Abstract

In fringe pattern analyses, the computational burden of implementing the arctangent function over an entire phase map is not trivial, hindering it from being used in real-time measurements. For overcoming this problem, this paper presents a general method for approximating the arctangent function. The domain of the arctangent function is split into a sequence of intervals. For each interval, approximation polynomials are determined in the maximum-norm sense. By applying these polynomials instead of the standard arctangent function to the fringe analyses, the efficiencies of phase evaluations are improved significantly. The accuracies and simplicities of the approximations have been analyzed numerically, and their validities have also been verified by using experimental results.

© 2007 Optical Society of America

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  1. M. Takeda, H. Ina, and S. Kobayashi, "Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry," J. Opt. Soc. Am. 72, 156-160 (1982).
    [CrossRef]
  2. M. Takeda and K. Mutoh, "Fourier transform profilometry for the automatic measurement of 3-D object shapes." Appl. Opt. 22, 3977-3982 (1983).
    [CrossRef] [PubMed]
  3. O. A. Skydan, M. J. Lalor, and D. R. Burton, "Technique for phase measurement and surface reconstruction by use of colored structured light," Appl. Opt. 41, 6104-6117 (2002).
    [CrossRef] [PubMed]
  4. C. J. Tay, C. Quan, H. M. Shang, T. Wu, and S. Wang, "New method for measuring dynamic response of small components by fringe projection," Opt. Eng. 42, 1715-1720 (2003).
    [CrossRef]
  5. Q. Zhang and X. Su, "High-speed optical measurement for the drumhead vibration," Opt. Express 13, 3110-3116 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-8-3110.
    [CrossRef] [PubMed]
  6. 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, 113601 (2005).
    [CrossRef]
  7. M. Takeda, Q. Gu, M. Kinoshita, H. Takai, and Y. Takahashi, "Frequency-multiplex Fourier-transform profilometry: a single-shot three-dimensional shape measurement of objects with large height discontinuities and/or surface isolations," App. Opt. 36, 5347-5354 (1997).
    [CrossRef]
  8. W-H. Su and H. Liu, "Calibration-based two-frequency projected fringe profilometry: a robust, accurate, and single-shot measurement for objects with large depth discontinuities," Opt. Express 14, 9178-9187 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-20-9178.
    [CrossRef] [PubMed]
  9. M. A. Herráez, D. R. Burton, and M. J. Lalor, "Accelerating fast Fourier transform and filtering operations in Fourier fringe analysis for accurate measurement of three-dimensional surfaces," Opt. Lasers Eng. 31, 135-145 (1999).
    [CrossRef]
  10. Y. Ichioka and M. Inuiya, "Direct Phase Detecting System," Appl. Opt. 11, 1507-1514 (1972).
    [CrossRef] [PubMed]
  11. K. H. Womack, "Interferometric phase measurement using spatial synchronous detection," Opt. Eng. 23, 391-395 (1984).
  12. S. Tang and Y. Y. Hung, "Fast profilometer for the automatic measurement of 3-D object shapes," Appl. Opt. 29, 3012-3018 (1990).
    [CrossRef] [PubMed]
  13. L. Mertz, "Real-time fringe-pattern analysis," Appl. Opt. 22, 1535-1539 (1983).
    [CrossRef] [PubMed]
  14. W. W. Macy, "Two-dimensional fringe-pattern analysis," Appl. Opt. 22, 3898-3901 (1983).
    [CrossRef] [PubMed]
  15. D. C. Williams, N. S. Nassar, J. E. Banyard, and M. S. Virdee, "Digital phase-step interferometry: a simplified approach," Opt. Laser Tech. 23, 147-150 (1991).
    [CrossRef]
  16. P. H. Chan and P. J. Bryanston-Cross, "Spatial phase stepping method of fringe-pattern analysis," Opt. Lasers Eng. 23, 343-354 (1995).
    [CrossRef]
  17. Y. Arai, S. Yokozeki, K. Shiraki, and T. Yamada, "High precision two-dimensional spatial fringe analysis method," J. Mod. Opt. 44, 739-751 (1997).
    [CrossRef]
  18. J. Kato, I. Yamaguchi, T. Nakamura, and S. Kuwashima, "Video-rate fringe analyzer based on phase-shifting electronic moiré patterns," Appl. Opt. 36, 8403-8412 (1997).
    [CrossRef]
  19. M. Kujawinska, "Spatial phase measurement methods," in Interferogram Analysis: Digital Fringe Pattern Measurement, D. W. Robinson and G. Reid, eds. (IOP, Bristol, UK, 1993), pp.141-193.
  20. Y. Morimoto and M. Fujigaki. "Real-time phase distribution analysis of fringe pattern," in International Conference on Applied Optical Metrology, P. K. Rastogi, and F. Gyímesi eds., Proc. SPIE,  3407, 34-39 (1998).
    [CrossRef]
  21. K. L. Baker and E. A. Stappaerts, "A single-shot pixellated phase-shifting interferometer utilizing a liquid-crystal spatial light modulator," Opt. Lett. 31, 733-735 (2006).
    [CrossRef] [PubMed]
  22. P. S. Huang, Q. Hu, and F-P. Chiang, "Color-encoded digital fringe projection technique for high-speed three-dimensional surface contouring," Opt. Eng. 38, 1065-1071 (1999).
    [CrossRef]
  23. C. Guan, L. G. Hassebrook, and D. L. Lau "Composite structured light pattern for three-dimensional video," Opt. Express 11, 406-417 (2003), http://www.opticsinfobase.org/abstract.cfm?URI=oe-11-5-406.
    [CrossRef] [PubMed]
  24. C. R. Coggrave and J. M. Huntley, "High-speed surface profilometer based on a spatial light modulator and pipeline image processor," Opt. Eng. 38, 1573-1581 (1999).
    [CrossRef]
  25. P. S. Huang, C. P. Zhang, and F. P. Chiang, "High-speed 3-D shape measurement based on digital fringe projection," Opt. Eng. 42, 163-168 (2003).
    [CrossRef]
  26. S. Zhang and S-T. Yau, "High-resolution, real-time 3D absolute coordinate measurement based on a phase-shifting method," Opt. Express 14, 2644-2649 (2006), http://www.opticsinfobase.org/abstract.cfm? URI=oe-14-7-2644.
    [CrossRef] [PubMed]
  27. S. Zhang, D. Royer, and S-T Yau, "GPU-assisted high-resolution, real-time 3-D shape measurement," Opt. Express,  14, 9120-9129 (2006), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-20-9120.
    [CrossRef] [PubMed]
  28. M. S. Mermelstein, D. L. Feldhun, and L. G. Shirley, "Video-rate surface profiling with acousto-optic accordion fringe interferometry," Opt. Eng. 39, 106-113 (2000).
    [CrossRef]
  29. C. Quan, C. J. Tay, X. Kang, X. Y. He, and H. M. Shang, "Shape measurement by use of liquid-crystal display, fringe projection with two-step phase-shifting," Appl. Opt. 42, 2329-2335 (2003).
    [CrossRef] [PubMed]
  30. S. Almazán-Cuéllar and D. Malacara-Hernández, "Two-step phase-shifting algorithm," Opt. Eng. 42, 3524-3531 (2003).
    [CrossRef]
  31. P. L. Wizinowich, "Phase shifting interferometry in the presence of vibration: a new algorithm and system," Appl. Opt. 29, 3271-3279 (1990).
    [CrossRef] [PubMed]
  32. H. A. Vrooman and A. A. M. Maas, "Image processing algorithms for the analysis of phase-shifted speckle interference patterns," Appl. Opt. 30, 1636-1641 (1991).
    [CrossRef] [PubMed]
  33. Z. Gao, S. Zhou, and Y. Hu, "High-speed fringe analysis by using stair-shaped virtual grating demodulation," Opt. Lasers Eng. 28, 411-422 (1997).
    [CrossRef]
  34. P. S. Huang and S. Zhang, "Fast three-step phase shifting algorithm," Appl. Opt. 45, 5086-5091 (2006).
    [CrossRef] [PubMed]
  35. R. Capelli, "Fast approximation to the arctangent," in Graphics Gems II, J. Arvo, ed. (AP Professional, Boston, USA, 1995), pp. 389-391.

2006 (4)

2005 (2)

Q. Zhang and X. Su, "High-speed optical measurement for the drumhead vibration," Opt. Express 13, 3110-3116 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-8-3110.
[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, 113601 (2005).
[CrossRef]

2003 (5)

C. J. Tay, C. Quan, H. M. Shang, T. Wu, and S. Wang, "New method for measuring dynamic response of small components by fringe projection," Opt. Eng. 42, 1715-1720 (2003).
[CrossRef]

P. S. Huang, C. P. Zhang, and F. P. Chiang, "High-speed 3-D shape measurement based on digital fringe projection," Opt. Eng. 42, 163-168 (2003).
[CrossRef]

C. Quan, C. J. Tay, X. Kang, X. Y. He, and H. M. Shang, "Shape measurement by use of liquid-crystal display, fringe projection with two-step phase-shifting," Appl. Opt. 42, 2329-2335 (2003).
[CrossRef] [PubMed]

S. Almazán-Cuéllar and D. Malacara-Hernández, "Two-step phase-shifting algorithm," Opt. Eng. 42, 3524-3531 (2003).
[CrossRef]

C. Guan, L. G. Hassebrook, and D. L. Lau "Composite structured light pattern for three-dimensional video," Opt. Express 11, 406-417 (2003), http://www.opticsinfobase.org/abstract.cfm?URI=oe-11-5-406.
[CrossRef] [PubMed]

2002 (1)

2000 (1)

M. S. Mermelstein, D. L. Feldhun, and L. G. Shirley, "Video-rate surface profiling with acousto-optic accordion fringe interferometry," Opt. Eng. 39, 106-113 (2000).
[CrossRef]

1999 (3)

C. R. Coggrave and J. M. Huntley, "High-speed surface profilometer based on a spatial light modulator and pipeline image processor," Opt. Eng. 38, 1573-1581 (1999).
[CrossRef]

P. S. Huang, Q. Hu, and F-P. Chiang, "Color-encoded digital fringe projection technique for high-speed three-dimensional surface contouring," Opt. Eng. 38, 1065-1071 (1999).
[CrossRef]

M. A. Herráez, D. R. Burton, and M. J. Lalor, "Accelerating fast Fourier transform and filtering operations in Fourier fringe analysis for accurate measurement of three-dimensional surfaces," Opt. Lasers Eng. 31, 135-145 (1999).
[CrossRef]

1998 (1)

Y. Morimoto and M. Fujigaki. "Real-time phase distribution analysis of fringe pattern," in International Conference on Applied Optical Metrology, P. K. Rastogi, and F. Gyímesi eds., Proc. SPIE,  3407, 34-39 (1998).
[CrossRef]

1997 (4)

Y. Arai, S. Yokozeki, K. Shiraki, and T. Yamada, "High precision two-dimensional spatial fringe analysis method," J. Mod. Opt. 44, 739-751 (1997).
[CrossRef]

J. Kato, I. Yamaguchi, T. Nakamura, and S. Kuwashima, "Video-rate fringe analyzer based on phase-shifting electronic moiré patterns," Appl. Opt. 36, 8403-8412 (1997).
[CrossRef]

M. Takeda, Q. Gu, M. Kinoshita, H. Takai, and Y. Takahashi, "Frequency-multiplex Fourier-transform profilometry: a single-shot three-dimensional shape measurement of objects with large height discontinuities and/or surface isolations," App. Opt. 36, 5347-5354 (1997).
[CrossRef]

Z. Gao, S. Zhou, and Y. Hu, "High-speed fringe analysis by using stair-shaped virtual grating demodulation," Opt. Lasers Eng. 28, 411-422 (1997).
[CrossRef]

1995 (1)

P. H. Chan and P. J. Bryanston-Cross, "Spatial phase stepping method of fringe-pattern analysis," Opt. Lasers Eng. 23, 343-354 (1995).
[CrossRef]

1991 (2)

D. C. Williams, N. S. Nassar, J. E. Banyard, and M. S. Virdee, "Digital phase-step interferometry: a simplified approach," Opt. Laser Tech. 23, 147-150 (1991).
[CrossRef]

H. A. Vrooman and A. A. M. Maas, "Image processing algorithms for the analysis of phase-shifted speckle interference patterns," Appl. Opt. 30, 1636-1641 (1991).
[CrossRef] [PubMed]

1990 (2)

1984 (1)

K. H. Womack, "Interferometric phase measurement using spatial synchronous detection," Opt. Eng. 23, 391-395 (1984).

1983 (3)

1982 (1)

1972 (1)

Almazán-Cuéllar, S.

S. Almazán-Cuéllar and D. Malacara-Hernández, "Two-step phase-shifting algorithm," Opt. Eng. 42, 3524-3531 (2003).
[CrossRef]

Arai, Y.

Y. Arai, S. Yokozeki, K. Shiraki, and T. Yamada, "High precision two-dimensional spatial fringe analysis method," J. Mod. Opt. 44, 739-751 (1997).
[CrossRef]

Baker, K. L.

Banyard, J. E.

D. C. Williams, N. S. Nassar, J. E. Banyard, and M. S. Virdee, "Digital phase-step interferometry: a simplified approach," Opt. Laser Tech. 23, 147-150 (1991).
[CrossRef]

Bryanston-Cross, P. J.

P. H. Chan and P. J. Bryanston-Cross, "Spatial phase stepping method of fringe-pattern analysis," Opt. Lasers Eng. 23, 343-354 (1995).
[CrossRef]

Burton, D. R.

O. A. Skydan, M. J. Lalor, and D. R. Burton, "Technique for phase measurement and surface reconstruction by use of colored structured light," Appl. Opt. 41, 6104-6117 (2002).
[CrossRef] [PubMed]

M. A. Herráez, D. R. Burton, and M. J. Lalor, "Accelerating fast Fourier transform and filtering operations in Fourier fringe analysis for accurate measurement of three-dimensional surfaces," Opt. Lasers Eng. 31, 135-145 (1999).
[CrossRef]

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, 113601 (2005).
[CrossRef]

Chan, P. H.

P. H. Chan and P. J. Bryanston-Cross, "Spatial phase stepping method of fringe-pattern analysis," Opt. Lasers Eng. 23, 343-354 (1995).
[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, 113601 (2005).
[CrossRef]

Chiang, F. P.

P. S. Huang, C. P. Zhang, and F. P. Chiang, "High-speed 3-D shape measurement based on digital fringe projection," Opt. Eng. 42, 163-168 (2003).
[CrossRef]

Chiang, F-P.

P. S. Huang, Q. Hu, and F-P. Chiang, "Color-encoded digital fringe projection technique for high-speed three-dimensional surface contouring," Opt. Eng. 38, 1065-1071 (1999).
[CrossRef]

Coggrave, C. R.

C. R. Coggrave and J. M. Huntley, "High-speed surface profilometer based on a spatial light modulator and pipeline image processor," Opt. Eng. 38, 1573-1581 (1999).
[CrossRef]

Feldhun, D. L.

M. S. Mermelstein, D. L. Feldhun, and L. G. Shirley, "Video-rate surface profiling with acousto-optic accordion fringe interferometry," Opt. Eng. 39, 106-113 (2000).
[CrossRef]

Fujigaki, M.

Y. Morimoto and M. Fujigaki. "Real-time phase distribution analysis of fringe pattern," in International Conference on Applied Optical Metrology, P. K. Rastogi, and F. Gyímesi eds., Proc. SPIE,  3407, 34-39 (1998).
[CrossRef]

Gao, Z.

Z. Gao, S. Zhou, and Y. Hu, "High-speed fringe analysis by using stair-shaped virtual grating demodulation," Opt. Lasers Eng. 28, 411-422 (1997).
[CrossRef]

Gu, Q.

M. Takeda, Q. Gu, M. Kinoshita, H. Takai, and Y. Takahashi, "Frequency-multiplex Fourier-transform profilometry: a single-shot three-dimensional shape measurement of objects with large height discontinuities and/or surface isolations," App. Opt. 36, 5347-5354 (1997).
[CrossRef]

Guan, C.

Hassebrook, L. G.

He, X. Y.

Herráez, M. A.

M. A. Herráez, D. R. Burton, and M. J. Lalor, "Accelerating fast Fourier transform and filtering operations in Fourier fringe analysis for accurate measurement of three-dimensional surfaces," Opt. Lasers Eng. 31, 135-145 (1999).
[CrossRef]

Hu, Q.

P. S. Huang, Q. Hu, and F-P. Chiang, "Color-encoded digital fringe projection technique for high-speed three-dimensional surface contouring," Opt. Eng. 38, 1065-1071 (1999).
[CrossRef]

Hu, Y.

Z. Gao, S. Zhou, and Y. Hu, "High-speed fringe analysis by using stair-shaped virtual grating demodulation," Opt. Lasers Eng. 28, 411-422 (1997).
[CrossRef]

Huang, P. S.

P. S. Huang and S. Zhang, "Fast three-step phase shifting algorithm," Appl. Opt. 45, 5086-5091 (2006).
[CrossRef] [PubMed]

P. S. Huang, C. P. Zhang, and F. P. Chiang, "High-speed 3-D shape measurement based on digital fringe projection," Opt. Eng. 42, 163-168 (2003).
[CrossRef]

P. S. Huang, Q. Hu, and F-P. Chiang, "Color-encoded digital fringe projection technique for high-speed three-dimensional surface contouring," Opt. Eng. 38, 1065-1071 (1999).
[CrossRef]

Hung, Y. Y.

Huntley, J. M.

C. R. Coggrave and J. M. Huntley, "High-speed surface profilometer based on a spatial light modulator and pipeline image processor," Opt. Eng. 38, 1573-1581 (1999).
[CrossRef]

Ichioka, Y.

Ina, H.

Inuiya, M.

Kang, X.

Kato, J.

Kinoshita, M.

M. Takeda, Q. Gu, M. Kinoshita, H. Takai, and Y. Takahashi, "Frequency-multiplex Fourier-transform profilometry: a single-shot three-dimensional shape measurement of objects with large height discontinuities and/or surface isolations," App. Opt. 36, 5347-5354 (1997).
[CrossRef]

Kobayashi, S.

Kuwashima, S.

Lalor, M. J.

O. A. Skydan, M. J. Lalor, and D. R. Burton, "Technique for phase measurement and surface reconstruction by use of colored structured light," Appl. Opt. 41, 6104-6117 (2002).
[CrossRef] [PubMed]

M. A. Herráez, D. R. Burton, and M. J. Lalor, "Accelerating fast Fourier transform and filtering operations in Fourier fringe analysis for accurate measurement of three-dimensional surfaces," Opt. Lasers Eng. 31, 135-145 (1999).
[CrossRef]

Lau, D. L.

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, 113601 (2005).
[CrossRef]

Liu, H.

Maas, A. A. M.

Macy, W. W.

Malacara-Hernández, D.

S. Almazán-Cuéllar and D. Malacara-Hernández, "Two-step phase-shifting algorithm," Opt. Eng. 42, 3524-3531 (2003).
[CrossRef]

Mermelstein, M. S.

M. S. Mermelstein, D. L. Feldhun, and L. G. Shirley, "Video-rate surface profiling with acousto-optic accordion fringe interferometry," Opt. Eng. 39, 106-113 (2000).
[CrossRef]

Mertz, L.

Morimoto, Y.

Y. Morimoto and M. Fujigaki. "Real-time phase distribution analysis of fringe pattern," in International Conference on Applied Optical Metrology, P. K. Rastogi, and F. Gyímesi eds., Proc. SPIE,  3407, 34-39 (1998).
[CrossRef]

Mutoh, K.

Nakamura, T.

Nassar, N. S.

D. C. Williams, N. S. Nassar, J. E. Banyard, and M. S. Virdee, "Digital phase-step interferometry: a simplified approach," Opt. Laser Tech. 23, 147-150 (1991).
[CrossRef]

Quan, C.

C. J. Tay, C. Quan, H. M. Shang, T. Wu, and S. Wang, "New method for measuring dynamic response of small components by fringe projection," Opt. Eng. 42, 1715-1720 (2003).
[CrossRef]

C. Quan, C. J. Tay, X. Kang, X. Y. He, and H. M. Shang, "Shape measurement by use of liquid-crystal display, fringe projection with two-step phase-shifting," Appl. Opt. 42, 2329-2335 (2003).
[CrossRef] [PubMed]

Royer, D.

Shang, H. M.

C. J. Tay, C. Quan, H. M. Shang, T. Wu, and S. Wang, "New method for measuring dynamic response of small components by fringe projection," Opt. Eng. 42, 1715-1720 (2003).
[CrossRef]

C. Quan, C. J. Tay, X. Kang, X. Y. He, and H. M. Shang, "Shape measurement by use of liquid-crystal display, fringe projection with two-step phase-shifting," Appl. Opt. 42, 2329-2335 (2003).
[CrossRef] [PubMed]

Shiraki, K.

Y. Arai, S. Yokozeki, K. Shiraki, and T. Yamada, "High precision two-dimensional spatial fringe analysis method," J. Mod. Opt. 44, 739-751 (1997).
[CrossRef]

Shirley, L. G.

M. S. Mermelstein, D. L. Feldhun, and L. G. Shirley, "Video-rate surface profiling with acousto-optic accordion fringe interferometry," Opt. Eng. 39, 106-113 (2000).
[CrossRef]

Skydan, O. A.

Stappaerts, E. A.

Su, W-H.

Su, X.

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, 113601 (2005).
[CrossRef]

Q. Zhang and X. Su, "High-speed optical measurement for the drumhead vibration," Opt. Express 13, 3110-3116 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-8-3110.
[CrossRef] [PubMed]

Takahashi, Y.

M. Takeda, Q. Gu, M. Kinoshita, H. Takai, and Y. Takahashi, "Frequency-multiplex Fourier-transform profilometry: a single-shot three-dimensional shape measurement of objects with large height discontinuities and/or surface isolations," App. Opt. 36, 5347-5354 (1997).
[CrossRef]

Takai, H.

M. Takeda, Q. Gu, M. Kinoshita, H. Takai, and Y. Takahashi, "Frequency-multiplex Fourier-transform profilometry: a single-shot three-dimensional shape measurement of objects with large height discontinuities and/or surface isolations," App. Opt. 36, 5347-5354 (1997).
[CrossRef]

Takeda, M.

M. Takeda, Q. Gu, M. Kinoshita, H. Takai, and Y. Takahashi, "Frequency-multiplex Fourier-transform profilometry: a single-shot three-dimensional shape measurement of objects with large height discontinuities and/or surface isolations," App. Opt. 36, 5347-5354 (1997).
[CrossRef]

M. Takeda and K. Mutoh, "Fourier transform profilometry for the automatic measurement of 3-D object shapes." Appl. Opt. 22, 3977-3982 (1983).
[CrossRef] [PubMed]

M. Takeda, H. Ina, and S. Kobayashi, "Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry," J. Opt. Soc. Am. 72, 156-160 (1982).
[CrossRef]

Tang, S.

Tay, C. J.

C. J. Tay, C. Quan, H. M. Shang, T. Wu, and S. Wang, "New method for measuring dynamic response of small components by fringe projection," Opt. Eng. 42, 1715-1720 (2003).
[CrossRef]

C. Quan, C. J. Tay, X. Kang, X. Y. He, and H. M. Shang, "Shape measurement by use of liquid-crystal display, fringe projection with two-step phase-shifting," Appl. Opt. 42, 2329-2335 (2003).
[CrossRef] [PubMed]

Virdee, M. S.

D. C. Williams, N. S. Nassar, J. E. Banyard, and M. S. Virdee, "Digital phase-step interferometry: a simplified approach," Opt. Laser Tech. 23, 147-150 (1991).
[CrossRef]

Vrooman, H. A.

Wang, S.

C. J. Tay, C. Quan, H. M. Shang, T. Wu, and S. Wang, "New method for measuring dynamic response of small components by fringe projection," Opt. Eng. 42, 1715-1720 (2003).
[CrossRef]

Williams, D. C.

D. C. Williams, N. S. Nassar, J. E. Banyard, and M. S. Virdee, "Digital phase-step interferometry: a simplified approach," Opt. Laser Tech. 23, 147-150 (1991).
[CrossRef]

Wizinowich, P. L.

Womack, K. H.

K. H. Womack, "Interferometric phase measurement using spatial synchronous detection," Opt. Eng. 23, 391-395 (1984).

Wu, T.

C. J. Tay, C. Quan, H. M. Shang, T. Wu, and S. Wang, "New method for measuring dynamic response of small components by fringe projection," Opt. Eng. 42, 1715-1720 (2003).
[CrossRef]

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, 113601 (2005).
[CrossRef]

Yamada, T.

Y. Arai, S. Yokozeki, K. Shiraki, and T. Yamada, "High precision two-dimensional spatial fringe analysis method," J. Mod. Opt. 44, 739-751 (1997).
[CrossRef]

Yamaguchi, I.

Yau, S-T

Yokozeki, S.

Y. Arai, S. Yokozeki, K. Shiraki, and T. Yamada, "High precision two-dimensional spatial fringe analysis method," J. Mod. Opt. 44, 739-751 (1997).
[CrossRef]

Zhang, C. P.

P. S. Huang, C. P. Zhang, and F. P. Chiang, "High-speed 3-D shape measurement based on digital fringe projection," Opt. Eng. 42, 163-168 (2003).
[CrossRef]

Zhang, Q.

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, 113601 (2005).
[CrossRef]

Q. Zhang and X. Su, "High-speed optical measurement for the drumhead vibration," Opt. Express 13, 3110-3116 (2005), http://www.opticsinfobase.org/abstract.cfm?URI=oe-13-8-3110.
[CrossRef] [PubMed]

Zhang, S.

Zhou, S.

Z. Gao, S. Zhou, and Y. Hu, "High-speed fringe analysis by using stair-shaped virtual grating demodulation," Opt. Lasers Eng. 28, 411-422 (1997).
[CrossRef]

App. Opt. (1)

M. Takeda, Q. Gu, M. Kinoshita, H. Takai, and Y. Takahashi, "Frequency-multiplex Fourier-transform profilometry: a single-shot three-dimensional shape measurement of objects with large height discontinuities and/or surface isolations," App. Opt. 36, 5347-5354 (1997).
[CrossRef]

Appl. Opt. (11)

M. Takeda and K. Mutoh, "Fourier transform profilometry for the automatic measurement of 3-D object shapes." Appl. Opt. 22, 3977-3982 (1983).
[CrossRef] [PubMed]

O. A. Skydan, M. J. Lalor, and D. R. Burton, "Technique for phase measurement and surface reconstruction by use of colored structured light," Appl. Opt. 41, 6104-6117 (2002).
[CrossRef] [PubMed]

S. Tang and Y. Y. Hung, "Fast profilometer for the automatic measurement of 3-D object shapes," Appl. Opt. 29, 3012-3018 (1990).
[CrossRef] [PubMed]

L. Mertz, "Real-time fringe-pattern analysis," Appl. Opt. 22, 1535-1539 (1983).
[CrossRef] [PubMed]

W. W. Macy, "Two-dimensional fringe-pattern analysis," Appl. Opt. 22, 3898-3901 (1983).
[CrossRef] [PubMed]

Y. Ichioka and M. Inuiya, "Direct Phase Detecting System," Appl. Opt. 11, 1507-1514 (1972).
[CrossRef] [PubMed]

J. Kato, I. Yamaguchi, T. Nakamura, and S. Kuwashima, "Video-rate fringe analyzer based on phase-shifting electronic moiré patterns," Appl. Opt. 36, 8403-8412 (1997).
[CrossRef]

C. Quan, C. J. Tay, X. Kang, X. Y. He, and H. M. Shang, "Shape measurement by use of liquid-crystal display, fringe projection with two-step phase-shifting," Appl. Opt. 42, 2329-2335 (2003).
[CrossRef] [PubMed]

P. L. Wizinowich, "Phase shifting interferometry in the presence of vibration: a new algorithm and system," Appl. Opt. 29, 3271-3279 (1990).
[CrossRef] [PubMed]

H. A. Vrooman and A. A. M. Maas, "Image processing algorithms for the analysis of phase-shifted speckle interference patterns," Appl. Opt. 30, 1636-1641 (1991).
[CrossRef] [PubMed]

P. S. Huang and S. Zhang, "Fast three-step phase shifting algorithm," Appl. Opt. 45, 5086-5091 (2006).
[CrossRef] [PubMed]

J. Mod. Opt. (1)

Y. Arai, S. Yokozeki, K. Shiraki, and T. Yamada, "High precision two-dimensional spatial fringe analysis method," J. Mod. Opt. 44, 739-751 (1997).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Eng. (8)

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, 113601 (2005).
[CrossRef]

C. J. Tay, C. Quan, H. M. Shang, T. Wu, and S. Wang, "New method for measuring dynamic response of small components by fringe projection," Opt. Eng. 42, 1715-1720 (2003).
[CrossRef]

K. H. Womack, "Interferometric phase measurement using spatial synchronous detection," Opt. Eng. 23, 391-395 (1984).

P. S. Huang, Q. Hu, and F-P. Chiang, "Color-encoded digital fringe projection technique for high-speed three-dimensional surface contouring," Opt. Eng. 38, 1065-1071 (1999).
[CrossRef]

C. R. Coggrave and J. M. Huntley, "High-speed surface profilometer based on a spatial light modulator and pipeline image processor," Opt. Eng. 38, 1573-1581 (1999).
[CrossRef]

P. S. Huang, C. P. Zhang, and F. P. Chiang, "High-speed 3-D shape measurement based on digital fringe projection," Opt. Eng. 42, 163-168 (2003).
[CrossRef]

M. S. Mermelstein, D. L. Feldhun, and L. G. Shirley, "Video-rate surface profiling with acousto-optic accordion fringe interferometry," Opt. Eng. 39, 106-113 (2000).
[CrossRef]

S. Almazán-Cuéllar and D. Malacara-Hernández, "Two-step phase-shifting algorithm," Opt. Eng. 42, 3524-3531 (2003).
[CrossRef]

Opt. Express (4)

Opt. Laser Tech. (1)

D. C. Williams, N. S. Nassar, J. E. Banyard, and M. S. Virdee, "Digital phase-step interferometry: a simplified approach," Opt. Laser Tech. 23, 147-150 (1991).
[CrossRef]

Opt. Lasers Eng. (3)

P. H. Chan and P. J. Bryanston-Cross, "Spatial phase stepping method of fringe-pattern analysis," Opt. Lasers Eng. 23, 343-354 (1995).
[CrossRef]

M. A. Herráez, D. R. Burton, and M. J. Lalor, "Accelerating fast Fourier transform and filtering operations in Fourier fringe analysis for accurate measurement of three-dimensional surfaces," Opt. Lasers Eng. 31, 135-145 (1999).
[CrossRef]

Z. Gao, S. Zhou, and Y. Hu, "High-speed fringe analysis by using stair-shaped virtual grating demodulation," Opt. Lasers Eng. 28, 411-422 (1997).
[CrossRef]

Opt. Lett. (1)

Proc. SPIE (1)

Y. Morimoto and M. Fujigaki. "Real-time phase distribution analysis of fringe pattern," in International Conference on Applied Optical Metrology, P. K. Rastogi, and F. Gyímesi eds., Proc. SPIE,  3407, 34-39 (1998).
[CrossRef]

Other (3)

S. Zhang and S-T. Yau, "High-resolution, real-time 3D absolute coordinate measurement based on a phase-shifting method," Opt. Express 14, 2644-2649 (2006), http://www.opticsinfobase.org/abstract.cfm? URI=oe-14-7-2644.
[CrossRef] [PubMed]

R. Capelli, "Fast approximation to the arctangent," in Graphics Gems II, J. Arvo, ed. (AP Professional, Boston, USA, 1995), pp. 389-391.

M. Kujawinska, "Spatial phase measurement methods," in Interferogram Analysis: Digital Fringe Pattern Measurement, D. W. Robinson and G. Reid, eds. (IOP, Bristol, UK, 1993), pp.141-193.

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Figures (12)

Fig. 1.
Fig. 1.

The 2π radian range of the arctangent function is segmented into four quadrants.

Fig. 2.
Fig. 2.

The approximation errors versus phases by using Eq. (15) with degrees being 1(solid line), 3 (dotted line) and 5 (dashed line), respectively.

Fig. 3.
Fig. 3.

(a) A fringe pattern and (b) its phase map reconstructed by using FTM and the standard arctangent function.

Fig. 4.
Fig. 4.

The approximation errors by using Eq. (15) with the degrees being (a) 1, (b) 3 and (c) 5.

Fig. 5.
Fig. 5.

The 2π radian range of the arctangent function is segmented into six sextants.

Fig. 6.
Fig. 6.

The approximation errors versus phases by using Eq. (17) with degrees being 1 (solid line), 3 (dotted line) and 5 (dashed line), respectively.

Fig. 7.
Fig. 7.

The approximation errors by using Eq. (17) with the degrees being (a) 1, (b) 3 and (c) 5.

Fig. 8.
Fig. 8.

The 2π radian range of the arctangent function is segmented into eight octants.

Fig. 9.
Fig. 9.

The approximation errors versus phases by using Eq. (18) with degrees being 2 (solid line), 3 (dotted line) and 4 (dashed line), respectively.

Fig. 10.
Fig. 10.

The approximation errors by using Eq. (18) with the degrees being (a) 2, (b) 3 and (c) 4.

Fig. 11.
Fig. 11.

The approximation errors versus phases by using Eq. (19) with degrees being 1 (solid line), 3 (dotted line) and 5 (dashed line), respectively.

Fig. 12.
Fig. 12.

The approximation errors by using Eq. (19) with the degrees being (a) 1, (b) 3 and (c) 5.

Tables (8)

Tables Icon

Table 1. Coefficients and Maximum Errors of the Polynomials when Approximating the Arctangent Function within the Interval [1, -1]

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Table 2. The Computational Operations Involved in Eq.15

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Table 3. Coefficients and Maximum Errors of the Polynomials when Approximating the Arctangent Function within the Interval [-√3/3, √3/3]

Tables Icon

Table 4. The Computational Operations Involved in Eq. (17)

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Table 5. Coefficients and Maximum Errors of the Polynomials when Approximating the Arctangent Function within the Interval [0, 1]

Tables Icon

Table 6. The Computational Operations Involved in Eq. (18)

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Table 7. Coefficients and Maximum Errors of the Polynomials when Approximating the Arctangent Function within the Interval [1 - √2, √2 - 1]

Tables Icon

Table 8. The Computational Operations Involved in Eq. (19)

Equations (19)

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

g x y = a x y + b x y cos [ u C x + v C y + Φ x y ] ,
g x y = a x y + c x y exp ( u C x + v C y ) + c * x y exp ( u C x v C y )
G x v = A x v + C u u C v v C + C * u + u C v + v C .
X x y = Re [ c x y ] ,
Y x y = Im [ c x y ]
r x y = Y x y X x y ,
Φ x y = Φ [ r x y ] = arctan [ r x y ] .
Ψ ( r ) = n = 0 N p n r n ,
Ψ ( r ) = p 0 + r ( p 1 + r ( p 2 + r ( p 3 + + r ( p N 1 + p N r ) ) ) ) ,
Ψ ( r 1 ) = n = 0 N p n r 1 n = arctan ( r 1 ) = Φ ( r 1 )
Ψ ( r 2 ) = n = 0 N p n r 2 n = arctan ( r 2 ) = Φ ( r 2 ) .
Φ Ψ = max r 1 r r 2 Φ ( r ) Ψ ( r ) .
Φ ( ξ k ) Ψ ( ξ k ) = arctan ( ξ k ) n = 0 N = p n ξ k n = ( 1 ) k η , for k = 1,2 , , N .
d [ Φ ( r ) Ψ ( r ) ] dr = 1 1 + r 2 n = 1 N np n r n 1 = 0 ,
Φ = { Ψ ( X Y ) if Y < X and Y X π 2 Ψ ( X Y ) if Y X and Y X π + Ψ ( Y X ) if Y X and Y < X π 2 Ψ ( X Y ) if Y < X and Y < X
Φ = { π 4 Ψ [ ( X Y ) ( X + Y ) ] if X 0 and Y 0 π 4 + Ψ [ ( X + Y ) ( X Y ) ] if X 0 and Y < 0 3 π 4 + Ψ [ ( X + Y ) ( X Y ) ] if X < 0 and Y 0 3 π 4 Ψ [ ( X Y ) ( X + Y ) ] if X < 0 and Y < 0
Φ = { π 2 Ψ ( X Y ) Y 0 and Y 3 X and Y 3 X 5 π 6 + Ψ [ ( 3 Y X ) ( Y 3 X ) ] Y 0 and Y 3 X and Y < 3 X π 6 + Ψ [ ( 3 Y X ) ( Y + 3 X ) ] Y 0 and Y < 3 X 5 π 6 + Ψ [ ( 3 Y + X ) ( Y 3 X ) ] Y < 0 and Y 3 X π 6 + Ψ [ ( 3 Y + X ) ( Y + 3 X ) ] Y < 0 and Y < 3 X and Y 3 X π 2 Ψ ( X Y ) Y < 0 and Y < 3 X and Y < 3 X ,
Φ = { π 2 Ψ ( X Y ) X 0 and Y 0 and Y X Ψ ( Y X ) X 0 and Y 0 and Y < X Ψ ( Y X ) X 0 and Y < 0 and Y X π 2 + Ψ ( X Y ) X 0 and Y < 0 and Y < X π 2 + Ψ ( X Y ) X < 0 and Y 0 and Y X π Ψ ( Y X ) X < 0 and Y 0 and Y < X π + Ψ ( Y X ) X < 0 and Y < 0 and Y X π 2 Ψ ( X Y ) X < 0 and Y < 0 and Y < X
Φ = { 3 π 8 Ψ { [ X ( 2 1 ) Y ] [ ( 2 1 ) X + Y ] } X 0 and Y 0 and Y X π 8 + Ψ { [ Y ( 2 1 ) X ] [ ( 2 1 ) Y + X ] } X 0 and Y 0 and Y < X π 8 ψ { [ Y + ( 2 1 ) X ] [ ( 2 1 ) Y X ] } X 0 and Y < 0 and Y X 3 π 8 + Ψ { [ X ( + 2 1 ) Y ] [ ( 2 1 ) X Y ] } X 0 and Y < 0 and Y < X 5 π 8 + Ψ { [ X ( 2 1 ) Y ] [ ( 2 1 ) X Y ] } X < 0 and Y 0 and Y X 7 π 8 Ψ { [ Y + ( 2 1 ) X ] [ ( 2 1 ) Y X ] } X < 0 and Y 0 and Y < X 7 π 8 + Ψ { [ Y ( 2 1 ) X ] [ ( 2 1 ) Y + X ] } X < 0 and Y < 0 and Y X 5 π 8 Ψ { [ X ( 2 1 ) Y ] [ ( 2 1 ) X + Y ] } X < 0 and Y < 0 and Y < X

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