944), 33 (2003).
[Crossref] [PubMed]

Wehner, R.

G. Horváth, A. Barta, I. Pomozi, B. Suhai, R. Hegedüs, S. Åkesson, B. Meyer-Rochow, and R. Wehner, “On the trail of Vikings with polarized skylight: experimental study of the atmospheric optical prerequisites allowing polarimetric navigation by Viking seafarers,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 772–782 (2011).
[Crossref] [PubMed]

R. Hegedüs, S. Åkesson, R. Wehner, and G. Horváth, “Could Vikings have navigated under foggy and cloudy conditions by skylight polarization? On the atmospheric optical prerequisites of polarimetric Viking navigation under foggy and cloudy skies,” Proc. R. Soc. A 463(2080), 1081–1095 (2007).
[Crossref]

I. Pomozi, G. Horváth, and R. Wehner, “How the clear-sky angle of polarization pattern continues underneath clouds: full-sky measurements and implications for animal orientation,” J. Exp. Biol. 204(Pt 17), 2933–2942 (2001).
[PubMed]

D. Lambrinos, R. Möller, T. Labhart, R. Pfeifer, and R. Wehner, “A mobile robot employing insect strategies for navigation,” Robot. Auton. Syst. 30(1), 39–64 (2000).
[Crossref]

D. Lambrinos, H. Kobayashi, R. Pfeifer, M. Maris, T. Labhart, and R. Wehner, “An autonomous agent navigating with a polarized light compass,” Adapt. Behav. 6(1), 131–161 (1997).
[Crossref]

S. Rossel and R. Wehner, “How bees analyse the polarization patterns in the sky,” J. Comp. Physiol. A Neuroethol. Sens. Neural Behav. Physiol. 154(5), 607–615 (1984).
[Crossref]

Wei, J.

Z. Fan, J. Gao, J. Wei, W. Zhang, and Y. Hu, “Investigation of atmospheric polarization measurement method imitating POL-neurons of cataglyphis,” Yiqi Yibiao Xuebao 29(4), 745–749 (2008).

Xian, Z.

T. Ma, X. Hu, L. Zhang, J. Lian, X. He, Y. Wang, and Z. Xian, “An evaluation of skylight polarization patterns for navigation,” Sensors (Basel) 15(3), 5895–5913 (2015).
[Crossref] [PubMed]

Xu, W.

Xue, C.

Yang, Y.

You, Z.

Yu, B.

S. Zhang, H. Liang, H. Zhu, D. Wang, and B. Yu, “A camera-based real-time polarization sensor and its application to mobile robot navigation,” in Proceedings of the 2014 IEEE International Conference on Robotics and Biomimetics (IEEE, 2014), pp. 271–276.
[Crossref]

Zhang, B.

Zhang, L.

T. Ma, X. Hu, L. Zhang, J. Lian, X. He, Y. Wang, and Z. Xian, “An evaluation of skylight polarization patterns for navigation,” Sensors (Basel) 15(3), 5895–5913 (2015).
[Crossref] [PubMed]

Zhang, N.

Zhang, Q.

J. K. Chu, K. C. Zhao, Q. Zhang, and T. C. Wang, “Construction and performance test of a novel polarization sensor for navigation,” Sensor. and Actuat. A. 148(1), 75–82 (2008).

Zhang, S.

S. Zhang, H. Liang, H. Zhu, D. Wang, and B. Yu, “A camera-based real-time polarization sensor and its application to mobile robot navigation,” in Proceedings of the 2014 IEEE International Conference on Robotics and Biomimetics (IEEE, 2014), pp. 271–276.
[Crossref]

Zhang, W.

W. Zhang, Y. Cao, X. Zhang, Y. Yang, and Y. Ning, “Angle of sky light polarization derived from digital images of the sky under various conditions,” Appl. Opt. 56(3), 587–595 (2017).
[Crossref] [PubMed]

Z. Fan, J. Gao, J. Wei, W. Zhang, and Y. Hu, “Investigation of atmospheric polarization measurement method imitating POL-neurons of cataglyphis,” Yiqi Yibiao Xuebao 29(4), 745–749 (2008).

Zhang, X.

Zhang, Y.

Zhao, H.

Zhao, K.

Zhao, K. C.

J. K. Chu, K. C. Zhao, Q. Zhang, and T. C. Wang, “Construction and performance test of a novel polarization sensor for navigation,” Sensor. and Actuat. A. 148(1), 75–82 (2008).

Zheng, Y.

L. Hao, Z. Kaichun, M. Qiang, and Y. Zheng, “Design and implementation of detection system for skylight polarized pattern using continuously spiniling polarization analyzer,” J. Astronautics (Chinese) 35(9), 1087–1094 (2014).

Zhi, W.

J. K. Chu, W. Wang, Y. Cui, W. Zhi, and Q. S. Gao, “Measurement for influence of aerosols on polarized sky radiance,” Optics and Precision Engineering 20(3), 520–526 (2012).
[Crossref]

Zhu, H.

S. Zhang, H. Liang, H. Zhu, D. Wang, and B. Yu, “A camera-based real-time polarization sensor and its application to mobile robot navigation,” in Proceedings of the 2014 IEEE International Conference on Robotics and Biomimetics (IEEE, 2014), pp. 271–276.
[Crossref]

Acta Opt. Sin. (1)

Q. Liu, J. K. Chu, J. Wang, and L. Guan, “Research and simulation Analysis of Atmospheric polarization properties under water cloud condition,” Acta Opt. Sin. 34(3), 0301004 (2014).
[Crossref]

Adapt. Behav. (1)

D. Lambrinos, H. Kobayashi, R. Pfeifer, M. Maris, T. Labhart, and R. Wehner, “An autonomous agent navigating with a polarized light compass,” Adapt. Behav. 6(1), 131–161 (1997).
[Crossref]

Appl. Opt. (5)

IEEE Sens. J. (1)

J. Chahl and A. Mizutani, “Biomimetic attitude and orientation sensors,” IEEE Sens. J. 12(2), 289–297 (2012).
[Crossref]

J. Appl. Remote Sens. (1)

N. J. Pust and J. A. Shaw, “Comparison of skylight polarization measurements and MODTRAN-P calculations,” J. Appl. Remote Sens. 5(1), 053529 (2011).
[Crossref]

J. Astronautics (Chinese) (1)

L. Hao, Z. Kaichun, M. Qiang, and Y. Zheng, “Design and implementation of detection system for skylight polarized pattern using continuously spiniling polarization analyzer,” J. Astronautics (Chinese) 35(9), 1087–1094 (2014).

J. Comp. Physiol. A Neuroethol. Sens. Neural Behav. Physiol. (3)

S. Rossel and R. Wehner, “How bees analyse the polarization patterns in the sky,” J. Comp. Physiol. A Neuroethol. Sens. Neural Behav. Physiol. 154(5), 607–615 (1984).
[Crossref]

T. Labhart, B. Hodel, and I. Valenzuela, “The physiology of the cricket’s compound eye with particular reference to the anatomically specialized dorsal rim area,” J. Comp. Physiol. A Neuroethol. Sens. Neural Behav. Physiol. 155(3), 289–296 (1984).
[Crossref]

K. Fent, “Polarized skylight orientation in the desert ant Cataglyphis,” J. Comp. Physiol. A Neuroethol. Sens. Neural Behav. Physiol. 158(2), 145–150 (1986).
[Crossref]

J. Exp. Biol. (3)

M. J. Henze and T. Labhart, “Haze, clouds and limited sky visibility: polarotactic orientation of crickets under difficult stimulus conditions,” J. Exp. Biol. 210(18), 3266–3276 (2007).
[Crossref] [PubMed]

I. Pomozi, G. Horváth, and R. Wehner, “How the clear-sky angle of polarization pattern continues underneath clouds: full-sky measurements and implications for animal orientation,” J. Exp. Biol. 204(Pt 17), 2933–2942 (2001).
[PubMed]

T. Labhart, “How polarization-sensitive interneurones of crickets perform at low degrees of polarization,” J. Exp. Biol. 199(Pt 7), 1467–1475 (1996).
[PubMed]

J. Opt. Soc. Am. A (2)

J. Quant. Spectrosc. Radiat. Transf. (1)

K. F. Evans and G. L. Stephens, “A new polarized atmospheric radiative transfer model,” J. Quant. Spectrosc. Radiat. Transf. 46(5), 413–423 (1991).
[Crossref]

J. Theor. Biol. (1)

A. Barta and G. Horváth, “Why is it advantageous for animals to detect celestial polarization in the ultraviolet? Skylight polarization under clouds and canopies is strongest in the UV,” J. Theor. Biol. 226(4), 429–437 (2004).
[Crossref] [PubMed]

Nature (1)

M. Dacke, D. E. Nilsson, C. H. Scholtz, M. Byrne, and E. J. Warrant, “Animal behaviour: Insect orientation to polarized moonlight,” Nature 424(6944), 33 (2003).
[Crossref] [PubMed]

Opt. Express (3)

Optics and Precision Engineering (1)

J. K. Chu, W. Wang, Y. Cui, W. Zhi, and Q. S. Gao, “Measurement for influence of aerosols on polarized sky radiance,” Optics and Precision Engineering 20(3), 520–526 (2012).
[Crossref]

Philos. Trans. R. Soc. Lond. B Biol. Sci. (3)

P. Kraft, C. Evangelista, M. Dacke, T. Labhart, and M. V. Srinivasan, “Honeybee navigation: following routes using polarized-light cues,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 703–708 (2011).
[Crossref] [PubMed]

M. Dacke, M. J. Byrne, E. Baird, C. H. Scholtz, and E. J. Warrant, “How dim is dim? Precision of the celestial compass in moonlight and sunlight,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 697–702 (2011).
[Crossref] [PubMed]

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[Crossref] [PubMed]

Proc. R. Soc. A (1)

R. Hegedüs, S. Åkesson, R. Wehner, and G. Horváth, “Could Vikings have navigated under foggy and cloudy conditions by skylight polarization? On the atmospheric optical prerequisites of polarimetric Viking navigation under foggy and cloudy skies,” Proc. R. Soc. A 463(2080), 1081–1095 (2007).
[Crossref]

Procedia Eng. (1)

M. Sarkar, D. S. S. Bello, C. V. Hoot, and A. Theuwissen, “Biologically inspired autonomous agent navigation using an integrated polarization analyzing CMOS image sensor,” Procedia Eng. 5(3), 673–676 (2010).
[Crossref]

Robot. Auton. Syst. (1)

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[Crossref]

Sensor. and Actuat. A. (1)

J. K. Chu, K. C. Zhao, Q. Zhang, and T. C. Wang, “Construction and performance test of a novel polarization sensor for navigation,” Sensor. and Actuat. A. 148(1), 75–82 (2008).

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H. Zhao and W. Xu, “A bionic polarization navigation sensor and its calibration method,” Sensors (Basel) 16(8), 1223 (2016).
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Yiqi Yibiao Xuebao (1)

Z. Fan, J. Gao, J. Wei, W. Zhang, and Y. Hu, “Investigation of atmospheric polarization measurement method imitating POL-neurons of cataglyphis,” Yiqi Yibiao Xuebao 29(4), 745–749 (2008).

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[Crossref]

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Figures (12)

Fig. 1
Fig. 1 The polarization of an arbitrary scattered skylight as experienced by an observer in original point O.
Fig. 2
Fig. 2 The patterns of polarization in the sky as experienced by an observer in original point O. S stands for the solar position and Z stands for the zenith. The polarization patterns only considered the Rayleigh scattering.
Fig. 3
Fig. 3 Photo of the experiment.
Fig. 4
Fig. 4 A typical gray scale image obtained by the camera, the center point of the map corresponds to the zenith point; the edge of the image corresponds to the horizon. There are some roofs of the tall buildings appear on the edge of the image. The black disk is a shading plate, which is used to shield the direct solar radiance to avoid overexposure. The arrow cross in the right part of the picture shows the direction information. Since the camera aimed at the zenith, the direction of the polarized image is opposite to that of the geographic coordinate system.
Fig. 5
Fig. 5 DOP and AOP patterns of the sky dome measured under different aerosol optical thickness. Group A: November 3, 2017, clear weather without air pollution. Group B: March 22, 2018, the air was slightly polluted by the haze. Group C: March 28, 2018, there was a sand storm and the air was heavily polluted.
Fig. 6
Fig. 6 The gray images of the sky dome under clear sky and three kinds of cloudy sky.
Fig. 7
Fig. 7 The DOP and AOP maps under different cloudy skies. Group A: November 3, 2017, clear sky. Group B: March 15, 2018, less cloudy sky. Group C: October 24, 2017, partly cloudy sky. Group D: April 3, 2018, overcast sky, the solar position is marked with a black disk.
Fig. 8
Fig. 8 A: The DOP map used for symmetry analysis. B: The mask image obtained from the DOP map, the white area stands for the valid data area, and the range of valid DOP is 0.05<DOP<0.9. C: The AOP image.
Fig. 9
Fig. 9 Coordinate system of the AOP image.
Fig. 10
Fig. 10 The results of symmetry scanning, the horizontal axis represents the scanning angle and the vertical axis represents the symmetry value. (A) The sparse scanning data with a step of 1°. The lowest symmetry value is marked out with the red dashed circle, and the dense scanning is around that angle range. (B) Dense scanning data with a step of 0.05°. The scanning data is represented with the discrete blue points, and the fitted curve is in red.
Fig. 11
Fig. 11 (A) The AOP map of skylight. (B) The gray image of skylight. The solar meridian calculated by our method is marked with white a bi-directional arrow.
Fig. 12
Fig. 12 The 476nm gray images of the sky dome in five different weather conditions. The red line indicates the solar meridian calculated by the symmetry scanning method. The numbers below the five images are the random errors during each scanning.

Tables (2)

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Table 1 AOT and solar zenith angle of the measurements

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Table 2 Conditions during the measurements

Equations (6)

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y c y =tanθ(x c x )
( x m x k , y m y k )(1,tanθ)
{ cosθ( y m c y )=sinθ( x m c x ) cosθ( x m x k )+sinθ( y m y k )=0
{ x k' =2 x m x k y k' =2 y m y k
s k =| AOP( x k , y k )AOP( x k' , y k' ) |
s θ = k=1 N s k

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