Abstract

In the phase measuring deflectometry, two groups of fringe patterns in orthogonal directions are usually applied to establish the correspondences between the pixel pairs on the screen and camera. Usually, 16 phase-shifting fringe patterns with different spatial frequencies are required in order to calculate the absolute phases in the conventional temporal phase unwrapping algorithms. This requirement makes the measurement inefficient and not robust against environmental noise. In this paper, an efficient phase retrieval strategy is developed, which requires only six fringe patterns. The modulating information in one-direction is obtained by first using four fringe patterns, and then it is applied to assist the phase calculation in the other direction, so that only two extra fringe patterns are needed. Subsequently the phases are unwrapped by using the geometric constraints of the software configurable optical test system without additional image acquisition. The measurement time is saved by 5/8, compared to the conventional methods. In this way, the influence of the low-frequency disturbances can be suppressed in the workshop condition. Experiments demonstrate that the proposed method can reliably retrieve the absolute phases, and it is of significance to improve the measuring efficiency and stability of in situ deflectometry.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. H. Ren, F. Gao, and X. Jiang, “Iterative optimization calibration method for stereo deflectometry,” Opt. Express 23(17), 22060–22068 (2015).
    [Crossref] [PubMed]
  2. S. Wan, X. Zhang, M. Xu, W. Wang, and X. Jiang, “Region-adaptive path planning for precision optical polishing with industrial robots,” Opt. Express 26(18), 23782–23795 (2018).
    [Crossref] [PubMed]
  3. L. Huang, C. S. Ng, and A. K. Asundi, “Dynamic three-dimensional sensing for specular surface with monoscopic fringe reflectometry,” Opt. Express 19(13), 12809–12814 (2011).
    [Crossref] [PubMed]
  4. Y. Liu, S. Huang, Z. Zhang, N. Gao, F. Gao, and X. Jiang, “Full-field 3D shape measurement of discontinuous specular objects by direct phase measuring deflectometry,” Sci. Rep. 7(1), 10293 (2017).
    [Crossref] [PubMed]
  5. P. Su, M. A. H. Khreishi, T. Su, R. Huang, M. Z. Dominguez, A. Maldonado, G. 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]
  6. C. Chen, N. Gao, X. Wang, Z. Zhang, F. Gao, and X. Jiang, “Generic exponential fringe model for alleviating phase error in phase measuring profilometry,” Opt. Lasers Eng. 110, 179–185 (2018).
    [Crossref]
  7. M. Zhao, L. Huang, Q. Zhang, X. Su, A. Asundi, and Q. Kemao, “Quality-guided phase unwrapping technique: comparison of quality maps and guiding strategies,” Appl. Opt. 50(33), 6214–6224 (2011).
    [Crossref] [PubMed]
  8. M. A. Herráez, D. R. Burton, M. J. Lalor, and M. A. Gdeisat, “Fast two-dimensional phase-unwrapping algorithm based on sorting by reliability following a noncontinuous path,” Appl. Opt. 41(35), 7437–7444 (2002).
    [Crossref] [PubMed]
  9. Z. Zhang, “Review of single-shot 3D shape measurement by phase calculation-based fringe projection techniques,” Opt. Lasers Eng. 50(8), 1097–1106 (2012).
    [Crossref]
  10. J. Tian, X. Peng, and X. Zhao, “A generalized temporal phase unwrapping algorithm for three-dimensional profilometry,” Opt. Lasers Eng. 46(4), 336–342 (2008).
    [Crossref]
  11. Y. Wang and S. Zhang, “Novel phase-coding method for absolute phase retrieval,” Opt. Lett. 37(11), 2067–2069 (2012).
    [Crossref] [PubMed]
  12. W. Lohry, V. Chen, and S. Zhang, “Absolute three-dimensional shape measurement using coded fringe patterns without phase unwrapping or projector calibration,” Opt. Express 22(2), 1287–1301 (2014).
    [Crossref] [PubMed]
  13. D. Dornfeld and D. E. Lee, Precision Manufacturing, Springer (2013).
  14. J. L. Flores, B. Bravo-Medina, and J. A. Ferrari, “One-frame two-dimensional deflectometry for phase retrieval by addition of orthogonal fringe patterns,” Appl. Opt. 52(26), 6537–6542 (2013).
    [Crossref] [PubMed]
  15. K. Zhong, Z. Li, Y. Shi, C. Wang, and Y. Lei, “Fast phase measurement profilometry for arbitrary shape objects without phase unwrapping,” Opt. Lasers Eng. 51(11), 1213–1222 (2013).
    [Crossref]
  16. Y. An, J. S. Hyun, and S. Zhang, “Pixel-wise absolute phase unwrapping using geometric constraints of structured light system,” Opt. Express 24(16), 18445–18459 (2016).
    [Crossref] [PubMed]
  17. C. Zuo, Q. Chen, G. Gu, S. Feng, F. Feng, R. Li, and G. Shen, “High-speed three-dimensional shape measurement for dynamic scenes using bi-frequency tripolar pulse-width-modulation fringe projection,” Opt. Lasers Eng. 51(8), 953–960 (2013).
    [Crossref]
  18. P. Su, Y. Wang, J. H. Burge, K. Kaznatcheev, and M. Idir, “Non-null full field X-ray mirror metrology using SCOTS: a reflection deflectometry approach,” Opt. Express 20(11), 12393–12406 (2012).
    [Crossref] [PubMed]
  19. BIPM, JCGM 100:2008, “Evaluation of measurement data — Guide to the expression of uncertainty in measurement,” http://www.bipm.org/en/publications/guides/gum.html .
  20. Z. Zhang, C. E. Towers, and D. P. Towers, “Time efficient color fringe projection system for 3D shape and color using optimum 3-frequency Selection,” Opt. Express 14(14), 6444–6455 (2006).
    [Crossref] [PubMed]
  21. Z. Niu, Y. Zhu, X. Zhang, X. Xu, S. Li, and W. Zhao, “Precision measurement of specular spherical surfaces based on monoscopic phase measuring deflectometry,” Proc. SPIE 10819, 43 (2018).
    [Crossref]
  22. J.-S. Hyun and S. Zhang, “Enhanced two-frequency phase-shifting method,” Appl. Opt. 55(16), 4395–4401 (2016).
    [Crossref] [PubMed]
  23. J. Gass, A. Dakoff, and M. K. Kim, “Phase imaging without 2π ambiguity by multiwavelength digital holography,” Opt. Lett. 28(13), 1141–1143 (2003).
    [Crossref] [PubMed]
  24. M. A. Herráez, D. R. Burton, M. J. Lalor, and M. A. Gdeisat, “Fast two-dimensional phase-unwrapping algorithm based on sorting by reliability following a noncontinuous path,” Appl. Opt. 41(35), 7437–7444 (2002).
    [Crossref] [PubMed]
  25. C. Schulze, D. Naidoo, D. Flamm, O. A. Schmidt, A. Forbes, and M. Duparré, “Wavefront reconstruction by modal decomposition,” Opt. Express 20(18), 19714–19725 (2012).
    [Crossref] [PubMed]

2018 (3)

S. Wan, X. Zhang, M. Xu, W. Wang, and X. Jiang, “Region-adaptive path planning for precision optical polishing with industrial robots,” Opt. Express 26(18), 23782–23795 (2018).
[Crossref] [PubMed]

C. Chen, N. Gao, X. Wang, Z. Zhang, F. Gao, and X. Jiang, “Generic exponential fringe model for alleviating phase error in phase measuring profilometry,” Opt. Lasers Eng. 110, 179–185 (2018).
[Crossref]

Z. Niu, Y. Zhu, X. Zhang, X. Xu, S. Li, and W. Zhao, “Precision measurement of specular spherical surfaces based on monoscopic phase measuring deflectometry,” Proc. SPIE 10819, 43 (2018).
[Crossref]

2017 (1)

Y. Liu, S. Huang, Z. Zhang, N. Gao, F. Gao, and X. Jiang, “Full-field 3D shape measurement of discontinuous specular objects by direct phase measuring deflectometry,” Sci. Rep. 7(1), 10293 (2017).
[Crossref] [PubMed]

2016 (2)

Y. An, J. S. Hyun, and S. Zhang, “Pixel-wise absolute phase unwrapping using geometric constraints of structured light system,” Opt. Express 24(16), 18445–18459 (2016).
[Crossref] [PubMed]

J.-S. Hyun and S. Zhang, “Enhanced two-frequency phase-shifting method,” Appl. Opt. 55(16), 4395–4401 (2016).
[Crossref] [PubMed]

2015 (1)

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

2014 (1)

W. Lohry, V. Chen, and S. Zhang, “Absolute three-dimensional shape measurement using coded fringe patterns without phase unwrapping or projector calibration,” Opt. Express 22(2), 1287–1301 (2014).
[Crossref] [PubMed]

2013 (4)

J. L. Flores, B. Bravo-Medina, and J. A. Ferrari, “One-frame two-dimensional deflectometry for phase retrieval by addition of orthogonal fringe patterns,” Appl. Opt. 52(26), 6537–6542 (2013).
[Crossref] [PubMed]

K. Zhong, Z. Li, Y. Shi, C. Wang, and Y. Lei, “Fast phase measurement profilometry for arbitrary shape objects without phase unwrapping,” Opt. Lasers Eng. 51(11), 1213–1222 (2013).
[Crossref]

C. Zuo, Q. Chen, G. Gu, S. Feng, F. Feng, R. Li, and G. Shen, “High-speed three-dimensional shape measurement for dynamic scenes using bi-frequency tripolar pulse-width-modulation fringe projection,” Opt. Lasers Eng. 51(8), 953–960 (2013).
[Crossref]

P. Su, M. A. H. Khreishi, T. Su, R. Huang, M. Z. Dominguez, A. Maldonado, G. 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 (4)

Z. Zhang, “Review of single-shot 3D shape measurement by phase calculation-based fringe projection techniques,” Opt. Lasers Eng. 50(8), 1097–1106 (2012).
[Crossref]

P. Su, Y. Wang, J. H. Burge, K. Kaznatcheev, and M. Idir, “Non-null full field X-ray mirror metrology using SCOTS: a reflection deflectometry approach,” Opt. Express 20(11), 12393–12406 (2012).
[Crossref] [PubMed]

Y. Wang and S. Zhang, “Novel phase-coding method for absolute phase retrieval,” Opt. Lett. 37(11), 2067–2069 (2012).
[Crossref] [PubMed]

C. Schulze, D. Naidoo, D. Flamm, O. A. Schmidt, A. Forbes, and M. Duparré, “Wavefront reconstruction by modal decomposition,” Opt. Express 20(18), 19714–19725 (2012).
[Crossref] [PubMed]

2011 (2)

M. Zhao, L. Huang, Q. Zhang, X. Su, A. Asundi, and Q. Kemao, “Quality-guided phase unwrapping technique: comparison of quality maps and guiding strategies,” Appl. Opt. 50(33), 6214–6224 (2011).
[Crossref] [PubMed]

L. Huang, C. S. Ng, and A. K. Asundi, “Dynamic three-dimensional sensing for specular surface with monoscopic fringe reflectometry,” Opt. Express 19(13), 12809–12814 (2011).
[Crossref] [PubMed]

2008 (1)

J. Tian, X. Peng, and X. Zhao, “A generalized temporal phase unwrapping algorithm for three-dimensional profilometry,” Opt. Lasers Eng. 46(4), 336–342 (2008).
[Crossref]

2006 (1)

Z. Zhang, C. E. Towers, and D. P. Towers, “Time efficient color fringe projection system for 3D shape and color using optimum 3-frequency Selection,” Opt. Express 14(14), 6444–6455 (2006).
[Crossref] [PubMed]

2003 (1)

J. Gass, A. Dakoff, and M. K. Kim, “Phase imaging without 2π ambiguity by multiwavelength digital holography,” Opt. Lett. 28(13), 1141–1143 (2003).
[Crossref] [PubMed]

2002 (2)

M. A. Herráez, D. R. Burton, M. J. Lalor, and M. A. Gdeisat, “Fast two-dimensional phase-unwrapping algorithm based on sorting by reliability following a noncontinuous path,” Appl. Opt. 41(35), 7437–7444 (2002).
[Crossref] [PubMed]

M. A. Herráez, D. R. Burton, M. J. Lalor, and M. A. Gdeisat, “Fast two-dimensional phase-unwrapping algorithm based on sorting by reliability following a noncontinuous path,” Appl. Opt. 41(35), 7437–7444 (2002).
[Crossref] [PubMed]

An, Y.

Y. An, J. S. Hyun, and S. Zhang, “Pixel-wise absolute phase unwrapping using geometric constraints of structured light system,” Opt. Express 24(16), 18445–18459 (2016).
[Crossref] [PubMed]

Asundi, A.

M. Zhao, L. Huang, Q. Zhang, X. Su, A. Asundi, and Q. Kemao, “Quality-guided phase unwrapping technique: comparison of quality maps and guiding strategies,” Appl. Opt. 50(33), 6214–6224 (2011).
[Crossref] [PubMed]

Asundi, A. K.

L. Huang, C. S. Ng, and A. K. Asundi, “Dynamic three-dimensional sensing for specular surface with monoscopic fringe reflectometry,” Opt. Express 19(13), 12809–12814 (2011).
[Crossref] [PubMed]

Bravo-Medina, B.

J. L. Flores, B. Bravo-Medina, and J. A. Ferrari, “One-frame two-dimensional deflectometry for phase retrieval by addition of orthogonal fringe patterns,” Appl. Opt. 52(26), 6537–6542 (2013).
[Crossref] [PubMed]

Burge, J. H.

P. Su, M. A. H. Khreishi, T. Su, R. Huang, M. Z. Dominguez, A. Maldonado, G. 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, Y. Wang, J. H. Burge, K. Kaznatcheev, and M. Idir, “Non-null full field X-ray mirror metrology using SCOTS: a reflection deflectometry approach,” Opt. Express 20(11), 12393–12406 (2012).
[Crossref] [PubMed]

Burton, D. R.

M. A. Herráez, D. R. Burton, M. J. Lalor, and M. A. Gdeisat, “Fast two-dimensional phase-unwrapping algorithm based on sorting by reliability following a noncontinuous path,” Appl. Opt. 41(35), 7437–7444 (2002).
[Crossref] [PubMed]

M. A. Herráez, D. R. Burton, M. J. Lalor, and M. A. Gdeisat, “Fast two-dimensional phase-unwrapping algorithm based on sorting by reliability following a noncontinuous path,” Appl. Opt. 41(35), 7437–7444 (2002).
[Crossref] [PubMed]

Butel, G.

P. Su, M. A. H. Khreishi, T. Su, R. Huang, M. Z. Dominguez, A. Maldonado, G. 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]

Chen, C.

C. Chen, N. Gao, X. Wang, Z. Zhang, F. Gao, and X. Jiang, “Generic exponential fringe model for alleviating phase error in phase measuring profilometry,” Opt. Lasers Eng. 110, 179–185 (2018).
[Crossref]

Chen, Q.

C. Zuo, Q. Chen, G. Gu, S. Feng, F. Feng, R. Li, and G. Shen, “High-speed three-dimensional shape measurement for dynamic scenes using bi-frequency tripolar pulse-width-modulation fringe projection,” Opt. Lasers Eng. 51(8), 953–960 (2013).
[Crossref]

Chen, V.

W. Lohry, V. Chen, and S. Zhang, “Absolute three-dimensional shape measurement using coded fringe patterns without phase unwrapping or projector calibration,” Opt. Express 22(2), 1287–1301 (2014).
[Crossref] [PubMed]

Dakoff, A.

J. Gass, A. Dakoff, and M. K. Kim, “Phase imaging without 2π ambiguity by multiwavelength digital holography,” Opt. Lett. 28(13), 1141–1143 (2003).
[Crossref] [PubMed]

Dominguez, M. Z.

P. Su, M. A. H. Khreishi, T. Su, R. Huang, M. Z. Dominguez, A. Maldonado, G. 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]

Duparré, M.

C. Schulze, D. Naidoo, D. Flamm, O. A. Schmidt, A. Forbes, and M. Duparré, “Wavefront reconstruction by modal decomposition,” Opt. Express 20(18), 19714–19725 (2012).
[Crossref] [PubMed]

Feng, F.

C. Zuo, Q. Chen, G. Gu, S. Feng, F. Feng, R. Li, and G. Shen, “High-speed three-dimensional shape measurement for dynamic scenes using bi-frequency tripolar pulse-width-modulation fringe projection,” Opt. Lasers Eng. 51(8), 953–960 (2013).
[Crossref]

Feng, S.

C. Zuo, Q. Chen, G. Gu, S. Feng, F. Feng, R. Li, and G. Shen, “High-speed three-dimensional shape measurement for dynamic scenes using bi-frequency tripolar pulse-width-modulation fringe projection,” Opt. Lasers Eng. 51(8), 953–960 (2013).
[Crossref]

Ferrari, J. A.

J. L. Flores, B. Bravo-Medina, and J. A. Ferrari, “One-frame two-dimensional deflectometry for phase retrieval by addition of orthogonal fringe patterns,” Appl. Opt. 52(26), 6537–6542 (2013).
[Crossref] [PubMed]

Flamm, D.

C. Schulze, D. Naidoo, D. Flamm, O. A. Schmidt, A. Forbes, and M. Duparré, “Wavefront reconstruction by modal decomposition,” Opt. Express 20(18), 19714–19725 (2012).
[Crossref] [PubMed]

Flores, J. L.

J. L. Flores, B. Bravo-Medina, and J. A. Ferrari, “One-frame two-dimensional deflectometry for phase retrieval by addition of orthogonal fringe patterns,” Appl. Opt. 52(26), 6537–6542 (2013).
[Crossref] [PubMed]

Forbes, A.

C. Schulze, D. Naidoo, D. Flamm, O. A. Schmidt, A. Forbes, and M. Duparré, “Wavefront reconstruction by modal decomposition,” Opt. Express 20(18), 19714–19725 (2012).
[Crossref] [PubMed]

Gao, F.

C. Chen, N. Gao, X. Wang, Z. Zhang, F. Gao, and X. Jiang, “Generic exponential fringe model for alleviating phase error in phase measuring profilometry,” Opt. Lasers Eng. 110, 179–185 (2018).
[Crossref]

Y. Liu, S. Huang, Z. Zhang, N. Gao, F. Gao, and X. Jiang, “Full-field 3D shape measurement of discontinuous specular objects by direct phase measuring deflectometry,” Sci. Rep. 7(1), 10293 (2017).
[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]

Gao, N.

C. Chen, N. Gao, X. Wang, Z. Zhang, F. Gao, and X. Jiang, “Generic exponential fringe model for alleviating phase error in phase measuring profilometry,” Opt. Lasers Eng. 110, 179–185 (2018).
[Crossref]

Y. Liu, S. Huang, Z. Zhang, N. Gao, F. Gao, and X. Jiang, “Full-field 3D shape measurement of discontinuous specular objects by direct phase measuring deflectometry,” Sci. Rep. 7(1), 10293 (2017).
[Crossref] [PubMed]

Gass, J.

J. Gass, A. Dakoff, and M. K. Kim, “Phase imaging without 2π ambiguity by multiwavelength digital holography,” Opt. Lett. 28(13), 1141–1143 (2003).
[Crossref] [PubMed]

Gdeisat, M. A.

M. A. Herráez, D. R. Burton, M. J. Lalor, and M. A. Gdeisat, “Fast two-dimensional phase-unwrapping algorithm based on sorting by reliability following a noncontinuous path,” Appl. Opt. 41(35), 7437–7444 (2002).
[Crossref] [PubMed]

M. A. Herráez, D. R. Burton, M. J. Lalor, and M. A. Gdeisat, “Fast two-dimensional phase-unwrapping algorithm based on sorting by reliability following a noncontinuous path,” Appl. Opt. 41(35), 7437–7444 (2002).
[Crossref] [PubMed]

Gu, G.

C. Zuo, Q. Chen, G. Gu, S. Feng, F. Feng, R. Li, and G. Shen, “High-speed three-dimensional shape measurement for dynamic scenes using bi-frequency tripolar pulse-width-modulation fringe projection,” Opt. Lasers Eng. 51(8), 953–960 (2013).
[Crossref]

Herráez, M. A.

M. A. Herráez, D. R. Burton, M. J. Lalor, and M. A. Gdeisat, “Fast two-dimensional phase-unwrapping algorithm based on sorting by reliability following a noncontinuous path,” Appl. Opt. 41(35), 7437–7444 (2002).
[Crossref] [PubMed]

M. A. Herráez, D. R. Burton, M. J. Lalor, and M. A. Gdeisat, “Fast two-dimensional phase-unwrapping algorithm based on sorting by reliability following a noncontinuous path,” Appl. Opt. 41(35), 7437–7444 (2002).
[Crossref] [PubMed]

Huang, L.

M. Zhao, L. Huang, Q. Zhang, X. Su, A. Asundi, and Q. Kemao, “Quality-guided phase unwrapping technique: comparison of quality maps and guiding strategies,” Appl. Opt. 50(33), 6214–6224 (2011).
[Crossref] [PubMed]

L. Huang, C. S. Ng, and A. K. Asundi, “Dynamic three-dimensional sensing for specular surface with monoscopic fringe reflectometry,” Opt. Express 19(13), 12809–12814 (2011).
[Crossref] [PubMed]

Huang, R.

P. Su, M. A. H. Khreishi, T. Su, R. Huang, M. Z. Dominguez, A. Maldonado, G. 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.

Y. Liu, S. Huang, Z. Zhang, N. Gao, F. Gao, and X. Jiang, “Full-field 3D shape measurement of discontinuous specular objects by direct phase measuring deflectometry,” Sci. Rep. 7(1), 10293 (2017).
[Crossref] [PubMed]

Hyun, J. S.

Y. An, J. S. Hyun, and S. Zhang, “Pixel-wise absolute phase unwrapping using geometric constraints of structured light system,” Opt. Express 24(16), 18445–18459 (2016).
[Crossref] [PubMed]

Hyun, J.-S.

J.-S. Hyun and S. Zhang, “Enhanced two-frequency phase-shifting method,” Appl. Opt. 55(16), 4395–4401 (2016).
[Crossref] [PubMed]

Idir, M.

P. Su, Y. Wang, J. H. Burge, K. Kaznatcheev, and M. Idir, “Non-null full field X-ray mirror metrology using SCOTS: a reflection deflectometry approach,” Opt. Express 20(11), 12393–12406 (2012).
[Crossref] [PubMed]

Jiang, X.

S. Wan, X. Zhang, M. Xu, W. Wang, and X. Jiang, “Region-adaptive path planning for precision optical polishing with industrial robots,” Opt. Express 26(18), 23782–23795 (2018).
[Crossref] [PubMed]

C. Chen, N. Gao, X. Wang, Z. Zhang, F. Gao, and X. Jiang, “Generic exponential fringe model for alleviating phase error in phase measuring profilometry,” Opt. Lasers Eng. 110, 179–185 (2018).
[Crossref]

Y. Liu, S. Huang, Z. Zhang, N. Gao, F. Gao, and X. Jiang, “Full-field 3D shape measurement of discontinuous specular objects by direct phase measuring deflectometry,” Sci. Rep. 7(1), 10293 (2017).
[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]

Kaznatcheev, K.

P. Su, Y. Wang, J. H. Burge, K. Kaznatcheev, and M. Idir, “Non-null full field X-ray mirror metrology using SCOTS: a reflection deflectometry approach,” Opt. Express 20(11), 12393–12406 (2012).
[Crossref] [PubMed]

Kemao, Q.

M. Zhao, L. Huang, Q. Zhang, X. Su, A. Asundi, and Q. Kemao, “Quality-guided phase unwrapping technique: comparison of quality maps and guiding strategies,” Appl. Opt. 50(33), 6214–6224 (2011).
[Crossref] [PubMed]

Khreishi, M. A. H.

P. Su, M. A. H. Khreishi, T. Su, R. Huang, M. Z. Dominguez, A. Maldonado, G. 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]

Kim, M. K.

J. Gass, A. Dakoff, and M. K. Kim, “Phase imaging without 2π ambiguity by multiwavelength digital holography,” Opt. Lett. 28(13), 1141–1143 (2003).
[Crossref] [PubMed]

Lalor, M. J.

M. A. Herráez, D. R. Burton, M. J. Lalor, and M. A. Gdeisat, “Fast two-dimensional phase-unwrapping algorithm based on sorting by reliability following a noncontinuous path,” Appl. Opt. 41(35), 7437–7444 (2002).
[Crossref] [PubMed]

M. A. Herráez, D. R. Burton, M. J. Lalor, and M. A. Gdeisat, “Fast two-dimensional phase-unwrapping algorithm based on sorting by reliability following a noncontinuous path,” Appl. Opt. 41(35), 7437–7444 (2002).
[Crossref] [PubMed]

Lei, Y.

K. Zhong, Z. Li, Y. Shi, C. Wang, and Y. Lei, “Fast phase measurement profilometry for arbitrary shape objects without phase unwrapping,” Opt. Lasers Eng. 51(11), 1213–1222 (2013).
[Crossref]

Li, R.

C. Zuo, Q. Chen, G. Gu, S. Feng, F. Feng, R. Li, and G. Shen, “High-speed three-dimensional shape measurement for dynamic scenes using bi-frequency tripolar pulse-width-modulation fringe projection,” Opt. Lasers Eng. 51(8), 953–960 (2013).
[Crossref]

Li, S.

Z. Niu, Y. Zhu, X. Zhang, X. Xu, S. Li, and W. Zhao, “Precision measurement of specular spherical surfaces based on monoscopic phase measuring deflectometry,” Proc. SPIE 10819, 43 (2018).
[Crossref]

Li, Z.

K. Zhong, Z. Li, Y. Shi, C. Wang, and Y. Lei, “Fast phase measurement profilometry for arbitrary shape objects without phase unwrapping,” Opt. Lasers Eng. 51(11), 1213–1222 (2013).
[Crossref]

Liu, Y.

Y. Liu, S. Huang, Z. Zhang, N. Gao, F. Gao, and X. Jiang, “Full-field 3D shape measurement of discontinuous specular objects by direct phase measuring deflectometry,” Sci. Rep. 7(1), 10293 (2017).
[Crossref] [PubMed]

Lohry, W.

W. Lohry, V. Chen, and S. Zhang, “Absolute three-dimensional shape measurement using coded fringe patterns without phase unwrapping or projector calibration,” Opt. Express 22(2), 1287–1301 (2014).
[Crossref] [PubMed]

Maldonado, A.

P. Su, M. A. H. Khreishi, T. Su, R. Huang, M. Z. Dominguez, A. Maldonado, G. 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]

Naidoo, D.

C. Schulze, D. Naidoo, D. Flamm, O. A. Schmidt, A. Forbes, and M. Duparré, “Wavefront reconstruction by modal decomposition,” Opt. Express 20(18), 19714–19725 (2012).
[Crossref] [PubMed]

Ng, C. S.

L. Huang, C. S. Ng, and A. K. Asundi, “Dynamic three-dimensional sensing for specular surface with monoscopic fringe reflectometry,” Opt. Express 19(13), 12809–12814 (2011).
[Crossref] [PubMed]

Niu, Z.

Z. Niu, Y. Zhu, X. Zhang, X. Xu, S. Li, and W. Zhao, “Precision measurement of specular spherical surfaces based on monoscopic phase measuring deflectometry,” Proc. SPIE 10819, 43 (2018).
[Crossref]

Parks, R. E.

P. Su, M. A. H. Khreishi, T. Su, R. Huang, M. Z. Dominguez, A. Maldonado, G. 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]

Peng, X.

J. Tian, X. Peng, and X. Zhao, “A generalized temporal phase unwrapping algorithm for three-dimensional profilometry,” Opt. Lasers Eng. 46(4), 336–342 (2008).
[Crossref]

Ren, H.

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

Schmidt, O. A.

C. Schulze, D. Naidoo, D. Flamm, O. A. Schmidt, A. Forbes, and M. Duparré, “Wavefront reconstruction by modal decomposition,” Opt. Express 20(18), 19714–19725 (2012).
[Crossref] [PubMed]

Schulze, C.

C. Schulze, D. Naidoo, D. Flamm, O. A. Schmidt, A. Forbes, and M. Duparré, “Wavefront reconstruction by modal decomposition,” Opt. Express 20(18), 19714–19725 (2012).
[Crossref] [PubMed]

Shen, G.

C. Zuo, Q. Chen, G. Gu, S. Feng, F. Feng, R. Li, and G. Shen, “High-speed three-dimensional shape measurement for dynamic scenes using bi-frequency tripolar pulse-width-modulation fringe projection,” Opt. Lasers Eng. 51(8), 953–960 (2013).
[Crossref]

Shi, Y.

K. Zhong, Z. Li, Y. Shi, C. Wang, and Y. Lei, “Fast phase measurement profilometry for arbitrary shape objects without phase unwrapping,” Opt. Lasers Eng. 51(11), 1213–1222 (2013).
[Crossref]

Su, P.

P. Su, M. A. H. Khreishi, T. Su, R. Huang, M. Z. Dominguez, A. Maldonado, G. 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, Y. Wang, J. H. Burge, K. Kaznatcheev, and M. Idir, “Non-null full field X-ray mirror metrology using SCOTS: a reflection deflectometry approach,” Opt. Express 20(11), 12393–12406 (2012).
[Crossref] [PubMed]

Su, T.

P. Su, M. A. H. Khreishi, T. Su, R. Huang, M. Z. Dominguez, A. Maldonado, G. 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.

M. Zhao, L. Huang, Q. Zhang, X. Su, A. Asundi, and Q. Kemao, “Quality-guided phase unwrapping technique: comparison of quality maps and guiding strategies,” Appl. Opt. 50(33), 6214–6224 (2011).
[Crossref] [PubMed]

Tian, J.

J. Tian, X. Peng, and X. Zhao, “A generalized temporal phase unwrapping algorithm for three-dimensional profilometry,” Opt. Lasers Eng. 46(4), 336–342 (2008).
[Crossref]

Towers, C. E.

Z. Zhang, C. E. Towers, and D. P. Towers, “Time efficient color fringe projection system for 3D shape and color using optimum 3-frequency Selection,” Opt. Express 14(14), 6444–6455 (2006).
[Crossref] [PubMed]

Towers, D. P.

Z. Zhang, C. E. Towers, and D. P. Towers, “Time efficient color fringe projection system for 3D shape and color using optimum 3-frequency Selection,” Opt. Express 14(14), 6444–6455 (2006).
[Crossref] [PubMed]

Wan, S.

S. Wan, X. Zhang, M. Xu, W. Wang, and X. Jiang, “Region-adaptive path planning for precision optical polishing with industrial robots,” Opt. Express 26(18), 23782–23795 (2018).
[Crossref] [PubMed]

Wang, C.

K. Zhong, Z. Li, Y. Shi, C. Wang, and Y. Lei, “Fast phase measurement profilometry for arbitrary shape objects without phase unwrapping,” Opt. Lasers Eng. 51(11), 1213–1222 (2013).
[Crossref]

Wang, W.

S. Wan, X. Zhang, M. Xu, W. Wang, and X. Jiang, “Region-adaptive path planning for precision optical polishing with industrial robots,” Opt. Express 26(18), 23782–23795 (2018).
[Crossref] [PubMed]

Wang, X.

C. Chen, N. Gao, X. Wang, Z. Zhang, F. Gao, and X. Jiang, “Generic exponential fringe model for alleviating phase error in phase measuring profilometry,” Opt. Lasers Eng. 110, 179–185 (2018).
[Crossref]

Wang, Y.

P. Su, M. A. H. Khreishi, T. Su, R. Huang, M. Z. Dominguez, A. Maldonado, G. 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, Y. Wang, J. H. Burge, K. Kaznatcheev, and M. Idir, “Non-null full field X-ray mirror metrology using SCOTS: a reflection deflectometry approach,” Opt. Express 20(11), 12393–12406 (2012).
[Crossref] [PubMed]

Y. Wang and S. Zhang, “Novel phase-coding method for absolute phase retrieval,” Opt. Lett. 37(11), 2067–2069 (2012).
[Crossref] [PubMed]

Xu, M.

S. Wan, X. Zhang, M. Xu, W. Wang, and X. Jiang, “Region-adaptive path planning for precision optical polishing with industrial robots,” Opt. Express 26(18), 23782–23795 (2018).
[Crossref] [PubMed]

Xu, X.

Z. Niu, Y. Zhu, X. Zhang, X. Xu, S. Li, and W. Zhao, “Precision measurement of specular spherical surfaces based on monoscopic phase measuring deflectometry,” Proc. SPIE 10819, 43 (2018).
[Crossref]

Zhang, Q.

M. Zhao, L. Huang, Q. Zhang, X. Su, A. Asundi, and Q. Kemao, “Quality-guided phase unwrapping technique: comparison of quality maps and guiding strategies,” Appl. Opt. 50(33), 6214–6224 (2011).
[Crossref] [PubMed]

Zhang, S.

Y. An, J. S. Hyun, and S. Zhang, “Pixel-wise absolute phase unwrapping using geometric constraints of structured light system,” Opt. Express 24(16), 18445–18459 (2016).
[Crossref] [PubMed]

J.-S. Hyun and S. Zhang, “Enhanced two-frequency phase-shifting method,” Appl. Opt. 55(16), 4395–4401 (2016).
[Crossref] [PubMed]

W. Lohry, V. Chen, and S. Zhang, “Absolute three-dimensional shape measurement using coded fringe patterns without phase unwrapping or projector calibration,” Opt. Express 22(2), 1287–1301 (2014).
[Crossref] [PubMed]

Y. Wang and S. Zhang, “Novel phase-coding method for absolute phase retrieval,” Opt. Lett. 37(11), 2067–2069 (2012).
[Crossref] [PubMed]

Zhang, X.

S. Wan, X. Zhang, M. Xu, W. Wang, and X. Jiang, “Region-adaptive path planning for precision optical polishing with industrial robots,” Opt. Express 26(18), 23782–23795 (2018).
[Crossref] [PubMed]

Z. Niu, Y. Zhu, X. Zhang, X. Xu, S. Li, and W. Zhao, “Precision measurement of specular spherical surfaces based on monoscopic phase measuring deflectometry,” Proc. SPIE 10819, 43 (2018).
[Crossref]

Zhang, Z.

C. Chen, N. Gao, X. Wang, Z. Zhang, F. Gao, and X. Jiang, “Generic exponential fringe model for alleviating phase error in phase measuring profilometry,” Opt. Lasers Eng. 110, 179–185 (2018).
[Crossref]

Y. Liu, S. Huang, Z. Zhang, N. Gao, F. Gao, and X. Jiang, “Full-field 3D shape measurement of discontinuous specular objects by direct phase measuring deflectometry,” Sci. Rep. 7(1), 10293 (2017).
[Crossref] [PubMed]

Z. Zhang, “Review of single-shot 3D shape measurement by phase calculation-based fringe projection techniques,” Opt. Lasers Eng. 50(8), 1097–1106 (2012).
[Crossref]

Z. Zhang, C. E. Towers, and D. P. Towers, “Time efficient color fringe projection system for 3D shape and color using optimum 3-frequency Selection,” Opt. Express 14(14), 6444–6455 (2006).
[Crossref] [PubMed]

Zhao, M.

M. Zhao, L. Huang, Q. Zhang, X. Su, A. Asundi, and Q. Kemao, “Quality-guided phase unwrapping technique: comparison of quality maps and guiding strategies,” Appl. Opt. 50(33), 6214–6224 (2011).
[Crossref] [PubMed]

Zhao, W.

Z. Niu, Y. Zhu, X. Zhang, X. Xu, S. Li, and W. Zhao, “Precision measurement of specular spherical surfaces based on monoscopic phase measuring deflectometry,” Proc. SPIE 10819, 43 (2018).
[Crossref]

Zhao, X.

J. Tian, X. Peng, and X. Zhao, “A generalized temporal phase unwrapping algorithm for three-dimensional profilometry,” Opt. Lasers Eng. 46(4), 336–342 (2008).
[Crossref]

Zhong, K.

K. Zhong, Z. Li, Y. Shi, C. Wang, and Y. Lei, “Fast phase measurement profilometry for arbitrary shape objects without phase unwrapping,” Opt. Lasers Eng. 51(11), 1213–1222 (2013).
[Crossref]

Zhu, Y.

Z. Niu, Y. Zhu, X. Zhang, X. Xu, S. Li, and W. Zhao, “Precision measurement of specular spherical surfaces based on monoscopic phase measuring deflectometry,” Proc. SPIE 10819, 43 (2018).
[Crossref]

Zuo, C.

C. Zuo, Q. Chen, G. Gu, S. Feng, F. Feng, R. Li, and G. Shen, “High-speed three-dimensional shape measurement for dynamic scenes using bi-frequency tripolar pulse-width-modulation fringe projection,” Opt. Lasers Eng. 51(8), 953–960 (2013).
[Crossref]

Appl. Opt. (5)

M. Zhao, L. Huang, Q. Zhang, X. Su, A. Asundi, and Q. Kemao, “Quality-guided phase unwrapping technique: comparison of quality maps and guiding strategies,” Appl. Opt. 50(33), 6214–6224 (2011).
[Crossref] [PubMed]

M. A. Herráez, D. R. Burton, M. J. Lalor, and M. A. Gdeisat, “Fast two-dimensional phase-unwrapping algorithm based on sorting by reliability following a noncontinuous path,” Appl. Opt. 41(35), 7437–7444 (2002).
[Crossref] [PubMed]

J. L. Flores, B. Bravo-Medina, and J. A. Ferrari, “One-frame two-dimensional deflectometry for phase retrieval by addition of orthogonal fringe patterns,” Appl. Opt. 52(26), 6537–6542 (2013).
[Crossref] [PubMed]

J.-S. Hyun and S. Zhang, “Enhanced two-frequency phase-shifting method,” Appl. Opt. 55(16), 4395–4401 (2016).
[Crossref] [PubMed]

M. A. Herráez, D. R. Burton, M. J. Lalor, and M. A. Gdeisat, “Fast two-dimensional phase-unwrapping algorithm based on sorting by reliability following a noncontinuous path,” Appl. Opt. 41(35), 7437–7444 (2002).
[Crossref] [PubMed]

Opt. Eng. (1)

P. Su, M. A. H. Khreishi, T. Su, R. Huang, M. Z. Dominguez, A. Maldonado, G. 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]

Opt. Express (8)

W. Lohry, V. Chen, and S. Zhang, “Absolute three-dimensional shape measurement using coded fringe patterns without phase unwrapping or projector calibration,” Opt. Express 22(2), 1287–1301 (2014).
[Crossref] [PubMed]

Y. An, J. S. Hyun, and S. Zhang, “Pixel-wise absolute phase unwrapping using geometric constraints of structured light system,” Opt. Express 24(16), 18445–18459 (2016).
[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]

S. Wan, X. Zhang, M. Xu, W. Wang, and X. Jiang, “Region-adaptive path planning for precision optical polishing with industrial robots,” Opt. Express 26(18), 23782–23795 (2018).
[Crossref] [PubMed]

L. Huang, C. S. Ng, and A. K. Asundi, “Dynamic three-dimensional sensing for specular surface with monoscopic fringe reflectometry,” Opt. Express 19(13), 12809–12814 (2011).
[Crossref] [PubMed]

C. Schulze, D. Naidoo, D. Flamm, O. A. Schmidt, A. Forbes, and M. Duparré, “Wavefront reconstruction by modal decomposition,” Opt. Express 20(18), 19714–19725 (2012).
[Crossref] [PubMed]

P. Su, Y. Wang, J. H. Burge, K. Kaznatcheev, and M. Idir, “Non-null full field X-ray mirror metrology using SCOTS: a reflection deflectometry approach,” Opt. Express 20(11), 12393–12406 (2012).
[Crossref] [PubMed]

Z. Zhang, C. E. Towers, and D. P. Towers, “Time efficient color fringe projection system for 3D shape and color using optimum 3-frequency Selection,” Opt. Express 14(14), 6444–6455 (2006).
[Crossref] [PubMed]

Opt. Lasers Eng. (5)

Z. Zhang, “Review of single-shot 3D shape measurement by phase calculation-based fringe projection techniques,” Opt. Lasers Eng. 50(8), 1097–1106 (2012).
[Crossref]

J. Tian, X. Peng, and X. Zhao, “A generalized temporal phase unwrapping algorithm for three-dimensional profilometry,” Opt. Lasers Eng. 46(4), 336–342 (2008).
[Crossref]

C. Zuo, Q. Chen, G. Gu, S. Feng, F. Feng, R. Li, and G. Shen, “High-speed three-dimensional shape measurement for dynamic scenes using bi-frequency tripolar pulse-width-modulation fringe projection,” Opt. Lasers Eng. 51(8), 953–960 (2013).
[Crossref]

K. Zhong, Z. Li, Y. Shi, C. Wang, and Y. Lei, “Fast phase measurement profilometry for arbitrary shape objects without phase unwrapping,” Opt. Lasers Eng. 51(11), 1213–1222 (2013).
[Crossref]

C. Chen, N. Gao, X. Wang, Z. Zhang, F. Gao, and X. Jiang, “Generic exponential fringe model for alleviating phase error in phase measuring profilometry,” Opt. Lasers Eng. 110, 179–185 (2018).
[Crossref]

Opt. Lett. (2)

Y. Wang and S. Zhang, “Novel phase-coding method for absolute phase retrieval,” Opt. Lett. 37(11), 2067–2069 (2012).
[Crossref] [PubMed]

J. Gass, A. Dakoff, and M. K. Kim, “Phase imaging without 2π ambiguity by multiwavelength digital holography,” Opt. Lett. 28(13), 1141–1143 (2003).
[Crossref] [PubMed]

Proc. SPIE (1)

Z. Niu, Y. Zhu, X. Zhang, X. Xu, S. Li, and W. Zhao, “Precision measurement of specular spherical surfaces based on monoscopic phase measuring deflectometry,” Proc. SPIE 10819, 43 (2018).
[Crossref]

Sci. Rep. (1)

Y. Liu, S. Huang, Z. Zhang, N. Gao, F. Gao, and X. Jiang, “Full-field 3D shape measurement of discontinuous specular objects by direct phase measuring deflectometry,” Sci. Rep. 7(1), 10293 (2017).
[Crossref] [PubMed]

Other (2)

D. Dornfeld and D. E. Lee, Precision Manufacturing, Springer (2013).

BIPM, JCGM 100:2008, “Evaluation of measurement data — Guide to the expression of uncertainty in measurement,” http://www.bipm.org/en/publications/guides/gum.html .

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Figures (10)

Fig. 1
Fig. 1 The SCOTS configuration.
Fig. 2
Fig. 2 Retracing in SCOTS.
Fig. 3
Fig. 3 Determination of fringe orders.
Fig. 4
Fig. 4 The actual deflectometric measuring system.
Fig. 5
Fig. 5 Results of demodulated phases using the proposed method. (a) Acquired v fringe pattern; (b) Acquired u fringe pattern; (c) Demodulated v phase map; (d) Demodulated u phase map.
Fig. 6
Fig. 6 Comparison of demodulating errors. (a) Phase difference between the two methods; (b) Phase-difference of the conventional method; (c) Phase-difference of the proposed method.
Fig. 7
Fig. 7 Results of phase unwrapping by reverse retracing. (a) Unwrapped v phases; (b) Unwrapped u phases.
Fig. 8
Fig. 8 Comparison between the unwrapping results of dual-frequency method and the proposed method. (a) Unwrapped phases of dual-frequency method; (b) A cross-section of (a); (c) Corrected unwrapped phases of dual-frequency method; (d) A cross-section of (c).
Fig. 9
Fig. 9 Reconstruction result of the concave surface using the proposed method. (a) Reconstructed surface; (b) Difference between the two methods.
Fig. 10
Fig. 10 Residual error of the proposed method.

Tables (2)

Tables Icon

Table 1 Notation of coordinate systems

Tables Icon

Table 2 Analysis of retracing error (Unit: mm)

Equations (15)

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s [ u c v c 1 ] = [ f u γ u 0 0 f v v 0 0 0 1 ] [ R w 2 c | T w 2 c ] [ x w y w z w 1 ] ,
[ x s y s ] = [ m 0 0 n ] ( [ u s v s ] [ d x d y ] ) .
w x ( x m , y m ) = x m x s d m 2 s + x m x c d m 2 c z m 2 s z ( x m , y m ) d m 2 s + z m 2 c z ( x m , y m ) d m 2 c w y ( x m , y m ) = y m y s d m 2 s + y m y c d m 2 c z m 2 s z ( x m , y m ) d m 2 s + z m 2 c z ( x m , y m ) d m 2 c ,
{ I v 1 ( x , y ) = I ' ( x , y ) + I ' ' ( x , y ) cos ( θ ( x , y ) ) I v 2 ( x , y ) = I ' ( x , y ) + I ' ' ( x , y ) cos ( θ ( x , y ) + π / 2 ) I v 3 ( x , y ) = I ' ( x , y ) + I ' ' ( x , y ) cos ( θ ( x , y ) + π ) I v 4 ( x , y ) = I ' ( x , y ) + I ' ' ( x , y ) cos ( θ ( x , y ) + 3 π / 2 ) ,
θ ' ( x , y ) = tan 1 I v 4 ( x , y ) I v 2 ( x , y ) I v 1 ( x , y ) I v 3 ( x , y ) .
I ' ( x , y ) = I v 1 ( x , y ) + I v 2 ( x , y ) + I v 3 ( x , y ) + I v 4 ( x , y ) 4 .
{ I u 1 ( x , y ) = I ' ( x , y ) + I ' ' ( x , y ) cos ( φ ( x , y ) ) I u 2 ( x , y ) = I ' ( x , y ) + I ' ' ( x , y ) cos ( φ ( x , y ) + π / 2 ) .
φ ' ( x , y ) = tan 1 I u 2 ( x , y ) I ' ( x , y ) I u 1 ( x , y ) I ' ( x , y ) .
{ U 2 ( tan θ ' ) = U 2 ( I v 2 ) + U 2 ( I v 4 ) ( I v 1 I v 3 ) 2 + ( I v 4 I v 2 ) 2 U 2 ( I v 1 ) + U 2 ( I v 3 ) ( I v 1 I v 3 ) 4 = σ 2 2 ( I " cos θ ' ) 2 + sin 2 θ ' σ 2 2 I " 2 cos 4 θ ' U 2 ( tan φ ' ) = U 2 ( I u 2 ) + U 2 ( I ' ) ( I u 1 I ' ) 2 + ( I u 2 I ' ) 2 U 2 ( I u 1 ) + U 2 ( I ' ) ( I u 1 I ' ) 4 = 5 σ 2 4 ( I " cos φ ' ) 2 + 5 sin 2 φ ' σ 2 4 I " 2 cos 4 φ ' .
( x ' s , y ' s , z ' s ) T = R w 2 s ( x ' s w , y ' s w , z ' s w ) T + T w 2 s .
[ u ' s v ' s ] = [ 1 m 0 0 1 n ] [ x ' s y ' s ] + [ d x d y ] .
K = round ( ϑ ϕ 2 π ) ,
x ' s w = x m ( z m 2 s z ( x m , y m ) + z m 2 c z ( x m , y m ) d m 2 c d m 2 s ) w x ( x m , y m ) + x m x c d m 2 c d m 2 s y ' s w = y m ( z m 2 s z ( x m , y m ) + z m 2 c z ( x m , y m ) d m 2 c d m 2 s ) w y ( x m , y m ) + y m y c d m 2 c d m 2 s .
Δ x ' s w = | x ' s w x m | Δ x m + | x ' s w z m 2 s | Δ z m 2 s + | x ' s w w ( x m , y m ) | Δ z ( x m , y m ) + | x ' s w z m 2 c | Δ z m 2 c + | x ' s w d m 2 c | Δ d m 2 c + | x ' s w d m 2 s | Δ d m 2 s + | x ' s w w x ( x m , y m ) | Δ w x ( x m , y m ) + | x ' s w x c | Δ x c .
min x i 1 2 [ p i q i ( x ) ] T [ p i q i ( x ) ] ,

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