Abstract

Many lens distortion models exist with several variations, and each distortion model is calibrated by using a different technique. If someone wants to correct lens distortion, choosing the right model could represent a very difficult task. Calibration depends on the chosen model, and some methods have unstable results. Normally, the distortion model containing radial, tangential, and prism distortion is used, but it does not represent high distortion accurately. The aim of this paper is to compare different lens distortion models to define the one that obtains better results under some conditions and to explore if some model can represent high and low distortion adequately. Also, we propose a calibration technique to calibrate several models under stable conditions. Since performance is hard conditioned with the calibration technique, the metric lens distortion calibration method is used to calibrate all the evaluated models.

© 2010 Optical Society of America

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  38. S. B. Kang, “Catadioptric self-calibration,” in Proceedings of the Conference on Computer Vision and Pattern Recognition (IEEE Computer Society, 2000), pp. 201–207.
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    [CrossRef]
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  41. C. Hughes, P. Denny, E. Jones, and M. Glavin, “Accuracy of fish-eye lens models,” Appl. Opt. 49, 3338–3347 (2010).
    [CrossRef] [PubMed]
  42. C. Hughes, R. McFeely, P. Denny, M. Glavin, and E. Jones, “Equidistant (fθ) fish-eye perspective with application in distortion centre estimation,” Image Vis. Comput. 28, 538–551 (2010).
    [CrossRef]
  43. C. Ricolfe-Viala and A. J. Sanchez-Salmeron, “Robust metric calibration of non-linear camera lens distortion,” Pattern Recogn. 43, 1688–1699 (2010).
    [CrossRef]
  44. J. Wang, F. Shi, J. Zhang, and Y. Liu, “A new calibration model of camera lens distortion,” Pattern Recogn. 41, 607–615(2008).
    [CrossRef]
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2010 (4)

C. Hughes, R. McFeely, P. Denny, M. Glavin, and E. Jones, “Equidistant (fθ) fish-eye perspective with application in distortion centre estimation,” Image Vis. Comput. 28, 538–551 (2010).
[CrossRef]

C. Ricolfe-Viala and A. J. Sanchez-Salmeron, “Robust metric calibration of non-linear camera lens distortion,” Pattern Recogn. 43, 1688–1699 (2010).
[CrossRef]

L. Huang, P. S. K. Chua, and A. Asundi, “Least-squares calibration method for fringe projection profilometry considering camera lens distortion,” Appl. Opt. 49, 1539–1548 (2010).
[CrossRef] [PubMed]

C. Hughes, P. Denny, E. Jones, and M. Glavin, “Accuracy of fish-eye lens models,” Appl. Opt. 49, 3338–3347 (2010).
[CrossRef] [PubMed]

2009 (3)

2008 (1)

J. Wang, F. Shi, J. Zhang, and Y. Liu, “A new calibration model of camera lens distortion,” Pattern Recogn. 41, 607–615(2008).
[CrossRef]

2007 (1)

R. Hartley and S. Kang, “Parameter-free radial distortion correction with center of distortion estimation,” IEEE Trans. Pattern Anal. Machine Intell. 29, 1309–1321 (2007).
[CrossRef]

2005 (1)

M. Ahmed and A. Farag, “Non-metric calibration of camera lens distortion: differential methods and robust estimation,” IEEE Trans. Image Process. 14, 1215–1230 (2005).
[CrossRef] [PubMed]

2001 (3)

H. Farid and A. C. Popescu, “Blind removal of lens distortion,” J. Opt. Soc. Am. A 18, 2072–2078 (2001).
[CrossRef]

S. S. Beauchemin and R. Bajcsy, “Modelling and removing radial and tangential distortions in spherical lenses, multi image analysis,” Lect. Notes Comput. Sci. 2032, 1–21 (2001).
[CrossRef]

F. Devernay and O. Faugeras, “Straight lines have to be straight,” Machine Vis. Apps. 13, 14–24 (2001).
[CrossRef]

2000 (2)

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

R. Swaminathan and S. Nayar, “Non-metric calibration of wide-angle lenses and polycameras,” IEEE Trans. Pattern Anal. Machine Intell. 22, 1172–1178 (2000).
[CrossRef]

1999 (1)

1997 (1)

B. Prescott and G. McLean, “Line-based correction of radial lens distortion,” Graph. Models Image Process. 59, 39–47 (1997).
[CrossRef]

1996 (1)

S. Shah and J. K. Aggarwal, “Intrinsic parameter calibration procedure for a (high distortion) fish-eye lens camera with distortion model and accuracy estimation,” Pattern Recogn. 29, 1775–1778 (1996).
[CrossRef]

1995 (1)

A. Basu and S. Licardie, “Alternative models for fish-eye lenses,” Pattern Recogn. Lett. 16, 433–441 (1995).
[CrossRef]

1992 (1)

J. Weng, P. Cohen, and M. Herniou, “Camera calibration with distortion models and accuracy evaluation,” IEEE Trans. Pattern Anal. Machine Intell. 14, 965–980 (1992).
[CrossRef]

1991 (1)

M. Penna, “Camera calibration: a quick and easy way to detection of scale factor,” IEEE Trans. Pattern Anal.Machine Intell. 13, 1240–1245 (1991).
[CrossRef]

1987 (1)

R. Tsai, “A versatile camera calibration technique for high-accuracy 3D machine vision metrology using off-the-self TV camera lenses,” IEEE J. Robotics Autom. RA-3, 323–344 (1987).
[CrossRef]

1971 (1)

D. C. Brown, “Close-range camera calibration,” Photogram. Eng. 37, 855–866 (1971).

1966 (1)

D. C. Brown, “Decentering distortion of lenses,” Photogram. Eng. 32, 444–462 (1966).

Aggarwal, J. K.

S. Shah and J. K. Aggarwal, “Intrinsic parameter calibration procedure for a (high distortion) fish-eye lens camera with distortion model and accuracy estimation,” Pattern Recogn. 29, 1775–1778 (1996).
[CrossRef]

Ahmed, M.

M. Ahmed and A. Farag, “Non-metric calibration of camera lens distortion: differential methods and robust estimation,” IEEE Trans. Image Process. 14, 1215–1230 (2005).
[CrossRef] [PubMed]

Asundi, A.

Bajcsy, R.

S. S. Beauchemin and R. Bajcsy, “Modelling and removing radial and tangential distortions in spherical lenses, multi image analysis,” Lect. Notes Comput. Sci. 2032, 1–21 (2001).
[CrossRef]

Barreto, J.

J. Barreto and K. Daniilidis, “Wide area multiple camera calibration and estimation of radial distortion,” in Proceedings of the 5th European Conference on Computer Vision (Springer, 2004).

Basu, A.

A. Basu and S. Licardie, “Alternative models for fish-eye lenses,” Pattern Recogn. Lett. 16, 433–441 (1995).
[CrossRef]

Beauchemin, S. S.

S. S. Beauchemin and R. Bajcsy, “Modelling and removing radial and tangential distortions in spherical lenses, multi image analysis,” Lect. Notes Comput. Sci. 2032, 1–21 (2001).
[CrossRef]

Becker, S.

S. Becker and V. Bove, “Semi-automatic 3D model extraction from uncalibrated 2D camera views,” in Proceedings of Visual Data Exploration and Analysis (1995), Vol. 2, pp. 447–461.

Bethel, J.

C. McGlone, E. Mikhail, and J. Bethel, Manual of Photogrammetry, 5th ed. (American Society of Photogrammetry and Remote Sensing, 2004).

Bove, V.

S. Becker and V. Bove, “Semi-automatic 3D model extraction from uncalibrated 2D camera views,” in Proceedings of Visual Data Exploration and Analysis (1995), Vol. 2, pp. 447–461.

Bradski, G.

G. Bradski and A. Kaehler, Learning OpenCV (O’Reilly Media, 2008).

Brown, D. C.

D. C. Brown, “Close-range camera calibration,” Photogram. Eng. 37, 855–866 (1971).

D. C. Brown, “Decentering distortion of lenses,” Photogram. Eng. 32, 444–462 (1966).

Chen, Y. Q.

L. Ma, Y. Q. Chen, and K. L. Moore, “Flexible camera calibration using a new analytical radial undistortion formula with application to mobile robot localization,” in Proceedings of IEEE International Symposium on Intelligent Control (IEEE, 2003).

Chua, P. S. K.

Claus, D.

D. Claus and A. Fitzgibbon, “A rational function lens distortion model for general cameras,” in Proceedings of the International Conference on Computer Vision and Pattern Recognition (IEEE Computer Society, 2005), pp. 213–219.

Cohen, P.

J. Weng, P. Cohen, and M. Herniou, “Camera calibration with distortion models and accuracy evaluation,” IEEE Trans. Pattern Anal. Machine Intell. 14, 965–980 (1992).
[CrossRef]

Cui, S.

Daniilidis, K.

C. Geyer and K. Daniilidis, “Structure and motion from uncalibrated catadioptric views,” in Proceedings of the International Conference on Computer Vision and Pattern Recognition (IEEE Computer Society, 2001), pp. 279–286.

J. Barreto and K. Daniilidis, “Wide area multiple camera calibration and estimation of radial distortion,” in Proceedings of the 5th European Conference on Computer Vision (Springer, 2004).

C. Geyer and K. Daniilidis, “A unifying theory for central panoramic systems and practical applications,” in Proceedings of the 3rd European Conference on Computer Vision (Springer, 2000, pp. 445–461.

Denny, P.

C. Hughes, R. McFeely, P. Denny, M. Glavin, and E. Jones, “Equidistant (fθ) fish-eye perspective with application in distortion centre estimation,” Image Vis. Comput. 28, 538–551 (2010).
[CrossRef]

C. Hughes, P. Denny, E. Jones, and M. Glavin, “Accuracy of fish-eye lens models,” Appl. Opt. 49, 3338–3347 (2010).
[CrossRef] [PubMed]

Devernay, F.

F. Devernay and O. Faugeras, “Straight lines have to be straight,” Machine Vis. Apps. 13, 14–24 (2001).
[CrossRef]

Dhome, M.

J. Lavest, M. Viala, and M. Dhome, “Do we really need accurate calibration pattern to achieve a reliable camera calibration,” in Proceedings of the 2nd European Conference on Computer Vision (Springer, 1998).

Du, H.

Farag, A.

M. Ahmed and A. Farag, “Non-metric calibration of camera lens distortion: differential methods and robust estimation,” IEEE Trans. Image Process. 14, 1215–1230 (2005).
[CrossRef] [PubMed]

Farid, H.

Faugeras, O.

F. Devernay and O. Faugeras, “Straight lines have to be straight,” Machine Vis. Apps. 13, 14–24 (2001).
[CrossRef]

Fitzgibbon, A.

A. Fitzgibbon, “Simultaneous linear estimation of multiple view geometry and lens distortion,” in Proceedings of the Conference on Computer Vision and Pattern Recognition (IEEE Computer Society, 2001), pp. 125–132.

D. Claus and A. Fitzgibbon, “A rational function lens distortion model for general cameras,” in Proceedings of the International Conference on Computer Vision and Pattern Recognition (IEEE Computer Society, 2005), pp. 213–219.

Geyer, C.

C. Geyer and K. Daniilidis, “A unifying theory for central panoramic systems and practical applications,” in Proceedings of the 3rd European Conference on Computer Vision (Springer, 2000, pp. 445–461.

C. Geyer and K. Daniilidis, “Structure and motion from uncalibrated catadioptric views,” in Proceedings of the International Conference on Computer Vision and Pattern Recognition (IEEE Computer Society, 2001), pp. 279–286.

Glavin, M.

C. Hughes, R. McFeely, P. Denny, M. Glavin, and E. Jones, “Equidistant (fθ) fish-eye perspective with application in distortion centre estimation,” Image Vis. Comput. 28, 538–551 (2010).
[CrossRef]

C. Hughes, P. Denny, E. Jones, and M. Glavin, “Accuracy of fish-eye lens models,” Appl. Opt. 49, 3338–3347 (2010).
[CrossRef] [PubMed]

Grossberg, M. D.

M. D. Grossberg and S. K. Nayar, “A general imaging model and a method for finding its parameters,” in Proceedings of the International Conference on Computer Vision (IEEE Computer Society, 2001), pp. 108–115.

Hartley, R.

R. Hartley and S. Kang, “Parameter-free radial distortion correction with center of distortion estimation,” IEEE Trans. Pattern Anal. Machine Intell. 29, 1309–1321 (2007).
[CrossRef]

Hartley, R. I.

R. I. Hartley and T. Saxena, “The cubic rational polynomial camera model,” in Proceedings DARPA Image Understanding Workshop (IEEE Computer Society, 1997), pp. 649–653.

Herniou, M.

J. Weng, P. Cohen, and M. Herniou, “Camera calibration with distortion models and accuracy evaluation,” IEEE Trans. Pattern Anal. Machine Intell. 14, 965–980 (1992).
[CrossRef]

Hlavac, H.

T. Svoboda, T. Pajdla, and H. Hlavac, “Epipolar geometry for panoramic cameras,” in Proceedings of the 2nd European Conference on Computer Vision (Springer, 1998), pp. 218–232.

Hu, Z.

X. Ying and Z. Hu, “Can we consider central catadioptric cameras and fisheye cameras within a unified imaging model?,” in Proceedings of the 5th European Conference on Computer Vision (Springer, 2004, pp. 442–455.

Huang, L.

Hughes, C.

C. Hughes, P. Denny, E. Jones, and M. Glavin, “Accuracy of fish-eye lens models,” Appl. Opt. 49, 3338–3347 (2010).
[CrossRef] [PubMed]

C. Hughes, R. McFeely, P. Denny, M. Glavin, and E. Jones, “Equidistant (fθ) fish-eye perspective with application in distortion centre estimation,” Image Vis. Comput. 28, 538–551 (2010).
[CrossRef]

Jones, E.

C. Hughes, R. McFeely, P. Denny, M. Glavin, and E. Jones, “Equidistant (fθ) fish-eye perspective with application in distortion centre estimation,” Image Vis. Comput. 28, 538–551 (2010).
[CrossRef]

C. Hughes, P. Denny, E. Jones, and M. Glavin, “Accuracy of fish-eye lens models,” Appl. Opt. 49, 3338–3347 (2010).
[CrossRef] [PubMed]

Kaehler, A.

G. Bradski and A. Kaehler, Learning OpenCV (O’Reilly Media, 2008).

Kang, S.

R. Hartley and S. Kang, “Parameter-free radial distortion correction with center of distortion estimation,” IEEE Trans. Pattern Anal. Machine Intell. 29, 1309–1321 (2007).
[CrossRef]

Kang, S. B.

S. B. Kang, “Catadioptric self-calibration,” in Proceedings of the Conference on Computer Vision and Pattern Recognition (IEEE Computer Society, 2000), pp. 201–207.

Lavest, J.

J. Lavest, M. Viala, and M. Dhome, “Do we really need accurate calibration pattern to achieve a reliable camera calibration,” in Proceedings of the 2nd European Conference on Computer Vision (Springer, 1998).

Licardie, S.

A. Basu and S. Licardie, “Alternative models for fish-eye lenses,” Pattern Recogn. Lett. 16, 433–441 (1995).
[CrossRef]

Liu, Y.

J. Wang, F. Shi, J. Zhang, and Y. Liu, “A new calibration model of camera lens distortion,” Pattern Recogn. 41, 607–615(2008).
[CrossRef]

Ma, L.

L. Ma, Y. Q. Chen, and K. L. Moore, “Flexible camera calibration using a new analytical radial undistortion formula with application to mobile robot localization,” in Proceedings of IEEE International Symposium on Intelligent Control (IEEE, 2003).

Maas, H.-G.

D. Schneider, E. Schwalbe, and H.-G. Maas, “Validation of geometric models for fisheye lenses,” ISPRS J. Photogram. Remote Sens. 64, 259–266 (2009).
[CrossRef]

Mallon, J.

J. Mallon and P. F. Whelan, “Precise radial un-distortion of images,” in Proceedings of the 17th International Conference on Pattern Recognition (IEEE Computer Society, 2004), pp. 18–21.
[CrossRef]

McFeely, R.

C. Hughes, R. McFeely, P. Denny, M. Glavin, and E. Jones, “Equidistant (fθ) fish-eye perspective with application in distortion centre estimation,” Image Vis. Comput. 28, 538–551 (2010).
[CrossRef]

McGlone, C.

C. McGlone, E. Mikhail, and J. Bethel, Manual of Photogrammetry, 5th ed. (American Society of Photogrammetry and Remote Sensing, 2004).

McLean, G.

B. Prescott and G. McLean, “Line-based correction of radial lens distortion,” Graph. Models Image Process. 59, 39–47 (1997).
[CrossRef]

Micusik, B.

B. Micusik and T. Pajdla, “Estimation of omnidirectional camera model from epipolar geometry,” in Proceedings of the International Conference on Computer Vision and Pattern Recognition (IEEE Computer Society, 2003), Vol. 1, pp. 485–490.

Mikhail, E.

C. McGlone, E. Mikhail, and J. Bethel, Manual of Photogrammetry, 5th ed. (American Society of Photogrammetry and Remote Sensing, 2004).

Moore, K. L.

L. Ma, Y. Q. Chen, and K. L. Moore, “Flexible camera calibration using a new analytical radial undistortion formula with application to mobile robot localization,” in Proceedings of IEEE International Symposium on Intelligent Control (IEEE, 2003).

Nayar, S.

R. Swaminathan and S. Nayar, “Non-metric calibration of wide-angle lenses and polycameras,” IEEE Trans. Pattern Anal. Machine Intell. 22, 1172–1178 (2000).
[CrossRef]

Nayar, S. K.

M. D. Grossberg and S. K. Nayar, “A general imaging model and a method for finding its parameters,” in Proceedings of the International Conference on Computer Vision (IEEE Computer Society, 2001), pp. 108–115.

Pajdla, T.

B. Micusik and T. Pajdla, “Estimation of omnidirectional camera model from epipolar geometry,” in Proceedings of the International Conference on Computer Vision and Pattern Recognition (IEEE Computer Society, 2003), Vol. 1, pp. 485–490.

T. Svoboda, T. Pajdla, and H. Hlavac, “Epipolar geometry for panoramic cameras,” in Proceedings of the 2nd European Conference on Computer Vision (Springer, 1998), pp. 218–232.

Park, S.

Penna, M.

M. Penna, “Camera calibration: a quick and easy way to detection of scale factor,” IEEE Trans. Pattern Anal.Machine Intell. 13, 1240–1245 (1991).
[CrossRef]

Popescu, A. C.

Prescott, B.

B. Prescott and G. McLean, “Line-based correction of radial lens distortion,” Graph. Models Image Process. 59, 39–47 (1997).
[CrossRef]

Ramalingam, S.

P. Sturm and S. Ramalingam, “A generic concept for camera calibration,” in Proceedings of the 5th European Conference on Computer Vision (Springer, 2004).

Ricolfe-Viala, C.

C. Ricolfe-Viala and A. J. Sanchez-Salmeron, “Robust metric calibration of non-linear camera lens distortion,” Pattern Recogn. 43, 1688–1699 (2010).
[CrossRef]

Sanchez-Salmeron, A. J.

C. Ricolfe-Viala and A. J. Sanchez-Salmeron, “Robust metric calibration of non-linear camera lens distortion,” Pattern Recogn. 43, 1688–1699 (2010).
[CrossRef]

Saxena, T.

R. I. Hartley and T. Saxena, “The cubic rational polynomial camera model,” in Proceedings DARPA Image Understanding Workshop (IEEE Computer Society, 1997), pp. 649–653.

Schneider, D.

D. Schneider, E. Schwalbe, and H.-G. Maas, “Validation of geometric models for fisheye lenses,” ISPRS J. Photogram. Remote Sens. 64, 259–266 (2009).
[CrossRef]

Schwalbe, E.

D. Schneider, E. Schwalbe, and H.-G. Maas, “Validation of geometric models for fisheye lenses,” ISPRS J. Photogram. Remote Sens. 64, 259–266 (2009).
[CrossRef]

Shah, S.

S. Shah and J. K. Aggarwal, “Intrinsic parameter calibration procedure for a (high distortion) fish-eye lens camera with distortion model and accuracy estimation,” Pattern Recogn. 29, 1775–1778 (1996).
[CrossRef]

Shi, F.

J. Wang, F. Shi, J. Zhang, and Y. Liu, “A new calibration model of camera lens distortion,” Pattern Recogn. 41, 607–615(2008).
[CrossRef]

Stein, G.

G. Stein, “Lens distortion calibration using point correspondences,” in Proceedings of the International Conference on Computer Vision and Pattern Recognition (IEEE Computer Society, 1997), pp. 602–608.

Sturm, P.

P. Sturm, “Mixing catadioptric and perspective cameras,” in Proceedings of the Workshop on Omnidirectional Vision (IEEE Computer Society, 2002), pp. 37–44.
[CrossRef]

P. Sturm and S. Ramalingam, “A generic concept for camera calibration,” in Proceedings of the 5th European Conference on Computer Vision (Springer, 2004).

Svoboda, T.

T. Svoboda, T. Pajdla, and H. Hlavac, “Epipolar geometry for panoramic cameras,” in Proceedings of the 2nd European Conference on Computer Vision (Springer, 1998), pp. 218–232.

Swaminathan, R.

R. Swaminathan and S. Nayar, “Non-metric calibration of wide-angle lenses and polycameras,” IEEE Trans. Pattern Anal. Machine Intell. 22, 1172–1178 (2000).
[CrossRef]

Tsai, R.

R. Tsai, “A versatile camera calibration technique for high-accuracy 3D machine vision metrology using off-the-self TV camera lenses,” IEEE J. Robotics Autom. RA-3, 323–344 (1987).
[CrossRef]

Viala, M.

J. Lavest, M. Viala, and M. Dhome, “Do we really need accurate calibration pattern to achieve a reliable camera calibration,” in Proceedings of the 2nd European Conference on Computer Vision (Springer, 1998).

Wang, J.

J. Wang, F. Shi, J. Zhang, and Y. Liu, “A new calibration model of camera lens distortion,” Pattern Recogn. 41, 607–615(2008).
[CrossRef]

Wang, W.

Wang, Z.

Weng, J.

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Figures (7)

Fig. 1
Fig. 1

Images captured with 2.7 mm and 8 mm lenses. (a) Image with 8 mm lens generates low distortion. (b) Image with 2.7 mm lens generates high distortion (c) Detected distorted points in the image and undistorted corrected points for low distortion. (d) Same for high distortion.

Fig. 2
Fig. 2

Calibration error of each model depending on the number of points. Calibration error is measured by evaluating the error function in Eq. (18) with undistorted points. The error function in Eq. (18) is zero for the corrected points that are used in the calibration process.

Fig. 3
Fig. 3

Time consumption depends on the number of points. Ninety-nine percent of the time is used for searching the true positions of distorted points in the corrected image. Calibration has been computed in an AMD Atlon Dual Core 5600 + 2.81   G H z with 2 G byte s of RAM implemented in MATLAB.

Fig. 4
Fig. 4

Low-distortion correction with different models. Images captured with 8 mm low-distortion lens. (a) Rational tangential model. (b) Logarithmic model. (c) Field-of-view model. (d) Division model. (e) Polynomial model. (f) Rational function model.

Fig. 5
Fig. 5

High-distortion correction with different models. Images captured with 2.7 mm high-distortion lens. (a) Rational tangential model. (b) Logarithmic model. (c) Field-of-view model. (d) Division model. (e) Polynomial model. (f) Rational function model.

Fig. 6
Fig. 6

Low-distortion correction error in different image areas with different models. Images captured with 8 mm low-distortion lens. (a) Rational tangential model. (b) Logarithmic model. (c) Field-of-view model. (d) Division model. (e) Polynomial model. (f) Rational function model.

Fig. 7
Fig. 7

High-distortion correction error in different image areas with different models. Images captured with 2.7 mm high-distortion lens. (a) Rational tangential model. (b) Logarithmic model. (c) Field-of-view model. (d) Division model. (e) Polynomial model. (f) Rational function model.

Tables (2)

Tables Icon

Table 1 Calibration Results a

Tables Icon

Table 2 Distortion Center Computed with the Model Parameters and Compared with the Principal Point Computed Alone a

Equations (32)

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

u p = u d δ u , v p = v d δ v ,
δ u = Δ u d · ( k 1 · r d 2 + k 2 · r d 4 + ) + p 1 ( 3 Δ u d 2 + Δ v d 2 ) + 2 p 2 · Δ u d · Δ v d + s 1 · r d 2 , δ v = Δ v d · ( k 1 · r d 2 + k 2 · r d 4 + ) + 2 p 1 · Δ u d · Δ v d + p 2 ( Δ u d 2 + 3 Δ v d 2 ) + s 2 · r d 2 .
r d = s · log ( 1 + λ · r p ) , α * = α .
u d = r d · cos α * , v d = r d · sin α * ,
r d = Δ u d 2 + Δ v d 2 ; α * = arctan ( Δ v d 2 / Δ u d 2 ) , r p = ( e r d / s 1 ) / λ ; α = α * , u p = r p · cos α ; v p = r p · sin α .
G ( r p ) = a 0 + a 1 · r p + a 1 · r p 2 + a 1 · r p n = i = 0 k a i · r p i ,
r d = a 0 + a 1 · r p + a 2 · r p 2 + a n · r p n = i = 0 k a a i · r p i , α d = b 0 + b 1 · α p + b 2 · α p 2 + b n · α p n = j = 0 k b b j · α p j ,
r d = 1 ω arctan ( 2 · r p · tan ω 2 ) , r p = tan ( r d · ω ) 2 · tan ω 2 .
r d = r p 1 + β 1 · r p 2 + β 2 · r p 4 + .
d ( u d , v d ) = [ a 11 · u d 2 + a 12 · u d · v d + a 13 · v d 2 + a 14 · u d + a 15 · v d + a 16 a 21 · u d 2 + a 22 · u d · v d + a 23 · v d 2 + a 24 · u d + a 25 · v d + a 26 a 31 · u d 2 + a 32 · u d · v d + a 33 · v d 2 + a 34 · u d + a 35 · v d + a 36 ] .
x ( u d , v d ) = [ u d 2 u d · v d v d 2 u d v d 1 ] T .
d ( u d , v d ) = A · x ( u d , v d ) ,
q p = ( u p , v p ) = ( a 1 T · x ( u d , v d ) a 3 T · x ( u d , v d ) , a 2 T · x ( u d , v d ) a 3 T · x ( u d , v d ) ) ,
CR ( q 1 d , q 2 d , q 3 d , q 4 d ) = s 13 · s 24 s 14 · s 23 = CR ( p 1 , p 2 , p 3 , p 4 ) ,
J CR = l = 1 n k = 1 m 3 CR ( q k , q k + 1 , l , q k + 2 , l , q k + 3 , l ) CR ( p 1 , p 2 , p 3 , p 4 ) .
a l · u i + b l · v i + c l = 0 ,
J ST = l = 1 n i = 1 m a l · u i + b l · v i + c l .
J CP = l = 1 n ( i = 1 m a l · u i + b l · v i + c l + k = 1 m 3 CR ( q k , q k + 1 , l , q k + 2 , l , q k + 3 , l ) CR ( p 1 , p 2 , p 3 , p 4 ) ) .
[ Δ u i , d · r i , d 2 Δ u i , d · r i , d 4 3 Δ u i , d 2 + Δ v i , d 2 2 · Δ u i , d · Δ v i , d r i , d 2 0 Δ v i , d · r i , d 2 Δ v i , d · r i , d 4 2 · Δ u i , d · Δ v i , d Δ u i , d 2 + 3 Δ v i , d 2 0 r i , d 2 ] · [ k 1 k 2 p 1 p 2 s 1 s 2 ] = [ δ u , i δ v , i ] .
J NLPD = i = 1 n · m ( δ u , i Δ u i , d · ( k 1 · r i , d 2 + k 2 · r i , d 4 ) + p 1 ( 3 Δ u i , d 2 + Δ v i , d 2 ) + 2 p 2 · Δ u i , d · Δ v i , d + s 1 · r i , d 2 + δ v , i Δ v i , d · ( k 1 · r i , d 2 + k 2 · r i , d 4 ) + 2 p 1 · Δ u i , d · Δ v i , d + p 2 ( Δ u i , d 2 + 3 Δ v i , d 2 ) + s 2 · r i , d 2 ) .
J FET = i = 1 n · m ( r i , d s · log ( 1 + λ · r i , p ) ) 2 .
J PFET = i = 1 n · m ( r i , d t = 0 k a ( a t · r i , p T ) ) 2 + i = 1 n · m ( α i , d t = 0 k b ( b t · α i , p T ) ) 2 ,
[ 1 r 1 , p r 1 , p 2 r 1 , p 3 r 1 , p 4 r 1 , p 5 1 r n · m , p r n · m , p 2 r n · m , p 3 r n · m , p 4 r n · m , p 5 ] · [ a 0 a 1 a 2 a 3 a 4 a 5 ] = [ r 1 , d r n · m , p ] .
J FOV = i = 1 n · m ( r d i 1 ω arctan ( 2 · r p i · tan ω 2 ) ) 2 .
r d = r p 1 + β · r p 2 .
[ r 1 , p 2 · r 1 , d r n · m , p 2 · r n · m , d ] · β = [ r 1 , d r 1 , p r n · m , d r n · m , p ] .
J DM = i = 1 n · m ( r i , d r i , p 1 + β · r i , p 2 ) 2 .
d ( u i , d , v i , d ) = A · x ( u i , d , v i , d ) .
q i , 0 = ( u i , 0 , v i , 0 ) = ( a 1 T · x ( u i , d , v i , d ) a 3 T · x ( u i , d , v i , d ) , a 2 T · x ( u i , d , v i , d ) a 3 T · x ( u i , d , v i , d ) ) .
a 3 T · x ( u i , d , v i , d ) · u i , 0 = a 1 T · x ( u i , d , v i , d ) , a 3 T · x ( u i , d , v i , d ) · v i , 0 = a 2 T · x ( u i , d , v i , d ) .
[ x ( u i , d , v i , d ) T 0 u i , 0 · x ( u i , d , v i , d ) T 0 x ( u i , d , v i , d ) T v i , 0 · x ( u i , d , v i , d ) T ] · [ a 1 a 2 a 3 ] = 0.
J RT = i = 1 n · m ( u i , 0 a 1 T · x ( u d , v d ) a 3 T · x ( u d , v d ) + v i , 0 a 2 T · x ( u i , d , v i , d ) a 3 T · x ( u i , d , v i , d ) ) .

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