Abstract

Many insects use the pattern of polarized light in the sky as a navigational cue. In this study, we use this sensory ability as a source of inspiration to create a computational orientation model based on an artificial neural network (POL-ANN). After a training phase using numerically generated sky polarization patterns, stable and convergent networks are obtained. We undertook a series of verification tests using four typical but different sky conditions and showed that the post-trained networks were able to make an accurate prediction of the direction of the sun. Comparisons between the proposed models and models based on the convolutional neural network (CNN) structure revealed the merits of the bio-inspired architecture. We further investigated the accuracy of the models based on two different (locust-like, broader; Drosophila-like, narrower) visual fields of the sky. We find that the accuracy of the computations depends on the overhead visual scene, specifically that wider fields of view perform better when information about the overhead polarization pattern is missing.

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

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    [Crossref]
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2019 (1)

A. Honkanen, A. Adden, J. da Silva Freitas, and S. Heinze, “The insect central complex and the neural basis of navigational strategies,” J. Exp. Biol. 222(PtSuppl 1), jeb188854 (2019).
[Crossref] [PubMed]

2018 (3)

H. Zhao, W. Xu, Y. Zhang, X. Li, H. Zhang, J. Xuan, and B. Jia, “Polarization patterns under different sky conditions and a navigation method based on the symmetry of the AOP map of skylight,” Opt. Express 26(22), 28589–28603 (2018).
[Crossref] [PubMed]

C. Fan, X. Hu, X. He, L. Zhang, and Y. Wang, “Multicamera polarized vision for the orientation with the skylight polarization patterns,” Opt. Eng. 57, 043101 (2018).
[Crossref]

S. B. Powell, R. Garnett, J. Marshall, C. Rizk, and V. Gruev, “Bioinspired polarization vision enables underwater geolocalization,” Sci. Adv. 4(4), eaao6841 (2018).
[Crossref] [PubMed]

2017 (4)

R. Wiltschko, “Navigation,” J. Comp. Physiol. A Neuroethol. Sens. Neural Behav. Physiol. 203(6-7), 455–463 (2017).
[Crossref] [PubMed]

J. R. Serres and F. Ruffier, “Optic flow-based collision-free strategies: From insects to robots,” Arthropod Struct. Dev. 46(5), 703–717 (2017).
[Crossref] [PubMed]

C. Lee, S. E. Yu, and D. Kim, “Landmark-based homing navigation using omnidirectional depth information,” Sensors (Basel) 17(8), 1928 (2017).
[Crossref] [PubMed]

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]

2016 (3)

J. Tang, N. Zhang, D. Li, F. Wang, B. Zhang, C. Wang, C. Shen, J. Ren, C. Xue, and J. Liu, “Novel robust skylight compass method based on full-sky polarization imaging under harsh conditions,” Opt. Express 24(14), 15834–15844 (2016).
[Crossref] [PubMed]

X. Wang, J. Gao, Z. G. Fan, and N. W. Roberts, “An analytical model for the celestial distribution of polarized light, accounting for polarization singularities, wavelength and atmospheric turbidity,” J. Opt. 18(6), 065601 (2016).
[Crossref]

P. T. Weir, M. J. Henze, C. Bleul, F. Baumann-Klausener, T. Labhart, and M. H. Dickinson, “Anatomical reconstruction and functional imaging reveal an ordered array of skylight polarization detectors in Drosophila,” J. Neurosci. 36(19), 5397–5404 (2016).
[Crossref] [PubMed]

2015 (6)

U. Homberg, “Sky compass orientation in desert locusts - evidence from field and laboratory studies,” Front. Behav. Neurosci. 9, 346 (2015).
[Crossref] [PubMed]

B. el Jundi, E. J. Warrant, M. J. Byrne, L. Khaldy, E. Baird, J. Smolka, and M. Dacke, “Neural coding underlying the cue preference for celestial orientation,” Proc. Natl. Acad. Sci. U. S. A. 112(36), 11395–11400 (2015).
[Crossref] [PubMed]

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]

W. Zhang, Y. Cao, X. Zhang, and Z. Liu, “Sky light polarization detection with linear polarizer triplet in light field camera inspired by insect vision,” Appl. Opt. 54(30), 8962–8970 (2015).
[Crossref] [PubMed]

J. Keshavan, G. Gremillion, H. Alvarez-Escobar, and J. S. Humbert, “Autonomous vision-based navigation of a quadrotor in corridor-like environments,” Int. J. Micro Air Veh. 7(2), 111–123 (2015).
[Crossref]

F. Schmeling, J. Tegtmeier, M. Kinoshita, and U. Homberg, “Photoreceptor projections and receptive fields in the dorsal rim area and main retina of the locust eye,” J. Comp. Physiol. A Neuroethol. Sens. Neural Behav. Physiol. 201(5), 427–440 (2015).
[Crossref] [PubMed]

2014 (2)

N. W. Roberts, M. J. How, M. L. Porter, S. E. Temple, R. L. Caldwell, S. B. Powell, V. Gruev, N. J. Marshall, and T. W. Cronin, “Animal polarization imaging and implications for optical processing,” Proc. IEEE 102(10), 1427–1434 (2014).
[Crossref]

Y. Wang, X. Hu, J. Lian, L. Zhang, Z. Xian, and T. Ma, “Design of a device for sky light polarization measurements,” Sensors (Basel) 14(8), 14916–14931 (2014).
[Crossref] [PubMed]

2013 (2)

M. J. How and N. J. Marshall, “Polarization distance: a framework for modelling object detection by polarization vision systems,” Proc. Biol. Sci. 281(1776), 20131632 (2013).
[Crossref] [PubMed]

M. Dacke, E. Baird, M. Byrne, C. H. Scholtz, and E. J. Warrant, “Dung beetles use the Milky Way for orientation,” Curr. Biol. 23(4), 298–300 (2013).
[Crossref] [PubMed]

2012 (1)

S. B. Karman, S. Z. Diah, and I. C. Gebeshuber, “Bio-inspired polarized skylight-based navigation sensors: A review,” Sensors (Basel) 12(11), 14232–14261 (2012).
[Crossref] [PubMed]

2011 (3)

N. W. Roberts, M. L. Porter, and T. W. Cronin, “The molecular basis of mechanisms underlying polarization vision,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 627–637 (2011).
[Crossref] [PubMed]

U. Homberg, S. Heinze, K. Pfeiffer, M. Kinoshita, and B. el Jundi, “Central neural coding of sky polarization in insects,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 680–687 (2011).
[Crossref] [PubMed]

A. Lerner, S. Sabbah, C. Erlick, and N. Shashar, “Navigation by light polarization in clear and turbid waters,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 671–679 (2011).
[Crossref] [PubMed]

2010 (1)

S. M. Reppert, R. J. Gegear, and C. Merlin, “Navigational mechanisms of migrating monarch butterflies,” Trends Neurosci. 33(9), 399–406 (2010).
[Crossref] [PubMed]

2007 (1)

S. Heinze and U. Homberg, “Maplike representation of celestial E-vector orientations in the brain of an insect,” Science 315(5814), 995–997 (2007).
[Crossref] [PubMed]

2004 (1)

T. S. Collett and P. Graham, “Animal navigation: path integration, visual landmarks and cognitive maps,” Curr. Biol. 14(12), R475–R477 (2004).
[Crossref] [PubMed]

2003 (1)

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

2002 (1)

J. F. Diego-Rasilla and R. M. Luengo, “Celestial orientation in the marbled newt (Triturus marmoratus),” J. Ethol. 20(2), 137–141 (2002).
[Crossref]

2000 (2)

M. O. Franz and H. A. Mallot, “Biomimetic robot navigation,” Robot. Auton. Syst. 30(1–2), 133–153 (2000).
[Crossref]

D. Lambrinos, R. Moller, T. Labhart, R. Pfeifer, and R. Wehner, “A mobile robot employing insect strategies for navigation,” Robot. Auton. Syst. 30(1–2), 39–64 (2000).
[Crossref]

1999 (1)

T. Labhart and E. P. Meyer, “Detectors for polarized skylight in insects: a survey of ommatidial specializations in the dorsal rim area of the compound eye,” Microsc. Res. Tech. 47(6), 368–379 (1999).
[Crossref] [PubMed]

1977 (1)

G. D. Bernard and R. Wehner, “Functional similarities between polarization vision and color vision,” Vision Res. 17(9), 1019–1028 (1977).
[Crossref] [PubMed]

1973 (1)

P. Duelli and R. Wehner, “The spectral sensitivity of polarized light orientation in Cataglyphis bicolor (Formicidae, Hymenoptera),” J. Comp. Physiol. A Neuroethol. Sens. Neural Behav. Physiol. 86(1), 37–53 (1973).
[Crossref]

Adden, A.

A. Honkanen, A. Adden, J. da Silva Freitas, and S. Heinze, “The insect central complex and the neural basis of navigational strategies,” J. Exp. Biol. 222(PtSuppl 1), jeb188854 (2019).
[Crossref] [PubMed]

Alvarez-Escobar, H.

J. Keshavan, G. Gremillion, H. Alvarez-Escobar, and J. S. Humbert, “Autonomous vision-based navigation of a quadrotor in corridor-like environments,” Int. J. Micro Air Veh. 7(2), 111–123 (2015).
[Crossref]

Baird, E.

B. el Jundi, E. J. Warrant, M. J. Byrne, L. Khaldy, E. Baird, J. Smolka, and M. Dacke, “Neural coding underlying the cue preference for celestial orientation,” Proc. Natl. Acad. Sci. U. S. A. 112(36), 11395–11400 (2015).
[Crossref] [PubMed]

M. Dacke, E. Baird, M. Byrne, C. H. Scholtz, and E. J. Warrant, “Dung beetles use the Milky Way for orientation,” Curr. Biol. 23(4), 298–300 (2013).
[Crossref] [PubMed]

Baumann-Klausener, F.

P. T. Weir, M. J. Henze, C. Bleul, F. Baumann-Klausener, T. Labhart, and M. H. Dickinson, “Anatomical reconstruction and functional imaging reveal an ordered array of skylight polarization detectors in Drosophila,” J. Neurosci. 36(19), 5397–5404 (2016).
[Crossref] [PubMed]

Bernard, G. D.

G. D. Bernard and R. Wehner, “Functional similarities between polarization vision and color vision,” Vision Res. 17(9), 1019–1028 (1977).
[Crossref] [PubMed]

Bleul, C.

P. T. Weir, M. J. Henze, C. Bleul, F. Baumann-Klausener, T. Labhart, and M. H. Dickinson, “Anatomical reconstruction and functional imaging reveal an ordered array of skylight polarization detectors in Drosophila,” J. Neurosci. 36(19), 5397–5404 (2016).
[Crossref] [PubMed]

Byrne, M.

M. Dacke, E. Baird, M. Byrne, C. H. Scholtz, and E. J. Warrant, “Dung beetles use the Milky Way for orientation,” Curr. Biol. 23(4), 298–300 (2013).
[Crossref] [PubMed]

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

Byrne, M. J.

B. el Jundi, E. J. Warrant, M. J. Byrne, L. Khaldy, E. Baird, J. Smolka, and M. Dacke, “Neural coding underlying the cue preference for celestial orientation,” Proc. Natl. Acad. Sci. U. S. A. 112(36), 11395–11400 (2015).
[Crossref] [PubMed]

Caldwell, R. L.

N. W. Roberts, M. J. How, M. L. Porter, S. E. Temple, R. L. Caldwell, S. B. Powell, V. Gruev, N. J. Marshall, and T. W. Cronin, “Animal polarization imaging and implications for optical processing,” Proc. IEEE 102(10), 1427–1434 (2014).
[Crossref]

Cao, Y.

Collett, T. S.

T. S. Collett and P. Graham, “Animal navigation: path integration, visual landmarks and cognitive maps,” Curr. Biol. 14(12), R475–R477 (2004).
[Crossref] [PubMed]

Cronin, T. W.

N. W. Roberts, M. J. How, M. L. Porter, S. E. Temple, R. L. Caldwell, S. B. Powell, V. Gruev, N. J. Marshall, and T. W. Cronin, “Animal polarization imaging and implications for optical processing,” Proc. IEEE 102(10), 1427–1434 (2014).
[Crossref]

N. W. Roberts, M. L. Porter, and T. W. Cronin, “The molecular basis of mechanisms underlying polarization vision,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 627–637 (2011).
[Crossref] [PubMed]

da Silva Freitas, J.

A. Honkanen, A. Adden, J. da Silva Freitas, and S. Heinze, “The insect central complex and the neural basis of navigational strategies,” J. Exp. Biol. 222(PtSuppl 1), jeb188854 (2019).
[Crossref] [PubMed]

Dacke, M.

B. el Jundi, E. J. Warrant, M. J. Byrne, L. Khaldy, E. Baird, J. Smolka, and M. Dacke, “Neural coding underlying the cue preference for celestial orientation,” Proc. Natl. Acad. Sci. U. S. A. 112(36), 11395–11400 (2015).
[Crossref] [PubMed]

M. Dacke, E. Baird, M. Byrne, C. H. Scholtz, and E. J. Warrant, “Dung beetles use the Milky Way for orientation,” Curr. Biol. 23(4), 298–300 (2013).
[Crossref] [PubMed]

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

Diah, S. Z.

S. B. Karman, S. Z. Diah, and I. C. Gebeshuber, “Bio-inspired polarized skylight-based navigation sensors: A review,” Sensors (Basel) 12(11), 14232–14261 (2012).
[Crossref] [PubMed]

Dickinson, M. H.

P. T. Weir, M. J. Henze, C. Bleul, F. Baumann-Klausener, T. Labhart, and M. H. Dickinson, “Anatomical reconstruction and functional imaging reveal an ordered array of skylight polarization detectors in Drosophila,” J. Neurosci. 36(19), 5397–5404 (2016).
[Crossref] [PubMed]

Diego-Rasilla, J. F.

J. F. Diego-Rasilla and R. M. Luengo, “Celestial orientation in the marbled newt (Triturus marmoratus),” J. Ethol. 20(2), 137–141 (2002).
[Crossref]

Duelli, P.

P. Duelli and R. Wehner, “The spectral sensitivity of polarized light orientation in Cataglyphis bicolor (Formicidae, Hymenoptera),” J. Comp. Physiol. A Neuroethol. Sens. Neural Behav. Physiol. 86(1), 37–53 (1973).
[Crossref]

el Jundi, B.

B. el Jundi, E. J. Warrant, M. J. Byrne, L. Khaldy, E. Baird, J. Smolka, and M. Dacke, “Neural coding underlying the cue preference for celestial orientation,” Proc. Natl. Acad. Sci. U. S. A. 112(36), 11395–11400 (2015).
[Crossref] [PubMed]

U. Homberg, S. Heinze, K. Pfeiffer, M. Kinoshita, and B. el Jundi, “Central neural coding of sky polarization in insects,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 680–687 (2011).
[Crossref] [PubMed]

Erlick, C.

A. Lerner, S. Sabbah, C. Erlick, and N. Shashar, “Navigation by light polarization in clear and turbid waters,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 671–679 (2011).
[Crossref] [PubMed]

Fan, C.

C. Fan, X. Hu, X. He, L. Zhang, and Y. Wang, “Multicamera polarized vision for the orientation with the skylight polarization patterns,” Opt. Eng. 57, 043101 (2018).
[Crossref]

Fan, Z. G.

X. Wang, J. Gao, Z. G. Fan, and N. W. Roberts, “An analytical model for the celestial distribution of polarized light, accounting for polarization singularities, wavelength and atmospheric turbidity,” J. Opt. 18(6), 065601 (2016).
[Crossref]

Franz, M. O.

M. O. Franz and H. A. Mallot, “Biomimetic robot navigation,” Robot. Auton. Syst. 30(1–2), 133–153 (2000).
[Crossref]

Gao, J.

X. Wang, J. Gao, Z. G. Fan, and N. W. Roberts, “An analytical model for the celestial distribution of polarized light, accounting for polarization singularities, wavelength and atmospheric turbidity,” J. Opt. 18(6), 065601 (2016).
[Crossref]

Garnett, R.

S. B. Powell, R. Garnett, J. Marshall, C. Rizk, and V. Gruev, “Bioinspired polarization vision enables underwater geolocalization,” Sci. Adv. 4(4), eaao6841 (2018).
[Crossref] [PubMed]

Gebeshuber, I. C.

S. B. Karman, S. Z. Diah, and I. C. Gebeshuber, “Bio-inspired polarized skylight-based navigation sensors: A review,” Sensors (Basel) 12(11), 14232–14261 (2012).
[Crossref] [PubMed]

Gegear, R. J.

S. M. Reppert, R. J. Gegear, and C. Merlin, “Navigational mechanisms of migrating monarch butterflies,” Trends Neurosci. 33(9), 399–406 (2010).
[Crossref] [PubMed]

Graham, P.

T. S. Collett and P. Graham, “Animal navigation: path integration, visual landmarks and cognitive maps,” Curr. Biol. 14(12), R475–R477 (2004).
[Crossref] [PubMed]

Graupe, D.

D. Graupe, Principles of Artificial Neural Networks (World Scientific, 2013).

Gremillion, G.

J. Keshavan, G. Gremillion, H. Alvarez-Escobar, and J. S. Humbert, “Autonomous vision-based navigation of a quadrotor in corridor-like environments,” Int. J. Micro Air Veh. 7(2), 111–123 (2015).
[Crossref]

Gruev, V.

S. B. Powell, R. Garnett, J. Marshall, C. Rizk, and V. Gruev, “Bioinspired polarization vision enables underwater geolocalization,” Sci. Adv. 4(4), eaao6841 (2018).
[Crossref] [PubMed]

N. W. Roberts, M. J. How, M. L. Porter, S. E. Temple, R. L. Caldwell, S. B. Powell, V. Gruev, N. J. Marshall, and T. W. Cronin, “Animal polarization imaging and implications for optical processing,” Proc. IEEE 102(10), 1427–1434 (2014).
[Crossref]

He, K.

K. He, X. Zhang, S. Ren, and J. Sun, “Deep residual learning for image recognition,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2016), pp. 770–778.

He, X.

C. Fan, X. Hu, X. He, L. Zhang, and Y. Wang, “Multicamera polarized vision for the orientation with the skylight polarization patterns,” Opt. Eng. 57, 043101 (2018).
[Crossref]

Heinze, S.

A. Honkanen, A. Adden, J. da Silva Freitas, and S. Heinze, “The insect central complex and the neural basis of navigational strategies,” J. Exp. Biol. 222(PtSuppl 1), jeb188854 (2019).
[Crossref] [PubMed]

U. Homberg, S. Heinze, K. Pfeiffer, M. Kinoshita, and B. el Jundi, “Central neural coding of sky polarization in insects,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 680–687 (2011).
[Crossref] [PubMed]

S. Heinze and U. Homberg, “Maplike representation of celestial E-vector orientations in the brain of an insect,” Science 315(5814), 995–997 (2007).
[Crossref] [PubMed]

Henze, M. J.

P. T. Weir, M. J. Henze, C. Bleul, F. Baumann-Klausener, T. Labhart, and M. H. Dickinson, “Anatomical reconstruction and functional imaging reveal an ordered array of skylight polarization detectors in Drosophila,” J. Neurosci. 36(19), 5397–5404 (2016).
[Crossref] [PubMed]

Homberg, U.

F. Schmeling, J. Tegtmeier, M. Kinoshita, and U. Homberg, “Photoreceptor projections and receptive fields in the dorsal rim area and main retina of the locust eye,” J. Comp. Physiol. A Neuroethol. Sens. Neural Behav. Physiol. 201(5), 427–440 (2015).
[Crossref] [PubMed]

U. Homberg, “Sky compass orientation in desert locusts - evidence from field and laboratory studies,” Front. Behav. Neurosci. 9, 346 (2015).
[Crossref] [PubMed]

U. Homberg, S. Heinze, K. Pfeiffer, M. Kinoshita, and B. el Jundi, “Central neural coding of sky polarization in insects,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 680–687 (2011).
[Crossref] [PubMed]

S. Heinze and U. Homberg, “Maplike representation of celestial E-vector orientations in the brain of an insect,” Science 315(5814), 995–997 (2007).
[Crossref] [PubMed]

Honkanen, A.

A. Honkanen, A. Adden, J. da Silva Freitas, and S. Heinze, “The insect central complex and the neural basis of navigational strategies,” J. Exp. Biol. 222(PtSuppl 1), jeb188854 (2019).
[Crossref] [PubMed]

How, M. J.

N. W. Roberts, M. J. How, M. L. Porter, S. E. Temple, R. L. Caldwell, S. B. Powell, V. Gruev, N. J. Marshall, and T. W. Cronin, “Animal polarization imaging and implications for optical processing,” Proc. IEEE 102(10), 1427–1434 (2014).
[Crossref]

M. J. How and N. J. Marshall, “Polarization distance: a framework for modelling object detection by polarization vision systems,” Proc. Biol. Sci. 281(1776), 20131632 (2013).
[Crossref] [PubMed]

Hu, X.

C. Fan, X. Hu, X. He, L. Zhang, and Y. Wang, “Multicamera polarized vision for the orientation with the skylight polarization patterns,” Opt. Eng. 57, 043101 (2018).
[Crossref]

Y. Wang, X. Hu, J. Lian, L. Zhang, Z. Xian, and T. Ma, “Design of a device for sky light polarization measurements,” Sensors (Basel) 14(8), 14916–14931 (2014).
[Crossref] [PubMed]

Huang, K.

Humbert, J. S.

J. Keshavan, G. Gremillion, H. Alvarez-Escobar, and J. S. Humbert, “Autonomous vision-based navigation of a quadrotor in corridor-like environments,” Int. J. Micro Air Veh. 7(2), 111–123 (2015).
[Crossref]

Jia, B.

Karman, S. B.

S. B. Karman, S. Z. Diah, and I. C. Gebeshuber, “Bio-inspired polarized skylight-based navigation sensors: A review,” Sensors (Basel) 12(11), 14232–14261 (2012).
[Crossref] [PubMed]

Keshavan, J.

J. Keshavan, G. Gremillion, H. Alvarez-Escobar, and J. S. Humbert, “Autonomous vision-based navigation of a quadrotor in corridor-like environments,” Int. J. Micro Air Veh. 7(2), 111–123 (2015).
[Crossref]

Khaldy, L.

B. el Jundi, E. J. Warrant, M. J. Byrne, L. Khaldy, E. Baird, J. Smolka, and M. Dacke, “Neural coding underlying the cue preference for celestial orientation,” Proc. Natl. Acad. Sci. U. S. A. 112(36), 11395–11400 (2015).
[Crossref] [PubMed]

Kim, D.

C. Lee, S. E. Yu, and D. Kim, “Landmark-based homing navigation using omnidirectional depth information,” Sensors (Basel) 17(8), 1928 (2017).
[Crossref] [PubMed]

Kinoshita, M.

F. Schmeling, J. Tegtmeier, M. Kinoshita, and U. Homberg, “Photoreceptor projections and receptive fields in the dorsal rim area and main retina of the locust eye,” J. Comp. Physiol. A Neuroethol. Sens. Neural Behav. Physiol. 201(5), 427–440 (2015).
[Crossref] [PubMed]

U. Homberg, S. Heinze, K. Pfeiffer, M. Kinoshita, and B. el Jundi, “Central neural coding of sky polarization in insects,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 680–687 (2011).
[Crossref] [PubMed]

Labhart, T.

P. T. Weir, M. J. Henze, C. Bleul, F. Baumann-Klausener, T. Labhart, and M. H. Dickinson, “Anatomical reconstruction and functional imaging reveal an ordered array of skylight polarization detectors in Drosophila,” J. Neurosci. 36(19), 5397–5404 (2016).
[Crossref] [PubMed]

D. Lambrinos, R. Moller, T. Labhart, R. Pfeifer, and R. Wehner, “A mobile robot employing insect strategies for navigation,” Robot. Auton. Syst. 30(1–2), 39–64 (2000).
[Crossref]

T. Labhart and E. P. Meyer, “Detectors for polarized skylight in insects: a survey of ommatidial specializations in the dorsal rim area of the compound eye,” Microsc. Res. Tech. 47(6), 368–379 (1999).
[Crossref] [PubMed]

Lambrinos, D.

D. Lambrinos, R. Moller, T. Labhart, R. Pfeifer, and R. Wehner, “A mobile robot employing insect strategies for navigation,” Robot. Auton. Syst. 30(1–2), 39–64 (2000).
[Crossref]

Lee, C.

C. Lee, S. E. Yu, and D. Kim, “Landmark-based homing navigation using omnidirectional depth information,” Sensors (Basel) 17(8), 1928 (2017).
[Crossref] [PubMed]

Lerner, A.

A. Lerner, S. Sabbah, C. Erlick, and N. Shashar, “Navigation by light polarization in clear and turbid waters,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 671–679 (2011).
[Crossref] [PubMed]

Li, D.

Li, X.

Lian, J.

Y. Wang, X. Hu, J. Lian, L. Zhang, Z. Xian, and T. Ma, “Design of a device for sky light polarization measurements,” Sensors (Basel) 14(8), 14916–14931 (2014).
[Crossref] [PubMed]

Liu, J.

Liu, Z.

Lu, H.

Luengo, R. M.

J. F. Diego-Rasilla and R. M. Luengo, “Celestial orientation in the marbled newt (Triturus marmoratus),” J. Ethol. 20(2), 137–141 (2002).
[Crossref]

Ma, T.

Y. Wang, X. Hu, J. Lian, L. Zhang, Z. Xian, and T. Ma, “Design of a device for sky light polarization measurements,” Sensors (Basel) 14(8), 14916–14931 (2014).
[Crossref] [PubMed]

Mallot, H. A.

M. O. Franz and H. A. Mallot, “Biomimetic robot navigation,” Robot. Auton. Syst. 30(1–2), 133–153 (2000).
[Crossref]

Marshall, J.

S. B. Powell, R. Garnett, J. Marshall, C. Rizk, and V. Gruev, “Bioinspired polarization vision enables underwater geolocalization,” Sci. Adv. 4(4), eaao6841 (2018).
[Crossref] [PubMed]

Marshall, N. J.

N. W. Roberts, M. J. How, M. L. Porter, S. E. Temple, R. L. Caldwell, S. B. Powell, V. Gruev, N. J. Marshall, and T. W. Cronin, “Animal polarization imaging and implications for optical processing,” Proc. IEEE 102(10), 1427–1434 (2014).
[Crossref]

M. J. How and N. J. Marshall, “Polarization distance: a framework for modelling object detection by polarization vision systems,” Proc. Biol. Sci. 281(1776), 20131632 (2013).
[Crossref] [PubMed]

Merlin, C.

S. M. Reppert, R. J. Gegear, and C. Merlin, “Navigational mechanisms of migrating monarch butterflies,” Trends Neurosci. 33(9), 399–406 (2010).
[Crossref] [PubMed]

Meyer, E. P.

T. Labhart and E. P. Meyer, “Detectors for polarized skylight in insects: a survey of ommatidial specializations in the dorsal rim area of the compound eye,” Microsc. Res. Tech. 47(6), 368–379 (1999).
[Crossref] [PubMed]

Moller, R.

D. Lambrinos, R. Moller, T. Labhart, R. Pfeifer, and R. Wehner, “A mobile robot employing insect strategies for navigation,” Robot. Auton. Syst. 30(1–2), 39–64 (2000).
[Crossref]

Nilsson, D. E.

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

Ning, Y.

Pfeifer, R.

D. Lambrinos, R. Moller, T. Labhart, R. Pfeifer, and R. Wehner, “A mobile robot employing insect strategies for navigation,” Robot. Auton. Syst. 30(1–2), 39–64 (2000).
[Crossref]

Pfeiffer, K.

U. Homberg, S. Heinze, K. Pfeiffer, M. Kinoshita, and B. el Jundi, “Central neural coding of sky polarization in insects,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 680–687 (2011).
[Crossref] [PubMed]

Porter, M. L.

N. W. Roberts, M. J. How, M. L. Porter, S. E. Temple, R. L. Caldwell, S. B. Powell, V. Gruev, N. J. Marshall, and T. W. Cronin, “Animal polarization imaging and implications for optical processing,” Proc. IEEE 102(10), 1427–1434 (2014).
[Crossref]

N. W. Roberts, M. L. Porter, and T. W. Cronin, “The molecular basis of mechanisms underlying polarization vision,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 627–637 (2011).
[Crossref] [PubMed]

Powell, S. B.

S. B. Powell, R. Garnett, J. Marshall, C. Rizk, and V. Gruev, “Bioinspired polarization vision enables underwater geolocalization,” Sci. Adv. 4(4), eaao6841 (2018).
[Crossref] [PubMed]

N. W. Roberts, M. J. How, M. L. Porter, S. E. Temple, R. L. Caldwell, S. B. Powell, V. Gruev, N. J. Marshall, and T. W. Cronin, “Animal polarization imaging and implications for optical processing,” Proc. IEEE 102(10), 1427–1434 (2014).
[Crossref]

Ren, J.

Ren, S.

K. He, X. Zhang, S. Ren, and J. Sun, “Deep residual learning for image recognition,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2016), pp. 770–778.

Reppert, S. M.

S. M. Reppert, R. J. Gegear, and C. Merlin, “Navigational mechanisms of migrating monarch butterflies,” Trends Neurosci. 33(9), 399–406 (2010).
[Crossref] [PubMed]

Rizk, C.

S. B. Powell, R. Garnett, J. Marshall, C. Rizk, and V. Gruev, “Bioinspired polarization vision enables underwater geolocalization,” Sci. Adv. 4(4), eaao6841 (2018).
[Crossref] [PubMed]

Roberts, N. W.

X. Wang, J. Gao, Z. G. Fan, and N. W. Roberts, “An analytical model for the celestial distribution of polarized light, accounting for polarization singularities, wavelength and atmospheric turbidity,” J. Opt. 18(6), 065601 (2016).
[Crossref]

N. W. Roberts, M. J. How, M. L. Porter, S. E. Temple, R. L. Caldwell, S. B. Powell, V. Gruev, N. J. Marshall, and T. W. Cronin, “Animal polarization imaging and implications for optical processing,” Proc. IEEE 102(10), 1427–1434 (2014).
[Crossref]

N. W. Roberts, M. L. Porter, and T. W. Cronin, “The molecular basis of mechanisms underlying polarization vision,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 627–637 (2011).
[Crossref] [PubMed]

Ruffier, F.

J. R. Serres and F. Ruffier, “Optic flow-based collision-free strategies: From insects to robots,” Arthropod Struct. Dev. 46(5), 703–717 (2017).
[Crossref] [PubMed]

Sabbah, S.

A. Lerner, S. Sabbah, C. Erlick, and N. Shashar, “Navigation by light polarization in clear and turbid waters,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 671–679 (2011).
[Crossref] [PubMed]

Schmeling, F.

F. Schmeling, J. Tegtmeier, M. Kinoshita, and U. Homberg, “Photoreceptor projections and receptive fields in the dorsal rim area and main retina of the locust eye,” J. Comp. Physiol. A Neuroethol. Sens. Neural Behav. Physiol. 201(5), 427–440 (2015).
[Crossref] [PubMed]

Scholtz, C. H.

M. Dacke, E. Baird, M. Byrne, C. H. Scholtz, and E. J. Warrant, “Dung beetles use the Milky Way for orientation,” Curr. Biol. 23(4), 298–300 (2013).
[Crossref] [PubMed]

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

Serres, J. R.

J. R. Serres and F. Ruffier, “Optic flow-based collision-free strategies: From insects to robots,” Arthropod Struct. Dev. 46(5), 703–717 (2017).
[Crossref] [PubMed]

Shashar, N.

A. Lerner, S. Sabbah, C. Erlick, and N. Shashar, “Navigation by light polarization in clear and turbid waters,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 671–679 (2011).
[Crossref] [PubMed]

Shen, C.

Smolka, J.

B. el Jundi, E. J. Warrant, M. J. Byrne, L. Khaldy, E. Baird, J. Smolka, and M. Dacke, “Neural coding underlying the cue preference for celestial orientation,” Proc. Natl. Acad. Sci. U. S. A. 112(36), 11395–11400 (2015).
[Crossref] [PubMed]

Sturzl, W.

W. Sturzl, “A lightweight single-camera polarization compass with covariance estimation,” in Proceedings of the IEEE International Conference on Computer Vision (IEEE, 2017), pp. 5363–5371.
[Crossref]

Sun, J.

K. He, X. Zhang, S. Ren, and J. Sun, “Deep residual learning for image recognition,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2016), pp. 770–778.

Tang, J.

Tegtmeier, J.

F. Schmeling, J. Tegtmeier, M. Kinoshita, and U. Homberg, “Photoreceptor projections and receptive fields in the dorsal rim area and main retina of the locust eye,” J. Comp. Physiol. A Neuroethol. Sens. Neural Behav. Physiol. 201(5), 427–440 (2015).
[Crossref] [PubMed]

Temple, S. E.

N. W. Roberts, M. J. How, M. L. Porter, S. E. Temple, R. L. Caldwell, S. B. Powell, V. Gruev, N. J. Marshall, and T. W. Cronin, “Animal polarization imaging and implications for optical processing,” Proc. IEEE 102(10), 1427–1434 (2014).
[Crossref]

Wang, C.

Wang, F.

Wang, X.

X. Wang, J. Gao, Z. G. Fan, and N. W. Roberts, “An analytical model for the celestial distribution of polarized light, accounting for polarization singularities, wavelength and atmospheric turbidity,” J. Opt. 18(6), 065601 (2016).
[Crossref]

Wang, Y.

C. Fan, X. Hu, X. He, L. Zhang, and Y. Wang, “Multicamera polarized vision for the orientation with the skylight polarization patterns,” Opt. Eng. 57, 043101 (2018).
[Crossref]

Y. Wang, X. Hu, J. Lian, L. Zhang, Z. Xian, and T. Ma, “Design of a device for sky light polarization measurements,” Sensors (Basel) 14(8), 14916–14931 (2014).
[Crossref] [PubMed]

Warrant, E. J.

B. el Jundi, E. J. Warrant, M. J. Byrne, L. Khaldy, E. Baird, J. Smolka, and M. Dacke, “Neural coding underlying the cue preference for celestial orientation,” Proc. Natl. Acad. Sci. U. S. A. 112(36), 11395–11400 (2015).
[Crossref] [PubMed]

M. Dacke, E. Baird, M. Byrne, C. H. Scholtz, and E. J. Warrant, “Dung beetles use the Milky Way for orientation,” Curr. Biol. 23(4), 298–300 (2013).
[Crossref] [PubMed]

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

Wehner, R.

D. Lambrinos, R. Moller, T. Labhart, R. Pfeifer, and R. Wehner, “A mobile robot employing insect strategies for navigation,” Robot. Auton. Syst. 30(1–2), 39–64 (2000).
[Crossref]

G. D. Bernard and R. Wehner, “Functional similarities between polarization vision and color vision,” Vision Res. 17(9), 1019–1028 (1977).
[Crossref] [PubMed]

P. Duelli and R. Wehner, “The spectral sensitivity of polarized light orientation in Cataglyphis bicolor (Formicidae, Hymenoptera),” J. Comp. Physiol. A Neuroethol. Sens. Neural Behav. Physiol. 86(1), 37–53 (1973).
[Crossref]

Weir, P. T.

P. T. Weir, M. J. Henze, C. Bleul, F. Baumann-Klausener, T. Labhart, and M. H. Dickinson, “Anatomical reconstruction and functional imaging reveal an ordered array of skylight polarization detectors in Drosophila,” J. Neurosci. 36(19), 5397–5404 (2016).
[Crossref] [PubMed]

Wiltschko, R.

R. Wiltschko, “Navigation,” J. Comp. Physiol. A Neuroethol. Sens. Neural Behav. Physiol. 203(6-7), 455–463 (2017).
[Crossref] [PubMed]

Xian, Z.

Y. Wang, X. Hu, J. Lian, L. Zhang, Z. Xian, and T. Ma, “Design of a device for sky light polarization measurements,” Sensors (Basel) 14(8), 14916–14931 (2014).
[Crossref] [PubMed]

Xu, W.

Xuan, J.

Xue, C.

Yang, Y.

You, Z.

Yu, S. E.

C. Lee, S. E. Yu, and D. Kim, “Landmark-based homing navigation using omnidirectional depth information,” Sensors (Basel) 17(8), 1928 (2017).
[Crossref] [PubMed]

Zhang, B.

Zhang, H.

Zhang, L.

C. Fan, X. Hu, X. He, L. Zhang, and Y. Wang, “Multicamera polarized vision for the orientation with the skylight polarization patterns,” Opt. Eng. 57, 043101 (2018).
[Crossref]

Y. Wang, X. Hu, J. Lian, L. Zhang, Z. Xian, and T. Ma, “Design of a device for sky light polarization measurements,” Sensors (Basel) 14(8), 14916–14931 (2014).
[Crossref] [PubMed]

Zhang, N.

Zhang, W.

Zhang, X.

Zhang, Y.

Zhao, H.

Zhao, K.

Appl. Opt. (2)

Arthropod Struct. Dev. (1)

J. R. Serres and F. Ruffier, “Optic flow-based collision-free strategies: From insects to robots,” Arthropod Struct. Dev. 46(5), 703–717 (2017).
[Crossref] [PubMed]

Curr. Biol. (2)

M. Dacke, E. Baird, M. Byrne, C. H. Scholtz, and E. J. Warrant, “Dung beetles use the Milky Way for orientation,” Curr. Biol. 23(4), 298–300 (2013).
[Crossref] [PubMed]

T. S. Collett and P. Graham, “Animal navigation: path integration, visual landmarks and cognitive maps,” Curr. Biol. 14(12), R475–R477 (2004).
[Crossref] [PubMed]

Front. Behav. Neurosci. (1)

U. Homberg, “Sky compass orientation in desert locusts - evidence from field and laboratory studies,” Front. Behav. Neurosci. 9, 346 (2015).
[Crossref] [PubMed]

Int. J. Micro Air Veh. (1)

J. Keshavan, G. Gremillion, H. Alvarez-Escobar, and J. S. Humbert, “Autonomous vision-based navigation of a quadrotor in corridor-like environments,” Int. J. Micro Air Veh. 7(2), 111–123 (2015).
[Crossref]

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

R. Wiltschko, “Navigation,” J. Comp. Physiol. A Neuroethol. Sens. Neural Behav. Physiol. 203(6-7), 455–463 (2017).
[Crossref] [PubMed]

P. Duelli and R. Wehner, “The spectral sensitivity of polarized light orientation in Cataglyphis bicolor (Formicidae, Hymenoptera),” J. Comp. Physiol. A Neuroethol. Sens. Neural Behav. Physiol. 86(1), 37–53 (1973).
[Crossref]

F. Schmeling, J. Tegtmeier, M. Kinoshita, and U. Homberg, “Photoreceptor projections and receptive fields in the dorsal rim area and main retina of the locust eye,” J. Comp. Physiol. A Neuroethol. Sens. Neural Behav. Physiol. 201(5), 427–440 (2015).
[Crossref] [PubMed]

J. Ethol. (1)

J. F. Diego-Rasilla and R. M. Luengo, “Celestial orientation in the marbled newt (Triturus marmoratus),” J. Ethol. 20(2), 137–141 (2002).
[Crossref]

J. Exp. Biol. (1)

A. Honkanen, A. Adden, J. da Silva Freitas, and S. Heinze, “The insect central complex and the neural basis of navigational strategies,” J. Exp. Biol. 222(PtSuppl 1), jeb188854 (2019).
[Crossref] [PubMed]

J. Neurosci. (1)

P. T. Weir, M. J. Henze, C. Bleul, F. Baumann-Klausener, T. Labhart, and M. H. Dickinson, “Anatomical reconstruction and functional imaging reveal an ordered array of skylight polarization detectors in Drosophila,” J. Neurosci. 36(19), 5397–5404 (2016).
[Crossref] [PubMed]

J. Opt. (1)

X. Wang, J. Gao, Z. G. Fan, and N. W. Roberts, “An analytical model for the celestial distribution of polarized light, accounting for polarization singularities, wavelength and atmospheric turbidity,” J. Opt. 18(6), 065601 (2016).
[Crossref]

Microsc. Res. Tech. (1)

T. Labhart and E. P. Meyer, “Detectors for polarized skylight in insects: a survey of ommatidial specializations in the dorsal rim area of the compound eye,” Microsc. Res. Tech. 47(6), 368–379 (1999).
[Crossref] [PubMed]

Nature (1)

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

Opt. Eng. (1)

C. Fan, X. Hu, X. He, L. Zhang, and Y. Wang, “Multicamera polarized vision for the orientation with the skylight polarization patterns,” Opt. Eng. 57, 043101 (2018).
[Crossref]

Opt. Express (3)

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

U. Homberg, S. Heinze, K. Pfeiffer, M. Kinoshita, and B. el Jundi, “Central neural coding of sky polarization in insects,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 680–687 (2011).
[Crossref] [PubMed]

N. W. Roberts, M. L. Porter, and T. W. Cronin, “The molecular basis of mechanisms underlying polarization vision,” Philos. Trans. R. Soc. Lond. B Biol. Sci. 366(1565), 627–637 (2011).
[Crossref] [PubMed]

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Figures (8)

Fig. 1
Fig. 1 The layouts of two different types of dorsal rim area (DRA) and their corresponding visual fields in space. (A) Type DRA-W with short but wide elliptical visual field. (B) Type DRA-L with elongated and narrow strip-like visual field.
Fig. 2
Fig. 2 The flow of polarization information. In the forward propagation process, original polarized skylight signals are first sampled by the DRA model. The outputs of the DRA model are taken as the input layer and transferred into the two hidden layers that have 3 and 16 neurons respectively. In the back-propagation process, the error signals are transferred back to the hidden layers and input layer from the output layer to adjust weights for the neurons in all of the layers. Where w 1 ij represents the vector of weights for the neurons between the input layer and the first hidden layer, w 2 jk represents the vector of weights for the neurons between the two hidden layers, w 3 k represents the vector of weights for the neurons between the second hidden layer and the output layer. The inputs and outputs for the first hidden layer, second hidden layer and output layer are S 1 j , S 2 k , S3 and O 1 j , O 2 k , O3 separately.
Fig. 3
Fig. 3 Representative images of simulated skylight polarization patterns. The first and second rows are Dop (degree of polarization) and Aop (angle of polarization) images respectively.
Fig. 4
Fig. 4 Calculated errors from each of the four DRA models. (A) Boxplot shows the minimum, first quartile, median, third quartile and maximum of errors in every DRA group. (B) Histograms show the occurrence rates of the errors and the total number of errors in every DRA group is 64. ‘W-1’ and ‘W-3′ represent the networks based on DRA-W model with receptors distributed in one circle and three circles respectively. While ‘L-1’ and ‘L-3′ have the same meaning for the networks based on DRA-L model.
Fig. 5
Fig. 5 Errors under four sky situations for different models. Images in the first row illustrate degree of polarization of the whole sky under four cases of different coverage, (A) un-obscured skylight polarization pattern, (B) skylight polarization pattern with one large obscured area, (C) skylight polarization pattern with several small missing patches, (D) skylight polarization pattern with randomly distributed missing points. The white areas in the calculated images represent the areas with missing information. For every sky situation, we used 64 calculated skylight polarization patterns for testing. Again, these test patterns were set at four different turbidities (3, 4, 6, 8), four different sun elevations (30°, 40°, 50°, 60°) and four different sun azimuths (30.5°, 118.5°, 233.5°, 324.5°). Boxplots show the minimum, first quartile, median, third quartile, maximum of errors and histograms show the occurrence rates of the errors. The third, fourth, fifth and sixth rows present results of the DRA-W-3 model, DRA-L-3 model, AlexNet model, and ResNet-50 model respectively.
Fig. 6
Fig. 6 Photo of the instrument for skylight polarization pattern measurements.
Fig. 7
Fig. 7 Representative images of measured skylight polarization patterns. The first and second rows are Dop (degree of polarization) and Aop (angle of polarization) images respectively.
Fig. 8
Fig. 8 Errors under real skylight polarization patterns. Similar to the tests with simulation data, some pixels of the images were removed to mimic different sky conditions, (A) un-obscured skylight polarization pattern, (B) skylight polarization pattern with one large obscured area, (C) skylight polarization pattern with several small missing patches, (D) skylight polarization pattern with randomly distributed missing points. Boxplots show the minimum, first quartile, median, third quartile, maximum of errors and histograms show the occurrence rates of the errors. The third, fourth, fifth and sixth rows present results of the DRA-W-3 model, DRA-L-3 model, AlexNet model, and ResNet-50 model respectively.

Equations (9)

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S n =KI( 1+dcos2(φ φ n ) )
P n =log( 1+dcos2(φ φ n ) 1dcos2(φ φ n ) )
P v =log( n=1 r I n (1+ d n cos2( φ n φ v )) n=1 r I n (1 d n cos2( φ n φ v )) )
M DRA =[ P 1 , P 2 , P 3 P v P m ].
{ O 1 j (n)=f( S 1 j (n) )=f( i=1 2m w 1 ij (n) P i (n) )j[1,3], O 2 k (n)=f( S 2 k (n) )=f( j=1 3 w 2 jk (n) O 1 j (n) )k[1,16], O3(n)=f( S3(n) )=g( k=1 16 w 3 k (n) O 2 k (n) ),
e(n)=d(n)O3(n).
E(n)= 1 2 e 2 (n).
{ δ3(n)= E(n) S3(n) =e(n) g ' ( S3(n) ), δ 2 k (n)= E(n) S 2 k (n) = f ' ( S 2 k (n) )δ3(n)w 3 k (n)k[1,16], δ 1 j (n)= E(n) S 1 j (n) = f ' ( S 1 j (n) ) k=1 16 ( δ 2 k (n)w 2 jk (n) ) j[1,3].
{ Δw 3 k (n)=ηδ3(n)O 2 k (n)k[1,16], Δw 2 jk (n)=ηδ 2 k (n)O 1 j (n)j[1,3],k[1,16], Δw 1 ij (n)=ηδ 1 j (n) P i (n)i[1,2m],j[1,3],