Abstract

In this paper, a calibration method for a camera with focus-related intrinsic parameters based on the thin-lens model is proposed to realize highly accurate measurement for small objects with extended depth of field (DOF). It mainly solves inaccurate calibration and small DOF problems at high magnification of vision systems. The mathematical camera model, initial camera calibration based on the iterative radial alignment constraint (IRAC) and optimization strategy are presented. The effectiveness, accuracy and practicality of the proposed calibration method are verified by both simulations and experiments. The root mean square errors of measured points in the 3D world coordinates with the proposed calibration method decrease from 22.02 μm to 1.66 μm when the magnification of the vision system increases from 0.12× to 0.66×. With the proposed calibration method based on the thin-lens model, accurate measurement, extended DOF and low calibration workload can be achieved.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2019 (3)

S. Zhang, B. Li, F. Ren, and R. Dong, “High-precision measurement of binocular telecentric vision system with novel calibration and matching methods,” IEEE Access 7, 54682–54692 (2019).
[Crossref]

Y. Hu, Q. Chen, S. Feng, T. Tao, A. Asundi, and C. Zuo, “A new microscopic telecentric stereo vision system-calibration, rectification, and three-dimensional reconstruction,” Opt. Lasers Eng. 113, 14–22 (2019).
[Crossref]

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[Crossref]

2018 (1)

L. Jiang, J. Zhang, B. Deng, H. Li, and L. Liu, “3d face reconstruction with geometry details from a single image,” IEEE Trans. on Image Process. 27(10), 4756–4770 (2018).
[Crossref]

2017 (2)

Z. Liu, Q. Wu, S. Wu, and X. Pan, “Flexible and accurate camera calibration using grid spherical images,” Opt. Express 25(13), 15269–15285 (2017).
[Crossref]

Z. Wang, J. Mills, W. Xiao, R. Huang, S. Zheng, and Z. Li, “A flexible, generic photogrammetric approach to zoom lens calibration,” Remote Sens. 9(3), 244 (2017).
[Crossref]

2016 (1)

2015 (4)

2014 (2)

A. Gallo, M. Muzzupappa, and F. Bruno, “3d reconstruction of small sized objects from a sequence of multi-focused images,” J. Cult. Herit. 15(2), 173–182 (2014).
[Crossref]

Y. Cui, F. Zhou, Y. Wang, L. Liu, and H. Gao, “Precise calibration of binocular vision system used for vision measurement,” Opt. Express 22(8), 9134–9149 (2014).
[Crossref]

2013 (2)

S. Shen, “Accurate multiple view 3d reconstruction using patch-based stereo for large-scale scenes,” IEEE Trans. on Image Process. 22(5), 1901–1914 (2013).
[Crossref]

B. Wu, H. Hu, Q. Zhu, and Y. Zhang, “A flexible method for zoom lens calibration and modeling using a planar checkerboard,” Photogramm. Eng. Remote Sens. 79(6), 555–571 (2013).
[Crossref]

2012 (3)

L. Alvarez, L. Gómez, and P. Henríquez, “Zoom dependent lens distortion mathematical models,” J. Math. Imaging Vis. 44(3), 480–490 (2012).
[Crossref]

E. Sanz-Ablanedo, J. H. Chandler, and R. Wackrow, “Parameterising internal camera geometry with focusing distance,” The Photogrammetric Record 27(138), 210–226 (2012).
[Crossref]

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[Crossref]

2011 (2)

K. Atsushi, H. Sueyasu, Y. Funayama, and T. Maekawa, “System for reconstruction of three-dimensional micro objects from multiple photographic images,” Comput. Des. 43(8), 1045–1055 (2011).
[Crossref]

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[Crossref]

2010 (1)

2009 (2)

Z. Ren and L. Cai, “Three-dimensional structure measurement of diamond crowns based on stereo vision,” Appl. Opt. 48(31), 5917–5932 (2009).
[Crossref]

M. Sarkis, C. T. Senft, and K. Diepold, “Calibrating an automatic zoom camera with moving least squares,” IEEE Trans. Automat. Sci. Eng. 6(3), 492–503 (2009).
[Crossref]

2008 (1)

X. Chen, J. Tao, and J. Ye, “Research and development of an accurate 3d shape measurement system based on fringe projection: Model analysis and performance evaluation,” Precis. Eng. 32(3), 215–221 (2008).
[Crossref]

2006 (1)

X. Ying and H. Zha, “Geometric interpretations of the relation between the image of the absolute conic and sphere images,” IEEE Trans. Pattern Anal. Machine Intell. 28(12), 2031–2036 (2006).
[Crossref]

2005 (1)

M. I. Lourakis, “A brief description of the levenberg-marquardt algorithm implemented by levmar,” Foundation of Research and Technology 4, 1–6 (2005).

2001 (1)

Y. S. Chen, S. W. Shih, Y. P. Hung, and C. S. Fuh, “Simple and efficient method of calibrating a motorized zoom lens,” Image and vision computing 19(14), 1099–1110 (2001).
[Crossref]

2000 (1)

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

1998 (1)

M. Bertozzi and A. Broggi, “GOLD: A parallel real-time stereo vision system for generic obstacle and lane detection,” IEEE Trans. on Image Process. 7(1), 62–81 (1998).
[Crossref]

1997 (1)

C. S. Fraser, “Digital camera self-calibration,” ISPRS journal of photogrammetry and remote sensing 52(4), 149–159 (1997).
[Crossref]

1994 (1)

G. Q. Wei and S. DeMa, “Implicit and explicit camera calibration: Theory and experiments,” IEEE Transactions on Pattern Analysis and Machine Intelligence 16(5), 469–480 (1994).
[Crossref]

1992 (1)

J. C. Wyant and K. Creath, “Basic wavefront aberration theory for optical metrology,” Applied optics and optical engineering 11, 28–39 (1992).

1988 (1)

R. K. Lenz and R. Y. Tsai, “Techniques for calibration of the scale factor and image center for high accuracy 3-d machine vision metrology,” IEEE Transactions on pattern analysis and machine intelligence 10(5), 713–720 (1988).
[Crossref]

1987 (1)

R. Tsai, “A versatile camera calibration technique for high-accuracy 3D machine vision metrology using off-the-shelf TV cameras and lenses,” IEEE J. Robot. Automat. 3(4), 323–344 (1987).
[Crossref]

1971 (1)

C. B. Duane, “Close-range camera calibration,” Photogramm. Eng 37, 855–866 (1971).

Alvarez, L.

L. Alvarez, L. Gómez, and P. Henríquez, “Zoom dependent lens distortion mathematical models,” J. Math. Imaging Vis. 44(3), 480–490 (2012).
[Crossref]

Asundi, A.

Y. Hu, Q. Chen, S. Feng, T. Tao, A. Asundi, and C. Zuo, “A new microscopic telecentric stereo vision system-calibration, rectification, and three-dimensional reconstruction,” Opt. Lasers Eng. 113, 14–22 (2019).
[Crossref]

Atienza, R.

R. Atienza and A. Zelinsky, “A practical zoom camera calibration technique: an application on active vision for human-robot interaction,” in Australian Conference on Robotics and Automation, (2001), pp. 85–90.

Atsushi, K.

K. Atsushi, H. Sueyasu, Y. Funayama, and T. Maekawa, “System for reconstruction of three-dimensional micro objects from multiple photographic images,” Comput. Des. 43(8), 1045–1055 (2011).
[Crossref]

Benesty, J.

J. Benesty, J. Chen, Y. Huang, and I. Cohen, Pearson correlation coefficient (Springer, 2009).

Bertozzi, M.

M. Bertozzi and A. Broggi, “GOLD: A parallel real-time stereo vision system for generic obstacle and lane detection,” IEEE Trans. on Image Process. 7(1), 62–81 (1998).
[Crossref]

Blaas, J.

X. Zhang, J. Blaas, C. Botha, P. Reischig, A. Bravin, and J. Dik, “Process for the 3d virtual reconstruction of a microcultural heritage artifact obtained by synchrotron radiation ct technology using open source and free software,” J. Cult. Herit. 13(2), 221–225 (2012).
[Crossref]

Botha, C.

X. Zhang, J. Blaas, C. Botha, P. Reischig, A. Bravin, and J. Dik, “Process for the 3d virtual reconstruction of a microcultural heritage artifact obtained by synchrotron radiation ct technology using open source and free software,” J. Cult. Herit. 13(2), 221–225 (2012).
[Crossref]

Bouguet, J.

J. Bouguet, “Camera calibration toolbox for matlab,” http://www.vision.caltech.edu/bouguetj/calib_doc/index.html .

Bravin, A.

X. Zhang, J. Blaas, C. Botha, P. Reischig, A. Bravin, and J. Dik, “Process for the 3d virtual reconstruction of a microcultural heritage artifact obtained by synchrotron radiation ct technology using open source and free software,” J. Cult. Herit. 13(2), 221–225 (2012).
[Crossref]

Broggi, A.

M. Bertozzi and A. Broggi, “GOLD: A parallel real-time stereo vision system for generic obstacle and lane detection,” IEEE Trans. on Image Process. 7(1), 62–81 (1998).
[Crossref]

Bruno, F.

A. Gallo, M. Muzzupappa, and F. Bruno, “3d reconstruction of small sized objects from a sequence of multi-focused images,” J. Cult. Herit. 15(2), 173–182 (2014).
[Crossref]

Cai, L.

Chandler, J. H.

E. Sanz-Ablanedo, J. H. Chandler, and R. Wackrow, “Parameterising internal camera geometry with focusing distance,” The Photogrammetric Record 27(138), 210–226 (2012).
[Crossref]

Chen, J.

J. Benesty, J. Chen, Y. Huang, and I. Cohen, Pearson correlation coefficient (Springer, 2009).

Chen, Q.

Y. Hu, Q. Chen, S. Feng, T. Tao, A. Asundi, and C. Zuo, “A new microscopic telecentric stereo vision system-calibration, rectification, and three-dimensional reconstruction,” Opt. Lasers Eng. 113, 14–22 (2019).
[Crossref]

Chen, X.

Z. Liu, Y. Yin, S. Liu, and X. Chen, “Extrinsic parameter calibration of stereo vision sensors using spot laser projector,” Appl. Opt. 55(25), 7098–7105 (2016).
[Crossref]

X. Chen, J. Tao, and J. Ye, “Research and development of an accurate 3d shape measurement system based on fringe projection: Model analysis and performance evaluation,” Precis. Eng. 32(3), 215–221 (2008).
[Crossref]

Chen, Y. S.

Y. S. Chen, S. W. Shih, Y. P. Hung, and C. S. Fuh, “Simple and efficient method of calibrating a motorized zoom lens,” Image and vision computing 19(14), 1099–1110 (2001).
[Crossref]

Cohen, I.

J. Benesty, J. Chen, Y. Huang, and I. Cohen, Pearson correlation coefficient (Springer, 2009).

Cornille, N.

N. Cornille, D. Garcia, M. A. Sutton, S. McNeill, and J. JOrteu, “Automated 3-d reconstruction using a scanning electron microscope,,” in SEM annual conf. exp. on experimental and applied mechanics, (2003).

Creath, K.

J. C. Wyant and K. Creath, “Basic wavefront aberration theory for optical metrology,” Applied optics and optical engineering 11, 28–39 (1992).

Cui, Y.

DeMa, S.

G. Q. Wei and S. DeMa, “Implicit and explicit camera calibration: Theory and experiments,” IEEE Transactions on Pattern Analysis and Machine Intelligence 16(5), 469–480 (1994).
[Crossref]

Deng, B.

L. Jiang, J. Zhang, B. Deng, H. Li, and L. Liu, “3d face reconstruction with geometry details from a single image,” IEEE Trans. on Image Process. 27(10), 4756–4770 (2018).
[Crossref]

Diepold, K.

M. Sarkis, C. T. Senft, and K. Diepold, “Calibrating an automatic zoom camera with moving least squares,” IEEE Trans. Automat. Sci. Eng. 6(3), 492–503 (2009).
[Crossref]

Dik, J.

X. Zhang, J. Blaas, C. Botha, P. Reischig, A. Bravin, and J. Dik, “Process for the 3d virtual reconstruction of a microcultural heritage artifact obtained by synchrotron radiation ct technology using open source and free software,” J. Cult. Herit. 13(2), 221–225 (2012).
[Crossref]

Dong, R.

S. Zhang, B. Li, F. Ren, and R. Dong, “High-precision measurement of binocular telecentric vision system with novel calibration and matching methods,” IEEE Access 7, 54682–54692 (2019).
[Crossref]

Duane, C. B.

C. B. Duane, “Close-range camera calibration,” Photogramm. Eng 37, 855–866 (1971).

Esparza, D. M. C.

J. G. R. Espino, J. Gonzalez-BarbosaJ, R. A. G. Loenzo, D. M. C. Esparza, and R. Gonzalez-Barbosa, “Vision system for 3d reconstruction with telecentric lens,” in Mexican Conference on Pattern Recognition, (2012), pp. 127–136.

Espino, J. G. R.

J. G. R. Espino, J. Gonzalez-BarbosaJ, R. A. G. Loenzo, D. M. C. Esparza, and R. Gonzalez-Barbosa, “Vision system for 3d reconstruction with telecentric lens,” in Mexican Conference on Pattern Recognition, (2012), pp. 127–136.

Fan, C.

Feng, S.

Y. Hu, Q. Chen, S. Feng, T. Tao, A. Asundi, and C. Zuo, “A new microscopic telecentric stereo vision system-calibration, rectification, and three-dimensional reconstruction,” Opt. Lasers Eng. 113, 14–22 (2019).
[Crossref]

Fitzgibbon, A. W.

B. Triggs, P. F. McLauchlan, R. I. Hartley, and A. W. Fitzgibbon, “Bundle adjustment–a modern synthesis,” in International workshop on vision algorithms, 298–372, (1999

Fraser, C. S.

C. S. Fraser, “Digital camera self-calibration,” ISPRS journal of photogrammetry and remote sensing 52(4), 149–159 (1997).
[Crossref]

Fuh, C. S.

Y. S. Chen, S. W. Shih, Y. P. Hung, and C. S. Fuh, “Simple and efficient method of calibrating a motorized zoom lens,” Image and vision computing 19(14), 1099–1110 (2001).
[Crossref]

Funayama, Y.

K. Atsushi, H. Sueyasu, Y. Funayama, and T. Maekawa, “System for reconstruction of three-dimensional micro objects from multiple photographic images,” Comput. Des. 43(8), 1045–1055 (2011).
[Crossref]

Gallo, A.

A. Gallo, M. Muzzupappa, and F. Bruno, “3d reconstruction of small sized objects from a sequence of multi-focused images,” J. Cult. Herit. 15(2), 173–182 (2014).
[Crossref]

Gao, H.

Garcia, D.

N. Cornille, D. Garcia, M. A. Sutton, S. McNeill, and J. JOrteu, “Automated 3-d reconstruction using a scanning electron microscope,,” in SEM annual conf. exp. on experimental and applied mechanics, (2003).

Gómez, L.

L. Alvarez, L. Gómez, and P. Henríquez, “Zoom dependent lens distortion mathematical models,” J. Math. Imaging Vis. 44(3), 480–490 (2012).
[Crossref]

Gonzalez-Barbosa, R.

J. G. R. Espino, J. Gonzalez-BarbosaJ, R. A. G. Loenzo, D. M. C. Esparza, and R. Gonzalez-Barbosa, “Vision system for 3d reconstruction with telecentric lens,” in Mexican Conference on Pattern Recognition, (2012), pp. 127–136.

Gonzalez-BarbosaJ, J.

J. G. R. Espino, J. Gonzalez-BarbosaJ, R. A. G. Loenzo, D. M. C. Esparza, and R. Gonzalez-Barbosa, “Vision system for 3d reconstruction with telecentric lens,” in Mexican Conference on Pattern Recognition, (2012), pp. 127–136.

Goodman, D. S.

K. Tarabanis, R. Y. Tsai, and D. S. Goodman, “Modeling of a computer-controlled zoom lens,” in IEEE International Conference on Robotics and Automation, (1992), pp. 1545–1551.

Hartley, R. I.

B. Triggs, P. F. McLauchlan, R. I. Hartley, and A. W. Fitzgibbon, “Bundle adjustment–a modern synthesis,” in International workshop on vision algorithms, 298–372, (1999

Henríquez, P.

L. Alvarez, L. Gómez, and P. Henríquez, “Zoom dependent lens distortion mathematical models,” J. Math. Imaging Vis. 44(3), 480–490 (2012).
[Crossref]

Hong, Y.

Hu, H.

B. Wu, H. Hu, Q. Zhu, and Y. Zhang, “A flexible method for zoom lens calibration and modeling using a planar checkerboard,” Photogramm. Eng. Remote Sens. 79(6), 555–571 (2013).
[Crossref]

Hu, Y.

Y. Hu, Q. Chen, S. Feng, T. Tao, A. Asundi, and C. Zuo, “A new microscopic telecentric stereo vision system-calibration, rectification, and three-dimensional reconstruction,” Opt. Lasers Eng. 113, 14–22 (2019).
[Crossref]

Huang, R.

Z. Wang, J. Mills, W. Xiao, R. Huang, S. Zheng, and Z. Li, “A flexible, generic photogrammetric approach to zoom lens calibration,” Remote Sens. 9(3), 244 (2017).
[Crossref]

S. Zheng, Z. Wang, and R. Huang, “Zoom lens calibration with zoom-and focus-related intrinsic parameters applied to bundle adjustment,” ISPRS journal of photogrammetry and remote sensing 102, 62–72 (2015).
[Crossref]

Huang, Y.

J. Benesty, J. Chen, Y. Huang, and I. Cohen, Pearson correlation coefficient (Springer, 2009).

Hung, Y. P.

Y. S. Chen, S. W. Shih, Y. P. Hung, and C. S. Fuh, “Simple and efficient method of calibrating a motorized zoom lens,” Image and vision computing 19(14), 1099–1110 (2001).
[Crossref]

Jia, Z.

Jiang, L.

L. Jiang, J. Zhang, B. Deng, H. Li, and L. Liu, “3d face reconstruction with geometry details from a single image,” IEEE Trans. on Image Process. 27(10), 4756–4770 (2018).
[Crossref]

JOrteu, J.

N. Cornille, D. Garcia, M. A. Sutton, S. McNeill, and J. JOrteu, “Automated 3-d reconstruction using a scanning electron microscope,,” in SEM annual conf. exp. on experimental and applied mechanics, (2003).

Kaneko, T.

A. Yamashita, A. Kawarago, T. Kaneko, and K. T. Miura, “Shape reconstruction and image restoration for non-flat surfaces of documents with a stereo vision system,” in International Conference on Pattern Recognition, (2004), pp. 482–485.

Karam, L. J.

N. D. Narvekar and L. J. Karam, “A no-reference image blur metric based on the cumulative probability of blur detection (CPBD),” IEEE Trans. on Image Process. 20(9), 2678–2683 (2011).
[Crossref]

Kawarago, A.

A. Yamashita, A. Kawarago, T. Kaneko, and K. T. Miura, “Shape reconstruction and image restoration for non-flat surfaces of documents with a stereo vision system,” in International Conference on Pattern Recognition, (2004), pp. 482–485.

Kingslake, R.

R. Kingslake, Optical System Design (Academic, 1983).

Lenz, R. K.

R. K. Lenz and R. Y. Tsai, “Techniques for calibration of the scale factor and image center for high accuracy 3-d machine vision metrology,” IEEE Transactions on pattern analysis and machine intelligence 10(5), 713–720 (1988).
[Crossref]

Li, B.

S. Zhang, B. Li, F. Ren, and R. Dong, “High-precision measurement of binocular telecentric vision system with novel calibration and matching methods,” IEEE Access 7, 54682–54692 (2019).
[Crossref]

B. Li and S. Zhang, “Flexible calibration method for microscopic structured light system using telecentric lens,” Opt. Express 23(20), 25795–25803 (2015).
[Crossref]

Li, H.

L. Jiang, J. Zhang, B. Deng, H. Li, and L. Liu, “3d face reconstruction with geometry details from a single image,” IEEE Trans. on Image Process. 27(10), 4756–4770 (2018).
[Crossref]

Li, Z.

Z. Wang, J. Mills, W. Xiao, R. Huang, S. Zheng, and Z. Li, “A flexible, generic photogrammetric approach to zoom lens calibration,” Remote Sens. 9(3), 244 (2017).
[Crossref]

Liao, J.

Liu, E.

Liu, L.

L. Jiang, J. Zhang, B. Deng, H. Li, and L. Liu, “3d face reconstruction with geometry details from a single image,” IEEE Trans. on Image Process. 27(10), 4756–4770 (2018).
[Crossref]

Y. Cui, F. Zhou, Y. Wang, L. Liu, and H. Gao, “Precise calibration of binocular vision system used for vision measurement,” Opt. Express 22(8), 9134–9149 (2014).
[Crossref]

Liu, S.

Liu, W.

Liu, Y.

Liu, Z.

Loenzo, R. A. G.

J. G. R. Espino, J. Gonzalez-BarbosaJ, R. A. G. Loenzo, D. M. C. Esparza, and R. Gonzalez-Barbosa, “Vision system for 3d reconstruction with telecentric lens,” in Mexican Conference on Pattern Recognition, (2012), pp. 127–136.

London, B.

B. London, J. Stone, and J. Upton, Photography (8th ed.) (Prentice Hall, 2005).

Lourakis, M. I.

M. I. Lourakis, “A brief description of the levenberg-marquardt algorithm implemented by levmar,” Foundation of Research and Technology 4, 1–6 (2005).

Lu, Z.

Maekawa, T.

K. Atsushi, H. Sueyasu, Y. Funayama, and T. Maekawa, “System for reconstruction of three-dimensional micro objects from multiple photographic images,” Comput. Des. 43(8), 1045–1055 (2011).
[Crossref]

McLauchlan, P. F.

B. Triggs, P. F. McLauchlan, R. I. Hartley, and A. W. Fitzgibbon, “Bundle adjustment–a modern synthesis,” in International workshop on vision algorithms, 298–372, (1999

McNeill, S.

N. Cornille, D. Garcia, M. A. Sutton, S. McNeill, and J. JOrteu, “Automated 3-d reconstruction using a scanning electron microscope,,” in SEM annual conf. exp. on experimental and applied mechanics, (2003).

Mills, J.

Z. Wang, J. Mills, W. Xiao, R. Huang, S. Zheng, and Z. Li, “A flexible, generic photogrammetric approach to zoom lens calibration,” Remote Sens. 9(3), 244 (2017).
[Crossref]

Miura, K. T.

A. Yamashita, A. Kawarago, T. Kaneko, and K. T. Miura, “Shape reconstruction and image restoration for non-flat surfaces of documents with a stereo vision system,” in International Conference on Pattern Recognition, (2004), pp. 482–485.

Muzzupappa, M.

A. Gallo, M. Muzzupappa, and F. Bruno, “3d reconstruction of small sized objects from a sequence of multi-focused images,” J. Cult. Herit. 15(2), 173–182 (2014).
[Crossref]

Narvekar, N. D.

N. D. Narvekar and L. J. Karam, “A no-reference image blur metric based on the cumulative probability of blur detection (CPBD),” IEEE Trans. on Image Process. 20(9), 2678–2683 (2011).
[Crossref]

Nedevschi, S.

S. Nedevschi, “High accuracy stereo vision system for far distance obstacle detection,,” in Intelligent Vehicles Symposium, (2004), pp. 292–297.

Pan, X.

Ray, S. F.

S. F. Ray, Applied Photographic Optics: Lenses and Optical Systems for Photography, Film, Video, Electronic and Digital Imaging (Focal, 2002).

Reischig, P.

X. Zhang, J. Blaas, C. Botha, P. Reischig, A. Bravin, and J. Dik, “Process for the 3d virtual reconstruction of a microcultural heritage artifact obtained by synchrotron radiation ct technology using open source and free software,” J. Cult. Herit. 13(2), 221–225 (2012).
[Crossref]

Ren, F.

S. Zhang, B. Li, F. Ren, and R. Dong, “High-precision measurement of binocular telecentric vision system with novel calibration and matching methods,” IEEE Access 7, 54682–54692 (2019).
[Crossref]

Ren, G.

Ren, Z.

Sanz-Ablanedo, E.

E. Sanz-Ablanedo, J. H. Chandler, and R. Wackrow, “Parameterising internal camera geometry with focusing distance,” The Photogrammetric Record 27(138), 210–226 (2012).
[Crossref]

Sarkis, M.

M. Sarkis, C. T. Senft, and K. Diepold, “Calibrating an automatic zoom camera with moving least squares,” IEEE Trans. Automat. Sci. Eng. 6(3), 492–503 (2009).
[Crossref]

Schroeder, D. J.

D. J. Schroeder, Astronomical Optics (2th ed.) (Academic, 1999).

Senft, C. T.

M. Sarkis, C. T. Senft, and K. Diepold, “Calibrating an automatic zoom camera with moving least squares,” IEEE Trans. Automat. Sci. Eng. 6(3), 492–503 (2009).
[Crossref]

Shen, S.

S. Shen, “Accurate multiple view 3d reconstruction using patch-based stereo for large-scale scenes,” IEEE Trans. on Image Process. 22(5), 1901–1914 (2013).
[Crossref]

Shih, S. W.

Y. S. Chen, S. W. Shih, Y. P. Hung, and C. S. Fuh, “Simple and efficient method of calibrating a motorized zoom lens,” Image and vision computing 19(14), 1099–1110 (2001).
[Crossref]

Stone, J.

B. London, J. Stone, and J. Upton, Photography (8th ed.) (Prentice Hall, 2005).

Sueyasu, H.

K. Atsushi, H. Sueyasu, Y. Funayama, and T. Maekawa, “System for reconstruction of three-dimensional micro objects from multiple photographic images,” Comput. Des. 43(8), 1045–1055 (2011).
[Crossref]

Sutton, M. A.

N. Cornille, D. Garcia, M. A. Sutton, S. McNeill, and J. JOrteu, “Automated 3-d reconstruction using a scanning electron microscope,,” in SEM annual conf. exp. on experimental and applied mechanics, (2003).

Tao, J.

X. Chen, J. Tao, and J. Ye, “Research and development of an accurate 3d shape measurement system based on fringe projection: Model analysis and performance evaluation,” Precis. Eng. 32(3), 215–221 (2008).
[Crossref]

Tao, T.

Y. Hu, Q. Chen, S. Feng, T. Tao, A. Asundi, and C. Zuo, “A new microscopic telecentric stereo vision system-calibration, rectification, and three-dimensional reconstruction,” Opt. Lasers Eng. 113, 14–22 (2019).
[Crossref]

Tarabanis, K.

K. Tarabanis, R. Y. Tsai, and D. S. Goodman, “Modeling of a computer-controlled zoom lens,” in IEEE International Conference on Robotics and Automation, (1992), pp. 1545–1551.

Triggs, B.

B. Triggs, P. F. McLauchlan, R. I. Hartley, and A. W. Fitzgibbon, “Bundle adjustment–a modern synthesis,” in International workshop on vision algorithms, 298–372, (1999

Tsai, R.

R. Tsai, “A versatile camera calibration technique for high-accuracy 3D machine vision metrology using off-the-shelf TV cameras and lenses,” IEEE J. Robot. Automat. 3(4), 323–344 (1987).
[Crossref]

Tsai, R. Y.

R. K. Lenz and R. Y. Tsai, “Techniques for calibration of the scale factor and image center for high accuracy 3-d machine vision metrology,” IEEE Transactions on pattern analysis and machine intelligence 10(5), 713–720 (1988).
[Crossref]

K. Tarabanis, R. Y. Tsai, and D. S. Goodman, “Modeling of a computer-controlled zoom lens,” in IEEE International Conference on Robotics and Automation, (1992), pp. 1545–1551.

Tsujimoto, T.

T. Tsujimoto, “Focus stacking image processing apparatus, imaging system, and image processing system,” U.S. Patent9224193 (Dec 29, 2015).

Upton, J.

B. London, J. Stone, and J. Upton, Photography (8th ed.) (Prentice Hall, 2005).

Wackrow, R.

E. Sanz-Ablanedo, J. H. Chandler, and R. Wackrow, “Parameterising internal camera geometry with focusing distance,” The Photogrammetric Record 27(138), 210–226 (2012).
[Crossref]

Wang, F.

Wang, L.

Wang, Y.

Wang, Z.

Z. Wang, J. Mills, W. Xiao, R. Huang, S. Zheng, and Z. Li, “A flexible, generic photogrammetric approach to zoom lens calibration,” Remote Sens. 9(3), 244 (2017).
[Crossref]

S. Zheng, Z. Wang, and R. Huang, “Zoom lens calibration with zoom-and focus-related intrinsic parameters applied to bundle adjustment,” ISPRS journal of photogrammetry and remote sensing 102, 62–72 (2015).
[Crossref]

Wei, G. Q.

G. Q. Wei and S. DeMa, “Implicit and explicit camera calibration: Theory and experiments,” IEEE Transactions on Pattern Analysis and Machine Intelligence 16(5), 469–480 (1994).
[Crossref]

Willson, R. G.

R. G. Willson, “Modeling and calibration of automated zoom lenses,” in Videometrics III, vol. 2350, (1994), pp. 170–187.

Wu, B.

B. Wu, H. Hu, Q. Zhu, and Y. Zhang, “A flexible method for zoom lens calibration and modeling using a planar checkerboard,” Photogramm. Eng. Remote Sens. 79(6), 555–571 (2013).
[Crossref]

Wu, Q.

Wu, S.

Wyant, J. C.

J. C. Wyant and K. Creath, “Basic wavefront aberration theory for optical metrology,” Applied optics and optical engineering 11, 28–39 (1992).

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Z. Wang, J. Mills, W. Xiao, R. Huang, S. Zheng, and Z. Li, “A flexible, generic photogrammetric approach to zoom lens calibration,” Remote Sens. 9(3), 244 (2017).
[Crossref]

Yamashita, A.

A. Yamashita, A. Kawarago, T. Kaneko, and K. T. Miura, “Shape reconstruction and image restoration for non-flat surfaces of documents with a stereo vision system,” in International Conference on Pattern Recognition, (2004), pp. 482–485.

Yang, J.

Ye, J.

X. Chen, J. Tao, and J. Ye, “Research and development of an accurate 3d shape measurement system based on fringe projection: Model analysis and performance evaluation,” Precis. Eng. 32(3), 215–221 (2008).
[Crossref]

Yin, Y.

Ying, X.

X. Ying and H. Zha, “Geometric interpretations of the relation between the image of the absolute conic and sphere images,” IEEE Trans. Pattern Anal. Machine Intell. 28(12), 2031–2036 (2006).
[Crossref]

Zelinsky, A.

R. Atienza and A. Zelinsky, “A practical zoom camera calibration technique: an application on active vision for human-robot interaction,” in Australian Conference on Robotics and Automation, (2001), pp. 85–90.

Zha, H.

X. Ying and H. Zha, “Geometric interpretations of the relation between the image of the absolute conic and sphere images,” IEEE Trans. Pattern Anal. Machine Intell. 28(12), 2031–2036 (2006).
[Crossref]

Zhang, J.

L. Jiang, J. Zhang, B. Deng, H. Li, and L. Liu, “3d face reconstruction with geometry details from a single image,” IEEE Trans. on Image Process. 27(10), 4756–4770 (2018).
[Crossref]

Zhang, S.

S. Zhang, B. Li, F. Ren, and R. Dong, “High-precision measurement of binocular telecentric vision system with novel calibration and matching methods,” IEEE Access 7, 54682–54692 (2019).
[Crossref]

B. Li and S. Zhang, “Flexible calibration method for microscopic structured light system using telecentric lens,” Opt. Express 23(20), 25795–25803 (2015).
[Crossref]

Zhang, X.

X. Zhang, J. Blaas, C. Botha, P. Reischig, A. Bravin, and J. Dik, “Process for the 3d virtual reconstruction of a microcultural heritage artifact obtained by synchrotron radiation ct technology using open source and free software,” J. Cult. Herit. 13(2), 221–225 (2012).
[Crossref]

Zhang, Y.

B. Wu, H. Hu, Q. Zhu, and Y. Zhang, “A flexible method for zoom lens calibration and modeling using a planar checkerboard,” Photogramm. Eng. Remote Sens. 79(6), 555–571 (2013).
[Crossref]

Zhang, Z.

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

Zhao, K.

Zheng, S.

Z. Wang, J. Mills, W. Xiao, R. Huang, S. Zheng, and Z. Li, “A flexible, generic photogrammetric approach to zoom lens calibration,” Remote Sens. 9(3), 244 (2017).
[Crossref]

S. Zheng, Z. Wang, and R. Huang, “Zoom lens calibration with zoom-and focus-related intrinsic parameters applied to bundle adjustment,” ISPRS journal of photogrammetry and remote sensing 102, 62–72 (2015).
[Crossref]

Zhou, F.

Zhu, Q.

B. Wu, H. Hu, Q. Zhu, and Y. Zhang, “A flexible method for zoom lens calibration and modeling using a planar checkerboard,” Photogramm. Eng. Remote Sens. 79(6), 555–571 (2013).
[Crossref]

Zuo, C.

Y. Hu, Q. Chen, S. Feng, T. Tao, A. Asundi, and C. Zuo, “A new microscopic telecentric stereo vision system-calibration, rectification, and three-dimensional reconstruction,” Opt. Lasers Eng. 113, 14–22 (2019).
[Crossref]

Appl. Opt. (3)

Applied optics and optical engineering (1)

J. C. Wyant and K. Creath, “Basic wavefront aberration theory for optical metrology,” Applied optics and optical engineering 11, 28–39 (1992).

Comput. Des. (1)

K. Atsushi, H. Sueyasu, Y. Funayama, and T. Maekawa, “System for reconstruction of three-dimensional micro objects from multiple photographic images,” Comput. Des. 43(8), 1045–1055 (2011).
[Crossref]

Foundation of Research and Technology (1)

M. I. Lourakis, “A brief description of the levenberg-marquardt algorithm implemented by levmar,” Foundation of Research and Technology 4, 1–6 (2005).

IEEE Access (1)

S. Zhang, B. Li, F. Ren, and R. Dong, “High-precision measurement of binocular telecentric vision system with novel calibration and matching methods,” IEEE Access 7, 54682–54692 (2019).
[Crossref]

IEEE J. Robot. Automat. (1)

R. Tsai, “A versatile camera calibration technique for high-accuracy 3D machine vision metrology using off-the-shelf TV cameras and lenses,” IEEE J. Robot. Automat. 3(4), 323–344 (1987).
[Crossref]

IEEE Trans. Automat. Sci. Eng. (1)

M. Sarkis, C. T. Senft, and K. Diepold, “Calibrating an automatic zoom camera with moving least squares,” IEEE Trans. Automat. Sci. Eng. 6(3), 492–503 (2009).
[Crossref]

IEEE Trans. on Image Process. (4)

N. D. Narvekar and L. J. Karam, “A no-reference image blur metric based on the cumulative probability of blur detection (CPBD),” IEEE Trans. on Image Process. 20(9), 2678–2683 (2011).
[Crossref]

L. Jiang, J. Zhang, B. Deng, H. Li, and L. Liu, “3d face reconstruction with geometry details from a single image,” IEEE Trans. on Image Process. 27(10), 4756–4770 (2018).
[Crossref]

M. Bertozzi and A. Broggi, “GOLD: A parallel real-time stereo vision system for generic obstacle and lane detection,” IEEE Trans. on Image Process. 7(1), 62–81 (1998).
[Crossref]

S. Shen, “Accurate multiple view 3d reconstruction using patch-based stereo for large-scale scenes,” IEEE Trans. on Image Process. 22(5), 1901–1914 (2013).
[Crossref]

IEEE Trans. Pattern Anal. Machine Intell. (2)

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

X. Ying and H. Zha, “Geometric interpretations of the relation between the image of the absolute conic and sphere images,” IEEE Trans. Pattern Anal. Machine Intell. 28(12), 2031–2036 (2006).
[Crossref]

IEEE Transactions on Pattern Analysis and Machine Intelligence (1)

G. Q. Wei and S. DeMa, “Implicit and explicit camera calibration: Theory and experiments,” IEEE Transactions on Pattern Analysis and Machine Intelligence 16(5), 469–480 (1994).
[Crossref]

R. K. Lenz and R. Y. Tsai, “Techniques for calibration of the scale factor and image center for high accuracy 3-d machine vision metrology,” IEEE Transactions on pattern analysis and machine intelligence 10(5), 713–720 (1988).
[Crossref]

Image and vision computing (1)

Y. S. Chen, S. W. Shih, Y. P. Hung, and C. S. Fuh, “Simple and efficient method of calibrating a motorized zoom lens,” Image and vision computing 19(14), 1099–1110 (2001).
[Crossref]

ISPRS journal of photogrammetry and remote sensing (2)

C. S. Fraser, “Digital camera self-calibration,” ISPRS journal of photogrammetry and remote sensing 52(4), 149–159 (1997).
[Crossref]

S. Zheng, Z. Wang, and R. Huang, “Zoom lens calibration with zoom-and focus-related intrinsic parameters applied to bundle adjustment,” ISPRS journal of photogrammetry and remote sensing 102, 62–72 (2015).
[Crossref]

J. Cult. Herit. (2)

A. Gallo, M. Muzzupappa, and F. Bruno, “3d reconstruction of small sized objects from a sequence of multi-focused images,” J. Cult. Herit. 15(2), 173–182 (2014).
[Crossref]

X. Zhang, J. Blaas, C. Botha, P. Reischig, A. Bravin, and J. Dik, “Process for the 3d virtual reconstruction of a microcultural heritage artifact obtained by synchrotron radiation ct technology using open source and free software,” J. Cult. Herit. 13(2), 221–225 (2012).
[Crossref]

J. Math. Imaging Vis. (1)

L. Alvarez, L. Gómez, and P. Henríquez, “Zoom dependent lens distortion mathematical models,” J. Math. Imaging Vis. 44(3), 480–490 (2012).
[Crossref]

Opt. Express (6)

Opt. Lasers Eng. (1)

Y. Hu, Q. Chen, S. Feng, T. Tao, A. Asundi, and C. Zuo, “A new microscopic telecentric stereo vision system-calibration, rectification, and three-dimensional reconstruction,” Opt. Lasers Eng. 113, 14–22 (2019).
[Crossref]

Photogramm. Eng (1)

C. B. Duane, “Close-range camera calibration,” Photogramm. Eng 37, 855–866 (1971).

Photogramm. Eng. Remote Sens. (1)

B. Wu, H. Hu, Q. Zhu, and Y. Zhang, “A flexible method for zoom lens calibration and modeling using a planar checkerboard,” Photogramm. Eng. Remote Sens. 79(6), 555–571 (2013).
[Crossref]

Precis. Eng. (1)

X. Chen, J. Tao, and J. Ye, “Research and development of an accurate 3d shape measurement system based on fringe projection: Model analysis and performance evaluation,” Precis. Eng. 32(3), 215–221 (2008).
[Crossref]

Remote Sens. (1)

Z. Wang, J. Mills, W. Xiao, R. Huang, S. Zheng, and Z. Li, “A flexible, generic photogrammetric approach to zoom lens calibration,” Remote Sens. 9(3), 244 (2017).
[Crossref]

The Photogrammetric Record (1)

E. Sanz-Ablanedo, J. H. Chandler, and R. Wackrow, “Parameterising internal camera geometry with focusing distance,” The Photogrammetric Record 27(138), 210–226 (2012).
[Crossref]

Other (16)

R. Atienza and A. Zelinsky, “A practical zoom camera calibration technique: an application on active vision for human-robot interaction,” in Australian Conference on Robotics and Automation, (2001), pp. 85–90.

R. Kingslake, Optical System Design (Academic, 1983).

K. Tarabanis, R. Y. Tsai, and D. S. Goodman, “Modeling of a computer-controlled zoom lens,” in IEEE International Conference on Robotics and Automation, (1992), pp. 1545–1551.

R. G. Willson, “Modeling and calibration of automated zoom lenses,” in Videometrics III, vol. 2350, (1994), pp. 170–187.

D. J. Schroeder, Astronomical Optics (2th ed.) (Academic, 1999).

S. F. Ray, Applied Photographic Optics: Lenses and Optical Systems for Photography, Film, Video, Electronic and Digital Imaging (Focal, 2002).

B. London, J. Stone, and J. Upton, Photography (8th ed.) (Prentice Hall, 2005).

AFSCOPE, “Afscope c301,” http://www.shunlioptotech.com/afscope/products/4829668.html .

A. Yamashita, A. Kawarago, T. Kaneko, and K. T. Miura, “Shape reconstruction and image restoration for non-flat surfaces of documents with a stereo vision system,” in International Conference on Pattern Recognition, (2004), pp. 482–485.

S. Nedevschi, “High accuracy stereo vision system for far distance obstacle detection,,” in Intelligent Vehicles Symposium, (2004), pp. 292–297.

N. Cornille, D. Garcia, M. A. Sutton, S. McNeill, and J. JOrteu, “Automated 3-d reconstruction using a scanning electron microscope,,” in SEM annual conf. exp. on experimental and applied mechanics, (2003).

T. Tsujimoto, “Focus stacking image processing apparatus, imaging system, and image processing system,” U.S. Patent9224193 (Dec 29, 2015).

J. G. R. Espino, J. Gonzalez-BarbosaJ, R. A. G. Loenzo, D. M. C. Esparza, and R. Gonzalez-Barbosa, “Vision system for 3d reconstruction with telecentric lens,” in Mexican Conference on Pattern Recognition, (2012), pp. 127–136.

J. Bouguet, “Camera calibration toolbox for matlab,” http://www.vision.caltech.edu/bouguetj/calib_doc/index.html .

B. Triggs, P. F. McLauchlan, R. I. Hartley, and A. W. Fitzgibbon, “Bundle adjustment–a modern synthesis,” in International workshop on vision algorithms, 298–372, (1999

J. Benesty, J. Chen, Y. Huang, and I. Cohen, Pearson correlation coefficient (Springer, 2009).

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Figures (9)

Fig. 1.
Fig. 1. (a) Properties against the working distance (WD) of a camera system. Red solid curve: magnification [28]; violet dashed curve: error of one pixel (reciprocal of magnification [29]); black dotted curve: depth of field [30];(b) reason of low calibration accuracy with small depth of field.
Fig. 2.
Fig. 2. Schematic diagram of the thin-lens model. The focal length or principal distance are variables in the thin-lens model, while parameters in the pin-hole model.
Fig. 3.
Fig. 3. Singularity and solution of the IRAC. (a.1) Errors of the RAC towards $k_{u_0},k_{v_0}$ with singularity. (a.2) Top view of a.1. (b.1) Errors of the RAC towards $k_{u_0},k_{v_0}$ without singularity. (b.2) Top view of b.1.
Fig. 4.
Fig. 4. Optimization Strategy. ${p_{f,i,j}} = {(u _{i,j},{v_{i,j}})^T}$ is the observed feature points in the frame buffer coordinates; $\widehat {{p_{f,i,j}}}({P_{W,i,j}},\textbf {M},\textbf {N},{\textbf {d}_\textbf {j}})$ is the projected feature points in the frame buffer coordinates with the camera parameters $\textbf {M},\textbf {N}$ and the displacement of the imaging sensor ${\textbf {d}_\textbf {j}}$ obtained by step motor; $m$ is the number of feature points at each $j^{th}$ position of the imaging sensor; $n$ is the number of positions of the imaging sensor; $\textbf {M} = ({k_\alpha },{k_\beta },{k_{{u_0}}},{k_{{v_0}}},o{m_i}(i = 1,2,3),{t_x},{t_y},{t_z},{a_i}(i = 1,2,3\cdots 12))$; $o{m_i}(i = 1,2,3)$ is the Rodrigues form of the rotation matrix ${r_i}(i = 1,2,3\cdots 9)$ to ensure its orthogonality; and $\textbf {N} = (\alpha ,\beta ,{u_0},{v_0})$.
Fig. 5.
Fig. 5. Performance of the proposed calibration method affected by different factors. (a) Magnitude of SD in the Gaussian noise (pixels). (b) Step accuracy of motors ($\mu m$). (c) Number of images. (d) The minimum magnification.
Fig. 6.
Fig. 6. Experimental setup. Checkerboard: a pattern of 5 x 4 squares each measuring 0.5 mm x 0.5 mm, produced in Nanosystem Fabrication Facility at HKUST by photolithography with $0.25\mu m$ accuracy; displacement platform: $1\mu m$ accuracy; thin-lens camera: AF301 from AFScope company [32], micro-step accuracy is 0.625 $\mu m$; and lens: 25mm, Edmund Optics #85357
Fig. 7.
Fig. 7. A total of eight images used for calibration. Magnification from 0.3538× to 0.6728×. The blue frames indicate the coincident field of view of the calibrated camera in a stereo vision system with two thin-lens cameras.
Fig. 8.
Fig. 8. 3D test fields for evaluation of measurement accuracy.
Fig. 9.
Fig. 9. Comparison of the 3D measurement accuracy of the proposed methods and prior methods. (a) Monofocal calibration + Interpolation [50]. (b) Monofocal calibration + BA [33]. (c) The proposed method. (d) Monofocal calibration results.

Tables (2)

Tables Icon

Table 1. Calibrated intrinsic parameters, extrinsic parameters and RMSE in pixels obtained respectively from initial estimation (initial) and bundle adjustment after 3 iterations (3 iters) with different numbers of images.

Tables Icon

Table 2. Calibrated intrinsic parameters, extrinsic parameters and RMSE in pixels with different sets of images.

Equations (25)

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

1 f + Z + 1 f + z = 1 f
P C = R P W + T
R = [ r 1 r 2 r 3 r 4 r 5 r 6 r 7 r 8 r 9 ] ,   T = [ t x t y t z ]
X u = p X , j Z C X C , Y u = p Y , j Z C Y C
A j = [ p X , j d x 0 C X , j 0 p Y , j d y C Y , j 0 0 1 ] = [ α + k α d j 0 u 0 + k u 0 d j 0 β + k β d j v 0 + k v 0 d j 0 0 1 ]
X d = X u / ( 1 + Δ x ) , Y d = Y u / ( 1 + Δ y )
Δ x = X u ( k 1 r 2 + k 2 r 4 ) + p 1 ( r 2 + 2 X u 2 ) + 2 p 2 X u Y u Δ y = Y u ( k 1 r 2 + k 2 r 4 ) + 2 p 1 X u Y u + p 2 ( r 2 + 2 Y u 2 ) r 2 = X u 2 + Y u 2
k 1 = a 1 P D j 2 + a 2 P D j + a 3 ,   k 2 = a 4 P D j 2 + a 5 P D j + a 6 p 1 = a 7 P D j 2 + a 8 P D j + a 9 ,   p 2 = a 10 P D j 2 + a 11 P D j + a 12
u = X d / d x + C X , j = X d / d x + u 0 + k u 0 d j , v = Y d / d x + C Y , j = Y d / d y + v 0 + k v 0 d j
X u Y u = X C Y C ( u C X , j ) d x / ( 1 + Δ x ) ( v C Y , j ) d y / ( 1 + Δ y ) = r 1 X W + r 2 Y W + r 3 Z W + t x r 4 X W + r 5 Y W + r 6 Z W + t y
[ Y u , i , j X W , i , j Y u , i , j Y W , i , j Y u , i , j Z W , i , j Y u , i , j X u , i , j X W , i , j X u , i , j Y W , i , j X u , i , j Z W , i , j ] [ r 1 t y r 2 t y r 3 t y t x t y r 4 t y r 5 t y r 6 t y ] T = X u , i , j
( u C X , j ) d x / ( 1 + Δ x ) ( r 4 X W + r 5 Y W + r 6 Z W + t y ) ( v C Y , j ) d y / ( 1 + Δ y ) ( r 1 X W + r 2 Y W + r 3 Z W + t x ) = 0
C X , j = C X , j + Δ C X , j = C X , j + Δ k u 0 d j C Y , j = C Y , j + Δ C Y , j = C Y , j + Δ k v 0 d j
( u C X , j ) d x / ( 1 + Δ x ) ( r 4 X W + r 5 Y W + r 6 Z W + t y ) ( v C Y , j ) d y / ( 1 + Δ y ) ( r 1 X W + r 2 Y W + r 3 Z W + t x ) [ Y u Z C α Y C d x / ( 1 + Δ x ) ] d j Δ k u 0 + [ X u Z C β + X C d y / ( 1 + Δ y ) ] d j Δ k v 0
r 1 = r 1 Δ C X α r 7 , r 2 = r 2 Δ C X α r 8 , r 3 = r 3 Δ C X α r 9 , r 4 = r 4 Δ C Y β r 7 , r 5 = r 5 Δ C Y β r 8 , r 6 = r 6 Δ C Y β r 9 , t x = t x Δ C X α t z , t y = t y Δ C Y β t z
Δ C X = m e a n ( Δ k u 0 d j ) = j n Δ k u 0 d j n , Δ C Y = m e a n ( Δ k v 0 d j ) = j n Δ k v 0 d j n
F = arg min k u 0 , k v 0 i = 1 m j = 1 n [ ( u i , j C X , j ) d x / ( 1 + Δ x i , j ) ( r 4 X W , i , j + r 5 Y W , i , j + r 6 Z W , i , j + t y ) ( v i , j C Y , j ) d y / ( 1 + Δ y i , j ) ( r 1 X W , i , j + r 2 Y W , i , j + r 3 Z W , i , j + t x ) ] 2
[ Y u , i , j X W , i , j Y u , i , j Y W , i , j Y u , i , j Z W , i , j X u , i , j X u , i , j X W , i , j X u , i , j Y W , i , j X u , i , j Z W , i , j ] [ r 1 t x r 2 t x r 3 t x t y t x r 4 t x r 5 t x r 6 t x ] T = Y u , i , j
[ X C d j 0 ( u u 0 k u 0 d j ) 0 Y C d j ( v v 0 k v 0 d j ) ] [ k α k β t z ] = [ ( r 7 X W + r 8 Y W + r 9 Z W ) ( u u 0 k u 0 d j ) α X C ( r 7 X W + r 8 Y W + r 9 Z W ) ( v v 0 k v 0 d j ) β Y C ]
u i , j d j X C , i , j Z C , i , j k α + k u 0 v i , j d j Y C , i , j Z C , i , j k β + k v 0
2 F = [ 2 F 2 k u 0 2 F k u 0 k v 0 2 F k v 0 k u 0 2 F 2 k v 0 ]
2 F 2 k u 0 = i = 1 m j = 1 n 2 d j 2 d x 2 ( 1 + Δ x i , j ) 2 Y C , i , j 2 2 F k u 0 k v 0 = i = 1 m j = 1 n 2 d j 2 d x d y X C , i , j Y C , i , j ( 1 + Δ x i , j ) ( 1 + Δ y i , j ) 2 F k v 0 k u 0 = i = 1 m j = 1 n 2 d j 2 d x d y X C , i , j Y C , i , j ( 1 + Δ x i , j ) ( 1 + Δ y i , j ) 2 F 2 k v 0 = i = 1 m j = 1 n 2 d j 2 d y 2 ( 1 + Δ y i , j ) 2 X C , i , j 2
a = 1 , b = i = 1 m j = 1 n 2 d j 2 d x 2 ( 1 + Δ x i , j ) 2 Y C , i , j 2 + 2 d j 2 d y 2 ( 1 + Δ y i , j ) 2 X C , i , j 2 c = [ i = 1 m j = 1 n 2 d j 2 d x 2 ( 1 + Δ x i , j ) 2 Y C , i , j 2 ] [ i = 1 m j = 1 n 2 d j 2 d y 2 ( 1 + Δ y i , j ) 2 X C , i , j 2 ] [ i = 1 m j = 1 n 2 d j 2 d x d y X C , i , j Y C , i , j ( 1 + Δ x i , j ) ( 1 + Δ y i , j ) ] 2
b 2 4 a c = [ i = 1 m j = 1 n 2 d j 2 d x 2 ( 1 + Δ x i , j ) 2 Y C , i , j 2 + 2 d j 2 d y 2 ( 1 + Δ y i , j ) 2 X C , i , j 2 ] 2 4 [ i = 1 m j = 1 n 2 d j 2 d x 2 ( 1 + Δ x i , j ) 2 Y C , i , j 2 ] [ i = 1 m j = 1 n 2 d j 2 d y 2 ( 1 + Δ y i , j ) 2 X C , i , j 2 ] + 4 [ i = 1 m j = 1 n 2 d j 2 d x d y X C , i , j Y C , i , j ( 1 + Δ x i , j ) ( 1 + Δ y i , j ) ] 2 = [ i = 1 m j = 1 n 2 d j 2 d x 2 ( 1 + Δ x i , j ) 2 Y C , i , j 2 2 d j 2 d y 2 ( 1 + Δ y i , j ) 2 X C , i , j 2 ] 2 + 4 [ i = 1 m j = 1 n 2 d j 2 d x d y X C , i , j Y C , i , j ( 1 + Δ x i , j ) ( 1 + Δ y i , j ) ] 2 > 0
4 a c = 4 [ i = 1 m j = 1 n 2 d j 2 d x 2 ( 1 + Δ x i , j ) 2 Y C , i , j 2 ] [ i = 1 m j = 1 n 2 d j 2 d y 2 ( 1 + Δ y i , j ) 2 X C , i , j 2 ] 4 [ i = 1 m j = 1 n 2 d j 2 d x d y X C , i , j Y C , i , j ( 1 + Δ x i , j ) ( 1 + Δ y i , j ) ] 2 = 4 i = 1 m j = 1 n k = i m l = j + 1 n [ ( 1 + Δ y i , j ) 2 X C , i , j 2 ( 1 + Δ x k , l ) 2 Y C , k , l 2 ( 1 + Δ y k , l ) 2 X C , k , l 2 ( 1 + Δ x i , j ) 2 Y C , i , j 2 ] 2 > 0

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