Abstract

Exact geometric calibration of optical devices like projectors or cameras is the basis for utilizing them in quantitative metrological applications. The common state-of-the-art photogrammetric pinhole-imaging-based models with supplemental polynomial corrections fail in the presence of nonsymmetric or high-spatial-frequency distortions and in describing caustics efficiently. These problems are solved by our vision ray calibration (VRC), which is proposed in this paper. The VRC takes an optical mapping system modeled as a black box and directly delivers corresponding vision rays for each mapped pixel. The underlying model, the calibration process, and examples are visualized and reviewed, demonstrating the potential of the VRC.

© 2010 Optical Society of America

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2010 (2)

A. K. Dunne, J. Mallon, and P. F. Whelan, “Efficient generic calibration method for general cameras with single centre of projection,” Comput. Vision Image Understand. 114, 220–233(2010).
[CrossRef]

P. Zhang, “Recent progresses on real-time 3D shape measurement using digital fringe projection techniques,” Opt. Lasers Eng. 48, 149–158 (2010).
[CrossRef]

2009 (1)

W. Jüptner and T. Bothe, “Sub-nanometer resolution for the inspection of reflective surfaces using white light,” Proc. SPIE 7405, 740502 (2009).
[CrossRef]

2008 (2)

2007 (5)

E. Savio, L. De Chiffre, and R. Schmitt, “Metrology of freeform shaped parts,” CIRP Ann. 56, 810–835 (2007).
[CrossRef]

J. Mallon and P. F. Whelan, “Which pattern? Biasing aspects of planar calibration patterns and detection methods,” Pattern Recogn. Lett. 28, 921–930 (2007).
[CrossRef]

C. Bräuer-Burchardt, “The influence of target distance to lens distortion variation,” Proc. SPIE 6617, 661708 (2007).
[CrossRef]

H. Louhichi, T. Fournel, J. M. Lavest, and H. Ben Aissia, “Self-calibration of Scheimpflug cameras: an easy protocol,” Meas. Sci. Technol. 18, 2616–2622 (2007).
[CrossRef]

R. Raskar, “Less Is more: coded computational photography,” Lect. Notes Comput. Sci. 4843, 1–12 (2007).
[CrossRef]

2006 (8)

P. Nayar, “Computational cameras: redefining the image,” Computer 39, 30–38 (2006).
[CrossRef]

P. Kuthirummal and P. K. Nayar, “Multiview radial catadioptric imaging for scene capture,” ACM Trans. Graphics 25, 916–923 (2006).
[CrossRef]

J. Kannala and P. Brandt, “A generic camera model and calibration method for conventional, wide-angle, and fish-eye lenses,” IEEE Trans. Pattern Anal. Mach. Intell. 28, 1335–1340 (2006).
[CrossRef] [PubMed]

J. Tardif, P. Sturm, and P. Roy, “Self-calibration of a general radially symmetric distortion model,” Lect. Notes Comput. Sci. 3954 , 186–199 (2006). 

L. Ma, Y. Chen, and K. L. Moore, “An analytical piecewise radial distortion model for precision camera calibration,” IEE Proc. Vision Image Signal Process. 153, 468–474 (2006).
[CrossRef]

F. Remondino and C. Fraser, “Digital camera calibration methods: considerations and comparisons,” ISPRS Symp. 36, 266–272 (2006).

G. Dai, “Zernike aberration coefficients transformed to and from Fourier series coefficients for wavefront representation,” Opt. Lett. 31, 501–503 (2006).
[CrossRef] [PubMed]

R. Swaminathan, M. D. Grossberg, and P. K. Nayar, “Non-single viewpoint catadioptric cameras: geometry and analysis,” Int. J. Comput. Vision 66, 211–229 (2006).
[CrossRef]

2005 (3)

M. D. Grossberg and S. K. Nayar, “The raxel imaging model and ray-based calibration,” Int. J. Comput. Vision 61, 119–137 (2005).
[CrossRef]

C. Bräuer-Burchardt, “Correcting lens distortion in 3D measuring systems using fringe projection,” Proc. SPIE 5962, 59620J (2005).
[CrossRef]

R. A. Hicks and R. K. Perline, “Equiresolution catadioptric sensors,” Appl. Opt. 44, 6108–6114 (2005).
[CrossRef] [PubMed]

2004 (4)

W. Li, T. Bothe, C. von Kopylow, and W. P. O. Jüptner, “Evaluation methods for gradient measurement techniques,” Proc. SPIE 5457, 300–311 (2004).
[CrossRef]

J. Yu and L. McMillan, “General linear cameras,” Lect. Notes Comput. Vision 3024, 14–27 (2004).

P. S. Sherif, W. T. Cathey, and E. R. Dowski, “Phase plate to extend the depth of field of incoherent hybrid imaging systems,” Appl. Opt. 43, 2709–2721 (2004).
[CrossRef] [PubMed]

R. Legarda-Saenz, T. Bothe, and W. P. Juptner, “Accurate procedure for the calibration of a structured light system,” Opt. Eng. 43, 464–471 (2004).
[CrossRef]

2003 (1)

A. Zomet, D. Feldman, P. Peleg, and D. Weinshall, “Mosaicing new views: the crossed-slits projection,” IEEE Trans. Pattern Anal. Mach. Intell. 25, 741–754 (2003).
[CrossRef]

2002 (3)

T. Pajdla, “Stereo with oblique cameras,” Int. J. Comput. Vision 47, 161–170 (2002).
[CrossRef]

P. M. Seitz and J. Kim, “The space of all stereo images,” Int. J. Comput. Vision 48, 21–21 (2002).
[CrossRef]

J. Burke, T. Bothe, W. Osten, and C. F. Hess, “Reverse engineering by fringe projection,” Proc. SPIE 4778, 312–324 (2002).
[CrossRef]

1987 (1)

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

Aach, T.

T. Stehle, D. Truhn, T. Aach, C. Trautwein, and J. Tischendorf, “Camera calibration for fish-eye lenses in endoscopy with an application to 3D reconstruction,” in Proceedings of the Fourth IEEE International Symposium on Biomedical Imaging: from Nano to Macro (IEEE, 2007), pp. 1176–1179.
[CrossRef]

Agrawal, A.

A. Veeraraghavan, R. Raskar, A. Agrawal, A. Mohan, and J. Tumblin, “Dappled photography: mask enhanced cameras for heterodyned light fields and coded aperture refocussing,” ACM Trans. Graphics, Vol. 26, paper 69 (2007).

Aissia, H. Ben

H. Louhichi, T. Fournel, J. M. Lavest, and H. Ben Aissia, “Self-calibration of Scheimpflug cameras: an easy protocol,” Meas. Sci. Technol. 18, 2616–2622 (2007).
[CrossRef]

Arakawa, T.

R. Sagawa, T. Sakai, T. Echigo, K. Yagi, M. Shiba, K. Higuchi, T. Arakawa, and Y. Yagi, “Omnidirectional vision attachment for medical endoscopes,” in Proceedings of the Eighth Workshop on Omnidirectional Vision, Camera Networks and Non-classical Cameras , R.Swaminathan, V.Caglioti, and A.Argyros, eds. (2008), http://hal.inria.fr/inria-00325318.
[PubMed]

Barreto, J.

J. Barreto, J. Santos, P. Menezes, and F. Fonseca, “Ray-based calibration of rigid medical endoscopes,” in Proceedings of the Eighth Workshop on Omnidirectional Vision, Camera Networks and Non-classical Cameras, R.Swaminathan, V.Caglioti, and A.Argyros, eds. (2008), http://hal.inria.fr/inria-00325318.
[PubMed]

Bauer, M.

Bothe, T.

W. Jüptner and T. Bothe, “Sub-nanometer resolution for the inspection of reflective surfaces using white light,” Proc. SPIE 7405, 740502 (2009).
[CrossRef]

W. Li, T. Bothe, C. von Kopylow, and W. P. O. Jüptner, “Evaluation methods for gradient measurement techniques,” Proc. SPIE 5457, 300–311 (2004).
[CrossRef]

R. Legarda-Saenz, T. Bothe, and W. P. Juptner, “Accurate procedure for the calibration of a structured light system,” Opt. Eng. 43, 464–471 (2004).
[CrossRef]

J. Burke, T. Bothe, W. Osten, and C. F. Hess, “Reverse engineering by fringe projection,” Proc. SPIE 4778, 312–324 (2002).
[CrossRef]

T. Bothe, A. Gesierich, W. Li, and M. Schulte, “Verfahren und vorrichtung zur kalibrierung einer optischen einrichtung,” German patent DE 10 2005 061 931 A1 (not-yet granted), BIAS (2005).

T. Bothe, Grundlegende untersuchungen zur formerfassung mit einem neuartigen prinzip der streifenprojektion und realisierung in einer kompakten 3D-kamera (BIAS Verlag, 2008).
[PubMed]

W. Li, M. Schulte, T. Bothe, C. von Kopylow, N. Köpp, and W. Jüptner, “Beam based calibration for optical imaging devices,” in Proceedings of the 3DTV Conference (IEEE, 2007), pp. 1–4.

Brandt, P.

J. Kannala and P. Brandt, “A generic camera model and calibration method for conventional, wide-angle, and fish-eye lenses,” IEEE Trans. Pattern Anal. Mach. Intell. 28, 1335–1340 (2006).
[CrossRef] [PubMed]

Bräuer-Burchardt, C.

C. Bräuer-Burchardt, “The influence of target distance to lens distortion variation,” Proc. SPIE 6617, 661708 (2007).
[CrossRef]

C. Bräuer-Burchardt, “Correcting lens distortion in 3D measuring systems using fringe projection,” Proc. SPIE 5962, 59620J (2005).
[CrossRef]

Bredif, M.

R. Ng, M. Levoy, M. Bredif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” Stanford Tech. Report CTSR 2005-02 (Stanford, 2005), http://graphics.stanford.edu:/papers/lfcamera/lfcam.

Burke, J.

J. Burke, T. Bothe, W. Osten, and C. F. Hess, “Reverse engineering by fringe projection,” Proc. SPIE 4778, 312–324 (2002).
[CrossRef]

Cathey, W. T.

Chen, Y.

L. Ma, Y. Chen, and K. L. Moore, “An analytical piecewise radial distortion model for precision camera calibration,” IEE Proc. Vision Image Signal Process. 153, 468–474 (2006).
[CrossRef]

Dai, G.

De Chiffre, L.

E. Savio, L. De Chiffre, and R. Schmitt, “Metrology of freeform shaped parts,” CIRP Ann. 56, 810–835 (2007).
[CrossRef]

Dowski, E. R.

Dunne, A. K.

A. K. Dunne, J. Mallon, and P. F. Whelan, “Efficient generic calibration method for general cameras with single centre of projection,” Comput. Vision Image Understand. 114, 220–233(2010).
[CrossRef]

A. K. Dunne, J. Mallon, and P. F. Whelan, “Efficient generic calibration method for general cameras with single centre of projection,” in Proceedings of the 11th International Conference on Computer Vision (IEEE, 2007), pp. 1–8.

Durrant-Whyte, H.

O. Frank, R. Katz, C. Tisse, and H. Durrant-Whyte, “Camera calibration for miniature, low-cost, wide-angle imaging systems,” in Proceedings of the 18th British Machine Vision Conference (British Machine Vision Association, 2007).
[CrossRef]

Duval, G.

R. Ng, M. Levoy, M. Bredif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” Stanford Tech. Report CTSR 2005-02 (Stanford, 2005), http://graphics.stanford.edu:/papers/lfcamera/lfcam.

Echigo, T.

R. Sagawa, T. Sakai, T. Echigo, K. Yagi, M. Shiba, K. Higuchi, T. Arakawa, and Y. Yagi, “Omnidirectional vision attachment for medical endoscopes,” in Proceedings of the Eighth Workshop on Omnidirectional Vision, Camera Networks and Non-classical Cameras , R.Swaminathan, V.Caglioti, and A.Argyros, eds. (2008), http://hal.inria.fr/inria-00325318.
[PubMed]

R. Sagawa, M. Takatsuji, T. Echigo, and Y. Yagi, “Calibration of lens distortion by structured-light scanning,” in 2005 IEEE/RSJ International Conference on Intelligent Robots and Systems (IEEE, 2005), pp. 832–837.
[CrossRef]

Feldman, D.

A. Zomet, D. Feldman, P. Peleg, and D. Weinshall, “Mosaicing new views: the crossed-slits projection,” IEEE Trans. Pattern Anal. Mach. Intell. 25, 741–754 (2003).
[CrossRef]

Fonseca, F.

J. Barreto, J. Santos, P. Menezes, and F. Fonseca, “Ray-based calibration of rigid medical endoscopes,” in Proceedings of the Eighth Workshop on Omnidirectional Vision, Camera Networks and Non-classical Cameras, R.Swaminathan, V.Caglioti, and A.Argyros, eds. (2008), http://hal.inria.fr/inria-00325318.
[PubMed]

Fournel, T.

H. Louhichi, T. Fournel, J. M. Lavest, and H. Ben Aissia, “Self-calibration of Scheimpflug cameras: an easy protocol,” Meas. Sci. Technol. 18, 2616–2622 (2007).
[CrossRef]

Frank, O.

O. Frank, R. Katz, C. Tisse, and H. Durrant-Whyte, “Camera calibration for miniature, low-cost, wide-angle imaging systems,” in Proceedings of the 18th British Machine Vision Conference (British Machine Vision Association, 2007).
[CrossRef]

Fraser, C.

F. Remondino and C. Fraser, “Digital camera calibration methods: considerations and comparisons,” ISPRS Symp. 36, 266–272 (2006).

Gesierich, A.

T. Bothe, A. Gesierich, W. Li, and M. Schulte, “Verfahren und vorrichtung zur kalibrierung einer optischen einrichtung,” German patent DE 10 2005 061 931 A1 (not-yet granted), BIAS (2005).

Griessbach, D.

Grossberg, M. D.

R. Swaminathan, M. D. Grossberg, and P. K. Nayar, “Non-single viewpoint catadioptric cameras: geometry and analysis,” Int. J. Comput. Vision 66, 211–229 (2006).
[CrossRef]

M. D. Grossberg and P. K. Nayar, “A general imaging model and a method for finding its parameters,” in Proceedings of the Eighth International Conference on Computer Vision, Vol.  2 (IEEE, 2001), pp. 108–115.

Grossberg, M.D.

M. D. Grossberg and S. K. Nayar, “The raxel imaging model and ray-based calibration,” Int. J. Comput. Vision 61, 119–137 (2005).
[CrossRef]

Hanrahan, P.

R. Ng, M. Levoy, M. Bredif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” Stanford Tech. Report CTSR 2005-02 (Stanford, 2005), http://graphics.stanford.edu:/papers/lfcamera/lfcam.

Hermerschmidt, A.

Hess, C. F.

J. Burke, T. Bothe, W. Osten, and C. F. Hess, “Reverse engineering by fringe projection,” Proc. SPIE 4778, 312–324 (2002).
[CrossRef]

Hicks, R. A.

Higuchi, K.

R. Sagawa, T. Sakai, T. Echigo, K. Yagi, M. Shiba, K. Higuchi, T. Arakawa, and Y. Yagi, “Omnidirectional vision attachment for medical endoscopes,” in Proceedings of the Eighth Workshop on Omnidirectional Vision, Camera Networks and Non-classical Cameras , R.Swaminathan, V.Caglioti, and A.Argyros, eds. (2008), http://hal.inria.fr/inria-00325318.
[PubMed]

Horowitz, M.

R. Ng, M. Levoy, M. Bredif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” Stanford Tech. Report CTSR 2005-02 (Stanford, 2005), http://graphics.stanford.edu:/papers/lfcamera/lfcam.

Juptner, W. P.

R. Legarda-Saenz, T. Bothe, and W. P. Juptner, “Accurate procedure for the calibration of a structured light system,” Opt. Eng. 43, 464–471 (2004).
[CrossRef]

Jüptner, W.

W. Jüptner and T. Bothe, “Sub-nanometer resolution for the inspection of reflective surfaces using white light,” Proc. SPIE 7405, 740502 (2009).
[CrossRef]

W. Li, M. Schulte, T. Bothe, C. von Kopylow, N. Köpp, and W. Jüptner, “Beam based calibration for optical imaging devices,” in Proceedings of the 3DTV Conference (IEEE, 2007), pp. 1–4.

Jüptner, W. P. O.

W. Li, T. Bothe, C. von Kopylow, and W. P. O. Jüptner, “Evaluation methods for gradient measurement techniques,” Proc. SPIE 5457, 300–311 (2004).
[CrossRef]

Kannala, J.

J. Kannala and P. Brandt, “A generic camera model and calibration method for conventional, wide-angle, and fish-eye lenses,” IEEE Trans. Pattern Anal. Mach. Intell. 28, 1335–1340 (2006).
[CrossRef] [PubMed]

Katz, R.

O. Frank, R. Katz, C. Tisse, and H. Durrant-Whyte, “Camera calibration for miniature, low-cost, wide-angle imaging systems,” in Proceedings of the 18th British Machine Vision Conference (British Machine Vision Association, 2007).
[CrossRef]

Kim, J.

P. M. Seitz and J. Kim, “The space of all stereo images,” Int. J. Comput. Vision 48, 21–21 (2002).
[CrossRef]

Köpp, N.

W. Li, M. Schulte, T. Bothe, C. von Kopylow, N. Köpp, and W. Jüptner, “Beam based calibration for optical imaging devices,” in Proceedings of the 3DTV Conference (IEEE, 2007), pp. 1–4.

Küger, S.

Kuthirummal, P.

P. Kuthirummal and P. K. Nayar, “Multiview radial catadioptric imaging for scene capture,” ACM Trans. Graphics 25, 916–923 (2006).
[CrossRef]

Lavest, J. M.

H. Louhichi, T. Fournel, J. M. Lavest, and H. Ben Aissia, “Self-calibration of Scheimpflug cameras: an easy protocol,” Meas. Sci. Technol. 18, 2616–2622 (2007).
[CrossRef]

Legarda-Saenz, R.

R. Legarda-Saenz, T. Bothe, and W. P. Juptner, “Accurate procedure for the calibration of a structured light system,” Opt. Eng. 43, 464–471 (2004).
[CrossRef]

Levoy, M.

R. Ng, M. Levoy, M. Bredif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” Stanford Tech. Report CTSR 2005-02 (Stanford, 2005), http://graphics.stanford.edu:/papers/lfcamera/lfcam.

Li, W.

W. Li, T. Bothe, C. von Kopylow, and W. P. O. Jüptner, “Evaluation methods for gradient measurement techniques,” Proc. SPIE 5457, 300–311 (2004).
[CrossRef]

T. Bothe, A. Gesierich, W. Li, and M. Schulte, “Verfahren und vorrichtung zur kalibrierung einer optischen einrichtung,” German patent DE 10 2005 061 931 A1 (not-yet granted), BIAS (2005).

W. Li, M. Schulte, T. Bothe, C. von Kopylow, N. Köpp, and W. Jüptner, “Beam based calibration for optical imaging devices,” in Proceedings of the 3DTV Conference (IEEE, 2007), pp. 1–4.

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]

Lodha, P. K.

S. Ramalingam, P. Sturm, and P. K. Lodha, “Towards complete generic camera calibration,” in Proceedings of Computer Society Conference on Computer Vision and Pattern Recognition, Vol.  1 (IEEE, 2005), pp. 1093–1098.

Louhichi, H.

H. Louhichi, T. Fournel, J. M. Lavest, and H. Ben Aissia, “Self-calibration of Scheimpflug cameras: an easy protocol,” Meas. Sci. Technol. 18, 2616–2622 (2007).
[CrossRef]

Ma, L.

L. Ma, Y. Chen, and K. L. Moore, “An analytical piecewise radial distortion model for precision camera calibration,” IEE Proc. Vision Image Signal Process. 153, 468–474 (2006).
[CrossRef]

Mallon, J.

A. K. Dunne, J. Mallon, and P. F. Whelan, “Efficient generic calibration method for general cameras with single centre of projection,” Comput. Vision Image Understand. 114, 220–233(2010).
[CrossRef]

J. Mallon and P. F. Whelan, “Which pattern? Biasing aspects of planar calibration patterns and detection methods,” Pattern Recogn. Lett. 28, 921–930 (2007).
[CrossRef]

A. K. Dunne, J. Mallon, and P. F. Whelan, “Efficient generic calibration method for general cameras with single centre of projection,” in Proceedings of the 11th International Conference on Computer Vision (IEEE, 2007), pp. 1–8.

McMillan, L.

J. Yu and L. McMillan, “General linear cameras,” Lect. Notes Comput. Vision 3024, 14–27 (2004).

Menezes, P.

J. Barreto, J. Santos, P. Menezes, and F. Fonseca, “Ray-based calibration of rigid medical endoscopes,” in Proceedings of the Eighth Workshop on Omnidirectional Vision, Camera Networks and Non-classical Cameras, R.Swaminathan, V.Caglioti, and A.Argyros, eds. (2008), http://hal.inria.fr/inria-00325318.
[PubMed]

Mohan, A.

A. Veeraraghavan, R. Raskar, A. Agrawal, A. Mohan, and J. Tumblin, “Dappled photography: mask enhanced cameras for heterodyned light fields and coded aperture refocussing,” ACM Trans. Graphics, Vol. 26, paper 69 (2007).

Moore, K. L.

L. Ma, Y. Chen, and K. L. Moore, “An analytical piecewise radial distortion model for precision camera calibration,” IEE Proc. Vision Image Signal Process. 153, 468–474 (2006).
[CrossRef]

Nagahara, H.

H. Nagahara, K. Yoshida, and M. Yachida, “An omnidirectional vision sensor with single view and constant resolution,” in Proceedings of the 11th International Conference on Computer Vision (IEEE, 2007), pp. 1–8.

Nayar, P.

P. Nayar, “Computational cameras: redefining the image,” Computer 39, 30–38 (2006).
[CrossRef]

Nayar, P. K.

P. Kuthirummal and P. K. Nayar, “Multiview radial catadioptric imaging for scene capture,” ACM Trans. Graphics 25, 916–923 (2006).
[CrossRef]

R. Swaminathan, M. D. Grossberg, and P. K. Nayar, “Non-single viewpoint catadioptric cameras: geometry and analysis,” Int. J. Comput. Vision 66, 211–229 (2006).
[CrossRef]

M. D. Grossberg and P. K. Nayar, “A general imaging model and a method for finding its parameters,” in Proceedings of the Eighth International Conference on Computer Vision, Vol.  2 (IEEE, 2001), pp. 108–115.

Nayar, S. K.

M. D. Grossberg and S. K. Nayar, “The raxel imaging model and ray-based calibration,” Int. J. Comput. Vision 61, 119–137 (2005).
[CrossRef]

Ng, R.

R. Ng, M. Levoy, M. Bredif, G. Duval, M. Horowitz, and P. Hanrahan, “Light field photography with a hand-held plenoptic camera,” Stanford Tech. Report CTSR 2005-02 (Stanford, 2005), http://graphics.stanford.edu:/papers/lfcamera/lfcam.

Orghidan, R.

R. Orghidan, “Catadioptric stereo based on structured light projection,” Ph.D. dissertation (Universitat de Giorona—EIA Electrònica, 2006).

Osten, W.

J. Burke, T. Bothe, W. Osten, and C. F. Hess, “Reverse engineering by fringe projection,” Proc. SPIE 4778, 312–324 (2002).
[CrossRef]

Pajdla, T.

T. Pajdla, “Stereo with oblique cameras,” Int. J. Comput. Vision 47, 161–170 (2002).
[CrossRef]

Peleg, P.

A. Zomet, D. Feldman, P. Peleg, and D. Weinshall, “Mosaicing new views: the crossed-slits projection,” IEEE Trans. Pattern Anal. Mach. Intell. 25, 741–754 (2003).
[CrossRef]

Perline, R. K.

Ramalingam, P.

P. Sturm and P. Ramalingam, “A generic concept for camera calibration,” in Proceedings of the European Conference on Computer Vision, Vol  2 (Springer, 2004), pp. 1–13.

Ramalingam, S.

S. Ramalingam, P. Sturm, and P. K. Lodha, “Towards complete generic camera calibration,” in Proceedings of Computer Society Conference on Computer Vision and Pattern Recognition, Vol.  1 (IEEE, 2005), pp. 1093–1098.

Raskar, R.

R. Raskar, “Less Is more: coded computational photography,” Lect. Notes Comput. Sci. 4843, 1–12 (2007).
[CrossRef]

A. Veeraraghavan, R. Raskar, A. Agrawal, A. Mohan, and J. Tumblin, “Dappled photography: mask enhanced cameras for heterodyned light fields and coded aperture refocussing,” ACM Trans. Graphics, Vol. 26, paper 69 (2007).

Remondino, F.

F. Remondino and C. Fraser, “Digital camera calibration methods: considerations and comparisons,” ISPRS Symp. 36, 266–272 (2006).

Roy, P.

J. Tardif, P. Sturm, and P. Roy, “Self-calibration of a general radially symmetric distortion model,” Lect. Notes Comput. Sci. 3954 , 186–199 (2006). 

Sagawa, R.

R. Sagawa, M. Takatsuji, T. Echigo, and Y. Yagi, “Calibration of lens distortion by structured-light scanning,” in 2005 IEEE/RSJ International Conference on Intelligent Robots and Systems (IEEE, 2005), pp. 832–837.
[CrossRef]

R. Sagawa, T. Sakai, T. Echigo, K. Yagi, M. Shiba, K. Higuchi, T. Arakawa, and Y. Yagi, “Omnidirectional vision attachment for medical endoscopes,” in Proceedings of the Eighth Workshop on Omnidirectional Vision, Camera Networks and Non-classical Cameras , R.Swaminathan, V.Caglioti, and A.Argyros, eds. (2008), http://hal.inria.fr/inria-00325318.
[PubMed]

Sakai, T.

R. Sagawa, T. Sakai, T. Echigo, K. Yagi, M. Shiba, K. Higuchi, T. Arakawa, and Y. Yagi, “Omnidirectional vision attachment for medical endoscopes,” in Proceedings of the Eighth Workshop on Omnidirectional Vision, Camera Networks and Non-classical Cameras , R.Swaminathan, V.Caglioti, and A.Argyros, eds. (2008), http://hal.inria.fr/inria-00325318.
[PubMed]

Santos, J.

J. Barreto, J. Santos, P. Menezes, and F. Fonseca, “Ray-based calibration of rigid medical endoscopes,” in Proceedings of the Eighth Workshop on Omnidirectional Vision, Camera Networks and Non-classical Cameras, R.Swaminathan, V.Caglioti, and A.Argyros, eds. (2008), http://hal.inria.fr/inria-00325318.
[PubMed]

Savio, E.

E. Savio, L. De Chiffre, and R. Schmitt, “Metrology of freeform shaped parts,” CIRP Ann. 56, 810–835 (2007).
[CrossRef]

Schäfer, B.

B. Schäfer, “Wavefront analysis: theory and practice,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science (Optical Society of America, 2003), paper CWQ4.

Schechner, Y.

T. Treibitz, Y. Schechner, and H. Singh, “Flat refractive geometry,” in Proceedings of the Conference on Computer Vision and Pattern Recognition (IEEE, 2008), pp. 1–8.

Scheele, M.

Schischmanow, A.

Schmitt, R.

E. Savio, L. De Chiffre, and R. Schmitt, “Metrology of freeform shaped parts,” CIRP Ann. 56, 810–835 (2007).
[CrossRef]

Schulte, M.

W. Li, M. Schulte, T. Bothe, C. von Kopylow, N. Köpp, and W. Jüptner, “Beam based calibration for optical imaging devices,” in Proceedings of the 3DTV Conference (IEEE, 2007), pp. 1–4.

T. Bothe, A. Gesierich, W. Li, and M. Schulte, “Verfahren und vorrichtung zur kalibrierung einer optischen einrichtung,” German patent DE 10 2005 061 931 A1 (not-yet granted), BIAS (2005).

Seitz, P. M.

P. M. Seitz and J. Kim, “The space of all stereo images,” Int. J. Comput. Vision 48, 21–21 (2002).
[CrossRef]

Sherif, P. S.

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]

Shiba, M.

R. Sagawa, T. Sakai, T. Echigo, K. Yagi, M. Shiba, K. Higuchi, T. Arakawa, and Y. Yagi, “Omnidirectional vision attachment for medical endoscopes,” in Proceedings of the Eighth Workshop on Omnidirectional Vision, Camera Networks and Non-classical Cameras , R.Swaminathan, V.Caglioti, and A.Argyros, eds. (2008), http://hal.inria.fr/inria-00325318.
[PubMed]

Singh, H.

T. Treibitz, Y. Schechner, and H. Singh, “Flat refractive geometry,” in Proceedings of the Conference on Computer Vision and Pattern Recognition (IEEE, 2008), pp. 1–8.

Stehle, T.

T. Stehle, D. Truhn, T. Aach, C. Trautwein, and J. Tischendorf, “Camera calibration for fish-eye lenses in endoscopy with an application to 3D reconstruction,” in Proceedings of the Fourth IEEE International Symposium on Biomedical Imaging: from Nano to Macro (IEEE, 2007), pp. 1176–1179.
[CrossRef]

Sturm, P.

J. Tardif, P. Sturm, and P. Roy, “Self-calibration of a general radially symmetric distortion model,” Lect. Notes Comput. Sci. 3954 , 186–199 (2006). 

S. Ramalingam, P. Sturm, and P. K. Lodha, “Towards complete generic camera calibration,” in Proceedings of Computer Society Conference on Computer Vision and Pattern Recognition, Vol.  1 (IEEE, 2005), pp. 1093–1098.

P. Sturm and P. Ramalingam, “A generic concept for camera calibration,” in Proceedings of the European Conference on Computer Vision, Vol  2 (Springer, 2004), pp. 1–13.

P. Sturm, “Multi-view geometry for general camera models,” in Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition, Vol.  1 (IEEE, 2005), pp. 206–212.

Swaminathan, R.

R. Swaminathan, M. D. Grossberg, and P. K. Nayar, “Non-single viewpoint catadioptric cameras: geometry and analysis,” Int. J. Comput. Vision 66, 211–229 (2006).
[CrossRef]

Takatsuji, M.

R. Sagawa, M. Takatsuji, T. Echigo, and Y. Yagi, “Calibration of lens distortion by structured-light scanning,” in 2005 IEEE/RSJ International Conference on Intelligent Robots and Systems (IEEE, 2005), pp. 832–837.
[CrossRef]

Tardif, J.

J. Tardif, P. Sturm, and P. Roy, “Self-calibration of a general radially symmetric distortion model,” Lect. Notes Comput. Sci. 3954 , 186–199 (2006). 

Tischendorf, J.

T. Stehle, D. Truhn, T. Aach, C. Trautwein, and J. Tischendorf, “Camera calibration for fish-eye lenses in endoscopy with an application to 3D reconstruction,” in Proceedings of the Fourth IEEE International Symposium on Biomedical Imaging: from Nano to Macro (IEEE, 2007), pp. 1176–1179.
[CrossRef]

Tisse, C.

O. Frank, R. Katz, C. Tisse, and H. Durrant-Whyte, “Camera calibration for miniature, low-cost, wide-angle imaging systems,” in Proceedings of the 18th British Machine Vision Conference (British Machine Vision Association, 2007).
[CrossRef]

Trautwein, C.

T. Stehle, D. Truhn, T. Aach, C. Trautwein, and J. Tischendorf, “Camera calibration for fish-eye lenses in endoscopy with an application to 3D reconstruction,” in Proceedings of the Fourth IEEE International Symposium on Biomedical Imaging: from Nano to Macro (IEEE, 2007), pp. 1176–1179.
[CrossRef]

Treibitz, T.

T. Treibitz, Y. Schechner, and H. Singh, “Flat refractive geometry,” in Proceedings of the Conference on Computer Vision and Pattern Recognition (IEEE, 2008), pp. 1–8.

Truhn, D.

T. Stehle, D. Truhn, T. Aach, C. Trautwein, and J. Tischendorf, “Camera calibration for fish-eye lenses in endoscopy with an application to 3D reconstruction,” in Proceedings of the Fourth IEEE International Symposium on Biomedical Imaging: from Nano to Macro (IEEE, 2007), pp. 1176–1179.
[CrossRef]

Tsai, R. Y.

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

Tumblin, J.

A. Veeraraghavan, R. Raskar, A. Agrawal, A. Mohan, and J. Tumblin, “Dappled photography: mask enhanced cameras for heterodyned light fields and coded aperture refocussing,” ACM Trans. Graphics, Vol. 26, paper 69 (2007).

Veeraraghavan, A.

A. Veeraraghavan, R. Raskar, A. Agrawal, A. Mohan, and J. Tumblin, “Dappled photography: mask enhanced cameras for heterodyned light fields and coded aperture refocussing,” ACM Trans. Graphics, Vol. 26, paper 69 (2007).

von Kopylow, C.

W. Li, T. Bothe, C. von Kopylow, and W. P. O. Jüptner, “Evaluation methods for gradient measurement techniques,” Proc. SPIE 5457, 300–311 (2004).
[CrossRef]

W. Li, M. Schulte, T. Bothe, C. von Kopylow, N. Köpp, and W. Jüptner, “Beam based calibration for optical imaging devices,” in Proceedings of the 3DTV Conference (IEEE, 2007), pp. 1–4.

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]

Weinshall, D.

A. Zomet, D. Feldman, P. Peleg, and D. Weinshall, “Mosaicing new views: the crossed-slits projection,” IEEE Trans. Pattern Anal. Mach. Intell. 25, 741–754 (2003).
[CrossRef]

Whelan, P. F.

A. K. Dunne, J. Mallon, and P. F. Whelan, “Efficient generic calibration method for general cameras with single centre of projection,” Comput. Vision Image Understand. 114, 220–233(2010).
[CrossRef]

J. Mallon and P. F. Whelan, “Which pattern? Biasing aspects of planar calibration patterns and detection methods,” Pattern Recogn. Lett. 28, 921–930 (2007).
[CrossRef]

A. K. Dunne, J. Mallon, and P. F. Whelan, “Efficient generic calibration method for general cameras with single centre of projection,” in Proceedings of the 11th International Conference on Computer Vision (IEEE, 2007), pp. 1–8.

Wolff, K.

K. Wolff, “Approximation of non projective mapping and their efficient application for a geometrically based 3D point determination using multiple views,” in Proceedings of Commission III ISPRS Congress Istanbul, 2004 (International Society for Photogrammetry and Remote Sensing, 2004), pp. 606–611.

Yachida, M.

H. Nagahara, K. Yoshida, and M. Yachida, “An omnidirectional vision sensor with single view and constant resolution,” in Proceedings of the 11th International Conference on Computer Vision (IEEE, 2007), pp. 1–8.

Yagi, K.

R. Sagawa, T. Sakai, T. Echigo, K. Yagi, M. Shiba, K. Higuchi, T. Arakawa, and Y. Yagi, “Omnidirectional vision attachment for medical endoscopes,” in Proceedings of the Eighth Workshop on Omnidirectional Vision, Camera Networks and Non-classical Cameras , R.Swaminathan, V.Caglioti, and A.Argyros, eds. (2008), http://hal.inria.fr/inria-00325318.
[PubMed]

Yagi, Y.

R. Sagawa, T. Sakai, T. Echigo, K. Yagi, M. Shiba, K. Higuchi, T. Arakawa, and Y. Yagi, “Omnidirectional vision attachment for medical endoscopes,” in Proceedings of the Eighth Workshop on Omnidirectional Vision, Camera Networks and Non-classical Cameras , R.Swaminathan, V.Caglioti, and A.Argyros, eds. (2008), http://hal.inria.fr/inria-00325318.
[PubMed]

R. Sagawa, M. Takatsuji, T. Echigo, and Y. Yagi, “Calibration of lens distortion by structured-light scanning,” in 2005 IEEE/RSJ International Conference on Intelligent Robots and Systems (IEEE, 2005), pp. 832–837.
[CrossRef]

Yoshida, K.

H. Nagahara, K. Yoshida, and M. Yachida, “An omnidirectional vision sensor with single view and constant resolution,” in Proceedings of the 11th International Conference on Computer Vision (IEEE, 2007), pp. 1–8.

Yu, J.

J. Yu and L. McMillan, “General linear cameras,” Lect. Notes Comput. Vision 3024, 14–27 (2004).

Zhang, 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]

Zhang, P.

P. Zhang, “Recent progresses on real-time 3D shape measurement using digital fringe projection techniques,” Opt. Lasers Eng. 48, 149–158 (2010).
[CrossRef]

Zomet, A.

A. Zomet, D. Feldman, P. Peleg, and D. Weinshall, “Mosaicing new views: the crossed-slits projection,” IEEE Trans. Pattern Anal. Mach. Intell. 25, 741–754 (2003).
[CrossRef]

ACM Trans. Graphics (1)

P. Kuthirummal and P. K. Nayar, “Multiview radial catadioptric imaging for scene capture,” ACM Trans. Graphics 25, 916–923 (2006).
[CrossRef]

Appl. Opt. (2)

CIRP Ann. (1)

E. Savio, L. De Chiffre, and R. Schmitt, “Metrology of freeform shaped parts,” CIRP Ann. 56, 810–835 (2007).
[CrossRef]

Comput. Vision Image Understand. (1)

A. K. Dunne, J. Mallon, and P. F. Whelan, “Efficient generic calibration method for general cameras with single centre of projection,” Comput. Vision Image Understand. 114, 220–233(2010).
[CrossRef]

Computer (1)

P. Nayar, “Computational cameras: redefining the image,” Computer 39, 30–38 (2006).
[CrossRef]

IEE Proc. Vision Image Signal Process. (1)

L. Ma, Y. Chen, and K. L. Moore, “An analytical piecewise radial distortion model for precision camera calibration,” IEE Proc. Vision Image Signal Process. 153, 468–474 (2006).
[CrossRef]

IEEE J. Robot. Autom. (1)

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

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

J. Kannala and P. Brandt, “A generic camera model and calibration method for conventional, wide-angle, and fish-eye lenses,” IEEE Trans. Pattern Anal. Mach. Intell. 28, 1335–1340 (2006).
[CrossRef] [PubMed]

A. Zomet, D. Feldman, P. Peleg, and D. Weinshall, “Mosaicing new views: the crossed-slits projection,” IEEE Trans. Pattern Anal. Mach. Intell. 25, 741–754 (2003).
[CrossRef]

Int. J. Comput. Vision (4)

T. Pajdla, “Stereo with oblique cameras,” Int. J. Comput. Vision 47, 161–170 (2002).
[CrossRef]

P. M. Seitz and J. Kim, “The space of all stereo images,” Int. J. Comput. Vision 48, 21–21 (2002).
[CrossRef]

R. Swaminathan, M. D. Grossberg, and P. K. Nayar, “Non-single viewpoint catadioptric cameras: geometry and analysis,” Int. J. Comput. Vision 66, 211–229 (2006).
[CrossRef]

M. D. Grossberg and S. K. Nayar, “The raxel imaging model and ray-based calibration,” Int. J. Comput. Vision 61, 119–137 (2005).
[CrossRef]

ISPRS Symp. (1)

F. Remondino and C. Fraser, “Digital camera calibration methods: considerations and comparisons,” ISPRS Symp. 36, 266–272 (2006).

Lect. Notes Comput. Sci. (1)

R. Raskar, “Less Is more: coded computational photography,” Lect. Notes Comput. Sci. 4843, 1–12 (2007).
[CrossRef]

Meas. Sci. Technol. (1)

H. Louhichi, T. Fournel, J. M. Lavest, and H. Ben Aissia, “Self-calibration of Scheimpflug cameras: an easy protocol,” Meas. Sci. Technol. 18, 2616–2622 (2007).
[CrossRef]

Opt. Eng. (1)

R. Legarda-Saenz, T. Bothe, and W. P. Juptner, “Accurate procedure for the calibration of a structured light system,” Opt. Eng. 43, 464–471 (2004).
[CrossRef]

Opt. Express (1)

Opt. Lasers Eng. (1)

P. Zhang, “Recent progresses on real-time 3D shape measurement using digital fringe projection techniques,” Opt. Lasers Eng. 48, 149–158 (2010).
[CrossRef]

Opt. Lett. (1)

Pattern Recogn. (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]

Pattern Recogn. Lett. (1)

J. Mallon and P. F. Whelan, “Which pattern? Biasing aspects of planar calibration patterns and detection methods,” Pattern Recogn. Lett. 28, 921–930 (2007).
[CrossRef]

Proc. SPIE (5)

C. Bräuer-Burchardt, “Correcting lens distortion in 3D measuring systems using fringe projection,” Proc. SPIE 5962, 59620J (2005).
[CrossRef]

W. Jüptner and T. Bothe, “Sub-nanometer resolution for the inspection of reflective surfaces using white light,” Proc. SPIE 7405, 740502 (2009).
[CrossRef]

W. Li, T. Bothe, C. von Kopylow, and W. P. O. Jüptner, “Evaluation methods for gradient measurement techniques,” Proc. SPIE 5457, 300–311 (2004).
[CrossRef]

C. Bräuer-Burchardt, “The influence of target distance to lens distortion variation,” Proc. SPIE 6617, 661708 (2007).
[CrossRef]

J. Burke, T. Bothe, W. Osten, and C. F. Hess, “Reverse engineering by fringe projection,” Proc. SPIE 4778, 312–324 (2002).
[CrossRef]

Other (23)

B. Schäfer, “Wavefront analysis: theory and practice,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science (Optical Society of America, 2003), paper CWQ4.

T. Bothe, A. Gesierich, W. Li, and M. Schulte, “Verfahren und vorrichtung zur kalibrierung einer optischen einrichtung,” German patent DE 10 2005 061 931 A1 (not-yet granted), BIAS (2005).

A. K. Dunne, J. Mallon, and P. F. Whelan, “Efficient generic calibration method for general cameras with single centre of projection,” in Proceedings of the 11th International Conference on Computer Vision (IEEE, 2007), pp. 1–8.

M. D. Grossberg and P. K. Nayar, “A general imaging model and a method for finding its parameters,” in Proceedings of the Eighth International Conference on Computer Vision, Vol.  2 (IEEE, 2001), pp. 108–115.

T. Bothe, Grundlegende untersuchungen zur formerfassung mit einem neuartigen prinzip der streifenprojektion und realisierung in einer kompakten 3D-kamera (BIAS Verlag, 2008).
[PubMed]

URL:www.fringeprocessor.com (2010).

H. Nagahara, K. Yoshida, and M. Yachida, “An omnidirectional vision sensor with single view and constant resolution,” in Proceedings of the 11th International Conference on Computer Vision (IEEE, 2007), pp. 1–8.

T. Stehle, D. Truhn, T. Aach, C. Trautwein, and J. Tischendorf, “Camera calibration for fish-eye lenses in endoscopy with an application to 3D reconstruction,” in Proceedings of the Fourth IEEE International Symposium on Biomedical Imaging: from Nano to Macro (IEEE, 2007), pp. 1176–1179.
[CrossRef]

O. Frank, R. Katz, C. Tisse, and H. Durrant-Whyte, “Camera calibration for miniature, low-cost, wide-angle imaging systems,” in Proceedings of the 18th British Machine Vision Conference (British Machine Vision Association, 2007).
[CrossRef]

K. Wolff, “Approximation of non projective mapping and their efficient application for a geometrically based 3D point determination using multiple views,” in Proceedings of Commission III ISPRS Congress Istanbul, 2004 (International Society for Photogrammetry and Remote Sensing, 2004), pp. 606–611.

W. Li, M. Schulte, T. Bothe, C. von Kopylow, N. Köpp, and W. Jüptner, “Beam based calibration for optical imaging devices,” in Proceedings of the 3DTV Conference (IEEE, 2007), pp. 1–4.

P. Sturm and P. Ramalingam, “A generic concept for camera calibration,” in Proceedings of the European Conference on Computer Vision, Vol  2 (Springer, 2004), pp. 1–13.

S. Ramalingam, P. Sturm, and P. K. Lodha, “Towards complete generic camera calibration,” in Proceedings of Computer Society Conference on Computer Vision and Pattern Recognition, Vol.  1 (IEEE, 2005), pp. 1093–1098.

R. Sagawa, M. Takatsuji, T. Echigo, and Y. Yagi, “Calibration of lens distortion by structured-light scanning,” in 2005 IEEE/RSJ International Conference on Intelligent Robots and Systems (IEEE, 2005), pp. 832–837.
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

Color-coded height map of a plane measured by fringe projection: The system’s short focal length lens showed strong distortions, which were not fully covered by state-of-the-art photogrammetric calibration, resulting in systematic errors. Heights in the range of 2.21 mm were generated, but the plane is known to have varying heights in the range of 0.5 mm . The low-frequency systematic errors by nonsymmetric distortion (implied by a line of dots) can be distinguished from high-frequency local distortions by lens inhomogeneities (marked by dashed circles).

Fig. 2
Fig. 2

Perspective pinhole model for optical mapping (one dimensional: y axis and j 0 omitted): An imaged control point p ̲ is connected by a ray v ̲ with origin O. Intersection of the ray with the plane in distance z = f delivers the image coordinate x = i . In the absence of distortions ( δ i = δ j = 0 ), equidistant image coordinates are delivered in the plane perpendicular to the optical axis z.

Fig. 3
Fig. 3

Caustic for nonsingle-viewpoint optics near the finite aperture (around point of closest ray intersection).

Fig. 4
Fig. 4

MVP ray with true direction t x , MVP and offset o x forced into an offset-free SVP description with modified direction t x , SVP (y axis omitted).

Fig. 5
Fig. 5

VRC design: calibration of imaging device D on example of vision ray v ̲ z . The calibration target is moved from position Pos 1 M , in each, measuring the piercing point of v ̲ z . By triangulation, the helping cameras H 1 and H 2 deliver corresponding 3D coordinates (dashed lines for target Pos 1 ). The 3D piercing points deliver the searched vision ray in a global coordinate system.

Fig. 6
Fig. 6

Experimental setup for VRC of device D using two helping cameras H 1 and H 2 on a rotation stage. Control point evaluation is performed for (a)  cameras by a monitor target and (b)  projectors by a scattering surface.

Fig. 7
Fig. 7

Calibration scheme for a large field of view: (a) rotation of the device with fixed calibration plane in distanced A, B, C and (b) result from device view is a filled view area that can be used for vision ray calculation by the measured 3D piercing points.

Fig. 8
Fig. 8

Calibrated lenses: (a) camera: Pentax f = 4.8 mm , f number set to k = 1.8 (thus, aperture : f / k = 2.7 mm ) on AVT Guppy ( 1392 × 1040 pixels); (b) projector: Pentax f = 12.5 mm , f number set to k = 1.4 (aperture : f / k = 8.9 mm ) in front of 14.3 × 10.8 mm LCD panel ( 1024 × 768 pixels).

Fig. 9
Fig. 9

Result images coded from minimum black to maximum white given as interval value [ min ; max ] .

Fig. 10
Fig. 10

Native VRC results of camera and projector after transformation into the NCS with fitting parameters: CAM [ ( i 0 , j 0 ) = ( 696.5 , 520.4 ) pixels, f = 964.27 pixels], PRJ [ ( i 0 , j 0 ) = ( 501.8 , 397.6 ) pixels, f = 908.26 pixels]: (a), (b) horizontal and vertical camera ray directions with given value range [black to white] in rad, which is additionally recalculated to (angle of view in degrees) via the arc tangent function; (c), (d) same for projector; (e), (f) horizontal and vertical camera ray offsets with given value range (black to white) in mm; (g), (h) same for projector.

Fig. 11
Fig. 11

Camera and projector distortions: forced SVP results (gradients and wavefront shape and curvature) after removal of first- and fifth-order fitting functions with given value range (black to white): (a), (b), (c) camera distortion (first-order fit) amplitude δ t , wavefront shape δ s , and curvature δ c . Additionally, δ t is recalculated into pixel unit distortions δ i j by multiplying with f after Eq. (20) (f given in Fig. 10); (d), (e), (f) same for projector; (g), (h), (i) camera distortion residuals (fifth-order fit) amplitude δ δ t , wavefront shape δ δ s , and curvature δ δ c . Additionally, δ δ t is recalculated into pixel unit distortions δ δ i j by multiplying with f after Eq. (20) (f given in Fig. 10); (j), (k), (l) same for projector.

Equations (20)

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Geometry : Intensity : ( ( i , j ) I ) Vision Ray ν ( p ̲ L ) .
( i j ) + ( δ i δ j ) = f · ( p x / p z p y / p z ) + ( i 0 j 0 ) .
t x tan ( α x ) = p x / p z , t y tan ( α y ) = p y / p z .
v ̲ z = ( o x o y 0 ) + z · ( t x t y 1 ) .
( ( x , y , z ) i , j ) M ( v ̲ z ) i , j .
( O x O y O z ) = ( o x ( i , j ) o y ( i , j ) 0 ) + z o · ( t x ( i , j ) t y ( i , j ) 1 ) .
( 1 0 t x ( i , j ) 0 1 t y ( i , j ) ) · ( O x O y O z ) = ( o x ( i , j ) o y ( i , j ) ) .
R ̲ ̲ = ( r 11 r 12 r 13 r 21 r 22 r 23 r 31 r 32 r 33 ) = ( R ̲ x R ̲ y R ̲ z ) = ( c 3 s 3 0 s 3 c 3 0 0 0 1 ) ( 1 0 0 0 c 2 s 2 0 s 2 c 2 ) ( c 1 s 1 0 s 1 c 1 0 0 0 1 ) .
( i j ) = f · ( R ̲ x · p ̲ g / R ̲ z · p ̲ g R ̲ y · p ̲ g / R ̲ z · p ̲ g ) + ( i 0 j 0 ) .
R ̲ z · r ̲ g z g · r 33 = r 31 r 33 q 31 x g z g t x + r 32 r 33 q 32 y g z g t y + 1.
i = t x ( q 1 1 · f + i 0 · q 3 1 ) h 11 + t y ( q 1 2 · f i 0 · q 3 2 ) h 1 2 + ( q 1 3 · f + i 0 ) h 1 3 i · t x · ( q 3 1 ) q 31 i · t y · ( q 3 2 ) q 32 , j = t x ( q 2 1 · f + j 0 · q 3 1 ) h 2 1 + t y ( q 2 2 · f j 0 · q 3 2 ) h 2 2 + ( q 2 3 · f + j 0 ) h 2 3 j · t x · ( q 3 1 ) q 31 j · t y · ( q 3 2 ) q 32 .
t 1 = q 31 q 32 , t 2 2 = q 31 2 + q 32 2 k t q 31 2 q 32 2 , t 3 = h 11 · q 31 h 12 · q 32 + h 13 · k t h 21 · q 31 + h 22 · q 32 + h 23 · k t , i 0 = h 11 · q 31 + h 12 · q 32 + h 13 1 + k t , j 0 = h 21 · q 31 + h 22 · q 32 + h 23 1 + k t , f = ( h 13 i 0 ) 2 + ( h 23 j 0 ) 2 t 2 2 ,
i = f · t x , j = f · t y .
t x SVP = o x + t x · z z = t x + o x z .
z = 1 m :     t x , SVP = t x , NCS + o x , NCS / m .
Δ triang = δ t x · 1 m .
t = t x , SVP 2 + t y , SVP 2 .
s ( i , j ) = ( 0 , 0 ) ( i , j )   along any path t x , SVP ( i * , j * ) d i * + t y , SVP ( i * , j * ) d j * .
c = 1 2 · ( t x , SVP x + t y , SVP y ) .
δ i = f · δ t x , δ j = f · δ t y , δ i j = f · δ t .

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