Hyper-accurate flexible calibration technique for fringe-projection-based three-dimensional imaging

Minh Vo; Zhaoyang Wang; Bing Pan; Tongyan Pan

doi:10.1364/OE.20.016926

Abstract

Fringe-projection-based (FPB) three-dimensional (3D) imaging technique has become one of the most prevalent methods for 3D shape measurement and 3D image acquisition, and an essential component of the technique is the calibration process. This paper presents a framework for hyper-accurate system calibration with flexible setup and inexpensive hardware. Owing to the crucial improvement in the camera calibration technique, an enhanced governing equation for 3D shape determination, and an advanced flexible system calibration technique as well as some practical considerations on accurate fringe phase retrieval, the novel FPB 3D imaging technique can achieve a relative measurement accuracy of 0.010%. The validity and practicality are verified by both simulation and experiments.

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References

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Author
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Publication

H. Du and Z. Wang, “Three-dimensional shape measurement with an arbitrarily arranged fringe projection profilometry system,” Opt. Lett. 32(16), 2438–2440 (2007).
[Crossref] [PubMed]
L. Huang, P. Chua, and A. Asundi, “Least-squares calibration method for fringe projection profilometry considering camera lens distortion,” Appl. Opt. 49(9), 1539–1548 (2010).
[Crossref] [PubMed]
Z. Wang, D. Nguyen, and J. Barnes, “Some practical considerations in fringe projection profilometry,” Opt. Lasers Eng. 48(2), 218–225 (2010).
[Crossref]
Z. Li, Y. Shi, C. Wang, and Y. Wang, “Accurate calibration method for a structured light system,” Opt. Eng. 47(5), 053604 (2008).
[Crossref]
X. Zhang and L. Zhu, “Projector calibration from the camera image point of view,” Opt. Eng. 48(11), 117208 (2009).
[Crossref]
Z. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 22(11), 1330–1334 (2000).
[Crossref]
J. Lavest, M. Viala, and M. Dhome, “Do we really need an accurate calibration pattern to achieve a reliable camera calibration?” in Proceedings European Conference on Computer Vision (1998), pp. 158–174.
A. Datta, J. Kim, and T. Kanade, “Accurate camera calibration using iterative refinement of control points,” in Proceedings IEEE International Conference on Computer Vision Workshops (IEEE, 2009), 1201–1208.
B. Pan, Q. Kemao, L. Huang, and A. Asundi, “Phase error analysis and compensation for nonsinusoidal waveforms in phase-shifting digital fringe projection profilometry,” Opt. Lett. 34(4), 416–418 (2009).
[Crossref] [PubMed]
J. Zhong and J. Weng, “Phase retrieval of optical fringe patterns from the ridge of a wavelet transform,” Opt. Lett. 30(19), 2560–2562 (2005).
[Crossref] [PubMed]
L. Xiong and S. Jia, “Phase-error analysis and elimination for nonsinusoidal waveforms in Hilbert transform digital-fringe projection profilometry,” Opt. Lett. 34(15), 2363–2365 (2009).
[Crossref] [PubMed]
J. Salvi, S. Fernandez, T. Pribanic, and X. Llado, “A state of art in structured light patterns for surface profilometry,” Pattern Recogn. 43(8), 2666–2680 (2010).
[Crossref]
T. Hoang, B. Pan, D. Nguyen, and Z. Wang, “Generic gamma correction for accuracy enhancement in fringe-projection profilometry,” Opt. Lett. 35(12), 1992–1994 (2010).
[Crossref] [PubMed]
D. Douxchamps and K. Chihara, “High-accuracy and robust localization of large control markers for geometric camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 31(2), 376–383 (2009).
[Crossref]
C. Engels, H. Stewenius, and D. Nister, “Bundle adjustment rules,” in Proceedings on Photogrammetric Computer Vision (2006), pp. 266–271.
J. Heikkila, “Moment and curvature preserving technique for accurate ellipse boundary detection,” in Proceedings IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 1998), pp. 734–737.
L. Luu, Z. Wang, M. Vo, T. Hoang, and J. Ma, “Accuracy enhancement of digital image correlation with B-spline interpolation,” Opt. Lett. 36(16), 3070–3072 (2011).
[Crossref] [PubMed]
B. Pan, K. Qian, H. Xie, and A. Asundi, “Two-dimensional digital image correlation for in-plane displacement and strain measurement: a review,” Meas. Sci. Technol. 20(6), 062001 (2009).
[Crossref]
B. Pan, H. Xie, and Z. Wang, “Equivalence of digital image correlation criteria for pattern matching,” Appl. Opt. 49(28), 5501–5509 (2010).
[Crossref] [PubMed]

2011 (1)

L. Luu, Z. Wang, M. Vo, T. Hoang, and J. Ma, “Accuracy enhancement of digital image correlation with B-spline interpolation,” Opt. Lett. 36(16), 3070–3072 (2011).
[Crossref] [PubMed]

2010 (5)

L. Huang, P. Chua, and A. Asundi, “Least-squares calibration method for fringe projection profilometry considering camera lens distortion,” Appl. Opt. 49(9), 1539–1548 (2010).
[Crossref] [PubMed]

T. Hoang, B. Pan, D. Nguyen, and Z. Wang, “Generic gamma correction for accuracy enhancement in fringe-projection profilometry,” Opt. Lett. 35(12), 1992–1994 (2010).
[Crossref] [PubMed]

B. Pan, H. Xie, and Z. Wang, “Equivalence of digital image correlation criteria for pattern matching,” Appl. Opt. 49(28), 5501–5509 (2010).
[Crossref] [PubMed]

Z. Wang, D. Nguyen, and J. Barnes, “Some practical considerations in fringe projection profilometry,” Opt. Lasers Eng. 48(2), 218–225 (2010).
[Crossref]

J. Salvi, S. Fernandez, T. Pribanic, and X. Llado, “A state of art in structured light patterns for surface profilometry,” Pattern Recogn. 43(8), 2666–2680 (2010).
[Crossref]

2009 (5)

D. Douxchamps and K. Chihara, “High-accuracy and robust localization of large control markers for geometric camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 31(2), 376–383 (2009).
[Crossref]

B. Pan, K. Qian, H. Xie, and A. Asundi, “Two-dimensional digital image correlation for in-plane displacement and strain measurement: a review,” Meas. Sci. Technol. 20(6), 062001 (2009).
[Crossref]

X. Zhang and L. Zhu, “Projector calibration from the camera image point of view,” Opt. Eng. 48(11), 117208 (2009).
[Crossref]

B. Pan, Q. Kemao, L. Huang, and A. Asundi, “Phase error analysis and compensation for nonsinusoidal waveforms in phase-shifting digital fringe projection profilometry,” Opt. Lett. 34(4), 416–418 (2009).
[Crossref] [PubMed]

L. Xiong and S. Jia, “Phase-error analysis and elimination for nonsinusoidal waveforms in Hilbert transform digital-fringe projection profilometry,” Opt. Lett. 34(15), 2363–2365 (2009).
[Crossref] [PubMed]

2008 (1)

Z. Li, Y. Shi, C. Wang, and Y. Wang, “Accurate calibration method for a structured light system,” Opt. Eng. 47(5), 053604 (2008).
[Crossref]

2007 (1)

H. Du and Z. Wang, “Three-dimensional shape measurement with an arbitrarily arranged fringe projection profilometry system,” Opt. Lett. 32(16), 2438–2440 (2007).
[Crossref] [PubMed]

2005 (1)

J. Zhong and J. Weng, “Phase retrieval of optical fringe patterns from the ridge of a wavelet transform,” Opt. Lett. 30(19), 2560–2562 (2005).
[Crossref] [PubMed]

2000 (1)

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

Asundi, A.

L. Huang, P. Chua, and A. Asundi, “Least-squares calibration method for fringe projection profilometry considering camera lens distortion,” Appl. Opt. 49(9), 1539–1548 (2010).
[Crossref] [PubMed]

B. Pan, Q. Kemao, L. Huang, and A. Asundi, “Phase error analysis and compensation for nonsinusoidal waveforms in phase-shifting digital fringe projection profilometry,” Opt. Lett. 34(4), 416–418 (2009).
[Crossref] [PubMed]

B. Pan, K. Qian, H. Xie, and A. Asundi, “Two-dimensional digital image correlation for in-plane displacement and strain measurement: a review,” Meas. Sci. Technol. 20(6), 062001 (2009).
[Crossref]

Barnes, J.

Z. Wang, D. Nguyen, and J. Barnes, “Some practical considerations in fringe projection profilometry,” Opt. Lasers Eng. 48(2), 218–225 (2010).
[Crossref]

Chihara, K.

D. Douxchamps and K. Chihara, “High-accuracy and robust localization of large control markers for geometric camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 31(2), 376–383 (2009).
[Crossref]

Chua, P.

L. Huang, P. Chua, and A. Asundi, “Least-squares calibration method for fringe projection profilometry considering camera lens distortion,” Appl. Opt. 49(9), 1539–1548 (2010).
[Crossref] [PubMed]

Datta, A.

A. Datta, J. Kim, and T. Kanade, “Accurate camera calibration using iterative refinement of control points,” in Proceedings IEEE International Conference on Computer Vision Workshops (IEEE, 2009), 1201–1208.

Dhome, M.

J. Lavest, M. Viala, and M. Dhome, “Do we really need an accurate calibration pattern to achieve a reliable camera calibration?” in Proceedings European Conference on Computer Vision (1998), pp. 158–174.

Douxchamps, D.

D. Douxchamps and K. Chihara, “High-accuracy and robust localization of large control markers for geometric camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 31(2), 376–383 (2009).
[Crossref]

Du, H.

H. Du and Z. Wang, “Three-dimensional shape measurement with an arbitrarily arranged fringe projection profilometry system,” Opt. Lett. 32(16), 2438–2440 (2007).
[Crossref] [PubMed]

Engels, C.

C. Engels, H. Stewenius, and D. Nister, “Bundle adjustment rules,” in Proceedings on Photogrammetric Computer Vision (2006), pp. 266–271.

Fernandez, S.

J. Salvi, S. Fernandez, T. Pribanic, and X. Llado, “A state of art in structured light patterns for surface profilometry,” Pattern Recogn. 43(8), 2666–2680 (2010).
[Crossref]

Heikkila, J.

J. Heikkila, “Moment and curvature preserving technique for accurate ellipse boundary detection,” in Proceedings IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 1998), pp. 734–737.

Kanade, T.

A. Datta, J. Kim, and T. Kanade, “Accurate camera calibration using iterative refinement of control points,” in Proceedings IEEE International Conference on Computer Vision Workshops (IEEE, 2009), 1201–1208.

Kemao, Q.

B. Pan, Q. Kemao, L. Huang, and A. Asundi, “Phase error analysis and compensation for nonsinusoidal waveforms in phase-shifting digital fringe projection profilometry,” Opt. Lett. 34(4), 416–418 (2009).
[Crossref] [PubMed]

Kim, J.

A. Datta, J. Kim, and T. Kanade, “Accurate camera calibration using iterative refinement of control points,” in Proceedings IEEE International Conference on Computer Vision Workshops (IEEE, 2009), 1201–1208.

Lavest, J.

J. Lavest, M. Viala, and M. Dhome, “Do we really need an accurate calibration pattern to achieve a reliable camera calibration?” in Proceedings European Conference on Computer Vision (1998), pp. 158–174.

Li, Z.

Z. Li, Y. Shi, C. Wang, and Y. Wang, “Accurate calibration method for a structured light system,” Opt. Eng. 47(5), 053604 (2008).
[Crossref]

Llado, X.

J. Salvi, S. Fernandez, T. Pribanic, and X. Llado, “A state of art in structured light patterns for surface profilometry,” Pattern Recogn. 43(8), 2666–2680 (2010).
[Crossref]

Luu, L.

L. Luu, Z. Wang, M. Vo, T. Hoang, and J. Ma, “Accuracy enhancement of digital image correlation with B-spline interpolation,” Opt. Lett. 36(16), 3070–3072 (2011).
[Crossref] [PubMed]

Ma, J.

L. Luu, Z. Wang, M. Vo, T. Hoang, and J. Ma, “Accuracy enhancement of digital image correlation with B-spline interpolation,” Opt. Lett. 36(16), 3070–3072 (2011).
[Crossref] [PubMed]

Nguyen, D.

T. Hoang, B. Pan, D. Nguyen, and Z. Wang, “Generic gamma correction for accuracy enhancement in fringe-projection profilometry,” Opt. Lett. 35(12), 1992–1994 (2010).
[Crossref] [PubMed]

Z. Wang, D. Nguyen, and J. Barnes, “Some practical considerations in fringe projection profilometry,” Opt. Lasers Eng. 48(2), 218–225 (2010).
[Crossref]

Nister, D.

C. Engels, H. Stewenius, and D. Nister, “Bundle adjustment rules,” in Proceedings on Photogrammetric Computer Vision (2006), pp. 266–271.

Pan, B.

B. Pan, H. Xie, and Z. Wang, “Equivalence of digital image correlation criteria for pattern matching,” Appl. Opt. 49(28), 5501–5509 (2010).
[Crossref] [PubMed]

T. Hoang, B. Pan, D. Nguyen, and Z. Wang, “Generic gamma correction for accuracy enhancement in fringe-projection profilometry,” Opt. Lett. 35(12), 1992–1994 (2010).
[Crossref] [PubMed]

B. Pan, Q. Kemao, L. Huang, and A. Asundi, “Phase error analysis and compensation for nonsinusoidal waveforms in phase-shifting digital fringe projection profilometry,” Opt. Lett. 34(4), 416–418 (2009).
[Crossref] [PubMed]

B. Pan, K. Qian, H. Xie, and A. Asundi, “Two-dimensional digital image correlation for in-plane displacement and strain measurement: a review,” Meas. Sci. Technol. 20(6), 062001 (2009).
[Crossref]

Pribanic, T.

J. Salvi, S. Fernandez, T. Pribanic, and X. Llado, “A state of art in structured light patterns for surface profilometry,” Pattern Recogn. 43(8), 2666–2680 (2010).
[Crossref]

Qian, K.

B. Pan, K. Qian, H. Xie, and A. Asundi, “Two-dimensional digital image correlation for in-plane displacement and strain measurement: a review,” Meas. Sci. Technol. 20(6), 062001 (2009).
[Crossref]

Salvi, J.

J. Salvi, S. Fernandez, T. Pribanic, and X. Llado, “A state of art in structured light patterns for surface profilometry,” Pattern Recogn. 43(8), 2666–2680 (2010).
[Crossref]

Shi, Y.

Z. Li, Y. Shi, C. Wang, and Y. Wang, “Accurate calibration method for a structured light system,” Opt. Eng. 47(5), 053604 (2008).
[Crossref]

Stewenius, H.

C. Engels, H. Stewenius, and D. Nister, “Bundle adjustment rules,” in Proceedings on Photogrammetric Computer Vision (2006), pp. 266–271.

Viala, M.

J. Lavest, M. Viala, and M. Dhome, “Do we really need an accurate calibration pattern to achieve a reliable camera calibration?” in Proceedings European Conference on Computer Vision (1998), pp. 158–174.

Vo, M.

L. Luu, Z. Wang, M. Vo, T. Hoang, and J. Ma, “Accuracy enhancement of digital image correlation with B-spline interpolation,” Opt. Lett. 36(16), 3070–3072 (2011).
[Crossref] [PubMed]

Wang, C.

Z. Li, Y. Shi, C. Wang, and Y. Wang, “Accurate calibration method for a structured light system,” Opt. Eng. 47(5), 053604 (2008).
[Crossref]

Wang, Y.

Z. Li, Y. Shi, C. Wang, and Y. Wang, “Accurate calibration method for a structured light system,” Opt. Eng. 47(5), 053604 (2008).
[Crossref]

Wang, Z.

L. Luu, Z. Wang, M. Vo, T. Hoang, and J. Ma, “Accuracy enhancement of digital image correlation with B-spline interpolation,” Opt. Lett. 36(16), 3070–3072 (2011).
[Crossref] [PubMed]

B. Pan, H. Xie, and Z. Wang, “Equivalence of digital image correlation criteria for pattern matching,” Appl. Opt. 49(28), 5501–5509 (2010).
[Crossref] [PubMed]

T. Hoang, B. Pan, D. Nguyen, and Z. Wang, “Generic gamma correction for accuracy enhancement in fringe-projection profilometry,” Opt. Lett. 35(12), 1992–1994 (2010).
[Crossref] [PubMed]

Z. Wang, D. Nguyen, and J. Barnes, “Some practical considerations in fringe projection profilometry,” Opt. Lasers Eng. 48(2), 218–225 (2010).
[Crossref]

H. Du and Z. Wang, “Three-dimensional shape measurement with an arbitrarily arranged fringe projection profilometry system,” Opt. Lett. 32(16), 2438–2440 (2007).
[Crossref] [PubMed]

Weng, J.

J. Zhong and J. Weng, “Phase retrieval of optical fringe patterns from the ridge of a wavelet transform,” Opt. Lett. 30(19), 2560–2562 (2005).
[Crossref] [PubMed]

Xie, H.

B. Pan, H. Xie, and Z. Wang, “Equivalence of digital image correlation criteria for pattern matching,” Appl. Opt. 49(28), 5501–5509 (2010).
[Crossref] [PubMed]

B. Pan, K. Qian, H. Xie, and A. Asundi, “Two-dimensional digital image correlation for in-plane displacement and strain measurement: a review,” Meas. Sci. Technol. 20(6), 062001 (2009).
[Crossref]

Xiong, L.

L. Xiong and S. Jia, “Phase-error analysis and elimination for nonsinusoidal waveforms in Hilbert transform digital-fringe projection profilometry,” Opt. Lett. 34(15), 2363–2365 (2009).
[Crossref] [PubMed]

Zhang, X.

X. Zhang and L. Zhu, “Projector calibration from the camera image point of view,” Opt. Eng. 48(11), 117208 (2009).
[Crossref]

Zhang, Z.

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

Zhong, J.

J. Zhong and J. Weng, “Phase retrieval of optical fringe patterns from the ridge of a wavelet transform,” Opt. Lett. 30(19), 2560–2562 (2005).
[Crossref] [PubMed]

Zhu, L.

X. Zhang and L. Zhu, “Projector calibration from the camera image point of view,” Opt. Eng. 48(11), 117208 (2009).
[Crossref]

Appl. Opt. (2)

L. Huang, P. Chua, and A. Asundi, “Least-squares calibration method for fringe projection profilometry considering camera lens distortion,” Appl. Opt. 49(9), 1539–1548 (2010).
[Crossref] [PubMed]

B. Pan, H. Xie, and Z. Wang, “Equivalence of digital image correlation criteria for pattern matching,” Appl. Opt. 49(28), 5501–5509 (2010).
[Crossref] [PubMed]

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

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

D. Douxchamps and K. Chihara, “High-accuracy and robust localization of large control markers for geometric camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 31(2), 376–383 (2009).
[Crossref]

Meas. Sci. Technol. (1)

B. Pan, K. Qian, H. Xie, and A. Asundi, “Two-dimensional digital image correlation for in-plane displacement and strain measurement: a review,” Meas. Sci. Technol. 20(6), 062001 (2009).
[Crossref]

Opt. Eng. (2)

Z. Li, Y. Shi, C. Wang, and Y. Wang, “Accurate calibration method for a structured light system,” Opt. Eng. 47(5), 053604 (2008).
[Crossref]

X. Zhang and L. Zhu, “Projector calibration from the camera image point of view,” Opt. Eng. 48(11), 117208 (2009).
[Crossref]

Opt. Lasers Eng. (1)

Z. Wang, D. Nguyen, and J. Barnes, “Some practical considerations in fringe projection profilometry,” Opt. Lasers Eng. 48(2), 218–225 (2010).
[Crossref]

Opt. Lett. (6)

B. Pan, Q. Kemao, L. Huang, and A. Asundi, “Phase error analysis and compensation for nonsinusoidal waveforms in phase-shifting digital fringe projection profilometry,” Opt. Lett. 34(4), 416–418 (2009).
[Crossref] [PubMed]

J. Zhong and J. Weng, “Phase retrieval of optical fringe patterns from the ridge of a wavelet transform,” Opt. Lett. 30(19), 2560–2562 (2005).
[Crossref] [PubMed]

L. Xiong and S. Jia, “Phase-error analysis and elimination for nonsinusoidal waveforms in Hilbert transform digital-fringe projection profilometry,” Opt. Lett. 34(15), 2363–2365 (2009).
[Crossref] [PubMed]

L. Luu, Z. Wang, M. Vo, T. Hoang, and J. Ma, “Accuracy enhancement of digital image correlation with B-spline interpolation,” Opt. Lett. 36(16), 3070–3072 (2011).
[Crossref] [PubMed]

H. Du and Z. Wang, “Three-dimensional shape measurement with an arbitrarily arranged fringe projection profilometry system,” Opt. Lett. 32(16), 2438–2440 (2007).
[Crossref] [PubMed]

T. Hoang, B. Pan, D. Nguyen, and Z. Wang, “Generic gamma correction for accuracy enhancement in fringe-projection profilometry,” Opt. Lett. 35(12), 1992–1994 (2010).
[Crossref] [PubMed]

Pattern Recogn. (1)

J. Salvi, S. Fernandez, T. Pribanic, and X. Llado, “A state of art in structured light patterns for surface profilometry,” Pattern Recogn. 43(8), 2666–2680 (2010).
[Crossref]

Other (4)

J. Lavest, M. Viala, and M. Dhome, “Do we really need an accurate calibration pattern to achieve a reliable camera calibration?” in Proceedings European Conference on Computer Vision (1998), pp. 158–174.

A. Datta, J. Kim, and T. Kanade, “Accurate camera calibration using iterative refinement of control points,” in Proceedings IEEE International Conference on Computer Vision Workshops (IEEE, 2009), 1201–1208.

C. Engels, H. Stewenius, and D. Nister, “Bundle adjustment rules,” in Proceedings on Photogrammetric Computer Vision (2006), pp. 266–271.

J. Heikkila, “Moment and curvature preserving technique for accurate ellipse boundary detection,” in Proceedings IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 1998), pp. 734–737.

Supplementary Material (1)

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Fig. 1 The conversion from raw image (left) to frontal image (middle) enables the correlation with the ring templates (right).

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Fig. 2 Representative images of system calibration: calibration board with projection fringes, unwrapped phase map, and out-of-reference-plane height map.

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Fig. 3 Localization errors of the control points obtained by: (a) Heikkila’s method, and (b) the frontal image correlation method.

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Fig. 4 Convergence of the proposed camera calibration scheme.

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Fig. 5 Errors of control point detection with different interpolation methods.

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Fig. 6 Reprojection error of camera calibration with different interpolation methods.

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Fig. 7 Reprojection errors at the control points obtained by: (a) the conventional method, and (b) the frontal image correlation method. The vector scales are different in the figures for clear illustration purpose.

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Fig. 8 Phase retrieval error as a function of: (a) number of images, and (b) gamma pre-encoded values.

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Fig. 9 The effect of pre-encoded gamma values on the captured fringe patterns.

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Fig. 10 3D imaging results of a plate with eight gage blocks.

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Fig. 11 A conch shell and its 3D images observed from five different views.

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Fig. 12 A printed circuit board, the 2D height map, and the 3D rendered surface.

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Fig. 13 A horse sculpture, a 2D height map in color, and the 3D images.

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Fig. 14 A lion toy and its 3D images ( Media 1).

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Tables (2)

Table 1 Retrieved calibration parameters and their accuracy assessment

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Table 2 Actual and measured heights of gage blocks

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Equations (16)

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

s {\begin{matrix} m \\ 1 \end{matrix}} = A [\begin{matrix} R & T \end{matrix}] {\begin{matrix} M \\ 1 \end{matrix}}, A = [\begin{matrix} α & γ & u_{0} \\ 0 & β & v_{0} \\ 0 & 0 & 1 \end{matrix}]

\begin{array}{r} x_{c n}^{'} = (1 + a_{0} r^{2} + a_{1} r^{4} + a_{2} r^{6}) x_{c n} + (p_{0} + r^{2} p_{2}) (r^{2} + 2 x_{c n}^{2}) + 2 (p_{1} + r^{2} p_{3}) w + s_{0} r^{2} + s_{2} r^{4} \\ y_{c n}^{'} = (1 + a_{0} r^{2} + a_{1} r^{4} + a_{2} r^{6}) y_{c n} + (p_{1} + r^{2} p_{3}) (r^{2} + 2 y_{c n}^{2}) + 2 (p_{0} + r^{2} p_{2}) w + s_{1} r^{2} + s_{3} r^{4} \\ r^{2} = x_{c n}^{2} + y_{c n}^{2}, w = x_{c n} y_{c n} \end{array}

{\begin{matrix} x_{c n} \\ y_{c n} \\ 1 \end{matrix}} = A^{- 1} {\begin{matrix} u \\ v \\ 1 \end{matrix}}, {\begin{matrix} x_{c n}^{'} \\ y_{c n}^{'} \\ 1 \end{matrix}} = A^{- 1} {\begin{matrix} u^{'} \\ v^{'} \\ 1 \end{matrix}}

S = \sum_{i = 1}^{k} \sum_{j = 1}^{l} {| | m_{i j} - P (A, φ, ρ_{i}, T_{i}, M_{j}) | |}^{2}

C = \sum_{i = 1}^{N} {[a f (x_{i}, y_{i}) + b - g (x_{i}^{'}, y_{i}^{'})]}^{2}

\begin{array}{l} x_{i}^{'} = x_{i} + ξ + s_{x} (x_{i} - x_{0}) \\ y_{i}^{'} = y_{i} + η + s_{y} (y_{i} - y_{0}) \end{array}

I_{0 i} (x, y) = I_{c} [1 + cos (2 π f x / W + δ_{i})]

I_{i} (u, v) = a (u, v) + \sum_{j = 1}^{p} b_{j} (u, v) cos {j [ϕ (u, v) + δ_{i}]}

ϕ^{w} (u, v) = arctan \frac{- \sum_{i = 1}^{N} sin (δ_{i}) I_{i} (u, v)}{\sum_{i = 1}^{N} cos (δ_{i}) I_{i} (u, v)}

\hat{I} = {\hat{I}}_{0}^{γ_{0}}

\hat{I} = {({\hat{I}}_{0}^{\frac{1}{γ_{p}}})}^{γ_{0}} = {\hat{I}}_{0}^{γ_{0} / γ_{p}}

ϕ_{i} (u, v) = ϕ_{i}^{w} (u, v) + 2 π \cdot I N T [\frac{ϕ_{i - 1}^{u w} \cdot (f_{i} / f_{i - 1}) - ϕ_{i}^{w}}{2 π}]

Z = \frac{1 + c_{1} ϕ + (c_{2} + c_{3} ϕ) u + (c_{4} + c_{5} ϕ) v}{d_{0} + d_{1} ϕ + (d_{2} + d_{3} ϕ) u + (d_{4} + d_{5} ϕ) v}

\begin{array}{l} Z & = & \frac{F_{c}}{F_{d}} \\ F_{c} & = & 1 + c_{1} ϕ + (c_{2} + c_{3} ϕ) u + (c_{4} + c_{5} ϕ) v + (c_{6} + c_{7} ϕ) u^{2} + (c_{8} + c_{9} ϕ) v^{2} \\ + (c_{10} + c_{11} ϕ) u v + (c_{12} + c_{13} ϕ) u^{2} v + (c_{14} + c_{15} ϕ) u v^{2} + (c_{16} + c_{17} ϕ) u^{2} v^{2} \\ F_{d} & = & d_{0} + d_{1} ϕ + (d_{2} + d_{3} ϕ) u + (d_{4} + d_{5} ϕ) v + (d_{6} + d_{7} ϕ) u^{2} + (d_{8} + d_{9} ϕ) v^{2} \\ + (d_{10} + d_{11} ϕ) u v + (d_{12} + d_{13} ϕ) u^{2} v + (d_{14} + d_{15} ϕ) u v^{2} + (d_{16} + d_{17} ϕ) u^{2} v^{2} \end{array}

Z_{i j} = \frac{A X_{c, i j} + B Y_{c, i j} + C Z_{c, i j} + 1}{\sqrt{A^{2} + B^{2} + C^{2}}}

S = \sum_{i = 1}^{k} {\sum_{j = 1}^{l} (\frac{F_{c}}{F_{d}} - Z_{i j})}^{2}

	True	Retrieved	Error(%)	Precision
α	5731^a	5730.9840^a	−0.0002791	0.00843
β	5731^a	5730.9835^a	−0.0002879	0.00838
γ	0.3260^a	0.3258^a	−0.06134	0.00110
u₀	1051^a	1051.0409^a	0.003891	0.0733
v₀	778^a	778.0414^a	0.005321	0.0895
a₀	0.1860	0.186038	0.02043	0.0000275
a₁	1.0314	1.028625	−0.2691	0.00176
a₂	22.00	22.0534	0.2427	0.0352
p₀	−0.00280	−0.00280505	0.1804	0.00000859
p₁	0.00259	0.00258619	−0.1471	0.0000104
p₂	−0.0030	−0.00300397	0.1322	0.0000695
p₃	0.0570	0.000569556	−0.07789	0.00000795
s₀	0.00520	0.00520503	0.09673	0.00000659
s₁	−0.00450	−0.00449727	−0.006007	0.0000785
s₂	0.0220	0.0219588	−0.178	0.0000846
s₃	−0.0270	−0.0270167	0.06185	0.0000704
θ_x	160.499^b	160.49904^b	0.00002492	0.0000176
θ_y	−12.0236^b	−12.023197^b	−0.003352	0.0000145
θ_z	1.35350^b	1.3535902^b	0.006664	0.00000263
T_x	−97.9818^c	−97.984577^c	−0.00283	0.00291
T_y	74.6039^c	74.606202^c	−0.00308	0.00217
T_z	1228.6703^c	1228.6658^c	−0.003662	0.00309

Gage	Actual	Mean measured	Error	Standard deviation
1	25.400	25.363	−0.037	0.033
2	19.050	19.048	−0.002	0.039
3	6.350	6.398	0.048	0.027
4	6.350	6.381	0.031	0.041
5	12.700	12.670	−0.030	0.032
6	15.875	15.898	0.023	0.036
7	9.525	9.478	−0.047	0.039
8	50.80	50.766	−0.034	0.033

	True	Retrieved	Error(%)	Precision
α	5731^a	5730.9840^a	−0.0002791	0.00843
β	5731^a	5730.9835^a	−0.0002879	0.00838
γ	0.3260^a	0.3258^a	−0.06134	0.00110
u₀	1051^a	1051.0409^a	0.003891	0.0733
v₀	778^a	778.0414^a	0.005321	0.0895
a₀	0.1860	0.186038	0.02043	0.0000275
a₁	1.0314	1.028625	−0.2691	0.00176
a₂	22.00	22.0534	0.2427	0.0352
p₀	−0.00280	−0.00280505	0.1804	0.00000859
p₁	0.00259	0.00258619	−0.1471	0.0000104
p₂	−0.0030	−0.00300397	0.1322	0.0000695
p₃	0.0570	0.000569556	−0.07789	0.00000795
s₀	0.00520	0.00520503	0.09673	0.00000659
s₁	−0.00450	−0.00449727	−0.006007	0.0000785
s₂	0.0220	0.0219588	−0.178	0.0000846
s₃	−0.0270	−0.0270167	0.06185	0.0000704
θ_x	160.499^b	160.49904^b	0.00002492	0.0000176
θ_y	−12.0236^b	−12.023197^b	−0.003352	0.0000145
θ_z	1.35350^b	1.3535902^b	0.006664	0.00000263
T_x	−97.9818^c	−97.984577^c	−0.00283	0.00291
T_y	74.6039^c	74.606202^c	−0.00308	0.00217
T_z	1228.6703^c	1228.6658^c	−0.003662	0.00309

Gage	Actual	Mean measured	Error	Standard deviation
1	25.400	25.363	−0.037	0.033
2	19.050	19.048	−0.002	0.039
3	6.350	6.398	0.048	0.027
4	6.350	6.381	0.031	0.041
5	12.700	12.670	−0.030	0.032
6	15.875	15.898	0.023	0.036
7	9.525	9.478	−0.047	0.039
8	50.80	50.766	−0.034	0.033

Abstract

References

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Appl. Opt. (2)

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

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