Abstract

In this paper, a strategy for the calibration and the measurement process of a multi-sensor fringe projection unit is presented. The objective is the development of an easy to use calibration and measurement procedure. Only one simple geometrical calibration target is needed and the calibration of the projection unit is not mandatory. To make the system ready for measurement tasks, a common world coordinate system is established. The geometrical camera calibration is derived with respect to the world frame. Note, that the cameras of the system are under Scheimpflug condition which is considered using a modified camera model. Furthermore an additional optimization step of the extrinsic camera parameters is presented to compensate the uncertainties of the calibration target. For completeness, a suitable calibration strategy for the projection unit is given, too. Additionally, the quality of the presented strategy is demonstrated by experimental data.

© 2015 Optical Society of America

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References

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  1. M. Kästner, Optische Geometrieprüfung Präzisionsgeschmiedeter Hochleistungsbauteile (Shaker Verlag, 2008).
  2. J. Burke, T. Bothe, W. Osten, and C. F. Hess, “Reverse engineering by fringe projection,” in “International Symposium on Optical Science and Technology,” (2002), pp. 312–324.
  3. H.-J. Przybilla, “Streifenprojektion–grundlagen, systeme und anwendungen,” Contributions from: 74th Society for Geodesy, Geoinformation, and Land Management (DVW) Seminar Terrestrical Laser Scanning–Terrestrisches-Laser-Scanning (TLS2007).
  4. T. Peng, Algorithms and Models for 3-D Shape Measurement Using Digital Fringe Projections (2007).
  5. F. Chen, G. M. Brown, and M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. 39, 10–22 (2000).
    [Crossref]
  6. S. S. Gorthi and P. Rastogi, “Fringe projection techniques: whither we are?”; Opt. Lasers Eng. 48, 133–140 (2010).
    [Crossref]
  7. Z. Zhang, “Review of single-shot 3d shape measurement by phase calculation-based fringe projection techniques,” Opt. Lasers Eng. 50, 1097–1106 (2012).
    [Crossref]
  8. Z. Zhang, S. Huang, S. Meng, F. Gao, and X. Jiang, “A simple, flexible and automatic 3d calibration method for a phase calculation-based fringe projection imaging system,” Opt. Express 21, 12218–12227 (2013).
    [Crossref] [PubMed]
  9. Y. Cai and X. Su, “Inverse projected-fringe technique based on multi projectors,” Opt. Lasers Eng. 45, 1028–1034 (2007).
    [Crossref]
  10. C. Munkelt, I. Schmidt, C. Bräuer-Burchardt, P. Kühmstedt, and G. Notni, “Cordless portable multi-view fringe projection system for 3d reconstruction,” in “Proceedings of IEEE Computer Vision and Pattern Recognition,” (IEEE, 2007), pp. 1–2.
  11. F. Chen, X. Chen, X. Xie, X. Feng, and L. Yang, “Full-field 3d measurement using multi-camera digital image correlation system,” Opt. Lasers Eng. 51, 1044–1052 (2013).
    [Crossref]
  12. A. K. Prasad and K. Jensen, “Scheimpflug stereocamera for particle image velocimetry in liquid flows,” Appl. Optics 34, 7092–7099 (1995).
    [Crossref]
  13. H. M. Merklinger, “Focusing the view camera,” Seaboard Printing Limited5 (1996).
  14. H. M. Merklinger, “Scheimpflug’s patent,” Photo Techniques 17, 56 (1996).
  15. H. Louhichi, T. Fournel, J. M. Lavest, and H. B. Aissia, “Camera self-calibration in scheimpflug condition for air flow investigation,” in “Advances in Visual Computing,” (Springer, Berlin Heidelberg, 2006), pp. 891–900.
  16. H. Louhichi, T. Fournel, J. M. Lavest, and H. B. Aissia, “Self-calibration of scheimpflug cameras: an easy protocol,” Meas. Sci. Technol. 18, 2616 (2007).
    [Crossref]
  17. A. Legarda, A. Izaguirre, N. Arana, and A. Iturrospe, “A new method for scheimpflug camera calibration,” in “Proceedings of 10th International Workshop on Electronics, Control, Measurement and Signals,” (2011), pp. 1–5.
  18. P. Fasogbon, L. Duvieubourg, P.-A. Lacaze, and L. Macaire, “Intrinsic camera calibration equipped with scheimpflug optical device,” in “Proceedings of the International Conference on Quality Control by Artificial Vision,” (2015), p. 953416.
  19. Z. Zhang, “A flexible new technique for camera calibration,” Pattern Analysis and Machine Intelligence, IEEE Transactions on  22, 1330–1334 (2000).
    [Crossref]
  20. R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision (Cambridge University, 2003).
  21. C. Reich, R. Ritter, and J. Thesing, “3-d shape measurement of complex objects by combining photogrammetry and fringe projection,” Opt. Eng. 39, 224–231 (2000).
    [Crossref]

2013 (2)

Z. Zhang, S. Huang, S. Meng, F. Gao, and X. Jiang, “A simple, flexible and automatic 3d calibration method for a phase calculation-based fringe projection imaging system,” Opt. Express 21, 12218–12227 (2013).
[Crossref] [PubMed]

F. Chen, X. Chen, X. Xie, X. Feng, and L. Yang, “Full-field 3d measurement using multi-camera digital image correlation system,” Opt. Lasers Eng. 51, 1044–1052 (2013).
[Crossref]

2012 (1)

Z. Zhang, “Review of single-shot 3d shape measurement by phase calculation-based fringe projection techniques,” Opt. Lasers Eng. 50, 1097–1106 (2012).
[Crossref]

2010 (1)

S. S. Gorthi and P. Rastogi, “Fringe projection techniques: whither we are?”; Opt. Lasers Eng. 48, 133–140 (2010).
[Crossref]

2007 (2)

Y. Cai and X. Su, “Inverse projected-fringe technique based on multi projectors,” Opt. Lasers Eng. 45, 1028–1034 (2007).
[Crossref]

H. Louhichi, T. Fournel, J. M. Lavest, and H. B. Aissia, “Self-calibration of scheimpflug cameras: an easy protocol,” Meas. Sci. Technol. 18, 2616 (2007).
[Crossref]

2000 (3)

Z. Zhang, “A flexible new technique for camera calibration,” Pattern Analysis and Machine Intelligence, IEEE Transactions on  22, 1330–1334 (2000).
[Crossref]

C. Reich, R. Ritter, and J. Thesing, “3-d shape measurement of complex objects by combining photogrammetry and fringe projection,” Opt. Eng. 39, 224–231 (2000).
[Crossref]

F. Chen, G. M. Brown, and M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. 39, 10–22 (2000).
[Crossref]

1996 (1)

H. M. Merklinger, “Scheimpflug’s patent,” Photo Techniques 17, 56 (1996).

1995 (1)

A. K. Prasad and K. Jensen, “Scheimpflug stereocamera for particle image velocimetry in liquid flows,” Appl. Optics 34, 7092–7099 (1995).
[Crossref]

Aissia, H. B.

H. Louhichi, T. Fournel, J. M. Lavest, and H. B. Aissia, “Self-calibration of scheimpflug cameras: an easy protocol,” Meas. Sci. Technol. 18, 2616 (2007).
[Crossref]

H. Louhichi, T. Fournel, J. M. Lavest, and H. B. Aissia, “Camera self-calibration in scheimpflug condition for air flow investigation,” in “Advances in Visual Computing,” (Springer, Berlin Heidelberg, 2006), pp. 891–900.

Arana, N.

A. Legarda, A. Izaguirre, N. Arana, and A. Iturrospe, “A new method for scheimpflug camera calibration,” in “Proceedings of 10th International Workshop on Electronics, Control, Measurement and Signals,” (2011), pp. 1–5.

Bothe, T.

J. Burke, T. Bothe, W. Osten, and C. F. Hess, “Reverse engineering by fringe projection,” in “International Symposium on Optical Science and Technology,” (2002), pp. 312–324.

Bräuer-Burchardt, C.

C. Munkelt, I. Schmidt, C. Bräuer-Burchardt, P. Kühmstedt, and G. Notni, “Cordless portable multi-view fringe projection system for 3d reconstruction,” in “Proceedings of IEEE Computer Vision and Pattern Recognition,” (IEEE, 2007), pp. 1–2.

Brown, G. M.

F. Chen, G. M. Brown, and M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. 39, 10–22 (2000).
[Crossref]

Burke, J.

J. Burke, T. Bothe, W. Osten, and C. F. Hess, “Reverse engineering by fringe projection,” in “International Symposium on Optical Science and Technology,” (2002), pp. 312–324.

Cai, Y.

Y. Cai and X. Su, “Inverse projected-fringe technique based on multi projectors,” Opt. Lasers Eng. 45, 1028–1034 (2007).
[Crossref]

Chen, F.

F. Chen, X. Chen, X. Xie, X. Feng, and L. Yang, “Full-field 3d measurement using multi-camera digital image correlation system,” Opt. Lasers Eng. 51, 1044–1052 (2013).
[Crossref]

F. Chen, G. M. Brown, and M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. 39, 10–22 (2000).
[Crossref]

Chen, X.

F. Chen, X. Chen, X. Xie, X. Feng, and L. Yang, “Full-field 3d measurement using multi-camera digital image correlation system,” Opt. Lasers Eng. 51, 1044–1052 (2013).
[Crossref]

Duvieubourg, L.

P. Fasogbon, L. Duvieubourg, P.-A. Lacaze, and L. Macaire, “Intrinsic camera calibration equipped with scheimpflug optical device,” in “Proceedings of the International Conference on Quality Control by Artificial Vision,” (2015), p. 953416.

Fasogbon, P.

P. Fasogbon, L. Duvieubourg, P.-A. Lacaze, and L. Macaire, “Intrinsic camera calibration equipped with scheimpflug optical device,” in “Proceedings of the International Conference on Quality Control by Artificial Vision,” (2015), p. 953416.

Feng, X.

F. Chen, X. Chen, X. Xie, X. Feng, and L. Yang, “Full-field 3d measurement using multi-camera digital image correlation system,” Opt. Lasers Eng. 51, 1044–1052 (2013).
[Crossref]

Fournel, T.

H. Louhichi, T. Fournel, J. M. Lavest, and H. B. Aissia, “Self-calibration of scheimpflug cameras: an easy protocol,” Meas. Sci. Technol. 18, 2616 (2007).
[Crossref]

H. Louhichi, T. Fournel, J. M. Lavest, and H. B. Aissia, “Camera self-calibration in scheimpflug condition for air flow investigation,” in “Advances in Visual Computing,” (Springer, Berlin Heidelberg, 2006), pp. 891–900.

Gao, F.

Gorthi, S. S.

S. S. Gorthi and P. Rastogi, “Fringe projection techniques: whither we are?”; Opt. Lasers Eng. 48, 133–140 (2010).
[Crossref]

Hartley, R.

R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision (Cambridge University, 2003).

Hess, C. F.

J. Burke, T. Bothe, W. Osten, and C. F. Hess, “Reverse engineering by fringe projection,” in “International Symposium on Optical Science and Technology,” (2002), pp. 312–324.

Huang, S.

Iturrospe, A.

A. Legarda, A. Izaguirre, N. Arana, and A. Iturrospe, “A new method for scheimpflug camera calibration,” in “Proceedings of 10th International Workshop on Electronics, Control, Measurement and Signals,” (2011), pp. 1–5.

Izaguirre, A.

A. Legarda, A. Izaguirre, N. Arana, and A. Iturrospe, “A new method for scheimpflug camera calibration,” in “Proceedings of 10th International Workshop on Electronics, Control, Measurement and Signals,” (2011), pp. 1–5.

Jensen, K.

A. K. Prasad and K. Jensen, “Scheimpflug stereocamera for particle image velocimetry in liquid flows,” Appl. Optics 34, 7092–7099 (1995).
[Crossref]

Jiang, X.

Kästner, M.

M. Kästner, Optische Geometrieprüfung Präzisionsgeschmiedeter Hochleistungsbauteile (Shaker Verlag, 2008).

Kühmstedt, P.

C. Munkelt, I. Schmidt, C. Bräuer-Burchardt, P. Kühmstedt, and G. Notni, “Cordless portable multi-view fringe projection system for 3d reconstruction,” in “Proceedings of IEEE Computer Vision and Pattern Recognition,” (IEEE, 2007), pp. 1–2.

Lacaze, P.-A.

P. Fasogbon, L. Duvieubourg, P.-A. Lacaze, and L. Macaire, “Intrinsic camera calibration equipped with scheimpflug optical device,” in “Proceedings of the International Conference on Quality Control by Artificial Vision,” (2015), p. 953416.

Lavest, J. M.

H. Louhichi, T. Fournel, J. M. Lavest, and H. B. Aissia, “Self-calibration of scheimpflug cameras: an easy protocol,” Meas. Sci. Technol. 18, 2616 (2007).
[Crossref]

H. Louhichi, T. Fournel, J. M. Lavest, and H. B. Aissia, “Camera self-calibration in scheimpflug condition for air flow investigation,” in “Advances in Visual Computing,” (Springer, Berlin Heidelberg, 2006), pp. 891–900.

Legarda, A.

A. Legarda, A. Izaguirre, N. Arana, and A. Iturrospe, “A new method for scheimpflug camera calibration,” in “Proceedings of 10th International Workshop on Electronics, Control, Measurement and Signals,” (2011), pp. 1–5.

Louhichi, H.

H. Louhichi, T. Fournel, J. M. Lavest, and H. B. Aissia, “Self-calibration of scheimpflug cameras: an easy protocol,” Meas. Sci. Technol. 18, 2616 (2007).
[Crossref]

H. Louhichi, T. Fournel, J. M. Lavest, and H. B. Aissia, “Camera self-calibration in scheimpflug condition for air flow investigation,” in “Advances in Visual Computing,” (Springer, Berlin Heidelberg, 2006), pp. 891–900.

Macaire, L.

P. Fasogbon, L. Duvieubourg, P.-A. Lacaze, and L. Macaire, “Intrinsic camera calibration equipped with scheimpflug optical device,” in “Proceedings of the International Conference on Quality Control by Artificial Vision,” (2015), p. 953416.

Meng, S.

Merklinger, H. M.

H. M. Merklinger, “Scheimpflug’s patent,” Photo Techniques 17, 56 (1996).

H. M. Merklinger, “Focusing the view camera,” Seaboard Printing Limited5 (1996).

Munkelt, C.

C. Munkelt, I. Schmidt, C. Bräuer-Burchardt, P. Kühmstedt, and G. Notni, “Cordless portable multi-view fringe projection system for 3d reconstruction,” in “Proceedings of IEEE Computer Vision and Pattern Recognition,” (IEEE, 2007), pp. 1–2.

Notni, G.

C. Munkelt, I. Schmidt, C. Bräuer-Burchardt, P. Kühmstedt, and G. Notni, “Cordless portable multi-view fringe projection system for 3d reconstruction,” in “Proceedings of IEEE Computer Vision and Pattern Recognition,” (IEEE, 2007), pp. 1–2.

Osten, W.

J. Burke, T. Bothe, W. Osten, and C. F. Hess, “Reverse engineering by fringe projection,” in “International Symposium on Optical Science and Technology,” (2002), pp. 312–324.

Peng, T.

T. Peng, Algorithms and Models for 3-D Shape Measurement Using Digital Fringe Projections (2007).

Prasad, A. K.

A. K. Prasad and K. Jensen, “Scheimpflug stereocamera for particle image velocimetry in liquid flows,” Appl. Optics 34, 7092–7099 (1995).
[Crossref]

Przybilla, H.-J.

H.-J. Przybilla, “Streifenprojektion–grundlagen, systeme und anwendungen,” Contributions from: 74th Society for Geodesy, Geoinformation, and Land Management (DVW) Seminar Terrestrical Laser Scanning–Terrestrisches-Laser-Scanning (TLS2007).

Rastogi, P.

S. S. Gorthi and P. Rastogi, “Fringe projection techniques: whither we are?”; Opt. Lasers Eng. 48, 133–140 (2010).
[Crossref]

Reich, C.

C. Reich, R. Ritter, and J. Thesing, “3-d shape measurement of complex objects by combining photogrammetry and fringe projection,” Opt. Eng. 39, 224–231 (2000).
[Crossref]

Ritter, R.

C. Reich, R. Ritter, and J. Thesing, “3-d shape measurement of complex objects by combining photogrammetry and fringe projection,” Opt. Eng. 39, 224–231 (2000).
[Crossref]

Schmidt, I.

C. Munkelt, I. Schmidt, C. Bräuer-Burchardt, P. Kühmstedt, and G. Notni, “Cordless portable multi-view fringe projection system for 3d reconstruction,” in “Proceedings of IEEE Computer Vision and Pattern Recognition,” (IEEE, 2007), pp. 1–2.

Song, M.

F. Chen, G. M. Brown, and M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. 39, 10–22 (2000).
[Crossref]

Su, X.

Y. Cai and X. Su, “Inverse projected-fringe technique based on multi projectors,” Opt. Lasers Eng. 45, 1028–1034 (2007).
[Crossref]

Thesing, J.

C. Reich, R. Ritter, and J. Thesing, “3-d shape measurement of complex objects by combining photogrammetry and fringe projection,” Opt. Eng. 39, 224–231 (2000).
[Crossref]

Xie, X.

F. Chen, X. Chen, X. Xie, X. Feng, and L. Yang, “Full-field 3d measurement using multi-camera digital image correlation system,” Opt. Lasers Eng. 51, 1044–1052 (2013).
[Crossref]

Yang, L.

F. Chen, X. Chen, X. Xie, X. Feng, and L. Yang, “Full-field 3d measurement using multi-camera digital image correlation system,” Opt. Lasers Eng. 51, 1044–1052 (2013).
[Crossref]

Zhang, Z.

Z. Zhang, S. Huang, S. Meng, F. Gao, and X. Jiang, “A simple, flexible and automatic 3d calibration method for a phase calculation-based fringe projection imaging system,” Opt. Express 21, 12218–12227 (2013).
[Crossref] [PubMed]

Z. Zhang, “Review of single-shot 3d shape measurement by phase calculation-based fringe projection techniques,” Opt. Lasers Eng. 50, 1097–1106 (2012).
[Crossref]

Z. Zhang, “A flexible new technique for camera calibration,” Pattern Analysis and Machine Intelligence, IEEE Transactions on  22, 1330–1334 (2000).
[Crossref]

Zisserman, A.

R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision (Cambridge University, 2003).

Appl. Optics (1)

A. K. Prasad and K. Jensen, “Scheimpflug stereocamera for particle image velocimetry in liquid flows,” Appl. Optics 34, 7092–7099 (1995).
[Crossref]

Meas. Sci. Technol. (1)

H. Louhichi, T. Fournel, J. M. Lavest, and H. B. Aissia, “Self-calibration of scheimpflug cameras: an easy protocol,” Meas. Sci. Technol. 18, 2616 (2007).
[Crossref]

Opt. Eng. (2)

F. Chen, G. M. Brown, and M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. 39, 10–22 (2000).
[Crossref]

C. Reich, R. Ritter, and J. Thesing, “3-d shape measurement of complex objects by combining photogrammetry and fringe projection,” Opt. Eng. 39, 224–231 (2000).
[Crossref]

Opt. Express (1)

Opt. Lasers Eng. (4)

Y. Cai and X. Su, “Inverse projected-fringe technique based on multi projectors,” Opt. Lasers Eng. 45, 1028–1034 (2007).
[Crossref]

S. S. Gorthi and P. Rastogi, “Fringe projection techniques: whither we are?”; Opt. Lasers Eng. 48, 133–140 (2010).
[Crossref]

Z. Zhang, “Review of single-shot 3d shape measurement by phase calculation-based fringe projection techniques,” Opt. Lasers Eng. 50, 1097–1106 (2012).
[Crossref]

F. Chen, X. Chen, X. Xie, X. Feng, and L. Yang, “Full-field 3d measurement using multi-camera digital image correlation system,” Opt. Lasers Eng. 51, 1044–1052 (2013).
[Crossref]

Pattern Analysis and Machine Intelligence (1)

Z. Zhang, “A flexible new technique for camera calibration,” Pattern Analysis and Machine Intelligence, IEEE Transactions on  22, 1330–1334 (2000).
[Crossref]

Photo Techniques (1)

H. M. Merklinger, “Scheimpflug’s patent,” Photo Techniques 17, 56 (1996).

Other (10)

H. Louhichi, T. Fournel, J. M. Lavest, and H. B. Aissia, “Camera self-calibration in scheimpflug condition for air flow investigation,” in “Advances in Visual Computing,” (Springer, Berlin Heidelberg, 2006), pp. 891–900.

H. M. Merklinger, “Focusing the view camera,” Seaboard Printing Limited5 (1996).

R. Hartley and A. Zisserman, Multiple View Geometry in Computer Vision (Cambridge University, 2003).

A. Legarda, A. Izaguirre, N. Arana, and A. Iturrospe, “A new method for scheimpflug camera calibration,” in “Proceedings of 10th International Workshop on Electronics, Control, Measurement and Signals,” (2011), pp. 1–5.

P. Fasogbon, L. Duvieubourg, P.-A. Lacaze, and L. Macaire, “Intrinsic camera calibration equipped with scheimpflug optical device,” in “Proceedings of the International Conference on Quality Control by Artificial Vision,” (2015), p. 953416.

C. Munkelt, I. Schmidt, C. Bräuer-Burchardt, P. Kühmstedt, and G. Notni, “Cordless portable multi-view fringe projection system for 3d reconstruction,” in “Proceedings of IEEE Computer Vision and Pattern Recognition,” (IEEE, 2007), pp. 1–2.

M. Kästner, Optische Geometrieprüfung Präzisionsgeschmiedeter Hochleistungsbauteile (Shaker Verlag, 2008).

J. Burke, T. Bothe, W. Osten, and C. F. Hess, “Reverse engineering by fringe projection,” in “International Symposium on Optical Science and Technology,” (2002), pp. 312–324.

H.-J. Przybilla, “Streifenprojektion–grundlagen, systeme und anwendungen,” Contributions from: 74th Society for Geodesy, Geoinformation, and Land Management (DVW) Seminar Terrestrical Laser Scanning–Terrestrisches-Laser-Scanning (TLS2007).

T. Peng, Algorithms and Models for 3-D Shape Measurement Using Digital Fringe Projections (2007).

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Figures (9)

Fig. 1
Fig. 1 The assembly of the fringe projection system.
Fig. 2
Fig. 2 The structure of the Scheimpflug cameras.
Fig. 3
Fig. 3 Schema of the calibration target.
Fig. 4
Fig. 4 Scheimpflug camera model part 1.
Fig. 5
Fig. 5 Scheimpflug camera model part 2.
Fig. 6
Fig. 6 Example of acquired calibration images.
Fig. 7
Fig. 7 The common world coordinate system.
Fig. 8
Fig. 8 Measurement principle.
Fig. 9
Fig. 9 Measurement example: camera image (left) and calculated object points (right).

Tables (5)

Tables Icon

Table 1 Intrinsic parameter values of the cameras (initialized and optimized)

Tables Icon

Table 2 Extrinsic parameters of the cameras in relation to the common world coordinate system

Tables Icon

Table 3 Optimized parameters of the cameras in relation to the common world coordinate system

Tables Icon

Table 4 Intrinsic parameter values of the projection unit (initialized and optimized)

Tables Icon

Table 5 Intrinsic unit

Equations (21)

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

P o b j = [ x 1 , o b j x 2 , o b j x 25 , o b j y 1 , o b j y 2 , o b j y 25 , o b j 0 0 0 ] = [ 0 5 10 0 20 0 0 0 5 20 0 0 0 0 0 ] .
s 1 [ x p e r , C C S y p e r , C C S f ] = [ f 0 0 0 f 0 0 0 f ] [ R O C t C O ] [ x o b j , O C S y o b j , O C S z o b j , O C S 1 ] ,
s 2 [ x ˜ p e r , C C S y ˜ p e r , C C S f ] = [ f 0 0 0 f 0 0 0 f ] [ R S , I C t S , C I ] ( [ x i m a , I C S y i m a , I C S 0 1 ] [ C x C y 0 0 ] ) ,
x ˜ p e r , C C S = x ˜ p e r , C C S ( 1 + k 1 r 2 + k 2 r 4 + k 3 r 6 ) + 2 p 1 x ˜ p e r , C C S y ˜ p e r , C C S + p 2 ( r 2 + 2 x ˜ 2 p e r , C C S ) , y ˜ p e r , C C S = y ˜ p e r , C C S ( 1 + k 1 r 2 + k 2 r 4 + k 3 r 6 ) + p 2 ( r 2 + 2 y ˜ 2 p e r , C C S ) + 2 p 1 y ˜ p e r , C C S x ˜ p e r , C C S ,
[ x o p t i y o p t i ] = [ x p e r , C C S y p e r , C C S ] [ x ˜ p e r , C C S y ˜ p e r , C C S ] .
s [ x m , k , i m a , I C S y m , k , i m a , I C S 1 ] = H k [ x m , o b j , O C S y m , o b j , O C S 1 ] ,
min ρ 1 k = 1 N m = 1 M ( x m , k , o p t i 2 + y m , k , o p t i 2 ) ,
min ρ 2 k = 1 20 m = 1 25 ( x m , k , o p t i 2 + y m , k , o p t i 2 ) ,
min ρ 3 k = 1 20 m = 1 25 ( x m , k , o p t i 2 + y m , k , o p t i 2 ) ,
min ρ 4 m = 1 25 ( x m , o p t i 2 + y m , o p t i 2 ) ,
R C W , w ( α w , β w , γ w ) = R O C , w 1 ( α w , β w , γ w ) , t W C , w = ( R C W , w t C O , w ) ,
r ( l ) = v b p + l v d v , v b p = t C W , w , v d v = R C W , w p ˜ p e r , C C S ,
min ρ 5 i = 1 25 ( p a d j , W C S r i ( l i ) )
min ρ 6 n = 1 N i = 1 I ( p n , a d j , W C S r i , n ( l i , n ) ) ,
s [ x P C S y P C S 1 ] = [ f p , x c p C x , p 0 f p , y C y , p 0 0 1 ] [ R p , W P t p , W P ] [ x a d j , W C S y a d j , W C S z a d j , W C S 1 ] ,
x ˜ P C S = x ˜ P C S ( 1 + k 1 , p r 2 + k 2 , p r 4 + k 3 , p r 6 ) + 2 p 1 , p x ˜ P C S y ˜ P C S + p 2 , p ( r 2 + 2 x ˜ 2 P C S ) ) , y ˜ P C S = y ˜ P C S ( 1 + k 1 , p r 2 + k 2 , p r 4 + k 3 , p r 6 ) + p 2 , p ( r 2 + 2 y ˜ 2 P C S ) + 2 p 1 , p y ˜ P C S x ˜ P C S ,
s [ x ˜ m , P C S y ˜ m , P C S 0 1 ] = H ˜ [ x m , a d j , W C S y m , a d j , W C S z m , a d j , W C S 1 ] ,
[ x o p t i , p y o p t i , p ] = [ x P C S y P C S ] [ x ˜ P C S y ˜ P C S ] ,
min ρ 7 n = 1 M ( x m , k , o p t i 2 + y m , k , o p t i 2 ) ,
R p , P W ( α w , β w , γ w ) = R p , W P 1 ( α p , β p , γ p ) , t p , W P = ( R p , P W t p , P W ) ,
r ( l ) p = v b p , p + l v d v , p , v b p , p = t p , P W , v d v , p = R p , P W p ˜ P C S , n o r m ;

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