Abstract

In this work, a model based method is applied to phase measuring deflectometry, named modal phase measuring deflectometry. The height and slopes of the surface under test are represented by mathematical models and updated by optimizing the model coefficients to minimize the discrepancy between the reprojection in ray tracing and the actual measurement. The pose of the screen relative to the camera is pre-calibrated and further optimized together with the shape coefficients of the surface under test. Simulations and experiments are conducted to demonstrate the feasibility of the proposed approach.

© 2016 Optical Society of America

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References

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2016 (1)

C. Zuo, L. Huang, M. Zhang, Q. Chen, and A. Asundi, “Temporal phase unwrapping algorithms for fringe projection profilometry: A comparative review,” Opt. Lasers Eng. 85, 84–103 (2016).
[Crossref]

2015 (7)

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

R. Huang, P. Su, J. H. Burge, L. Huang, and M. Idir, “High-accuracy aspheric X-ray mirror metrology using Software Configurable Optical Test System/deflectometry,” Opt. Eng. 54(8), 084103 (2015).
[Crossref]

I. Mochi and K. A. Goldberg, “Modal wavefront reconstruction from its gradient,” Appl. Opt. 54(12), 3780–3785 (2015).
[Crossref]

M. Liu, R. Hartley, and M. Salzmann, “Mirror Surface Reconstruction from a Single Image,” IEEE Trans. Pattern Anal. Mach. Intell. 37(4), 760–773 (2015).
[Crossref] [PubMed]

P. de Groot, “Principles of interference microscopy for the measurement of surface topography,” Adv. Opt. Photonics 7(1), 1–65 (2015).
[Crossref]

T. Su, A. Maldonado, P. Su, and J. H. Burge, “Instrument transfer function of slope measuring deflectometry systems,” Appl. Opt. 54(10), 2981–2990 (2015).
[Crossref] [PubMed]

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]

2014 (2)

Y. Liu, E. Olesch, Z. Yang, and G. Häusler, “Fast and accurate deflectometry with crossed fringes,” Adv. Opt. Technol. 3, 441–445 (2014).

P. C. L. Stephenson, “Recurrence relations for the Cartesian derivatives of the Zernike polynomials,” J. Opt. Soc. Am. A 31(4), 708–715 (2014).
[Crossref] [PubMed]

2013 (2)

2012 (6)

L. Huang, C. S. Ng, and A. K. Asundi, “Fast full-field out-of-plane deformation measurement using fringe reflectometry,” Opt. Lasers Eng. 50(4), 529–533 (2012).
[Crossref]

W. Li, M. Sandner, A. Gesierich, and J. Burke, “Absolute optical surface measurement with deflectometry,” Proc. SPIE 8494, 84940 (2012).

C. Faber, E. Olesch, R. Krobot, and G. Häusler, “Deflectometry challenges interferometry: the competition gets tougher!” Proc. SPIE 8493, 84930R (2012).

Y.-L. Xiao, X. Su, and W. Chen, “Flexible geometrical calibration for fringe-reflection 3D measurement,” Opt. Lett. 37(4), 620–622 (2012).
[Crossref] [PubMed]

F. Dai, F. Tang, X. Wang, O. Sasaki, and P. Feng, “Modal wavefront reconstruction based on Zernike polynomials for lateral shearing interferometry: comparisons of existing algorithms,” Appl. Opt. 51(21), 5028–5037 (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]

2011 (2)

L. Huang and A. K. Asundi, “Phase invalidity identification framework with the temporal phase unwrapping method,” Meas. Sci. Technol. 22(3), 035304 (2011).
[Crossref]

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]

2010 (1)

2008 (3)

2005 (1)

M. Petz and R. Tutsch, “Reflection grating photogrammetry: a technique for absolute shape measurement of specular free-form surfaces,” Proc. SPIE 5869, 58691 (2005).

2004 (2)

T. Bothe, W. Li, C. von Kopylow, and W. P. O. Jüptner, “High-resolution 3D shape measurement on specular surfaces by fringe reflection,” Proc. SPIE 5457, 411–422 (2004).

M. C. Knauer, J. Kaminski, and G. Häusler, “Phase measuring deflectometry: a new approach to measure specular free-form surfaces,” Proc. SPIE 5457, 366–376 (2004).

2000 (1)

F. Chen, G. M. Brown, and M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. 39(1), 10–22 (2000).
[Crossref]

1996 (1)

Angel, R. P.

Asundi, A.

C. Zuo, L. Huang, M. Zhang, Q. Chen, and A. Asundi, “Temporal phase unwrapping algorithms for fringe projection profilometry: A comparative review,” Opt. Lasers Eng. 85, 84–103 (2016).
[Crossref]

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]

Asundi, A. K.

L. Huang, C. S. Ng, and A. K. Asundi, “Fast full-field out-of-plane deformation measurement using fringe reflectometry,” Opt. Lasers Eng. 50(4), 529–533 (2012).
[Crossref]

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]

L. Huang and A. K. Asundi, “Phase invalidity identification framework with the temporal phase unwrapping method,” Meas. Sci. Technol. 22(3), 035304 (2011).
[Crossref]

Bothe, T.

T. Bothe, W. Li, C. von Kopylow, and W. P. O. Jüptner, “High-resolution 3D shape measurement on specular surfaces by fringe reflection,” Proc. SPIE 5457, 411–422 (2004).

Brown, G. M.

F. Chen, G. M. Brown, and M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. 39(1), 10–22 (2000).
[Crossref]

Burge, J. H.

Burke, J.

W. Li, M. Sandner, A. Gesierich, and J. Burke, “Absolute optical surface measurement with deflectometry,” Proc. SPIE 8494, 84940 (2012).

Chen, F.

F. Chen, G. M. Brown, and M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. 39(1), 10–22 (2000).
[Crossref]

Chen, Q.

C. Zuo, L. Huang, M. Zhang, Q. Chen, and A. Asundi, “Temporal phase unwrapping algorithms for fringe projection profilometry: A comparative review,” Opt. Lasers Eng. 85, 84–103 (2016).
[Crossref]

Chen, W.

Dai, F.

Dai, G.

de Groot, P.

P. de Groot, “Principles of interference microscopy for the measurement of surface topography,” Adv. Opt. Photonics 7(1), 1–65 (2015).
[Crossref]

Ettl, S.

Faber, C.

C. Faber, E. Olesch, R. Krobot, and G. Häusler, “Deflectometry challenges interferometry: the competition gets tougher!” Proc. SPIE 8493, 84930R (2012).

D. Sprenger, C. Faber, M. Seraphim, and G. Häusler, “UV-Deflectometry: No parasitic reflections,” in Proc. 112th Annual Meeting of the DGaOA (2010), A19.

Feng, P.

Gao, F.

Gesierich, A.

W. Li, M. Sandner, A. Gesierich, and J. Burke, “Absolute optical surface measurement with deflectometry,” Proc. SPIE 8494, 84940 (2012).

Goldberg, K. A.

Hartley, R.

M. Liu, R. Hartley, and M. Salzmann, “Mirror Surface Reconstruction from a Single Image,” IEEE Trans. Pattern Anal. Mach. Intell. 37(4), 760–773 (2015).
[Crossref] [PubMed]

Häusler, G.

Y. Liu, E. Olesch, Z. Yang, and G. Häusler, “Fast and accurate deflectometry with crossed fringes,” Adv. Opt. Technol. 3, 441–445 (2014).

C. Faber, E. Olesch, R. Krobot, and G. Häusler, “Deflectometry challenges interferometry: the competition gets tougher!” Proc. SPIE 8493, 84930R (2012).

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

M. C. Knauer, J. Kaminski, and G. Häusler, “Phase measuring deflectometry: a new approach to measure specular free-form surfaces,” Proc. SPIE 5457, 366–376 (2004).

D. Sprenger, C. Faber, M. Seraphim, and G. Häusler, “UV-Deflectometry: No parasitic reflections,” in Proc. 112th Annual Meeting of the DGaOA (2010), A19.

Huang, L.

C. Zuo, L. Huang, M. Zhang, Q. Chen, and A. Asundi, “Temporal phase unwrapping algorithms for fringe projection profilometry: A comparative review,” Opt. Lasers Eng. 85, 84–103 (2016).
[Crossref]

R. Huang, P. Su, J. H. Burge, L. Huang, and M. Idir, “High-accuracy aspheric X-ray mirror metrology using Software Configurable Optical Test System/deflectometry,” Opt. Eng. 54(8), 084103 (2015).
[Crossref]

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]

L. Huang, C. S. Ng, and A. K. Asundi, “Fast full-field out-of-plane deformation measurement using fringe reflectometry,” Opt. Lasers Eng. 50(4), 529–533 (2012).
[Crossref]

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]

L. Huang and A. K. Asundi, “Phase invalidity identification framework with the temporal phase unwrapping method,” Meas. Sci. Technol. 22(3), 035304 (2011).
[Crossref]

Huang, R.

R. Huang, P. Su, J. H. Burge, L. Huang, and M. Idir, “High-accuracy aspheric X-ray mirror metrology using Software Configurable Optical Test System/deflectometry,” Opt. Eng. 54(8), 084103 (2015).
[Crossref]

R. Huang, P. Su, J. H. Burge, and M. Idir, “X-ray mirror metrology using SCOTS/deflectometry,” Proc. SPIE 8848, 88480G (2013).
[Crossref]

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]

R. Huang, P. Su, J. H. Burge, L. Huang, and M. Idir, “High-accuracy aspheric X-ray mirror metrology using Software Configurable Optical Test System/deflectometry,” Opt. Eng. 54(8), 084103 (2015).
[Crossref]

R. Huang, P. Su, J. H. Burge, and M. Idir, “X-ray mirror metrology using SCOTS/deflectometry,” Proc. SPIE 8848, 88480G (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]

Jiang, X.

Jing, H.

Jüptner, W. P. O.

T. Bothe, W. Li, C. von Kopylow, and W. P. O. Jüptner, “High-resolution 3D shape measurement on specular surfaces by fringe reflection,” Proc. SPIE 5457, 411–422 (2004).

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]

M. C. Knauer, J. Kaminski, and G. Häusler, “Phase measuring deflectometry: a new approach to measure specular free-form surfaces,” Proc. SPIE 5457, 366–376 (2004).

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]

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]

Knauer, M. C.

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

M. C. Knauer, J. Kaminski, and G. Häusler, “Phase measuring deflectometry: a new approach to measure specular free-form surfaces,” Proc. SPIE 5457, 366–376 (2004).

Krobot, R.

C. Faber, E. Olesch, R. Krobot, and G. Häusler, “Deflectometry challenges interferometry: the competition gets tougher!” Proc. SPIE 8493, 84930R (2012).

Li, W.

W. Li, M. Sandner, A. Gesierich, and J. Burke, “Absolute optical surface measurement with deflectometry,” Proc. SPIE 8494, 84940 (2012).

T. Bothe, W. Li, C. von Kopylow, and W. P. O. Jüptner, “High-resolution 3D shape measurement on specular surfaces by fringe reflection,” Proc. SPIE 5457, 411–422 (2004).

Liu, M.

M. Liu, R. Hartley, and M. Salzmann, “Mirror Surface Reconstruction from a Single Image,” IEEE Trans. Pattern Anal. Mach. Intell. 37(4), 760–773 (2015).
[Crossref] [PubMed]

Liu, Y.

Maldonado, A.

Mochi, I.

Ng, C. S.

L. Huang, C. S. Ng, and A. K. Asundi, “Fast full-field out-of-plane deformation measurement using fringe reflectometry,” Opt. Lasers Eng. 50(4), 529–533 (2012).
[Crossref]

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]

Olesch, E.

Y. Liu, E. Olesch, Z. Yang, and G. Häusler, “Fast and accurate deflectometry with crossed fringes,” Adv. Opt. Technol. 3, 441–445 (2014).

C. Faber, E. Olesch, R. Krobot, and G. Häusler, “Deflectometry challenges interferometry: the competition gets tougher!” Proc. SPIE 8493, 84930R (2012).

Ou, Z.

Parks, R. E.

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]

Petz, M.

M. Petz and R. Tutsch, “Reflection grating photogrammetry: a technique for absolute shape measurement of specular free-form surfaces,” Proc. SPIE 5869, 58691 (2005).

Ren, H.

Salzmann, M.

M. Liu, R. Hartley, and M. Salzmann, “Mirror Surface Reconstruction from a Single Image,” IEEE Trans. Pattern Anal. Mach. Intell. 37(4), 760–773 (2015).
[Crossref] [PubMed]

Sandner, M.

W. Li, M. Sandner, A. Gesierich, and J. Burke, “Absolute optical surface measurement with deflectometry,” Proc. SPIE 8494, 84940 (2012).

Sasaki, O.

Seraphim, M.

D. Sprenger, C. Faber, M. Seraphim, and G. Häusler, “UV-Deflectometry: No parasitic reflections,” in Proc. 112th Annual Meeting of the DGaOA (2010), A19.

Song, M.

F. Chen, G. M. Brown, and M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. 39(1), 10–22 (2000).
[Crossref]

Sprenger, D.

D. Sprenger, C. Faber, M. Seraphim, and G. Häusler, “UV-Deflectometry: No parasitic reflections,” in Proc. 112th Annual Meeting of the DGaOA (2010), A19.

Stephenson, P. C. L.

Su, P.

Su, T.

Su, X.

Tang, F.

Tang, Y.

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]

Tutsch, R.

M. Petz and R. Tutsch, “Reflection grating photogrammetry: a technique for absolute shape measurement of specular free-form surfaces,” Proc. SPIE 5869, 58691 (2005).

von Kopylow, C.

T. Bothe, W. Li, C. von Kopylow, and W. P. O. Jüptner, “High-resolution 3D shape measurement on specular surfaces by fringe reflection,” Proc. SPIE 5457, 411–422 (2004).

Wang, L.

Wang, X.

Wang, Y.

Wu, Y.

Xiao, Y.-L.

Yang, Z.

Y. Liu, E. Olesch, Z. Yang, and G. Häusler, “Fast and accurate deflectometry with crossed fringes,” Adv. Opt. Technol. 3, 441–445 (2014).

Yue, H.

Zhang, M.

C. Zuo, L. Huang, M. Zhang, Q. Chen, and A. Asundi, “Temporal phase unwrapping algorithms for fringe projection profilometry: A comparative review,” Opt. Lasers Eng. 85, 84–103 (2016).
[Crossref]

Zhao, B.

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]

Zhou, 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]

Zuo, C.

C. Zuo, L. Huang, M. Zhang, Q. Chen, and A. Asundi, “Temporal phase unwrapping algorithms for fringe projection profilometry: A comparative review,” Opt. Lasers Eng. 85, 84–103 (2016).
[Crossref]

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]

Adv. Opt. Photonics (1)

P. de Groot, “Principles of interference microscopy for the measurement of surface topography,” Adv. Opt. Photonics 7(1), 1–65 (2015).
[Crossref]

Adv. Opt. Technol. (1)

Y. Liu, E. Olesch, Z. Yang, and G. Häusler, “Fast and accurate deflectometry with crossed fringes,” Adv. Opt. Technol. 3, 441–445 (2014).

Appl. Opt. (5)

IEEE Trans. Pattern Anal. Mach. Intell. (1)

M. Liu, R. Hartley, and M. Salzmann, “Mirror Surface Reconstruction from a Single Image,” IEEE Trans. Pattern Anal. Mach. Intell. 37(4), 760–773 (2015).
[Crossref] [PubMed]

J. Opt. Soc. Am. A (2)

Meas. Sci. Technol. (1)

L. Huang and A. K. Asundi, “Phase invalidity identification framework with the temporal phase unwrapping method,” Meas. Sci. Technol. 22(3), 035304 (2011).
[Crossref]

Opt. Eng. (2)

R. Huang, P. Su, J. H. Burge, L. Huang, and M. Idir, “High-accuracy aspheric X-ray mirror metrology using Software Configurable Optical Test System/deflectometry,” Opt. Eng. 54(8), 084103 (2015).
[Crossref]

F. Chen, G. M. Brown, and M. Song, “Overview of three-dimensional shape measurement using optical methods,” Opt. Eng. 39(1), 10–22 (2000).
[Crossref]

Opt. Express (5)

Opt. Lasers Eng. (4)

L. Huang, C. S. Ng, and A. K. Asundi, “Fast full-field out-of-plane deformation measurement using fringe reflectometry,” Opt. Lasers Eng. 50(4), 529–533 (2012).
[Crossref]

C. Zuo, L. Huang, M. Zhang, Q. Chen, and A. Asundi, “Temporal phase unwrapping algorithms for fringe projection profilometry: A comparative review,” Opt. Lasers Eng. 85, 84–103 (2016).
[Crossref]

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]

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]

Opt. Lett. (1)

Proc. SPIE (6)

W. Li, M. Sandner, A. Gesierich, and J. Burke, “Absolute optical surface measurement with deflectometry,” Proc. SPIE 8494, 84940 (2012).

R. Huang, P. Su, J. H. Burge, and M. Idir, “X-ray mirror metrology using SCOTS/deflectometry,” Proc. SPIE 8848, 88480G (2013).
[Crossref]

M. C. Knauer, J. Kaminski, and G. Häusler, “Phase measuring deflectometry: a new approach to measure specular free-form surfaces,” Proc. SPIE 5457, 366–376 (2004).

M. Petz and R. Tutsch, “Reflection grating photogrammetry: a technique for absolute shape measurement of specular free-form surfaces,” Proc. SPIE 5869, 58691 (2005).

T. Bothe, W. Li, C. von Kopylow, and W. P. O. Jüptner, “High-resolution 3D shape measurement on specular surfaces by fringe reflection,” Proc. SPIE 5457, 411–422 (2004).

C. Faber, E. Olesch, R. Krobot, and G. Häusler, “Deflectometry challenges interferometry: the competition gets tougher!” Proc. SPIE 8493, 84930R (2012).

Other (4)

E. Olesch, C. Faber, and G. Häusler, “Deflectometric Self-Calibration for arbitrary specular surfaces,” in Proc. 112th Annual Meeting of the DGaOA (DGaO, 2011).

F. Brunet, “Contributions to parametric image registration and 3d surface reconstruction,” European Ph. D. in Computer Vision, Université dAuvergne, Clérmont-Ferrand, France, and Technische Universitat Munchen, Germany (2010).

L. Miaomiao, R. Hartley, and M. Salzmann, “Mirror Surface Reconstruction from a Single Image,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (CVPR) (IEEE, 2013), 129–136.

D. Sprenger, C. Faber, M. Seraphim, and G. Häusler, “UV-Deflectometry: No parasitic reflections,” in Proc. 112th Annual Meeting of the DGaOA (2010), A19.

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Figures (17)

Fig. 1
Fig. 1 Sketches of PMD can be drawn either in physical principle with light from the screen (a) or in ray tracing with probe from camera (b).
Fig. 2
Fig. 2 The MPMD approach can be applied to multi-camera PMD.
Fig. 3
Fig. 3 Simulated PMD systems: (a) mono-PMD, (b) stereo-PMD, and (c) the SUT.
Fig. 4
Fig. 4 Error evaluation. Absolute height (a), absolute height error (b), shape with piston, tip and tilt terms removed (c) and shape error (d).
Fig. 5
Fig. 5 Shape errors by using MPMD in different PMD systems: (a-c) mono-PMD, (d-f) stereo-PMD, and with different models: (a, d) Chebyshev, (b, e) Zernike, (c, f) B-spline.
Fig. 6
Fig. 6 The calibration error of the screen pose is added in simulation.
Fig. 7
Fig. 7 Large error occurs if inaccurate geometric calibration is trusted in the optimization with Chebyshev polynomials in the mono-PMD system. (a) Reconstructed shape, (b) reprojection on the screen, (c) reprojection residuals, (d) vectors of reprojection residuals, and (e) shape error.
Fig. 8
Fig. 8 Shape error significantly reduces, when the inaccurate pose of screen is used as an initial guess in the optimization with Chebyshev polynomials in the mono-PMD system. (a) Reconstructed shape, (b) reprojection on the screen, (c) reprojection residuals, (d) vectors of reprojection residuals, and (e) shape error.
Fig. 9
Fig. 9 The shape of SUT is successfully reconstruction in a form of Chebyshev polynomials in the stereo-PMD system. (a) Reconstructed shape, (b) reprojection of the two camera rays on the screen, (c) reprojection residuals, (d) vectors of reprojection residuals, and (e) shape error.
Fig. 10
Fig. 10 Experiment setup mainly consists of two cameras, one LCD screen, and the SUT.
Fig. 11
Fig. 11 System geometry is pre-calibrated as the initial values for the optimization.
Fig. 12
Fig. 12 Two typical captured fringe patterns with x- (a) and y- (b) phases, and the corresponding coordinates on screen in x- (c) and y- (d) directions.
Fig. 13
Fig. 13 Iterative optimization with mono-PMD data starts from the initial guess (a), reduces the reprojection residuals (b), and finalizes the result with convergence (c).
Fig. 14
Fig. 14 Iterative reconstruction with stereo-PMD data starts from the initial guess (a), reduces reprojection residuals of both left and right rays (b), and ends up with a final result (c).
Fig. 15
Fig. 15 Comparison of reconstruction results from mono-PMD in the 1st test (a) and the 2nd test (d), stereo-PMD in the 1st test (b) and the 2nd test (e). Height differences (c) = (a) – (b), (f) = (d) – (e), (g) = (a) – (d), and (h) = (b) – (e).
Fig. 16
Fig. 16 Pre-calibration is still important although MPMD relaxes the calibration. A good calibration (a) offers better initial values for a better reconstruction with less shape error (b), comparing to a poor calibration (c) with its corresponding large reconstruction error (d).
Fig. 17
Fig. 17 Number of model coefficients may influence the reconstruction accuracy. Reprojection residuals (a-b) are smaller and the reconstructed shape (c) is more accurate with less shape error (d) with 289 orders of Zernike polynomials than those (e-h) with 36 orders of Zernike polynomials

Equations (10)

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z c = w Τ c,
z c x n = w x Τ c,
z c y n = w y Τ c,
{ z c x n = z c x c x c x n + z c y c y c x n = z c x c ( z c + x n z c x n )+ y n z c y c z c x n z c y n = z c x c x c y n + z c y c y c y n = x n z c x c z c y n + z c y c ( z c + y n z c y n ) .
z c x c = z c x n z c + x n z c x n + y n z c y n ,
z c y c = z c y n z c + x n z c x n + y n z c y n .
r=p2 p,n n,
[ ω ^ , T ^ , c ^ ]=arg min ω,T,c n=1 N m ^ n ( ω,T,c ) m n 2 .
[ ω ^ , T ^ , c ^ ]=arg min ω,T,c k=1 K n=1 N m ^ k,n ( ω,T,c ) m k,n ( ω,T,c ) 2 ,
z= 1 10 ( cos( 2πx 200+x )+sin( 2πy 100 ) )+ 10 5 x 2 +2× 10 5 y 2 , 100x100 100y100 .

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