Abstract

Polarization navigation is a promising orientation-determination method inspired by insects’ foraging behavior that offers the advantages of autonomous and high precision. In this paper, using the solar meridian as an azimuth reference is proposed. The model of the distribution pattern of the polarized skylight projected onto an imaging sensor is analyzed. The sufficient features of the solar meridian are proven. According to these features, an angle algorithm for an imaging polarization navigation sensor based on a machine-vision algorithm is proposed. In consideration of noise in images, the relation between the measured angle and the noise in images is modeled. This model cannot only optimize the threshold tolerance R in the algorithm but also describe the effects of several primary factors that can affect the measuring precision. In the simulation test, the measurement accuracy was better than 0.34°. When the algorithm was tested on the polarization-detection system, the measurement accuracy was better than 0.37°.

© 2015 Optical Society of America

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References

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  1. S. Rössel and R. Wehner, “Polarization vision in bees,” Nature 323(6084), 128–131 (1986).
    [Crossref]
  2. T. Labhart and K. Keller, “Fine structure and growth of the polarization-sensitive dorsal rim area in the compound eye of larval crickets,” Naturwissenschaften 79(11), 527–529 (1992).
    [Crossref]
  3. 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]
  4. 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]
  5. 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]
  6. 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,” P. Roy. Soc. A: Math. Phy. 463(2080), 1081–1095 (2007).
    [Crossref]
  7. J. Chahl and A. Mizutani, “Biomimetic attitude and orientation sensors,” IEEE Sens. J. 12(2), 289–297 (2012).
    [Crossref]
  8. J. Chu, K. Zhao, T. Wang, and Q. Zhang, “Research on a novel polarization sensor for navigation,” in Proceedings of IEEE Conference on Information Acquisition (IEEE, 2007), pp. 241–246.
    [Crossref]
  9. J. Chu, K. Zhao, Q. Zhang, and T. Wang, “Construction and performance test of a novel polarization sensor for navigation,” Sens. Actuators A Phys. 148(1), 75–82 (2008).
    [Crossref]
  10. 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]
  11. W. Stürzl and N. Carey, “A fisheye camera system for polarisation detection on UAVs,” in Proceedings of Computer Vision–ECCV 2012. Workshops and Demonstrations (Springer, 2012), pp. 431–440.
  12. D. Miyazaki, M. Ammar, R. Kawakami, and K. Ikeuchi, “Estimating sunlight polarization using a fish-eye lens,” IPSJ J. 49, 1234–1246 (2008).
  13. D. H. Ballard, “Generalizing the Hough transform to detect arbitrary shapes,” Pattern Recognit. 13(2), 111–122 (1981).
    [Crossref]
  14. J. Princen, J. Illingworth, and J. Kittler, “Hypothesis testing: a framework for analyzing and optimizing Hough transform performance,” IEEE Trans. Pattern Anal. Mach. Intell. 16(4), 329–341 (1994).
    [Crossref]
  15. F. Goudail and A. Bénière, “Estimation precision of the degree of linear polarization and of the angle of polarization in the presence of different sources of noise,” Appl. Opt. 49(4), 683–693 (2010).
    [Crossref] [PubMed]
  16. H. Lu, K. Zhao, and Z. You, “Automatic detection system for skyligth polarized pattern,” Opt. Precis. Eng. 21(2), 239–245 (2013).
    [Crossref]
  17. L. Le Cam, “An approximation theorem for the Poisson binomial distribution,” Pac. J. Math. 10(4), 1181–1197 (1960).
    [Crossref]
  18. Y. Hong, “On computing the distribution function for the Poisson binomial distribution,” Comput. Stat. Data Anal. 59, 41–51 (2013).
    [Crossref]
  19. J. Chu, W. Wang, Y. Cui, W. Zhi, and Q. Gao, “Measurement for influence of aerosols on polarized sky radiance,” Opt. Precis. Eng. 20(3), 520–526.
  20. EMVA, “EMVA-1288-3.0, ” http://www.emva.org/cms/upload/Standards/Stadard_1288/EMVA1288-3.0.pdf . accessed 6/6/14.
  21. H. Lu, K. Zhao, Z. You, and Q. Ma, “Design and implementation of detection system for skylight polarized pattern using continuously spinning polarization analyzer,” J. Astronaut. 35(9), 1087–1094.
  22. Z. Zhang, “A flexible new technique for camera calibration,” IEEE T. Pattern Anal. 22(11), 1330–1334 (2000).
    [Crossref]

2013 (2)

H. Lu, K. Zhao, and Z. You, “Automatic detection system for skyligth polarized pattern,” Opt. Precis. Eng. 21(2), 239–245 (2013).
[Crossref]

Y. Hong, “On computing the distribution function for the Poisson binomial distribution,” Comput. Stat. Data Anal. 59, 41–51 (2013).
[Crossref]

2012 (1)

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

2010 (1)

2008 (2)

J. Chu, K. Zhao, Q. Zhang, and T. Wang, “Construction and performance test of a novel polarization sensor for navigation,” Sens. Actuators A Phys. 148(1), 75–82 (2008).
[Crossref]

D. Miyazaki, M. Ammar, R. Kawakami, and K. Ikeuchi, “Estimating sunlight polarization using a fish-eye lens,” IPSJ J. 49, 1234–1246 (2008).

2007 (2)

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,” P. Roy. Soc. A: Math. Phy. 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]

2004 (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 (2)

Z. Zhang, “A flexible new technique for camera calibration,” IEEE T. Pattern Anal. 22(11), 1330–1334 (2000).
[Crossref]

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]

1994 (1)

J. Princen, J. Illingworth, and J. Kittler, “Hypothesis testing: a framework for analyzing and optimizing Hough transform performance,” IEEE Trans. Pattern Anal. Mach. Intell. 16(4), 329–341 (1994).
[Crossref]

1992 (1)

T. Labhart and K. Keller, “Fine structure and growth of the polarization-sensitive dorsal rim area in the compound eye of larval crickets,” Naturwissenschaften 79(11), 527–529 (1992).
[Crossref]

1986 (1)

S. Rössel and R. Wehner, “Polarization vision in bees,” Nature 323(6084), 128–131 (1986).
[Crossref]

1981 (1)

D. H. Ballard, “Generalizing the Hough transform to detect arbitrary shapes,” Pattern Recognit. 13(2), 111–122 (1981).
[Crossref]

1960 (1)

L. Le Cam, “An approximation theorem for the Poisson binomial distribution,” Pac. J. Math. 10(4), 1181–1197 (1960).
[Crossref]

Åkesson, S.

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,” P. Roy. Soc. A: Math. Phy. 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]

Ammar, M.

D. Miyazaki, M. Ammar, R. Kawakami, and K. Ikeuchi, “Estimating sunlight polarization using a fish-eye lens,” IPSJ J. 49, 1234–1246 (2008).

Ballard, D. H.

D. H. Ballard, “Generalizing the Hough transform to detect arbitrary shapes,” Pattern Recognit. 13(2), 111–122 (1981).
[Crossref]

Bénière, A.

Chahl, J.

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

Chu, J.

J. Chu, K. Zhao, Q. Zhang, and T. Wang, “Construction and performance test of a novel polarization sensor for navigation,” Sens. Actuators A Phys. 148(1), 75–82 (2008).
[Crossref]

J. Chu, K. Zhao, T. Wang, and Q. Zhang, “Research on a novel polarization sensor for navigation,” in Proceedings of IEEE Conference on Information Acquisition (IEEE, 2007), pp. 241–246.
[Crossref]

Goudail, F.

Hegedüs, R.

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, 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,” P. Roy. Soc. A: Math. Phy. 463(2080), 1081–1095 (2007).
[Crossref]

Hong, Y.

Y. Hong, “On computing the distribution function for the Poisson binomial distribution,” Comput. Stat. Data Anal. 59, 41–51 (2013).
[Crossref]

Horváth, G.

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,” P. Roy. Soc. A: Math. Phy. 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]

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]

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]

Ikeuchi, K.

D. Miyazaki, M. Ammar, R. Kawakami, and K. Ikeuchi, “Estimating sunlight polarization using a fish-eye lens,” IPSJ J. 49, 1234–1246 (2008).

Illingworth, J.

J. Princen, J. Illingworth, and J. Kittler, “Hypothesis testing: a framework for analyzing and optimizing Hough transform performance,” IEEE Trans. Pattern Anal. Mach. Intell. 16(4), 329–341 (1994).
[Crossref]

Kawakami, R.

D. Miyazaki, M. Ammar, R. Kawakami, and K. Ikeuchi, “Estimating sunlight polarization using a fish-eye lens,” IPSJ J. 49, 1234–1246 (2008).

Keller, K.

T. Labhart and K. Keller, “Fine structure and growth of the polarization-sensitive dorsal rim area in the compound eye of larval crickets,” Naturwissenschaften 79(11), 527–529 (1992).
[Crossref]

Kittler, J.

J. Princen, J. Illingworth, and J. Kittler, “Hypothesis testing: a framework for analyzing and optimizing Hough transform performance,” IEEE Trans. Pattern Anal. Mach. Intell. 16(4), 329–341 (1994).
[Crossref]

Labhart, T.

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]

T. Labhart and K. Keller, “Fine structure and growth of the polarization-sensitive dorsal rim area in the compound eye of larval crickets,” Naturwissenschaften 79(11), 527–529 (1992).
[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]

Le Cam, L.

L. Le Cam, “An approximation theorem for the Poisson binomial distribution,” Pac. J. Math. 10(4), 1181–1197 (1960).
[Crossref]

Lu, H.

H. Lu, K. Zhao, and Z. You, “Automatic detection system for skyligth polarized pattern,” Opt. Precis. Eng. 21(2), 239–245 (2013).
[Crossref]

Miyazaki, D.

D. Miyazaki, M. Ammar, R. Kawakami, and K. Ikeuchi, “Estimating sunlight polarization using a fish-eye lens,” IPSJ J. 49, 1234–1246 (2008).

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]

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]

Pomozi, I.

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]

Princen, J.

J. Princen, J. Illingworth, and J. Kittler, “Hypothesis testing: a framework for analyzing and optimizing Hough transform performance,” IEEE Trans. Pattern Anal. Mach. Intell. 16(4), 329–341 (1994).
[Crossref]

Rössel, S.

S. Rössel and R. Wehner, “Polarization vision in bees,” Nature 323(6084), 128–131 (1986).
[Crossref]

Suhai, B.

Wang, T.

J. Chu, K. Zhao, Q. Zhang, and T. Wang, “Construction and performance test of a novel polarization sensor for navigation,” Sens. Actuators A Phys. 148(1), 75–82 (2008).
[Crossref]

J. Chu, K. Zhao, T. Wang, and Q. Zhang, “Research on a novel polarization sensor for navigation,” in Proceedings of IEEE Conference on Information Acquisition (IEEE, 2007), pp. 241–246.
[Crossref]

Wehner, R.

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,” P. Roy. Soc. A: Math. Phy. 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]

S. Rössel and R. Wehner, “Polarization vision in bees,” Nature 323(6084), 128–131 (1986).
[Crossref]

You, Z.

H. Lu, K. Zhao, and Z. You, “Automatic detection system for skyligth polarized pattern,” Opt. Precis. Eng. 21(2), 239–245 (2013).
[Crossref]

Zhang, Q.

J. Chu, K. Zhao, Q. Zhang, and T. Wang, “Construction and performance test of a novel polarization sensor for navigation,” Sens. Actuators A Phys. 148(1), 75–82 (2008).
[Crossref]

J. Chu, K. Zhao, T. Wang, and Q. Zhang, “Research on a novel polarization sensor for navigation,” in Proceedings of IEEE Conference on Information Acquisition (IEEE, 2007), pp. 241–246.
[Crossref]

Zhang, Z.

Z. Zhang, “A flexible new technique for camera calibration,” IEEE T. Pattern Anal. 22(11), 1330–1334 (2000).
[Crossref]

Zhao, K.

H. Lu, K. Zhao, and Z. You, “Automatic detection system for skyligth polarized pattern,” Opt. Precis. Eng. 21(2), 239–245 (2013).
[Crossref]

J. Chu, K. Zhao, Q. Zhang, and T. Wang, “Construction and performance test of a novel polarization sensor for navigation,” Sens. Actuators A Phys. 148(1), 75–82 (2008).
[Crossref]

J. Chu, K. Zhao, T. Wang, and Q. Zhang, “Research on a novel polarization sensor for navigation,” in Proceedings of IEEE Conference on Information Acquisition (IEEE, 2007), pp. 241–246.
[Crossref]

Appl. Opt. (1)

Comput. Stat. Data Anal. (1)

Y. Hong, “On computing the distribution function for the Poisson binomial distribution,” Comput. Stat. Data Anal. 59, 41–51 (2013).
[Crossref]

IEEE Sens. J. (1)

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

IEEE T. Pattern Anal. (1)

Z. Zhang, “A flexible new technique for camera calibration,” IEEE T. Pattern Anal. 22(11), 1330–1334 (2000).
[Crossref]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

J. Princen, J. Illingworth, and J. Kittler, “Hypothesis testing: a framework for analyzing and optimizing Hough transform performance,” IEEE Trans. Pattern Anal. Mach. Intell. 16(4), 329–341 (1994).
[Crossref]

IPSJ J. (1)

D. Miyazaki, M. Ammar, R. Kawakami, and K. Ikeuchi, “Estimating sunlight polarization using a fish-eye lens,” IPSJ J. 49, 1234–1246 (2008).

J. Exp. Biol. (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]

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

Nature (1)

S. Rössel and R. Wehner, “Polarization vision in bees,” Nature 323(6084), 128–131 (1986).
[Crossref]

Naturwissenschaften (1)

T. Labhart and K. Keller, “Fine structure and growth of the polarization-sensitive dorsal rim area in the compound eye of larval crickets,” Naturwissenschaften 79(11), 527–529 (1992).
[Crossref]

Opt. Precis. Eng. (1)

H. Lu, K. Zhao, and Z. You, “Automatic detection system for skyligth polarized pattern,” Opt. Precis. Eng. 21(2), 239–245 (2013).
[Crossref]

P. Roy. Soc. A: Math. Phy. (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,” P. Roy. Soc. A: Math. Phy. 463(2080), 1081–1095 (2007).
[Crossref]

Pac. J. Math. (1)

L. Le Cam, “An approximation theorem for the Poisson binomial distribution,” Pac. J. Math. 10(4), 1181–1197 (1960).
[Crossref]

Pattern Recognit. (1)

D. H. Ballard, “Generalizing the Hough transform to detect arbitrary shapes,” Pattern Recognit. 13(2), 111–122 (1981).
[Crossref]

Robot. Auton. Syst. (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]

Sens. Actuators A Phys. (1)

J. Chu, K. Zhao, Q. Zhang, and T. Wang, “Construction and performance test of a novel polarization sensor for navigation,” Sens. Actuators A Phys. 148(1), 75–82 (2008).
[Crossref]

Other (5)

J. Chu, K. Zhao, T. Wang, and Q. Zhang, “Research on a novel polarization sensor for navigation,” in Proceedings of IEEE Conference on Information Acquisition (IEEE, 2007), pp. 241–246.
[Crossref]

W. Stürzl and N. Carey, “A fisheye camera system for polarisation detection on UAVs,” in Proceedings of Computer Vision–ECCV 2012. Workshops and Demonstrations (Springer, 2012), pp. 431–440.

J. Chu, W. Wang, Y. Cui, W. Zhi, and Q. Gao, “Measurement for influence of aerosols on polarized sky radiance,” Opt. Precis. Eng. 20(3), 520–526.

EMVA, “EMVA-1288-3.0, ” http://www.emva.org/cms/upload/Standards/Stadard_1288/EMVA1288-3.0.pdf . accessed 6/6/14.

H. Lu, K. Zhao, Z. You, and Q. Ma, “Design and implementation of detection system for skylight polarized pattern using continuously spinning polarization analyzer,” J. Astronaut. 35(9), 1087–1094.

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Figures (17)

Fig. 1
Fig. 1 Principle of Rayleigh scattering.
Fig. 2
Fig. 2 Geometric relationships of solar vector and beams measured in the sensor.
Fig. 3
Fig. 3 Procedure of the algorithm from angle of E-vector image to the solar azimuth.
Fig. 4
Fig. 4 Relationships between key parameters and error-distribution interval.
Fig. 5
Fig. 5 Pinhole camera model.
Fig. 6
Fig. 6 Effect on the resulting distribution interval of the maximal DoLP, SNR, and solar zenith. (a) The result interval vs. maximal DoLP when the solar-zenith angle is 40°, the tolerance R is 0.02°, and the SNR of the image is 90 dB. (b) The expectation of peak vs. the SNR of the image when the solar-zenith angle is 40°, the tolerance R is 0.02°, and the maximal DoLP is 20%. (c) The result interval vs. the solar-zenith angle when the tolerance R is 0.02° and the SNR of image is 90 dB.
Fig. 7
Fig. 7 Expectation curve in the parameter space under different R values. (a) R = 0.02, (b) R = 0.1, and (c) R = 0.2.
Fig. 8
Fig. 8 1-D HT for a null image no matter what the point values are.
Fig. 9
Fig. 9 Example of effect of quantization of the HT.
Fig. 10
Fig. 10 Comparison between the algorithm with and without the filter under the condition of DoLP 20% and solar zenith 30°. (a) Without filter and (b) with filter.
Fig. 11
Fig. 11 Simulation error of algorithm without filter under the condition of DoLP 20% and solar zenith 30°.
Fig. 12
Fig. 12 Simulation error of algorithm with filter under the condition of DoLP 20% and solar zenith 30°.
Fig. 13
Fig. 13 Comparison of measurement angle and true azimuth.
Fig. 14
Fig. 14 Measurement error.
Fig. 15
Fig. 15 Polarization-detection system.
Fig. 16
Fig. 16 Measurement error of experiment.
Fig. 17
Fig. 17 Measured azimuth and calculated zenith of experiment.

Tables (2)

Tables Icon

Table 1 Parameters of image

Tables Icon

Table 2 Performance of polarization-detection system

Equations (25)

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

tanα= cos( h p )sin( h s )sin( h p )cos( h s )cos( φ s φ p ) cos( h s )sin( φ s φ p ) ,( h p π 2 ),
sin( φ s φ p )= h xy × s xy h xy s xy = y h cos φ s x h sin φ s x h 2 + y h 2 .
tanα= ( cos( h p )sin( h s )sin( h p )cos( h s )cos( φ s φ p ) ) x h 2 + y h 2 cos( h s )( y h cos φ s x h sin φ s ) .
lim α90° ( y h cos φ s x h sin φ s )=0.
{ imaBin( x,y )=1,( | p_angle( x,y ) π 2 |<R ) imaBin( x,y )=0,( the others ) ,
H( Ω )= i=1 n k( X i ,Ω )
H( θ )= i=1 n k( X i ,θ ) X i =( x i , y i ),
k( X i ,θ )={ 1( θ=arctan y i x i ) 0( others ) .
h p =arccot( i 2 + j 2 f ) φ p =arg( i,j ) tan α i,j = cos( h p )sin( h s )sin( h p )cos( h s )cos( φ s φ p ) cos( h s )sin( φ s φ p ) .
σ i,j 2 =Var[ α ]= ( σ S 0 ) 2 2N P i,j 2 ,
B i,j = π 2 R π 2 +R f X i,j ( x ) dx,
f X i,j ( x )= 1 σ i,j 2π exp( ( x α i,j ) 2 2 σ i,j 2 ),
N θ = i=1 n j=1 n B i,j θ=arctan j i .
μ= k=1 n p k ,
σ 2 = k=1 n p k ( 1 p k ),
γ= 1 σ 3 i=1 n p k ( 1 p k )( 12 p k ),
μ θ = B i,j ifθ=arctan j i ,
σ θ 2 = B i,j ( 1 B i,j ) ifθ=arctan j i ,
γ θ = 1 σ θ 3 B i,j ( 1 B i,j )( 12 B i,j ) ifθ=arctan j i .
F N ( k )G( k+0.5 μ i,j σ i,j ),
p( N θ 0 isn't the maximum| Z θ 0 =z )= F N θ 0 ( z1 ).
F Z θ 0 ( z )=P( Z θ 0 z )= θ=1 n F N θ ( z ) ,( θ θ 0 ).
f Z θ 0 ( z )={ F Z θ 0 (z)z=1 F Z θ 0 ( z ) F Z θ 0 ( z1 )z>1 .
p( N θ 0 isn't the maximum )= p( Z θ 0 =z )p( N θ 0 isn't the maximum|Z θ 0 =z ) = z f Z θ 0 ( z ) F N θ 0 ( z1 ).
p( N θ 0 is the maximum )=1 z f Z θ 0 ( z ) F N θ 0 ( z1 ) .

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