Abstract

To increase the field of view (FOV), combining multiple time-delayed and integrated charge-coupled devices (TDI-CCD) into the camera and the pushbroom imaging modality are traditionally used with high-resolution optical satellites. It is becoming increasingly labor- and cost-intensive to build and maintain a calibration field with high resolution and broad coverage. This paper introduces a simple and feasible on-orbit geometric self-calibration approach for high-resolution multi-TDI-CCD optical satellites based on three-view stereoscopic images. With the aid of the a priori geometric constraint of tie points in the triple-overlap regions of stereoscopic images, as well as tie points between adjacent single TDI-CCD images (STIs), high accuracy calibration of all TDI-CCD detectors can be achieved using a small number of absolute ground control points (GCPs) covering the selected primary STI. This method greatly reduces the demand on the calibration field and thus is more time-, effort- and cost-effective. Experimental results indicated that the proposed self-calibration approach is effective for increasing the relative internal accuracy without the limitations associated with using a traditional reference calibration field, which could have great significance for future super-high-resolution optical satellites.

© 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]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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2018 (1)

G. Zhang, K. Xu, Q. J. Zhang, and D. R. Li, “Correction of Pushbroom Satellite Imagery Interior Distortions Independent of Ground Control Points,” Remote Sens. 10(2), 98 (2018).
[Crossref]

2017 (2)

B. Guan, Y. Shang, and Q. Yu, “Planar self-calibration for stereo cameras with radial distortion,” Appl. Opt. 56(33), 9257–9267 (2017).
[Crossref] [PubMed]

M. Wang, Y. F. Cheng, X. L. Chang, S. Y. Jin, and Y. Zhu, “On-orbit geometric calibration and geometric quality assessment for the high-resolution geostationary optical satellite gaofen4,” ISPRS J. Photogramm. Remote Sens. 125, 63–77 (2017).
[Crossref]

2016 (3)

2015 (2)

J. Li, F. Xing, T. Sun, and Z. You, “Efficient assessment method of on-board modulation transfer function of optical remote sensing sensors,” Opt. Express 23(5), 6187–6208 (2015).
[Crossref] [PubMed]

M. T. Zheng, Y. J. Zhang, J. F. Zhu, and X. D. Xiong, “Self-Calibration adjustment of CBERS-02B long-strip imagery,” IEEE Trans. Geosci. Remote Sens. 53(7), 3847–3854 (2015).
[Crossref]

2014 (2)

K. C. Di, Y. L. Liu, B. Liu, M. Peng, and W. M. Hu, “A Self-Calibration Bundle Adjustment Method for Photogrammetric Processing of Chang E-2 Stereo Lunar Imagery,” IEEE Trans. Geosci. Remote Sens. 52(9), 5432–5442 (2014).
[Crossref]

M. Wang, B. Yang, F. Hu, and X. Zang, “On-Orbit geometric calibration model and its applications for high-resolution optical satellite imagery,” Remote Sens. 6(12), 4391–4408 (2014).
[Crossref]

2011 (2)

2009 (1)

T. Tadono, M. Shimada, H. Murakami, and J. Takaku, “Calibration of PRISM and AVNIR-2 onboard ALOS—Daichi,” IEEE Trans. Geosci. Remote Sens. 47(12), 4042–4050 (2009).
[Crossref]

2007 (1)

A. Gruen, S. Kocaman, and K. Wolff, “Calibration and validation of early ALOS/PRISM images,” J. Jpn. Soc. Photogram. Rem. Sens. 46(1), 24–38 (2007).

2006 (1)

2005 (1)

2004 (1)

D. G. Lowe, “Distinctive image features from scale-invariant keypoints,” Int. J. Comput. Vis. 60(2), 91–110 (2004).
[Crossref]

2003 (1)

C. S. Fraser and H. B. Hanley, “Bias compensation in rational functions for IKONOS satellite imagery,” Photogramm. Eng. Remote Sensing 69(1), 53–57 (2003).
[Crossref]

2002 (1)

E. Malis and R. Cipolla, “Camera self-calibration from unknown planar structures enforcing the multiview constraints between collineations,” IEEE Trans. Pattern Anal. Mach. Intell. 24(9), 1268–1272 (2002).
[Crossref]

1992 (1)

O. D. Faugeras, Q. T. Luong, and S. J. Maybank, “Camera self-calibration: Theory and experiments,” Computer Vision—ECCV, 588(12), 321–334 (1992).

Chang, X. L.

M. Wang, Y. F. Cheng, X. L. Chang, S. Y. Jin, and Y. Zhu, “On-orbit geometric calibration and geometric quality assessment for the high-resolution geostationary optical satellite gaofen4,” ISPRS J. Photogramm. Remote Sens. 125, 63–77 (2017).
[Crossref]

Chen, Y. S.

Chen, Z.

Cheng, Y.

Cheng, Y. F.

M. Wang, Y. F. Cheng, X. L. Chang, S. Y. Jin, and Y. Zhu, “On-orbit geometric calibration and geometric quality assessment for the high-resolution geostationary optical satellite gaofen4,” ISPRS J. Photogramm. Remote Sens. 125, 63–77 (2017).
[Crossref]

Cipolla, R.

E. Malis and R. Cipolla, “Camera self-calibration from unknown planar structures enforcing the multiview constraints between collineations,” IEEE Trans. Pattern Anal. Mach. Intell. 24(9), 1268–1272 (2002).
[Crossref]

Di, K. C.

K. C. Di, Y. L. Liu, B. Liu, M. Peng, and W. M. Hu, “A Self-Calibration Bundle Adjustment Method for Photogrammetric Processing of Chang E-2 Stereo Lunar Imagery,” IEEE Trans. Geosci. Remote Sens. 52(9), 5432–5442 (2014).
[Crossref]

Dong, S.

Eisa, M. M.

Faugeras, O. D.

O. D. Faugeras, Q. T. Luong, and S. J. Maybank, “Camera self-calibration: Theory and experiments,” Computer Vision—ECCV, 588(12), 321–334 (1992).

Fraser, C. S.

C. S. Fraser and H. B. Hanley, “Bias compensation in rational functions for IKONOS satellite imagery,” Photogramm. Eng. Remote Sensing 69(1), 53–57 (2003).
[Crossref]

Grodecki, J.

J. Grodecki, “IKONOS geometric calibrations,” Proceedings of the ASPRS 2005 Annual Conference (2005).

Gruen, A.

A. Gruen, S. Kocaman, and K. Wolff, “Calibration and validation of early ALOS/PRISM images,” J. Jpn. Soc. Photogram. Rem. Sens. 46(1), 24–38 (2007).

Guan, B.

Hanley, H. B.

C. S. Fraser and H. B. Hanley, “Bias compensation in rational functions for IKONOS satellite imagery,” Photogramm. Eng. Remote Sensing 69(1), 53–57 (2003).
[Crossref]

He, X.

Hsu, W. H.

Hu, F.

M. Wang, B. Yang, F. Hu, and X. Zang, “On-Orbit geometric calibration model and its applications for high-resolution optical satellite imagery,” Remote Sens. 6(12), 4391–4408 (2014).
[Crossref]

Hu, W. M.

K. C. Di, Y. L. Liu, B. Liu, M. Peng, and W. M. Hu, “A Self-Calibration Bundle Adjustment Method for Photogrammetric Processing of Chang E-2 Stereo Lunar Imagery,” IEEE Trans. Geosci. Remote Sens. 52(9), 5432–5442 (2014).
[Crossref]

Hu, Z.

Jin, S.

Jin, S. Y.

M. Wang, Y. F. Cheng, X. L. Chang, S. Y. Jin, and Y. Zhu, “On-orbit geometric calibration and geometric quality assessment for the high-resolution geostationary optical satellite gaofen4,” ISPRS J. Photogramm. Remote Sens. 125, 63–77 (2017).
[Crossref]

M. Wang, Y. Zhu, S. Y. Jin, J. Pan, and Q. S. Zhu, “Correction of zy-3 image distortion caused by satellite jitter via virtual steady reimaging using attitude data,” ISPRS J. Photogramm. Remote Sens. 119, 108–123 (2016).
[Crossref]

Kocaman, S.

A. Gruen, S. Kocaman, and K. Wolff, “Calibration and validation of early ALOS/PRISM images,” J. Jpn. Soc. Photogram. Rem. Sens. 46(1), 24–38 (2007).

Kuang, H.

Li, D. R.

G. Zhang, K. Xu, Q. J. Zhang, and D. R. Li, “Correction of Pushbroom Satellite Imagery Interior Distortions Independent of Ground Control Points,” Remote Sens. 10(2), 98 (2018).
[Crossref]

Li, J.

Liu, B.

K. C. Di, Y. L. Liu, B. Liu, M. Peng, and W. M. Hu, “A Self-Calibration Bundle Adjustment Method for Photogrammetric Processing of Chang E-2 Stereo Lunar Imagery,” IEEE Trans. Geosci. Remote Sens. 52(9), 5432–5442 (2014).
[Crossref]

Liu, Y. L.

K. C. Di, Y. L. Liu, B. Liu, M. Peng, and W. M. Hu, “A Self-Calibration Bundle Adjustment Method for Photogrammetric Processing of Chang E-2 Stereo Lunar Imagery,” IEEE Trans. Geosci. Remote Sens. 52(9), 5432–5442 (2014).
[Crossref]

Lowe, D. G.

D. G. Lowe, “Distinctive image features from scale-invariant keypoints,” Int. J. Comput. Vis. 60(2), 91–110 (2004).
[Crossref]

Luong, Q. T.

O. D. Faugeras, Q. T. Luong, and S. J. Maybank, “Camera self-calibration: Theory and experiments,” Computer Vision—ECCV, 588(12), 321–334 (1992).

Malis, E.

E. Malis and R. Cipolla, “Camera self-calibration from unknown planar structures enforcing the multiview constraints between collineations,” IEEE Trans. Pattern Anal. Mach. Intell. 24(9), 1268–1272 (2002).
[Crossref]

Maybank, S. J.

O. D. Faugeras, Q. T. Luong, and S. J. Maybank, “Camera self-calibration: Theory and experiments,” Computer Vision—ECCV, 588(12), 321–334 (1992).

Murakami, H.

T. Tadono, M. Shimada, H. Murakami, and J. Takaku, “Calibration of PRISM and AVNIR-2 onboard ALOS—Daichi,” IEEE Trans. Geosci. Remote Sens. 47(12), 4042–4050 (2009).
[Crossref]

Pan, J.

M. Wang, Y. Zhu, S. Y. Jin, J. Pan, and Q. S. Zhu, “Correction of zy-3 image distortion caused by satellite jitter via virtual steady reimaging using attitude data,” ISPRS J. Photogramm. Remote Sens. 119, 108–123 (2016).
[Crossref]

Peng, M.

K. C. Di, Y. L. Liu, B. Liu, M. Peng, and W. M. Hu, “A Self-Calibration Bundle Adjustment Method for Photogrammetric Processing of Chang E-2 Stereo Lunar Imagery,” IEEE Trans. Geosci. Remote Sens. 52(9), 5432–5442 (2014).
[Crossref]

Shang, Y.

Shao, X.

Shimada, M.

T. Tadono, M. Shimada, H. Murakami, and J. Takaku, “Calibration of PRISM and AVNIR-2 onboard ALOS—Daichi,” IEEE Trans. Geosci. Remote Sens. 47(12), 4042–4050 (2009).
[Crossref]

Su, H.

Sun, T.

Tadono, T.

T. Tadono, M. Shimada, H. Murakami, and J. Takaku, “Calibration of PRISM and AVNIR-2 onboard ALOS—Daichi,” IEEE Trans. Geosci. Remote Sens. 47(12), 4042–4050 (2009).
[Crossref]

Takaku, J.

T. Tadono, M. Shimada, H. Murakami, and J. Takaku, “Calibration of PRISM and AVNIR-2 onboard ALOS—Daichi,” IEEE Trans. Geosci. Remote Sens. 47(12), 4042–4050 (2009).
[Crossref]

Tan, Z.

Teng, C. H.

Wang, D.

Wang, M.

M. Wang, Y. F. Cheng, X. L. Chang, S. Y. Jin, and Y. Zhu, “On-orbit geometric calibration and geometric quality assessment for the high-resolution geostationary optical satellite gaofen4,” ISPRS J. Photogramm. Remote Sens. 125, 63–77 (2017).
[Crossref]

M. Wang, Y. Zhu, S. Y. Jin, J. Pan, and Q. S. Zhu, “Correction of zy-3 image distortion caused by satellite jitter via virtual steady reimaging using attitude data,” ISPRS J. Photogramm. Remote Sens. 119, 108–123 (2016).
[Crossref]

M. Wang, Y. Cheng, B. Yang, S. Jin, and H. Su, “On-orbit calibration approach for optical navigation camera in deep space exploration,” Opt. Express 24(5), 5536–5554 (2016).
[Crossref] [PubMed]

M. Wang, B. Yang, F. Hu, and X. Zang, “On-Orbit geometric calibration model and its applications for high-resolution optical satellite imagery,” Remote Sens. 6(12), 4391–4408 (2014).
[Crossref]

Wolff, K.

A. Gruen, S. Kocaman, and K. Wolff, “Calibration and validation of early ALOS/PRISM images,” J. Jpn. Soc. Photogram. Rem. Sens. 46(1), 24–38 (2007).

Xing, F.

Xiong, X. D.

M. T. Zheng, Y. J. Zhang, J. F. Zhu, and X. D. Xiong, “Self-Calibration adjustment of CBERS-02B long-strip imagery,” IEEE Trans. Geosci. Remote Sens. 53(7), 3847–3854 (2015).
[Crossref]

Xu, K.

G. Zhang, K. Xu, Q. J. Zhang, and D. R. Li, “Correction of Pushbroom Satellite Imagery Interior Distortions Independent of Ground Control Points,” Remote Sens. 10(2), 98 (2018).
[Crossref]

Yang, B.

M. Wang, Y. Cheng, B. Yang, S. Jin, and H. Su, “On-orbit calibration approach for optical navigation camera in deep space exploration,” Opt. Express 24(5), 5536–5554 (2016).
[Crossref] [PubMed]

M. Wang, B. Yang, F. Hu, and X. Zang, “On-Orbit geometric calibration model and its applications for high-resolution optical satellite imagery,” Remote Sens. 6(12), 4391–4408 (2014).
[Crossref]

Yilmaztürk, F.

You, Z.

Yu, Q.

Zang, X.

M. Wang, B. Yang, F. Hu, and X. Zang, “On-Orbit geometric calibration model and its applications for high-resolution optical satellite imagery,” Remote Sens. 6(12), 4391–4408 (2014).
[Crossref]

Zhang, G.

G. Zhang, K. Xu, Q. J. Zhang, and D. R. Li, “Correction of Pushbroom Satellite Imagery Interior Distortions Independent of Ground Control Points,” Remote Sens. 10(2), 98 (2018).
[Crossref]

Zhang, Q. J.

G. Zhang, K. Xu, Q. J. Zhang, and D. R. Li, “Correction of Pushbroom Satellite Imagery Interior Distortions Independent of Ground Control Points,” Remote Sens. 10(2), 98 (2018).
[Crossref]

Zhang, T.

Zhang, Y. J.

M. T. Zheng, Y. J. Zhang, J. F. Zhu, and X. D. Xiong, “Self-Calibration adjustment of CBERS-02B long-strip imagery,” IEEE Trans. Geosci. Remote Sens. 53(7), 3847–3854 (2015).
[Crossref]

Zheng, M. T.

M. T. Zheng, Y. J. Zhang, J. F. Zhu, and X. D. Xiong, “Self-Calibration adjustment of CBERS-02B long-strip imagery,” IEEE Trans. Geosci. Remote Sens. 53(7), 3847–3854 (2015).
[Crossref]

Zhu, J. F.

M. T. Zheng, Y. J. Zhang, J. F. Zhu, and X. D. Xiong, “Self-Calibration adjustment of CBERS-02B long-strip imagery,” IEEE Trans. Geosci. Remote Sens. 53(7), 3847–3854 (2015).
[Crossref]

Zhu, Q. S.

M. Wang, Y. Zhu, S. Y. Jin, J. Pan, and Q. S. Zhu, “Correction of zy-3 image distortion caused by satellite jitter via virtual steady reimaging using attitude data,” ISPRS J. Photogramm. Remote Sens. 119, 108–123 (2016).
[Crossref]

Zhu, Y.

M. Wang, Y. F. Cheng, X. L. Chang, S. Y. Jin, and Y. Zhu, “On-orbit geometric calibration and geometric quality assessment for the high-resolution geostationary optical satellite gaofen4,” ISPRS J. Photogramm. Remote Sens. 125, 63–77 (2017).
[Crossref]

M. Wang, Y. Zhu, S. Y. Jin, J. Pan, and Q. S. Zhu, “Correction of zy-3 image distortion caused by satellite jitter via virtual steady reimaging using attitude data,” ISPRS J. Photogramm. Remote Sens. 119, 108–123 (2016).
[Crossref]

Appl. Opt. (3)

Computer Vision—ECCV (1)

O. D. Faugeras, Q. T. Luong, and S. J. Maybank, “Camera self-calibration: Theory and experiments,” Computer Vision—ECCV, 588(12), 321–334 (1992).

IEEE Trans. Geosci. Remote Sens. (3)

K. C. Di, Y. L. Liu, B. Liu, M. Peng, and W. M. Hu, “A Self-Calibration Bundle Adjustment Method for Photogrammetric Processing of Chang E-2 Stereo Lunar Imagery,” IEEE Trans. Geosci. Remote Sens. 52(9), 5432–5442 (2014).
[Crossref]

M. T. Zheng, Y. J. Zhang, J. F. Zhu, and X. D. Xiong, “Self-Calibration adjustment of CBERS-02B long-strip imagery,” IEEE Trans. Geosci. Remote Sens. 53(7), 3847–3854 (2015).
[Crossref]

T. Tadono, M. Shimada, H. Murakami, and J. Takaku, “Calibration of PRISM and AVNIR-2 onboard ALOS—Daichi,” IEEE Trans. Geosci. Remote Sens. 47(12), 4042–4050 (2009).
[Crossref]

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

E. Malis and R. Cipolla, “Camera self-calibration from unknown planar structures enforcing the multiview constraints between collineations,” IEEE Trans. Pattern Anal. Mach. Intell. 24(9), 1268–1272 (2002).
[Crossref]

Int. J. Comput. Vis. (1)

D. G. Lowe, “Distinctive image features from scale-invariant keypoints,” Int. J. Comput. Vis. 60(2), 91–110 (2004).
[Crossref]

ISPRS J. Photogramm. Remote Sens. (2)

M. Wang, Y. Zhu, S. Y. Jin, J. Pan, and Q. S. Zhu, “Correction of zy-3 image distortion caused by satellite jitter via virtual steady reimaging using attitude data,” ISPRS J. Photogramm. Remote Sens. 119, 108–123 (2016).
[Crossref]

M. Wang, Y. F. Cheng, X. L. Chang, S. Y. Jin, and Y. Zhu, “On-orbit geometric calibration and geometric quality assessment for the high-resolution geostationary optical satellite gaofen4,” ISPRS J. Photogramm. Remote Sens. 125, 63–77 (2017).
[Crossref]

J. Jpn. Soc. Photogram. Rem. Sens. (1)

A. Gruen, S. Kocaman, and K. Wolff, “Calibration and validation of early ALOS/PRISM images,” J. Jpn. Soc. Photogram. Rem. Sens. 46(1), 24–38 (2007).

Opt. Express (5)

Photogramm. Eng. Remote Sensing (1)

C. S. Fraser and H. B. Hanley, “Bias compensation in rational functions for IKONOS satellite imagery,” Photogramm. Eng. Remote Sensing 69(1), 53–57 (2003).
[Crossref]

Remote Sens. (2)

G. Zhang, K. Xu, Q. J. Zhang, and D. R. Li, “Correction of Pushbroom Satellite Imagery Interior Distortions Independent of Ground Control Points,” Remote Sens. 10(2), 98 (2018).
[Crossref]

M. Wang, B. Yang, F. Hu, and X. Zang, “On-Orbit geometric calibration model and its applications for high-resolution optical satellite imagery,” Remote Sens. 6(12), 4391–4408 (2014).
[Crossref]

Other (2)

J. Grodecki, “IKONOS geometric calibrations,” Proceedings of the ASPRS 2005 Annual Conference (2005).

J. M. Delevit, D. Greslou, V. Amberg, C. Dechoz, F. de Lussy, L. Lebegue, C. Latry, S. Artigues, and L. Bernard, “Attitude assessment using Pleiades-HR capabilities,” in Proceedings of the International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences (2012), 525–530.
[Crossref]

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Figures (9)

Fig. 1
Fig. 1 Flow diagram of the self-calibration approach: (a) appropriate triple-overlap relationship, (b) distribution of GCPs that used for the calibration of the primary STI, (c) distribution of tie points that used for the calibration of the external relative reference coordinate systems, (d) distribution of tie points that used for the calibration of the internal parameters of the secondary STIs.
Fig. 2
Fig. 2 Flow chart of the self-calibration approach.
Fig. 3
Fig. 3 Defining the mapping relationship: a schematic diagram.
Fig. 4
Fig. 4 Schematic diagram of the intersection model of tie points.
Fig. 5
Fig. 5 Estimation of a 0 and b 0 : (a) systematic translation of the TDI-CCD detector, (b) distribution of tie points and the computation sequence of the internal parameters of the secondary STIs.
Fig. 6
Fig. 6 Installation structure and configuration of the Gaofen-2 (GF2) double-camera imaging system.
Fig. 7
Fig. 7 Overlap of the three stereoscopic GF2 satellite images.
Fig. 8
Fig. 8 Internal line of sight distortion of each TDI-CCD detector after external calibration.
Fig. 9
Fig. 9 Overlap of the simulated stereoscopic GF2 satellite images.

Tables (9)

Tables Icon

Table 1 Detailed information regarding the satellite imagery data.

Tables Icon

Table 2 Calculated installation angles and internal parameters of the primary TDI-CCD 3 detector.

Tables Icon

Table 3 Positioning accuracy of image 2 before and after internal calibration.

Tables Icon

Table 4 Mosaic accuracy between adjacent STIs of image 2.

Tables Icon

Table 5 Relative positioning accuracy using four GCPs based on the calibrated internal parameters.

Tables Icon

Table 6 Internal parameters of the primary TDI-CCD 3 detector.

Tables Icon

Table 7 Installation angles of the three stereoscopic images.

Tables Icon

Table 8 Positioning accuracy before and after internal calibration of image 1.

Tables Icon

Table 9 Mosaic accuracy between adjacent STIs of image 1.

Equations (16)

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[ tan ψ x tan ψ y 1 ]=λ R body camera R J2000 body R WGS84 J2000 ( [ X Y Z ] WGS84 [ X s Y s Z s ] WGS84 ).
{ tan ψ x = a 0 + a 1 s+ a 2 s 2 + a 3 s 3 tan ψ y = b 0 + b 1 s+ b 2 s 2 + b 3 s 3
{ ( W x W y W z )= R J2000 body R WGS84 J2000 ( [ X Y Z ] WGS84 [ X s Y s Z s ] WGS84 ). R body camera ( roll,pitch,yaw )=[ A 1 B 1 C 1 A 2 B 2 C 2 A 3 B 3 C 3 ]
{ F= A 1 W x + B 1 W y + C 1 W z A 3 W x + B 3 W y + C 3 W z tan( ψ x ( s,l ) ) G= A 2 W x + B 2 W y + C 2 W z A 3 W x + B 3 W y + C 3 W z tan( ψ y ( s,l ) )
R i E = H i E Δ X E
R i E = [ F( X E , X I 0 ) G( X E , X I 0 ) ] i , H i E =[ F i X E G i X E ]=[ F i roll F i pitch F i yaw G i roll G i pitch G i yaw ],Δ X E =[ Δroll Δpitch Δyaw ]
{ f= A 1 W x + B 1 W y + C 1 W z A 3 W x + B 3 W y + C 3 W z =tan( ψ x ( s,l ) ) g= A 2 W x + B 2 W y + C 2 W z A 3 W x + B 3 W y + C 3 W z =tan( ψ y ( s,l ) )
R i I = H i I X I
R i I = [ f( X E ) g( X E ) ] i , H i I = [ d( tan( ψ x (s,l)) ) d X I d( tan( ψ y (s,l)) ) d X I ] i = [ dtan ψ x d a 0 dtan ψ x d a 3 dtan ψ x d b 0 dtan ψ x d b 3 dtan ψ y d a 0 dtan ψ y d a 3 dtan ψ y d b 0 dtan ψ y d b 3 ] i , X I = [ a 0 ,, a 3 , b 0 ,, b 3 ] T
v image1,x,i =tan( ψ x ( s i , l i ) )+ U x ( X i , Y i , Z i ) U z ( X i , Y i , Z i ) , v image2,x,i =tan( ψ x ( s i , l i ) )+ U x ( X i , Y i , Z i ) U z ( X i , Y i , Z i ) , v image3,x,i =tan( ψ x ( s i , l i ) )+ U x ( X i , Y i , Z i ) U z ( X i , Y i , Z i ) v image1,y,i =tan( ψ y ( s i , l i ) )+ U y ( X i , Y i , Z i ) U z ( X i , Y i , Z i ) , v image2,y,i =tan( ψ y ( s i , l i ) )+ U y ( X i , Y i , Z i ) U z ( X i , Y i , Z i ) , v image3,y,i =tan( ψ y ( s i , l i ) )+ U y ( X i , Y i , Z i ) U z ( X i , Y i , Z i )
V k = A k x k + B k t k L k P k
{ tan ψ x = a 0 + a 1 s tan ψ y = b 0 + b 1 s
[ A k T P k A k A k T P k B k B k T P k A k B k T P k B k ][ x k t k ]=[ A k T P k L k B k T P k L k ]
x k = U 1 Q
U= A k T P k A k A k T P k B k ( B k T P k B k ) 1 B k T P k A k
Q= A k T P k L k A k T P k B k ( B k T P k B k ) 1 B k T P k L k

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