Abstract

Many vision measurement systems, especially in outdoor engineering, usually utilize glass ports to protect sensors against environmental influences. The refraction caused by glass ports lead to measurement errors in traditional single viewpoint model. Most existing methods only deal with the refraction that happens once, and the glass ports are required to be perpendicular to cameras or the orientations of glass ports are obtained by auxiliary equipment. This paper proposes a corrected 3D reconstruction model based on refraction geometry, which can be used for any number of glass ports with any orientations. The orientation of each glass port is obtained only using refracted and unrefracted images of the same scene, which doesn’t need any auxiliary equipment. A series of validation experiments are performed. An existing image rectification method is used to make a comparison. The proposed method is also employed in a train wheelset profile measurement application, which proves that the method is effective in actual applications.

© 2017 Optical Society of America

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References

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  1. T. Treibitz, Y. Schechner, C. Kunz, and H. Singh, “Flat refractive geometry,” IEEE Trans. Pattern Anal. Mach. Intell. 34(1), 51–65 (2012).
    [Crossref] [PubMed]
  2. M. A. Haile and P. G. Ifju, “Application of elastic image registration and refraction correction for non-contact underwater strain measurement,” Strain 48(2), 136–142 (2012).
    [Crossref]
  3. D. Samper, J. Santolaria, A. C. Majarena, and J. J. Aguilar, “Correction of the refraction phenomenon in photogrammetric measurement systems,” Metrol. Meas. Syst. 20(4), 601–612 (2013).
    [Crossref]
  4. H. Du, M. G. Li, and J. Meng, “Study on the reconstruction method of stereo vision in glass flume,” Adv. Eng. Softw. 94, 14–19 (2016).
    [Crossref]
  5. L. Kang, L. Wu, and Y. Yang, “Two-view underwater structure and motion for cameras under flat refractive interfaces,” in 12th European Conference on Computer Vision (Springer, 2012), pp. 303–316.
    [Crossref]
  6. B. K. Seo, J. Park, and J. I. Park, “3D trajectory reconstruction under refraction at a cylindrical surface,” in Proceedings of IEEE International Conference on Image Processing (IEEE, 2015), pp. 2660–2664.
    [Crossref]
  7. Y. Wang, S. Negahdaripour, and M. D. Aykin, “Calibration and 3D reconstruction of underwater objects with non-single-view projection model by structured light stereo imaging,” Appl. Opt. 55(24), 6564–6575 (2016).
    [Crossref] [PubMed]
  8. X. Ke, M. A. Sutton, S. M. Lessner, and M. Yost, “Robust stereo vision and calibration methodology for accurate three-dimensional digital image correlation measurements on submerged objects,” J. Strain Anal. Eng. 43(8), 689–704 (2008).
    [Crossref]
  9. R. Ferreira, J. P. Costeira, and J. A. Santos, “Stereo reconstruction of a submerged scene,” in 2nd Iberian Conference on Pattern Recognition and Image Analysis (Springer, 2005), pp. 102–109.
  10. Y. J. Chang and T. H. Chen, “Multi-view 3D reconstruction for scenes under the refractive plane with known vertical direction,” in Proceedings of IEEE International Conference on Computer Vision, (IEEE, 2011), pp. 351–358.
    [Crossref]
  11. E. J. Moore, “Underwater photogrammetry,” Photogramm. Rec. 8(48), 748–763 (1976).
    [Crossref]
  12. H. R. Suiter, “Correction of underwater pincushion distortion by a compensating camera lens,” Proc. SPIE 8357, 83571R (2012).
    [Crossref]
  13. M. Shortis, “Calibration techniques for accurate measurements by underwater camera systems,” Sensors (Basel) 15(12), 30810–30826 (2015).
    [Crossref] [PubMed]
  14. N. J. W. Morris and K. N. Kutulakos, “Dynamic refraction stereo,” IEEE Trans. Pattern Anal. Mach. Intell. 33(8), 1518–1531 (2011).
    [Crossref] [PubMed]
  15. MERMEC Group, “Profile and diameter,” http://www.mermecgroup.com/inspect/train-monitoring/87/wheel-parameters.php .
  16. K. L. D. Labs, Inc., “Wheel profile measurement,” http://www.kldlabs.com/index.php?s=wheel+profile+measurement .
  17. R. Li, C. Tao, and W. Zou, “An underwater digital photogrammetric system for fishery geomatics,” In XVIIIth ISPRS Congress Technical Commission V: Close Range Techniques and Machine Vision (ISPRS, 1996) pp. 319–323.
  18. Z. Y. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 22(11), 1330–1334 (2000).
    [Crossref]
  19. L. Kang, L. Wu, and Y. H. Yang, “Experimental study of the influence of refraction on underwater three-dimensional reconstruction using the SVP camera model,” Appl. Opt. 51(31), 7591–7603 (2012).
    [Crossref] [PubMed]
  20. J. H. Sun, G. J. Zhang, and Q. Z. Liu, “Universal method for calibrating structured-light vision sensor on the spot,” J. Mech. Eng. 45(03), 174–177 (2009).
    [Crossref]
  21. Z. Gong, J. Sun, and G. Zhang, “Dynamic measurement for the diameter of a train wheel based on structured-light vision,” Sensors (Basel) 16(4), 564 (2016).
    [Crossref] [PubMed]

2016 (3)

H. Du, M. G. Li, and J. Meng, “Study on the reconstruction method of stereo vision in glass flume,” Adv. Eng. Softw. 94, 14–19 (2016).
[Crossref]

Z. Gong, J. Sun, and G. Zhang, “Dynamic measurement for the diameter of a train wheel based on structured-light vision,” Sensors (Basel) 16(4), 564 (2016).
[Crossref] [PubMed]

Y. Wang, S. Negahdaripour, and M. D. Aykin, “Calibration and 3D reconstruction of underwater objects with non-single-view projection model by structured light stereo imaging,” Appl. Opt. 55(24), 6564–6575 (2016).
[Crossref] [PubMed]

2015 (1)

M. Shortis, “Calibration techniques for accurate measurements by underwater camera systems,” Sensors (Basel) 15(12), 30810–30826 (2015).
[Crossref] [PubMed]

2013 (1)

D. Samper, J. Santolaria, A. C. Majarena, and J. J. Aguilar, “Correction of the refraction phenomenon in photogrammetric measurement systems,” Metrol. Meas. Syst. 20(4), 601–612 (2013).
[Crossref]

2012 (4)

H. R. Suiter, “Correction of underwater pincushion distortion by a compensating camera lens,” Proc. SPIE 8357, 83571R (2012).
[Crossref]

T. Treibitz, Y. Schechner, C. Kunz, and H. Singh, “Flat refractive geometry,” IEEE Trans. Pattern Anal. Mach. Intell. 34(1), 51–65 (2012).
[Crossref] [PubMed]

M. A. Haile and P. G. Ifju, “Application of elastic image registration and refraction correction for non-contact underwater strain measurement,” Strain 48(2), 136–142 (2012).
[Crossref]

L. Kang, L. Wu, and Y. H. Yang, “Experimental study of the influence of refraction on underwater three-dimensional reconstruction using the SVP camera model,” Appl. Opt. 51(31), 7591–7603 (2012).
[Crossref] [PubMed]

2011 (1)

N. J. W. Morris and K. N. Kutulakos, “Dynamic refraction stereo,” IEEE Trans. Pattern Anal. Mach. Intell. 33(8), 1518–1531 (2011).
[Crossref] [PubMed]

2009 (1)

J. H. Sun, G. J. Zhang, and Q. Z. Liu, “Universal method for calibrating structured-light vision sensor on the spot,” J. Mech. Eng. 45(03), 174–177 (2009).
[Crossref]

2008 (1)

X. Ke, M. A. Sutton, S. M. Lessner, and M. Yost, “Robust stereo vision and calibration methodology for accurate three-dimensional digital image correlation measurements on submerged objects,” J. Strain Anal. Eng. 43(8), 689–704 (2008).
[Crossref]

2000 (1)

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

1976 (1)

E. J. Moore, “Underwater photogrammetry,” Photogramm. Rec. 8(48), 748–763 (1976).
[Crossref]

Aguilar, J. J.

D. Samper, J. Santolaria, A. C. Majarena, and J. J. Aguilar, “Correction of the refraction phenomenon in photogrammetric measurement systems,” Metrol. Meas. Syst. 20(4), 601–612 (2013).
[Crossref]

Aykin, M. D.

Chang, Y. J.

Y. J. Chang and T. H. Chen, “Multi-view 3D reconstruction for scenes under the refractive plane with known vertical direction,” in Proceedings of IEEE International Conference on Computer Vision, (IEEE, 2011), pp. 351–358.
[Crossref]

Chen, T. H.

Y. J. Chang and T. H. Chen, “Multi-view 3D reconstruction for scenes under the refractive plane with known vertical direction,” in Proceedings of IEEE International Conference on Computer Vision, (IEEE, 2011), pp. 351–358.
[Crossref]

Du, H.

H. Du, M. G. Li, and J. Meng, “Study on the reconstruction method of stereo vision in glass flume,” Adv. Eng. Softw. 94, 14–19 (2016).
[Crossref]

Gong, Z.

Z. Gong, J. Sun, and G. Zhang, “Dynamic measurement for the diameter of a train wheel based on structured-light vision,” Sensors (Basel) 16(4), 564 (2016).
[Crossref] [PubMed]

Haile, M. A.

M. A. Haile and P. G. Ifju, “Application of elastic image registration and refraction correction for non-contact underwater strain measurement,” Strain 48(2), 136–142 (2012).
[Crossref]

Ifju, P. G.

M. A. Haile and P. G. Ifju, “Application of elastic image registration and refraction correction for non-contact underwater strain measurement,” Strain 48(2), 136–142 (2012).
[Crossref]

Kang, L.

Ke, X.

X. Ke, M. A. Sutton, S. M. Lessner, and M. Yost, “Robust stereo vision and calibration methodology for accurate three-dimensional digital image correlation measurements on submerged objects,” J. Strain Anal. Eng. 43(8), 689–704 (2008).
[Crossref]

Kunz, C.

T. Treibitz, Y. Schechner, C. Kunz, and H. Singh, “Flat refractive geometry,” IEEE Trans. Pattern Anal. Mach. Intell. 34(1), 51–65 (2012).
[Crossref] [PubMed]

Kutulakos, K. N.

N. J. W. Morris and K. N. Kutulakos, “Dynamic refraction stereo,” IEEE Trans. Pattern Anal. Mach. Intell. 33(8), 1518–1531 (2011).
[Crossref] [PubMed]

Lessner, S. M.

X. Ke, M. A. Sutton, S. M. Lessner, and M. Yost, “Robust stereo vision and calibration methodology for accurate three-dimensional digital image correlation measurements on submerged objects,” J. Strain Anal. Eng. 43(8), 689–704 (2008).
[Crossref]

Li, M. G.

H. Du, M. G. Li, and J. Meng, “Study on the reconstruction method of stereo vision in glass flume,” Adv. Eng. Softw. 94, 14–19 (2016).
[Crossref]

Li, R.

R. Li, C. Tao, and W. Zou, “An underwater digital photogrammetric system for fishery geomatics,” In XVIIIth ISPRS Congress Technical Commission V: Close Range Techniques and Machine Vision (ISPRS, 1996) pp. 319–323.

Liu, Q. Z.

J. H. Sun, G. J. Zhang, and Q. Z. Liu, “Universal method for calibrating structured-light vision sensor on the spot,” J. Mech. Eng. 45(03), 174–177 (2009).
[Crossref]

Majarena, A. C.

D. Samper, J. Santolaria, A. C. Majarena, and J. J. Aguilar, “Correction of the refraction phenomenon in photogrammetric measurement systems,” Metrol. Meas. Syst. 20(4), 601–612 (2013).
[Crossref]

Meng, J.

H. Du, M. G. Li, and J. Meng, “Study on the reconstruction method of stereo vision in glass flume,” Adv. Eng. Softw. 94, 14–19 (2016).
[Crossref]

Moore, E. J.

E. J. Moore, “Underwater photogrammetry,” Photogramm. Rec. 8(48), 748–763 (1976).
[Crossref]

Morris, N. J. W.

N. J. W. Morris and K. N. Kutulakos, “Dynamic refraction stereo,” IEEE Trans. Pattern Anal. Mach. Intell. 33(8), 1518–1531 (2011).
[Crossref] [PubMed]

Negahdaripour, S.

Park, J.

B. K. Seo, J. Park, and J. I. Park, “3D trajectory reconstruction under refraction at a cylindrical surface,” in Proceedings of IEEE International Conference on Image Processing (IEEE, 2015), pp. 2660–2664.
[Crossref]

Park, J. I.

B. K. Seo, J. Park, and J. I. Park, “3D trajectory reconstruction under refraction at a cylindrical surface,” in Proceedings of IEEE International Conference on Image Processing (IEEE, 2015), pp. 2660–2664.
[Crossref]

Samper, D.

D. Samper, J. Santolaria, A. C. Majarena, and J. J. Aguilar, “Correction of the refraction phenomenon in photogrammetric measurement systems,” Metrol. Meas. Syst. 20(4), 601–612 (2013).
[Crossref]

Santolaria, J.

D. Samper, J. Santolaria, A. C. Majarena, and J. J. Aguilar, “Correction of the refraction phenomenon in photogrammetric measurement systems,” Metrol. Meas. Syst. 20(4), 601–612 (2013).
[Crossref]

Schechner, Y.

T. Treibitz, Y. Schechner, C. Kunz, and H. Singh, “Flat refractive geometry,” IEEE Trans. Pattern Anal. Mach. Intell. 34(1), 51–65 (2012).
[Crossref] [PubMed]

Seo, B. K.

B. K. Seo, J. Park, and J. I. Park, “3D trajectory reconstruction under refraction at a cylindrical surface,” in Proceedings of IEEE International Conference on Image Processing (IEEE, 2015), pp. 2660–2664.
[Crossref]

Shortis, M.

M. Shortis, “Calibration techniques for accurate measurements by underwater camera systems,” Sensors (Basel) 15(12), 30810–30826 (2015).
[Crossref] [PubMed]

Singh, H.

T. Treibitz, Y. Schechner, C. Kunz, and H. Singh, “Flat refractive geometry,” IEEE Trans. Pattern Anal. Mach. Intell. 34(1), 51–65 (2012).
[Crossref] [PubMed]

Suiter, H. R.

H. R. Suiter, “Correction of underwater pincushion distortion by a compensating camera lens,” Proc. SPIE 8357, 83571R (2012).
[Crossref]

Sun, J.

Z. Gong, J. Sun, and G. Zhang, “Dynamic measurement for the diameter of a train wheel based on structured-light vision,” Sensors (Basel) 16(4), 564 (2016).
[Crossref] [PubMed]

Sun, J. H.

J. H. Sun, G. J. Zhang, and Q. Z. Liu, “Universal method for calibrating structured-light vision sensor on the spot,” J. Mech. Eng. 45(03), 174–177 (2009).
[Crossref]

Sutton, M. A.

X. Ke, M. A. Sutton, S. M. Lessner, and M. Yost, “Robust stereo vision and calibration methodology for accurate three-dimensional digital image correlation measurements on submerged objects,” J. Strain Anal. Eng. 43(8), 689–704 (2008).
[Crossref]

Tao, C.

R. Li, C. Tao, and W. Zou, “An underwater digital photogrammetric system for fishery geomatics,” In XVIIIth ISPRS Congress Technical Commission V: Close Range Techniques and Machine Vision (ISPRS, 1996) pp. 319–323.

Treibitz, T.

T. Treibitz, Y. Schechner, C. Kunz, and H. Singh, “Flat refractive geometry,” IEEE Trans. Pattern Anal. Mach. Intell. 34(1), 51–65 (2012).
[Crossref] [PubMed]

Wang, Y.

Wu, L.

Yang, Y. H.

Yost, M.

X. Ke, M. A. Sutton, S. M. Lessner, and M. Yost, “Robust stereo vision and calibration methodology for accurate three-dimensional digital image correlation measurements on submerged objects,” J. Strain Anal. Eng. 43(8), 689–704 (2008).
[Crossref]

Zhang, G.

Z. Gong, J. Sun, and G. Zhang, “Dynamic measurement for the diameter of a train wheel based on structured-light vision,” Sensors (Basel) 16(4), 564 (2016).
[Crossref] [PubMed]

Zhang, G. J.

J. H. Sun, G. J. Zhang, and Q. Z. Liu, “Universal method for calibrating structured-light vision sensor on the spot,” J. Mech. Eng. 45(03), 174–177 (2009).
[Crossref]

Zhang, Z. Y.

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

Zou, W.

R. Li, C. Tao, and W. Zou, “An underwater digital photogrammetric system for fishery geomatics,” In XVIIIth ISPRS Congress Technical Commission V: Close Range Techniques and Machine Vision (ISPRS, 1996) pp. 319–323.

Adv. Eng. Softw. (1)

H. Du, M. G. Li, and J. Meng, “Study on the reconstruction method of stereo vision in glass flume,” Adv. Eng. Softw. 94, 14–19 (2016).
[Crossref]

Appl. Opt. (2)

IEEE Trans. Pattern Anal. Mach. Intell. (3)

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

T. Treibitz, Y. Schechner, C. Kunz, and H. Singh, “Flat refractive geometry,” IEEE Trans. Pattern Anal. Mach. Intell. 34(1), 51–65 (2012).
[Crossref] [PubMed]

N. J. W. Morris and K. N. Kutulakos, “Dynamic refraction stereo,” IEEE Trans. Pattern Anal. Mach. Intell. 33(8), 1518–1531 (2011).
[Crossref] [PubMed]

J. Mech. Eng. (1)

J. H. Sun, G. J. Zhang, and Q. Z. Liu, “Universal method for calibrating structured-light vision sensor on the spot,” J. Mech. Eng. 45(03), 174–177 (2009).
[Crossref]

J. Strain Anal. Eng. (1)

X. Ke, M. A. Sutton, S. M. Lessner, and M. Yost, “Robust stereo vision and calibration methodology for accurate three-dimensional digital image correlation measurements on submerged objects,” J. Strain Anal. Eng. 43(8), 689–704 (2008).
[Crossref]

Metrol. Meas. Syst. (1)

D. Samper, J. Santolaria, A. C. Majarena, and J. J. Aguilar, “Correction of the refraction phenomenon in photogrammetric measurement systems,” Metrol. Meas. Syst. 20(4), 601–612 (2013).
[Crossref]

Photogramm. Rec. (1)

E. J. Moore, “Underwater photogrammetry,” Photogramm. Rec. 8(48), 748–763 (1976).
[Crossref]

Proc. SPIE (1)

H. R. Suiter, “Correction of underwater pincushion distortion by a compensating camera lens,” Proc. SPIE 8357, 83571R (2012).
[Crossref]

Sensors (Basel) (2)

M. Shortis, “Calibration techniques for accurate measurements by underwater camera systems,” Sensors (Basel) 15(12), 30810–30826 (2015).
[Crossref] [PubMed]

Z. Gong, J. Sun, and G. Zhang, “Dynamic measurement for the diameter of a train wheel based on structured-light vision,” Sensors (Basel) 16(4), 564 (2016).
[Crossref] [PubMed]

Strain (1)

M. A. Haile and P. G. Ifju, “Application of elastic image registration and refraction correction for non-contact underwater strain measurement,” Strain 48(2), 136–142 (2012).
[Crossref]

Other (7)

L. Kang, L. Wu, and Y. Yang, “Two-view underwater structure and motion for cameras under flat refractive interfaces,” in 12th European Conference on Computer Vision (Springer, 2012), pp. 303–316.
[Crossref]

B. K. Seo, J. Park, and J. I. Park, “3D trajectory reconstruction under refraction at a cylindrical surface,” in Proceedings of IEEE International Conference on Image Processing (IEEE, 2015), pp. 2660–2664.
[Crossref]

R. Ferreira, J. P. Costeira, and J. A. Santos, “Stereo reconstruction of a submerged scene,” in 2nd Iberian Conference on Pattern Recognition and Image Analysis (Springer, 2005), pp. 102–109.

Y. J. Chang and T. H. Chen, “Multi-view 3D reconstruction for scenes under the refractive plane with known vertical direction,” in Proceedings of IEEE International Conference on Computer Vision, (IEEE, 2011), pp. 351–358.
[Crossref]

MERMEC Group, “Profile and diameter,” http://www.mermecgroup.com/inspect/train-monitoring/87/wheel-parameters.php .

K. L. D. Labs, Inc., “Wheel profile measurement,” http://www.kldlabs.com/index.php?s=wheel+profile+measurement .

R. Li, C. Tao, and W. Zou, “An underwater digital photogrammetric system for fishery geomatics,” In XVIIIth ISPRS Congress Technical Commission V: Close Range Techniques and Machine Vision (ISPRS, 1996) pp. 319–323.

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Figures (15)

Fig. 1
Fig. 1 Refractive ray paths of two cases. (a) The glass port is perpendicular. (b) The glass port is titled. α is the incident angle, β is the refractive angle, γ is the emergent angle, h is the thickness of the glass port, P is the real objective point, and P’ is the pseudo objective point.
Fig. 2
Fig. 2 Refraction in general 3D space.
Fig. 3
Fig. 3 Expansion effect of refraction.
Fig. 4
Fig. 4 Experimental platform.
Fig. 5
Fig. 5 Refracted images.
Fig. 6
Fig. 6 Determination of expansion center. (a) The glass port is perpendicular. (b) The glass port is tilted.
Fig. 7
Fig. 7 Image for 3D reconstruction and measurement. (a) The glass port is perpendicular. (b) The glass port is tilted. The four sides of the target are indicated by the red lines (L, R, T, and B), and the two diagonals are indicated by the green lines (LS and RS).
Fig. 8
Fig. 8 3D reconstruction errors of a perpendicular glass port. (a) For the SVP model with unrefracted parameters, the RMS error is 0.12 mm. (b) For the SVP model with refracted parameters, the RMS error is 0.22 mm. (c) For the proposed method with unrefracted parameters, the RMS error is 0.01 mm.
Fig. 9
Fig. 9 3D reconstruction errors of a tilted glass port. (a) For the SVP model with unrefracted parameters, the RMS error is 0.22 mm. (b) For the SVP model with refracted parameters, the RMS error is 0.88 mm. (c) For the proposed method with unrefracted parameters, the RMS error is 0.02 mm.
Fig. 10
Fig. 10 3D reconstruction errors of two glass ports. (a) For the SVP model with unrefracted parameters, the RMS error is 0.29 mm. (b) For the SVP model with refracted parameters, the RMS error is 0.84 mm. (c) For the proposed method with unrefracted parameters, the RMS error is 0.05 mm.
Fig. 11
Fig. 11 Images for calibration and 3D reconstruction. (a) 10 images for calibration. (b) Image for 3D reconstruction.
Fig. 12
Fig. 12 Results of structured light vision experiment.
Fig. 13
Fig. 13 3D reconstruction errors. (a) In the proposed method, the RMS error is 0.01 mm. (b) In the image registration method, the RMS error is 0.01 mm.
Fig. 14
Fig. 14 A wheelset profile measurement system with glass ports and optical filters.
Fig. 15
Fig. 15 Reconstruction result of wheelset profiles. (a) Profiles constructed by SVP method. (b) Profiles constructed by the proposed method.

Tables (4)

Tables Icon

Table 1 Parameters and tilt angles of camera.

Tables Icon

Table 2 Measurement errors of sides and diagonals of target in experiment with one glass port.

Tables Icon

Table 3 Measurement errors of sides and diagonals of target in experiment with two glass ports.

Tables Icon

Table 4 Results of flange height and flange thickness of the wheel.

Equations (22)

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

n air sinα= n glass sinβ= n air sinγ,
y=tanαx.
y=tanβx+( tanαtanβ ) x 0 .
y=tanγx+( tanβtanα )h=tanαx+( tanβtanα )h.
y=tanα( xh )+( sinα ( n glass / n air ) 2 sin 2 α )h.
y=tan( θ+απ/2 )x.
y=tan( β+θπ/2 )x+ ( tan( α+θπ/2 )tan( β+θπ/2 ) )tanθ x 0 tanθtan( α+θπ/2 ) .
y=tan( θ+απ/2 )x+ tan( β+θπ/2 )tan( α+θπ/2 ) cosθ( tanθtan( β+θπ/2 ) ) h.
y=tan( θ+απ/2 )x+( tanθ+cot( α+θ ) tanθ+ cosθ ( n glass / n air ) 2 sin 2 α sinαsinθ sinαcosθ+sinθ ( n glass / n air ) 2 sin 2 α 1 ) h cosθ .
y=tanφx+( tanθtanφ tanθ+ cosθ ( n glass / n air ) 2 cos 2 ( θφ ) cos( θφ )sinθ cos( θφ )cosθ+sinθ ( n glass / n air ) 2 cos 2 ( θφ ) 1 ) h cosθ .
( ( 1 cosα ( n glass / n air ) 2 sin 2 α )h,0 ).
( x i , y i , z i )=( 1 cosα ( n glass / n air ) 2 sin 2 α )h N N .
x x i I x = y y i I y = z z i 1 .
A=[ f/ dX u 0 f/ dY v 0 1 ],
s[ u u v u 1 ]=A[ x c y c z c ],
[ u d v d ]=( 1+ k 1 ( u u 2 + v u 2 )+ k 2 ( u u 2 + v u 2 ) 2 )[ u u v u ],
IR =( u u u 0 f/ dX , v u v 0 f/ dY ,1 ).
( v wg v og )x+( u og u wg )y+( u wg u og ) v og ( v wg v og ) u og =0.
arg min (ue,ve) i=1 K ( ( v wg i v og i ) u e +( u og i u wg i ) v e +( u wg i u og i ) v og i ( v wg i v og i ) u og i ( v wg i v og i ),( u og i u wg i ) 2 ) 2 ,
N =( u e u 0 f/ dX , v e v 0 f/ dY ,1 ).
( x i m , y i m , z i m )=( 1 cos< IR , N m > ( n glass m / n air m ) 2 sin 2 < IR , N m > ) h m N m N m +( x i m1 , y i m1 , z i m1 ).
x x i m I x = y y i m I y = z z i m 1 .

Metrics