Abstract

Dynamic full-field three-dimensional sensing of specular reflective surfaces can be conveniently implemented with fringe reflection technique. A monoscopic fringe reflectometric system can be adopted as a simple measuring setup. With the assistance of the windowed Fourier ridges method as an advanced fringe demodulation technique, only one cross grating is needed to reconstruct the three-dimensional surface shape changes. A suitable calibration enables determination of the actual three-dimensional surface profile. Experimental results of water wave variations are shown to demonstrate the feasibility of the proposed approach.

© 2011 OSA

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References

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  1. Y. Tang, X. Su, Y. Liu, and H. Jing, “3D shape measurement of the aspheric mirror by advanced phase measuring deflectometry,” Opt. Express 16(19), 15090–15096 (2008).
    [CrossRef] [PubMed]
  2. Y. Tang, X. Su, F. Wu, and Y. Liu, “A novel phase measuring deflectometry for aspheric mirror test,” Opt. Express 17(22), 19778–19784 (2009).
    [CrossRef] [PubMed]
  3. W. Zhao, X. Su, Y. Liu, and Q. Zhang, “Testing an aspheric mirror based on phase measuring deflectometry,” Opt. Eng. 48(10), 103603 (2009).
    [CrossRef]
  4. S. Werling, M. Mai, M. Heizmann, and J. Beyerer, “Inspection of specular and partially specular surfaces,” Metro. Measure. Syst. 16(3) 415–431 (2009).
  5. J. Balzer and S. Werling, “Principles of Shape from Specular Reflection,” Measurement 43(10), 1305–1317 (2010).
    [CrossRef]
  6. Y. M. Zhang, H. S. Song, and G. Saeed, “Observation of a dynamic specular weld pool surface,” Meas. Sci. Technol. 17(6), L9–L12 (2006).
    [CrossRef]
  7. T. Bothe, W. Li, C. von Kopylow, and W. P. O. Jüptner, “High-resolution 3D shape measurement on specular surfaces by fringe reflection,” in Optical Metrology in Production Engineering, (SPIE, 2004), 411–422.
  8. M. C. Knauer, J. Kaminski, and G. Häusler, “Phase measuring deflectometry: a new approach to measure specular free-form surfaces,” in Optical Metrology in Production Engineering, (SPIE, 2004), 366–376.
  9. M. Petz, and R. Tutsch, “Reflection grating photogrammetry: a technique for absolute shape measurement of specular free-form surfaces,” in Optical Manufacturing and Testing VI, (SPIE, 2005), 58691D1–58691D12.
  10. X. Su and Q. Zhang, “Dynamic 3-D shape measurement method: A review,” Opt. Lasers Eng. 48(2), 191–204 (2010).
    [CrossRef]
  11. Q.-C. Zhang and X.-Y. Su, “An optical measurement of vortex shape at a free surface,” Opt. Laser Technol. 34(2), 107–113 (2002).
    [CrossRef]
  12. Q. Zhang and X. Su, “High-speed optical measurement for the drumhead vibration,” Opt. Express 13(8), 3110–3116 (2005).
    [CrossRef] [PubMed]
  13. Q. Zhang, X. Su, Y. Cao, Y. Li, L. Xiang, and W. Chen, “Optical 3-D shape and deformation measurement of rotating blades using stroboscopic structured illumination,” Opt. Eng. 44(11), 113601 (2005).
    [CrossRef]
  14. Y. Watanabe, T. Komuro, and M. Ishikawa, “955-fps Real-time shape measurement of a moving/deforming object using high-speed vision for numerous-point analysis,” In ICRA (2007), 3192–3197.
  15. L.-C. Chen, Y.-T. Huang, X.-L. Nguyen, J.-L. Chen, and C.-C. Chang, “Dynamic out-of-plane profilometry for nano-scale full-field characterization of MEMS using stroboscopic interferometry with novel signal deconvolution algorithm,” Opt. Lasers Eng. 47(2), 237–251 (2009).
    [CrossRef]
  16. S. Zhang and P. S. Huang, “High-resolution, real-time three-dimensional shape measurement,” Opt. Eng. 45(12), 123601 (2006).
    [CrossRef]
  17. S. Zhang, “Recent progresses on real-time 3D shape measurement using digital fringe projection techniques,” Opt. Lasers Eng. 48(2), 149–158 (2010).
    [CrossRef]
  18. Z. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 22(11), 1330–1334 (2000).
    [CrossRef]
  19. J. Y. Bouguet, Camera calibration toolbox for matlab, please see http://www.vision.caltech.edu/bouguetj/calib_doc/ .
  20. W. H. Southwell, “Wave-front estimation from wave-front slope measurements,” J. Opt. Soc. Am. 70(8), 998–1006 (1980).
    [CrossRef]
  21. W. Li, T. Bothe, C. von Kopylow, and W. P. O. Jüptner, “Evaluation methods for gradient measurement techniques,” in Optical Metrology in Production Engineering, (SPIE, 2004), 300–311.
  22. X. Su and W. Chen, “Fourier transform profilometry: A review,” Opt. Lasers Eng. 35(5), 263–284 (2001).
    [CrossRef]
  23. Q. Kemao, “Windowed Fourier transform for fringe pattern analysis,” Appl. Opt. 43(13), 2695–2702 (2004).
    [CrossRef] [PubMed]
  24. Z. Wang and H. Ma, “Advanced continuous wavelet transform algorithm for digital interferogram analysis and processing,” Opt. Eng. 45(4), 045601 (2006).
    [CrossRef]
  25. L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Lasers Eng. 48(2), 141–148 (2010).
    [CrossRef]

2010

J. Balzer and S. Werling, “Principles of Shape from Specular Reflection,” Measurement 43(10), 1305–1317 (2010).
[CrossRef]

X. Su and Q. Zhang, “Dynamic 3-D shape measurement method: A review,” Opt. Lasers Eng. 48(2), 191–204 (2010).
[CrossRef]

S. Zhang, “Recent progresses on real-time 3D shape measurement using digital fringe projection techniques,” Opt. Lasers Eng. 48(2), 149–158 (2010).
[CrossRef]

L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Lasers Eng. 48(2), 141–148 (2010).
[CrossRef]

2009

L.-C. Chen, Y.-T. Huang, X.-L. Nguyen, J.-L. Chen, and C.-C. Chang, “Dynamic out-of-plane profilometry for nano-scale full-field characterization of MEMS using stroboscopic interferometry with novel signal deconvolution algorithm,” Opt. Lasers Eng. 47(2), 237–251 (2009).
[CrossRef]

W. Zhao, X. Su, Y. Liu, and Q. Zhang, “Testing an aspheric mirror based on phase measuring deflectometry,” Opt. Eng. 48(10), 103603 (2009).
[CrossRef]

S. Werling, M. Mai, M. Heizmann, and J. Beyerer, “Inspection of specular and partially specular surfaces,” Metro. Measure. Syst. 16(3) 415–431 (2009).

Y. Tang, X. Su, F. Wu, and Y. Liu, “A novel phase measuring deflectometry for aspheric mirror test,” Opt. Express 17(22), 19778–19784 (2009).
[CrossRef] [PubMed]

2008

2006

Y. M. Zhang, H. S. Song, and G. Saeed, “Observation of a dynamic specular weld pool surface,” Meas. Sci. Technol. 17(6), L9–L12 (2006).
[CrossRef]

S. Zhang and P. S. Huang, “High-resolution, real-time three-dimensional shape measurement,” Opt. Eng. 45(12), 123601 (2006).
[CrossRef]

Z. Wang and H. Ma, “Advanced continuous wavelet transform algorithm for digital interferogram analysis and processing,” Opt. Eng. 45(4), 045601 (2006).
[CrossRef]

2005

Q. Zhang, X. Su, Y. Cao, Y. Li, L. Xiang, and W. Chen, “Optical 3-D shape and deformation measurement of rotating blades using stroboscopic structured illumination,” Opt. Eng. 44(11), 113601 (2005).
[CrossRef]

Q. Zhang and X. Su, “High-speed optical measurement for the drumhead vibration,” Opt. Express 13(8), 3110–3116 (2005).
[CrossRef] [PubMed]

2004

2002

Q.-C. Zhang and X.-Y. Su, “An optical measurement of vortex shape at a free surface,” Opt. Laser Technol. 34(2), 107–113 (2002).
[CrossRef]

2001

X. Su and W. Chen, “Fourier transform profilometry: A review,” Opt. Lasers Eng. 35(5), 263–284 (2001).
[CrossRef]

2000

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

1980

Asundi, A. K.

L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Lasers Eng. 48(2), 141–148 (2010).
[CrossRef]

Balzer, J.

J. Balzer and S. Werling, “Principles of Shape from Specular Reflection,” Measurement 43(10), 1305–1317 (2010).
[CrossRef]

Beyerer, J.

S. Werling, M. Mai, M. Heizmann, and J. Beyerer, “Inspection of specular and partially specular surfaces,” Metro. Measure. Syst. 16(3) 415–431 (2009).

Cao, Y.

Q. Zhang, X. Su, Y. Cao, Y. Li, L. Xiang, and W. Chen, “Optical 3-D shape and deformation measurement of rotating blades using stroboscopic structured illumination,” Opt. Eng. 44(11), 113601 (2005).
[CrossRef]

Chang, C.-C.

L.-C. Chen, Y.-T. Huang, X.-L. Nguyen, J.-L. Chen, and C.-C. Chang, “Dynamic out-of-plane profilometry for nano-scale full-field characterization of MEMS using stroboscopic interferometry with novel signal deconvolution algorithm,” Opt. Lasers Eng. 47(2), 237–251 (2009).
[CrossRef]

Chen, J.-L.

L.-C. Chen, Y.-T. Huang, X.-L. Nguyen, J.-L. Chen, and C.-C. Chang, “Dynamic out-of-plane profilometry for nano-scale full-field characterization of MEMS using stroboscopic interferometry with novel signal deconvolution algorithm,” Opt. Lasers Eng. 47(2), 237–251 (2009).
[CrossRef]

Chen, L.-C.

L.-C. Chen, Y.-T. Huang, X.-L. Nguyen, J.-L. Chen, and C.-C. Chang, “Dynamic out-of-plane profilometry for nano-scale full-field characterization of MEMS using stroboscopic interferometry with novel signal deconvolution algorithm,” Opt. Lasers Eng. 47(2), 237–251 (2009).
[CrossRef]

Chen, W.

Q. Zhang, X. Su, Y. Cao, Y. Li, L. Xiang, and W. Chen, “Optical 3-D shape and deformation measurement of rotating blades using stroboscopic structured illumination,” Opt. Eng. 44(11), 113601 (2005).
[CrossRef]

X. Su and W. Chen, “Fourier transform profilometry: A review,” Opt. Lasers Eng. 35(5), 263–284 (2001).
[CrossRef]

Heizmann, M.

S. Werling, M. Mai, M. Heizmann, and J. Beyerer, “Inspection of specular and partially specular surfaces,” Metro. Measure. Syst. 16(3) 415–431 (2009).

Huang, L.

L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Lasers Eng. 48(2), 141–148 (2010).
[CrossRef]

Huang, P. S.

S. Zhang and P. S. Huang, “High-resolution, real-time three-dimensional shape measurement,” Opt. Eng. 45(12), 123601 (2006).
[CrossRef]

Huang, Y.-T.

L.-C. Chen, Y.-T. Huang, X.-L. Nguyen, J.-L. Chen, and C.-C. Chang, “Dynamic out-of-plane profilometry for nano-scale full-field characterization of MEMS using stroboscopic interferometry with novel signal deconvolution algorithm,” Opt. Lasers Eng. 47(2), 237–251 (2009).
[CrossRef]

Jing, H.

Kemao, Q.

L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Lasers Eng. 48(2), 141–148 (2010).
[CrossRef]

Q. Kemao, “Windowed Fourier transform for fringe pattern analysis,” Appl. Opt. 43(13), 2695–2702 (2004).
[CrossRef] [PubMed]

Li, Y.

Q. Zhang, X. Su, Y. Cao, Y. Li, L. Xiang, and W. Chen, “Optical 3-D shape and deformation measurement of rotating blades using stroboscopic structured illumination,” Opt. Eng. 44(11), 113601 (2005).
[CrossRef]

Liu, Y.

Ma, H.

Z. Wang and H. Ma, “Advanced continuous wavelet transform algorithm for digital interferogram analysis and processing,” Opt. Eng. 45(4), 045601 (2006).
[CrossRef]

Mai, M.

S. Werling, M. Mai, M. Heizmann, and J. Beyerer, “Inspection of specular and partially specular surfaces,” Metro. Measure. Syst. 16(3) 415–431 (2009).

Nguyen, X.-L.

L.-C. Chen, Y.-T. Huang, X.-L. Nguyen, J.-L. Chen, and C.-C. Chang, “Dynamic out-of-plane profilometry for nano-scale full-field characterization of MEMS using stroboscopic interferometry with novel signal deconvolution algorithm,” Opt. Lasers Eng. 47(2), 237–251 (2009).
[CrossRef]

Pan, B.

L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Lasers Eng. 48(2), 141–148 (2010).
[CrossRef]

Saeed, G.

Y. M. Zhang, H. S. Song, and G. Saeed, “Observation of a dynamic specular weld pool surface,” Meas. Sci. Technol. 17(6), L9–L12 (2006).
[CrossRef]

Song, H. S.

Y. M. Zhang, H. S. Song, and G. Saeed, “Observation of a dynamic specular weld pool surface,” Meas. Sci. Technol. 17(6), L9–L12 (2006).
[CrossRef]

Southwell, W. H.

Su, X.

X. Su and Q. Zhang, “Dynamic 3-D shape measurement method: A review,” Opt. Lasers Eng. 48(2), 191–204 (2010).
[CrossRef]

W. Zhao, X. Su, Y. Liu, and Q. Zhang, “Testing an aspheric mirror based on phase measuring deflectometry,” Opt. Eng. 48(10), 103603 (2009).
[CrossRef]

Y. Tang, X. Su, F. Wu, and Y. Liu, “A novel phase measuring deflectometry for aspheric mirror test,” Opt. Express 17(22), 19778–19784 (2009).
[CrossRef] [PubMed]

Y. Tang, X. Su, Y. Liu, and H. Jing, “3D shape measurement of the aspheric mirror by advanced phase measuring deflectometry,” Opt. Express 16(19), 15090–15096 (2008).
[CrossRef] [PubMed]

Q. Zhang, X. Su, Y. Cao, Y. Li, L. Xiang, and W. Chen, “Optical 3-D shape and deformation measurement of rotating blades using stroboscopic structured illumination,” Opt. Eng. 44(11), 113601 (2005).
[CrossRef]

Q. Zhang and X. Su, “High-speed optical measurement for the drumhead vibration,” Opt. Express 13(8), 3110–3116 (2005).
[CrossRef] [PubMed]

X. Su and W. Chen, “Fourier transform profilometry: A review,” Opt. Lasers Eng. 35(5), 263–284 (2001).
[CrossRef]

Su, X.-Y.

Q.-C. Zhang and X.-Y. Su, “An optical measurement of vortex shape at a free surface,” Opt. Laser Technol. 34(2), 107–113 (2002).
[CrossRef]

Tang, Y.

Wang, Z.

Z. Wang and H. Ma, “Advanced continuous wavelet transform algorithm for digital interferogram analysis and processing,” Opt. Eng. 45(4), 045601 (2006).
[CrossRef]

Werling, S.

J. Balzer and S. Werling, “Principles of Shape from Specular Reflection,” Measurement 43(10), 1305–1317 (2010).
[CrossRef]

S. Werling, M. Mai, M. Heizmann, and J. Beyerer, “Inspection of specular and partially specular surfaces,” Metro. Measure. Syst. 16(3) 415–431 (2009).

Wu, F.

Xiang, L.

Q. Zhang, X. Su, Y. Cao, Y. Li, L. Xiang, and W. Chen, “Optical 3-D shape and deformation measurement of rotating blades using stroboscopic structured illumination,” Opt. Eng. 44(11), 113601 (2005).
[CrossRef]

Zhang, Q.

X. Su and Q. Zhang, “Dynamic 3-D shape measurement method: A review,” Opt. Lasers Eng. 48(2), 191–204 (2010).
[CrossRef]

W. Zhao, X. Su, Y. Liu, and Q. Zhang, “Testing an aspheric mirror based on phase measuring deflectometry,” Opt. Eng. 48(10), 103603 (2009).
[CrossRef]

Q. Zhang, X. Su, Y. Cao, Y. Li, L. Xiang, and W. Chen, “Optical 3-D shape and deformation measurement of rotating blades using stroboscopic structured illumination,” Opt. Eng. 44(11), 113601 (2005).
[CrossRef]

Q. Zhang and X. Su, “High-speed optical measurement for the drumhead vibration,” Opt. Express 13(8), 3110–3116 (2005).
[CrossRef] [PubMed]

Zhang, Q.-C.

Q.-C. Zhang and X.-Y. Su, “An optical measurement of vortex shape at a free surface,” Opt. Laser Technol. 34(2), 107–113 (2002).
[CrossRef]

Zhang, S.

S. Zhang, “Recent progresses on real-time 3D shape measurement using digital fringe projection techniques,” Opt. Lasers Eng. 48(2), 149–158 (2010).
[CrossRef]

S. Zhang and P. S. Huang, “High-resolution, real-time three-dimensional shape measurement,” Opt. Eng. 45(12), 123601 (2006).
[CrossRef]

Zhang, Y. M.

Y. M. Zhang, H. S. Song, and G. Saeed, “Observation of a dynamic specular weld pool surface,” Meas. Sci. Technol. 17(6), L9–L12 (2006).
[CrossRef]

Zhang, Z.

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

Zhao, W.

W. Zhao, X. Su, Y. Liu, and Q. Zhang, “Testing an aspheric mirror based on phase measuring deflectometry,” Opt. Eng. 48(10), 103603 (2009).
[CrossRef]

Appl. Opt.

IEEE Trans. Pattern Anal. Mach. Intell.

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

J. Opt. Soc. Am.

Meas. Sci. Technol.

Y. M. Zhang, H. S. Song, and G. Saeed, “Observation of a dynamic specular weld pool surface,” Meas. Sci. Technol. 17(6), L9–L12 (2006).
[CrossRef]

Measurement

J. Balzer and S. Werling, “Principles of Shape from Specular Reflection,” Measurement 43(10), 1305–1317 (2010).
[CrossRef]

Metro. Measure. Syst.

S. Werling, M. Mai, M. Heizmann, and J. Beyerer, “Inspection of specular and partially specular surfaces,” Metro. Measure. Syst. 16(3) 415–431 (2009).

Opt. Eng.

W. Zhao, X. Su, Y. Liu, and Q. Zhang, “Testing an aspheric mirror based on phase measuring deflectometry,” Opt. Eng. 48(10), 103603 (2009).
[CrossRef]

S. Zhang and P. S. Huang, “High-resolution, real-time three-dimensional shape measurement,” Opt. Eng. 45(12), 123601 (2006).
[CrossRef]

Q. Zhang, X. Su, Y. Cao, Y. Li, L. Xiang, and W. Chen, “Optical 3-D shape and deformation measurement of rotating blades using stroboscopic structured illumination,” Opt. Eng. 44(11), 113601 (2005).
[CrossRef]

Z. Wang and H. Ma, “Advanced continuous wavelet transform algorithm for digital interferogram analysis and processing,” Opt. Eng. 45(4), 045601 (2006).
[CrossRef]

Opt. Express

Opt. Laser Technol.

Q.-C. Zhang and X.-Y. Su, “An optical measurement of vortex shape at a free surface,” Opt. Laser Technol. 34(2), 107–113 (2002).
[CrossRef]

Opt. Lasers Eng.

L.-C. Chen, Y.-T. Huang, X.-L. Nguyen, J.-L. Chen, and C.-C. Chang, “Dynamic out-of-plane profilometry for nano-scale full-field characterization of MEMS using stroboscopic interferometry with novel signal deconvolution algorithm,” Opt. Lasers Eng. 47(2), 237–251 (2009).
[CrossRef]

S. Zhang, “Recent progresses on real-time 3D shape measurement using digital fringe projection techniques,” Opt. Lasers Eng. 48(2), 149–158 (2010).
[CrossRef]

X. Su and Q. Zhang, “Dynamic 3-D shape measurement method: A review,” Opt. Lasers Eng. 48(2), 191–204 (2010).
[CrossRef]

L. Huang, Q. Kemao, B. Pan, and A. K. Asundi, “Comparison of Fourier transform, windowed Fourier transform, and wavelet transform methods for phase extraction from a single fringe pattern in fringe projection profilometry,” Opt. Lasers Eng. 48(2), 141–148 (2010).
[CrossRef]

X. Su and W. Chen, “Fourier transform profilometry: A review,” Opt. Lasers Eng. 35(5), 263–284 (2001).
[CrossRef]

Other

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

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

M. Petz, and R. Tutsch, “Reflection grating photogrammetry: a technique for absolute shape measurement of specular free-form surfaces,” in Optical Manufacturing and Testing VI, (SPIE, 2005), 58691D1–58691D12.

J. Y. Bouguet, Camera calibration toolbox for matlab, please see http://www.vision.caltech.edu/bouguetj/calib_doc/ .

W. Li, T. Bothe, C. von Kopylow, and W. P. O. Jüptner, “Evaluation methods for gradient measurement techniques,” in Optical Metrology in Production Engineering, (SPIE, 2004), 300–311.

Y. Watanabe, T. Komuro, and M. Ishikawa, “955-fps Real-time shape measurement of a moving/deforming object using high-speed vision for numerous-point analysis,” In ICRA (2007), 3192–3197.

Supplementary Material (1)

» Media 1: AVI (3378 KB)     

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Figures (7)

Fig. 1
Fig. 1

Basic setup for specular surface measurement with monoscopic fringe reflectometry.

Fig. 2
Fig. 2

Principle of surface normal determination.

Fig. 3
Fig. 3

Measurement result of a concave mirror. (a) Slope in x-direction p(x, y, z), (b) slope in y-direction q(x, y, z), and (c) integrated height distribution z(x, y).

Fig. 4
Fig. 4

Two-directional fringe pattern. (a) Displayed image, (b) typical image captured by camera, and (c) amplitude (shown with its natural logarithm) of Fourier spectrum of (b).

Fig. 5
Fig. 5

Typical experimental result at t = 0.05s. (a) Captured fringe pattern, (b) amplitude of Fourier spectrum shown with its natural logarithm value, (c) and (d) retrieved wrapped phase maps in x- and y-directions with the 2D WFR method (with boundary region removed).

Fig. 6
Fig. 6

Reflectometric result at t = 0.05s. (a) Slope in x-direction p(x, y, z), (b) slope in y-direction q(x, y, z), and (c) height distribution z(x, y) from integration.

Fig. 7
Fig. 7

Some reconstructed 3D shapes of dynamic water wave at different moments (Media 1). (a) t = 0.00 s, (b) t = 0.05 s, (c) t = 0.10 s, (d) t = 0.15 s, (e) t = 0.20 s, (f) t = 0.25 s, (g) t = 0.30 s, (h) t = 0.35 s, and (i) t = 0.40 s.

Equations (5)

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

L = X S .
x : = X X , a n d l : = L L .
N = ( x + l ) = : ( N x N y N z ) .
z ( x , y ) = 1 N z [ N x N y ] = : [ p ( x , y , z ) q ( x , y , z ) ] ,
f ( S x S , S y S ) = 255 × [ 1 2 + 1 4 cos ( 2 π S x S p x ) + 1 4 cos ( 2 π S y S p y ) ] ,

Metrics