Abstract

The phase measurement deflectometry considering the refraction effect is presented to measure the mirror surface in this paper. In the context of the conventional phase measurement deflectometry, the biplanar structure of the system constructed by spatial multiplexing of a screen or a half mirror with two screens is a compromise of traditional display technology, while they suffer from complex calibration process and low accuracy. To improve the system compactness and efficiency, a novel measurement model consisting of a transparent screen and an ordinary screen is used to determine the incident light. To compensate for the measurement errors caused by transparent screen refraction, the refraction of the transparent screen is characterized by two physical parameters, which can be calibrated thanks to the multi-stereo vision technique. Then, the improved mirror calibration method with the refraction model is proposed to determine the posed relationship of the system. After that, the three-dimensional (3D) information of mirror surface is restored by the radial basis function interpolation with the optimized refraction parameters and posed relationships from the gradient data which is transformed from the normal information. Higher measurement efficiency, higher measurement accuracy and more compactness of the proposed measurement method are verified by the experimental results.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
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References

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2018 (4)

P. Zhao, N. Gao, Z. Zhang, F. Gao, and X. Jiang, “Performance analysis and evaluation of direct phase measuring deflectometry,” Opt. Lasers Eng. 103, 24–33 (2018).
[Crossref]

S. Huang, Y. Liu, N. Gao, Z. Zhang, F. Gao, and X. Jiang, “Distance calibration between reference plane and screen in direct phase measuring deflectometry,” Sensors (Basel) 18(2), 144 (2018).
[Crossref] [PubMed]

Y. Xu, F. Gao, Z. Zhang, and X. Jiang, “A holistic calibration method with iterative distortion compensation for stereo deflectometry,” Opt. Lasers Eng. 106, 111–118 (2018).
[Crossref]

C. Li, X. Zhang, and D. Tu, “Posed relationship calibration with parallel mirror reflection for stereo deflectometry,” Opt. Eng. 57(3), 034103 (2018).
[Crossref]

2017 (2)

C. Li, X. Zhang, D. Tu, J. Jia, W. Cui, and C. Zhang, “Deflectometry measurement method of single-camera monitoring,” Acta Opt. Sin. 37(10), 1012007 (2017).
[Crossref]

Z. H. Zhang, J. Guo, Y. M. Wang, S. J. Huang, N. Gao, and Y. J. Xiao, “Parallel-alignment and correction of two displays in three-dimensional measuring system of specular surfaces,” Opt. Precis. Eng. 2, 002 (2017).

2015 (5)

2014 (1)

L. Xiao, X.-G. Xia, and W. Wang, “Multi-stage robust chinese remainder theorem,” IEEE Trans. Signal Process. 62(18), 4772–4785 (2014).
[Crossref]

2013 (1)

P. Su, M. Khreishi, T. Su, R. Huang, M. Z. Dominguez, A. V. Maldonado, G. P. Butel, Y. Wang, R. E. Parks, and J. H. Burge, “Aspheric and freeform surfaces metrology with software configurable optical test system: a computerized reverse hartmann test,” Opt. Eng. 53(3), 031305 (2013).
[Crossref]

2012 (2)

Y.-L. Xiao, X. Su, W. Chen, and Y. Liu, “Three-dimensional shape measurement of aspheric mirrors with fringe reflection photogrammetry,” Appl. Opt. 51(4), 457–464 (2012).
[Crossref] [PubMed]

X. Y. S. X. C. Wenjing, “Fringe reflection photogrammetry based on pose estimation with free planar mirror reflection,” Acta Opt. Sinica 5, 013 (2012).

2011 (1)

2010 (2)

H. Guo, P. Feng, and T. Tao, “Specular surface measurement by using least squares light tracking technique,” Opt. Lasers Eng. 48(2), 166–171 (2010).
[Crossref]

P. Su, R. E. Parks, L. Wang, R. P. Angel, and J. H. Burge, “Software configurable optical test system: a computerized reverse Hartmann test,” Appl. Opt. 49(23), 4404–4412 (2010).
[Crossref] [PubMed]

2008 (4)

2007 (1)

S. Ettl, J. Kaminski, and G. Häusler, “Generalized hermite interpolation with radial basis functions considering only gradient data,” Curve Surf. Fitting: Avignon 2006, 141–149 (2007).

1988 (1)

R. H. Byrd, R. B. Schnabel, and G. A. Shultz, “Approximate solution of the trust region problem by minimization over two-dimensional subspaces,” Math. Program. 40(40), 247–263 (1988).
[Crossref]

1983 (1)

T. Steihaug, “The conjugate gradient method and trust regions in large scale optimization,” SIAM J. Numer. Anal. 20(3), 626–637 (1983).
[Crossref]

Angel, R. P.

Asundi, A.

L. Huang, M. Idir, C. Zuo, K. Kaznatcheev, L. Zhou, and A. Asundi, “Comparison of two-dimensional integration methods for shape reconstruction from gradient data,” Opt. Lasers Eng. 64, 1–11 (2015).
[Crossref]

Burge, J. H.

P. Su, M. Khreishi, T. Su, R. Huang, M. Z. Dominguez, A. V. Maldonado, G. P. Butel, Y. Wang, R. E. Parks, and J. H. Burge, “Aspheric and freeform surfaces metrology with software configurable optical test system: a computerized reverse hartmann test,” Opt. Eng. 53(3), 031305 (2013).
[Crossref]

P. Su, R. E. Parks, L. Wang, R. P. Angel, and J. H. Burge, “Software configurable optical test system: a computerized reverse Hartmann test,” Appl. Opt. 49(23), 4404–4412 (2010).
[Crossref] [PubMed]

Butel, G. P.

P. Su, M. Khreishi, T. Su, R. Huang, M. Z. Dominguez, A. V. Maldonado, G. P. Butel, Y. Wang, R. E. Parks, and J. H. Burge, “Aspheric and freeform surfaces metrology with software configurable optical test system: a computerized reverse hartmann test,” Opt. Eng. 53(3), 031305 (2013).
[Crossref]

Byrd, R. H.

R. H. Byrd, R. B. Schnabel, and G. A. Shultz, “Approximate solution of the trust region problem by minimization over two-dimensional subspaces,” Math. Program. 40(40), 247–263 (1988).
[Crossref]

Chen, W.

Cui, W.

C. Li, X. Zhang, D. Tu, J. Jia, W. Cui, and C. Zhang, “Deflectometry measurement method of single-camera monitoring,” Acta Opt. Sin. 37(10), 1012007 (2017).
[Crossref]

Dominguez, M. Z.

P. Su, M. Khreishi, T. Su, R. Huang, M. Z. Dominguez, A. V. Maldonado, G. P. Butel, Y. Wang, R. E. Parks, and J. H. Burge, “Aspheric and freeform surfaces metrology with software configurable optical test system: a computerized reverse hartmann test,” Opt. Eng. 53(3), 031305 (2013).
[Crossref]

Ettl, S.

S. Ettl, J. Kaminski, M. C. Knauer, and G. Häusler, “Shape reconstruction from gradient data,” Appl. Opt. 47(12), 2091–2097 (2008).
[Crossref] [PubMed]

S. Ettl, J. Kaminski, and G. Häusler, “Generalized hermite interpolation with radial basis functions considering only gradient data,” Curve Surf. Fitting: Avignon 2006, 141–149 (2007).

Feng, P.

H. Guo, P. Feng, and T. Tao, “Specular surface measurement by using least squares light tracking technique,” Opt. Lasers Eng. 48(2), 166–171 (2010).
[Crossref]

Gao, F.

S. Huang, Y. Liu, N. Gao, Z. Zhang, F. Gao, and X. Jiang, “Distance calibration between reference plane and screen in direct phase measuring deflectometry,” Sensors (Basel) 18(2), 144 (2018).
[Crossref] [PubMed]

P. Zhao, N. Gao, Z. Zhang, F. Gao, and X. Jiang, “Performance analysis and evaluation of direct phase measuring deflectometry,” Opt. Lasers Eng. 103, 24–33 (2018).
[Crossref]

Y. Xu, F. Gao, Z. Zhang, and X. Jiang, “A holistic calibration method with iterative distortion compensation for stereo deflectometry,” Opt. Lasers Eng. 106, 111–118 (2018).
[Crossref]

H. Ren, F. Gao, and X. Jiang, “Iterative optimization calibration method for stereo deflectometry,” Opt. Express 23(17), 22060–22068 (2015).
[Crossref] [PubMed]

H. Ren, F. Gao, and X. Jiang, “Improvement of high-order least-squares integration method for stereo deflectometry,” Appl. Opt. 54(34), 10249–10255 (2015).
[Crossref] [PubMed]

Gao, N.

P. Zhao, N. Gao, Z. Zhang, F. Gao, and X. Jiang, “Performance analysis and evaluation of direct phase measuring deflectometry,” Opt. Lasers Eng. 103, 24–33 (2018).
[Crossref]

S. Huang, Y. Liu, N. Gao, Z. Zhang, F. Gao, and X. Jiang, “Distance calibration between reference plane and screen in direct phase measuring deflectometry,” Sensors (Basel) 18(2), 144 (2018).
[Crossref] [PubMed]

Z. H. Zhang, J. Guo, Y. M. Wang, S. J. Huang, N. Gao, and Y. J. Xiao, “Parallel-alignment and correction of two displays in three-dimensional measuring system of specular surfaces,” Opt. Precis. Eng. 2, 002 (2017).

Guo, H.

H. Guo, P. Feng, and T. Tao, “Specular surface measurement by using least squares light tracking technique,” Opt. Lasers Eng. 48(2), 166–171 (2010).
[Crossref]

Guo, J.

Z. H. Zhang, J. Guo, Y. M. Wang, S. J. Huang, N. Gao, and Y. J. Xiao, “Parallel-alignment and correction of two displays in three-dimensional measuring system of specular surfaces,” Opt. Precis. Eng. 2, 002 (2017).

Häusler, G.

Hu, S.

Huang, L.

L. Huang, M. Idir, C. Zuo, K. Kaznatcheev, L. Zhou, and A. Asundi, “Comparison of two-dimensional integration methods for shape reconstruction from gradient data,” Opt. Lasers Eng. 64, 1–11 (2015).
[Crossref]

Huang, R.

P. Su, M. Khreishi, T. Su, R. Huang, M. Z. Dominguez, A. V. Maldonado, G. P. Butel, Y. Wang, R. E. Parks, and J. H. Burge, “Aspheric and freeform surfaces metrology with software configurable optical test system: a computerized reverse hartmann test,” Opt. Eng. 53(3), 031305 (2013).
[Crossref]

Huang, S.

S. Huang, Y. Liu, N. Gao, Z. Zhang, F. Gao, and X. Jiang, “Distance calibration between reference plane and screen in direct phase measuring deflectometry,” Sensors (Basel) 18(2), 144 (2018).
[Crossref] [PubMed]

Huang, S. J.

Z. H. Zhang, J. Guo, Y. M. Wang, S. J. Huang, N. Gao, and Y. J. Xiao, “Parallel-alignment and correction of two displays in three-dimensional measuring system of specular surfaces,” Opt. Precis. Eng. 2, 002 (2017).

Idir, M.

L. Huang, M. Idir, C. Zuo, K. Kaznatcheev, L. Zhou, and A. Asundi, “Comparison of two-dimensional integration methods for shape reconstruction from gradient data,” Opt. Lasers Eng. 64, 1–11 (2015).
[Crossref]

Jia, J.

C. Li, X. Zhang, D. Tu, J. Jia, W. Cui, and C. Zhang, “Deflectometry measurement method of single-camera monitoring,” Acta Opt. Sin. 37(10), 1012007 (2017).
[Crossref]

Jiang, X.

Y. Xu, F. Gao, Z. Zhang, and X. Jiang, “A holistic calibration method with iterative distortion compensation for stereo deflectometry,” Opt. Lasers Eng. 106, 111–118 (2018).
[Crossref]

S. Huang, Y. Liu, N. Gao, Z. Zhang, F. Gao, and X. Jiang, “Distance calibration between reference plane and screen in direct phase measuring deflectometry,” Sensors (Basel) 18(2), 144 (2018).
[Crossref] [PubMed]

P. Zhao, N. Gao, Z. Zhang, F. Gao, and X. Jiang, “Performance analysis and evaluation of direct phase measuring deflectometry,” Opt. Lasers Eng. 103, 24–33 (2018).
[Crossref]

H. Ren, F. Gao, and X. Jiang, “Improvement of high-order least-squares integration method for stereo deflectometry,” Appl. Opt. 54(34), 10249–10255 (2015).
[Crossref] [PubMed]

H. Ren, F. Gao, and X. Jiang, “Iterative optimization calibration method for stereo deflectometry,” Opt. Express 23(17), 22060–22068 (2015).
[Crossref] [PubMed]

Jing, H.

Kaminski, J.

S. Ettl, J. Kaminski, M. C. Knauer, and G. Häusler, “Shape reconstruction from gradient data,” Appl. Opt. 47(12), 2091–2097 (2008).
[Crossref] [PubMed]

S. Ettl, J. Kaminski, and G. Häusler, “Generalized hermite interpolation with radial basis functions considering only gradient data,” Curve Surf. Fitting: Avignon 2006, 141–149 (2007).

Kaznatcheev, K.

L. Huang, M. Idir, C. Zuo, K. Kaznatcheev, L. Zhou, and A. Asundi, “Comparison of two-dimensional integration methods for shape reconstruction from gradient data,” Opt. Lasers Eng. 64, 1–11 (2015).
[Crossref]

Khreishi, M.

P. Su, M. Khreishi, T. Su, R. Huang, M. Z. Dominguez, A. V. Maldonado, G. P. Butel, Y. Wang, R. E. Parks, and J. H. Burge, “Aspheric and freeform surfaces metrology with software configurable optical test system: a computerized reverse hartmann test,” Opt. Eng. 53(3), 031305 (2013).
[Crossref]

Knauer, M. C.

Leitz, K.-H.

Li, C.

C. Li, X. Zhang, and D. Tu, “Posed relationship calibration with parallel mirror reflection for stereo deflectometry,” Opt. Eng. 57(3), 034103 (2018).
[Crossref]

C. Li, X. Zhang, D. Tu, J. Jia, W. Cui, and C. Zhang, “Deflectometry measurement method of single-camera monitoring,” Acta Opt. Sin. 37(10), 1012007 (2017).
[Crossref]

Li, Y. F.

D. Xu, Y. F. Li, and M. Tan, “A general recursive linear method and unique solution pattern design for the perspective-n-point problem,” Image Vis. Comput. 26(6), 740–750 (2008).
[Crossref]

Liu, Y.

Maldonado, A. V.

P. Su, M. Khreishi, T. Su, R. Huang, M. Z. Dominguez, A. V. Maldonado, G. P. Butel, Y. Wang, R. E. Parks, and J. H. Burge, “Aspheric and freeform surfaces metrology with software configurable optical test system: a computerized reverse hartmann test,” Opt. Eng. 53(3), 031305 (2013).
[Crossref]

Parks, R. E.

P. Su, M. Khreishi, T. Su, R. Huang, M. Z. Dominguez, A. V. Maldonado, G. P. Butel, Y. Wang, R. E. Parks, and J. H. Burge, “Aspheric and freeform surfaces metrology with software configurable optical test system: a computerized reverse hartmann test,” Opt. Eng. 53(3), 031305 (2013).
[Crossref]

P. Su, R. E. Parks, L. Wang, R. P. Angel, and J. H. Burge, “Software configurable optical test system: a computerized reverse Hartmann test,” Appl. Opt. 49(23), 4404–4412 (2010).
[Crossref] [PubMed]

Ren, H.

Richter, C.

Schnabel, R. B.

R. H. Byrd, R. B. Schnabel, and G. A. Shultz, “Approximate solution of the trust region problem by minimization over two-dimensional subspaces,” Math. Program. 40(40), 247–263 (1988).
[Crossref]

Shultz, G. A.

R. H. Byrd, R. B. Schnabel, and G. A. Shultz, “Approximate solution of the trust region problem by minimization over two-dimensional subspaces,” Math. Program. 40(40), 247–263 (1988).
[Crossref]

Steihaug, T.

T. Steihaug, “The conjugate gradient method and trust regions in large scale optimization,” SIAM J. Numer. Anal. 20(3), 626–637 (1983).
[Crossref]

Su, P.

P. Su, M. Khreishi, T. Su, R. Huang, M. Z. Dominguez, A. V. Maldonado, G. P. Butel, Y. Wang, R. E. Parks, and J. H. Burge, “Aspheric and freeform surfaces metrology with software configurable optical test system: a computerized reverse hartmann test,” Opt. Eng. 53(3), 031305 (2013).
[Crossref]

P. Su, R. E. Parks, L. Wang, R. P. Angel, and J. H. Burge, “Software configurable optical test system: a computerized reverse Hartmann test,” Appl. Opt. 49(23), 4404–4412 (2010).
[Crossref] [PubMed]

Su, T.

P. Su, M. Khreishi, T. Su, R. Huang, M. Z. Dominguez, A. V. Maldonado, G. P. Butel, Y. Wang, R. E. Parks, and J. H. Burge, “Aspheric and freeform surfaces metrology with software configurable optical test system: a computerized reverse hartmann test,” Opt. Eng. 53(3), 031305 (2013).
[Crossref]

Su, X.

Tan, M.

D. Xu, Y. F. Li, and M. Tan, “A general recursive linear method and unique solution pattern design for the perspective-n-point problem,” Image Vis. Comput. 26(6), 740–750 (2008).
[Crossref]

Tang, S.

S. Tang, X. Zhang, and D. Tu, “Micro-phase measuring profilometry: its sensitivity analysis and phase unwrapping,” Opt. Lasers Eng. 72, 47–57 (2015).
[Crossref]

Tang, Y.

Tao, T.

H. Guo, P. Feng, and T. Tao, “Specular surface measurement by using least squares light tracking technique,” Opt. Lasers Eng. 48(2), 166–171 (2010).
[Crossref]

Tao, X.

Tu, D.

C. Li, X. Zhang, and D. Tu, “Posed relationship calibration with parallel mirror reflection for stereo deflectometry,” Opt. Eng. 57(3), 034103 (2018).
[Crossref]

C. Li, X. Zhang, D. Tu, J. Jia, W. Cui, and C. Zhang, “Deflectometry measurement method of single-camera monitoring,” Acta Opt. Sin. 37(10), 1012007 (2017).
[Crossref]

S. Tang, X. Zhang, and D. Tu, “Micro-phase measuring profilometry: its sensitivity analysis and phase unwrapping,” Opt. Lasers Eng. 72, 47–57 (2015).
[Crossref]

Wang, L.

Wang, W.

L. Xiao, X.-G. Xia, and W. Wang, “Multi-stage robust chinese remainder theorem,” IEEE Trans. Signal Process. 62(18), 4772–4785 (2014).
[Crossref]

Wang, Y.

P. Su, M. Khreishi, T. Su, R. Huang, M. Z. Dominguez, A. V. Maldonado, G. P. Butel, Y. Wang, R. E. Parks, and J. H. Burge, “Aspheric and freeform surfaces metrology with software configurable optical test system: a computerized reverse hartmann test,” Opt. Eng. 53(3), 031305 (2013).
[Crossref]

Wang, Y. M.

Z. H. Zhang, J. Guo, Y. M. Wang, S. J. Huang, N. Gao, and Y. J. Xiao, “Parallel-alignment and correction of two displays in three-dimensional measuring system of specular surfaces,” Opt. Precis. Eng. 2, 002 (2017).

Wenjing, X. Y. S. X. C.

X. Y. S. X. C. Wenjing, “Fringe reflection photogrammetry based on pose estimation with free planar mirror reflection,” Acta Opt. Sinica 5, 013 (2012).

Xia, X.-G.

L. Xiao, X.-G. Xia, and W. Wang, “Multi-stage robust chinese remainder theorem,” IEEE Trans. Signal Process. 62(18), 4772–4785 (2014).
[Crossref]

Xiao, L.

L. Xiao, X.-G. Xia, and W. Wang, “Multi-stage robust chinese remainder theorem,” IEEE Trans. Signal Process. 62(18), 4772–4785 (2014).
[Crossref]

Xiao, Y. J.

Z. H. Zhang, J. Guo, Y. M. Wang, S. J. Huang, N. Gao, and Y. J. Xiao, “Parallel-alignment and correction of two displays in three-dimensional measuring system of specular surfaces,” Opt. Precis. Eng. 2, 002 (2017).

Xiao, Y.-L.

Xu, D.

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C. Li, X. Zhang, D. Tu, J. Jia, W. Cui, and C. Zhang, “Deflectometry measurement method of single-camera monitoring,” Acta Opt. Sin. 37(10), 1012007 (2017).
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S. Tang, X. Zhang, and D. Tu, “Micro-phase measuring profilometry: its sensitivity analysis and phase unwrapping,” Opt. Lasers Eng. 72, 47–57 (2015).
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Y. Xu, F. Gao, Z. Zhang, and X. Jiang, “A holistic calibration method with iterative distortion compensation for stereo deflectometry,” Opt. Lasers Eng. 106, 111–118 (2018).
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P. Zhao, N. Gao, Z. Zhang, F. Gao, and X. Jiang, “Performance analysis and evaluation of direct phase measuring deflectometry,” Opt. Lasers Eng. 103, 24–33 (2018).
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S. Huang, Y. Liu, N. Gao, Z. Zhang, F. Gao, and X. Jiang, “Distance calibration between reference plane and screen in direct phase measuring deflectometry,” Sensors (Basel) 18(2), 144 (2018).
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Zhang, Z. H.

Z. H. Zhang, J. Guo, Y. M. Wang, S. J. Huang, N. Gao, and Y. J. Xiao, “Parallel-alignment and correction of two displays in three-dimensional measuring system of specular surfaces,” Opt. Precis. Eng. 2, 002 (2017).

Zhao, P.

P. Zhao, N. Gao, Z. Zhang, F. Gao, and X. Jiang, “Performance analysis and evaluation of direct phase measuring deflectometry,” Opt. Lasers Eng. 103, 24–33 (2018).
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C. Li, X. Zhang, D. Tu, J. Jia, W. Cui, and C. Zhang, “Deflectometry measurement method of single-camera monitoring,” Acta Opt. Sin. 37(10), 1012007 (2017).
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X. Y. S. X. C. Wenjing, “Fringe reflection photogrammetry based on pose estimation with free planar mirror reflection,” Acta Opt. Sinica 5, 013 (2012).

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L. Huang, M. Idir, C. Zuo, K. Kaznatcheev, L. Zhou, and A. Asundi, “Comparison of two-dimensional integration methods for shape reconstruction from gradient data,” Opt. Lasers Eng. 64, 1–11 (2015).
[Crossref]

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[Crossref]

S. Tang, X. Zhang, and D. Tu, “Micro-phase measuring profilometry: its sensitivity analysis and phase unwrapping,” Opt. Lasers Eng. 72, 47–57 (2015).
[Crossref]

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Sensors (Basel) (1)

S. Huang, Y. Liu, N. Gao, Z. Zhang, F. Gao, and X. Jiang, “Distance calibration between reference plane and screen in direct phase measuring deflectometry,” Sensors (Basel) 18(2), 144 (2018).
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Figures (12)

Fig. 1
Fig. 1 The phase measurement deflectometry without refraction model.
Fig. 2
Fig. 2 The phase measurement deflectometry with refraction model.
Fig. 3
Fig. 3 Light propagation refraction model.
Fig. 4
Fig. 4 Refraction parameters solution.
Fig. 5
Fig. 5 The posed relationship calibration with refraction model.
Fig. 6
Fig. 6 The system of phase measurement deflectometry with refraction model. (a) The system, (b) two screens are observed from the subjective perspective (as shown in (a)), the brighter text on the back screen and the darker text on the front screen, (c) the fringe image projected by front screen captured by camera, (d) the fringe image projected by back screen captured by camera.
Fig. 7
Fig. 7 The absolute phase. (a) and (b) The absolute phase from fringes projected by front screen in the x- and y-directions of two mirror positions. (c) and (d) The absolute phase from fringes projected by back screen in the x- and y-directions of two mirror positions.
Fig. 8
Fig. 8 The absolute phase. (a), (b) and (c) The absolute phase from fringes projected by front screen in the x- and y-directions of three mirror positions. (d), (e) and (f) The absolute phase from fringes projected by back screen in the x- and y-directions of two mirror positions.
Fig. 9
Fig. 9 Calculated gradient data and error. (a) Gradient in the x-direction, (b) gradient error in the x-direction, (c) gradient in the y-direction, and (d) gradient error in the y-direction.
Fig. 10
Fig. 10 Reconstruction mirror surface by the radial basis interpolation function with optimized refraction parameters and error. (a) Reconstruction mirror surface and (b) reconstruction error of mirror surface.
Fig. 11
Fig. 11 Reconstruction mirror surface by the radial basis interpolation function without refraction parameters and error. (a) Reconstruction mirror surface and (b) reconstruction error of mirror surface.
Fig. 12
Fig. 12 Reconstruction of spherical mirror surface by the radial basis interpolation function with optimized refraction parameters and error. (a) Reconstruction of spherical mirror surface and (b) reconstruction error of spherical mirror surface.

Equations (10)

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[ X c p f 1 ]=[ R f T f 0 1 ][ X f p f 1 ] , [ X c p b 1 ]=[ R b T b 0 1 ][ X b p b 1 ]
n i = X c p f X c p b || X c p f X c p b ||
n= n i + n r
g=[ g x g y ]=[ n x n z n y n z ] , n=[ n x n y n z ]
p f '=M+|M p f '|i
p f = p f '+| p f ' p f |t
p= p f +| p f p|i
t=ηi+β n f
[η, d f ]=min( k=1 N p ( 1 N j=1 N i [ M k M kj ] ) 2 )
[ R * , T * ]=min( i=1 g j=1 k || p ^ (R,T, n i , d i , M ij , d f ,η) p(R,T, n i , d i , M ij , d f ,η)||)

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