Abstract

Phase measuring profilometry is a method of structured light illumination whose three-dimensional reconstructions are susceptible to error from nonunitary gamma in the associated optical devices. While the effects of this distortion diminish with an increasing number of employed phase-shifted patterns, gamma distortion may be unavoidable in real-time systems where the number of projected patterns is limited by the presence of target motion. A mathematical model is developed for predicting the effects of nonunitary gamma on phase measuring profilometry, while also introducing an accurate gamma calibration method and two strategies for minimizing gamma’s effect on phase determination. These phase correction strategies include phase corrections with and without gamma calibration. With the reduction in noise, for three-step phase measuring profilometry, analysis of the root mean squared error of the corrected phase will show a 60× reduction in phase error when the proposed gamma calibration is performed versus 33× reduction without calibration.

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. Salvi, J. Pages, and J. Batlle, “Pattern codification strategies in structured light systems,” Pattern Recogn. 37, 827-849 (2004).
    [CrossRef]
  2. S. Zhang, D. Royer, and S. T. Yau, “High-resolution, real-time-geometry video acquisition system,” in International Conference on Computer Graphics and Interactive Techniques. ACM SIGGRAPH 2006 (ACM, 2006), Emerging Technologies, article 14.
  3. C. Guan, L. G. Hassebrook, and D. L. Lau, “Composite structured light pattern for three-dimensional video,” Opt. Express 11, 406-417 (2003).
    [CrossRef] [PubMed]
  4. V. Srinivasan, H. C. Liu, and M. Halioua, “Automated phase-measuring profilometry of 3-D diffuse objects,” Appl. Opt. 23, 3105-3108 (1984).
    [CrossRef] [PubMed]
  5. X. Su, G. von Bally, and D. Vukicevic, “Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun. 98, 141-150 (1993).
    [CrossRef]
  6. C. Rathjen, “Statistical properties of phase-shift algorithms,” J. Opt. Soc. Am. A 12, 1997-2008 (1995).
    [CrossRef]
  7. J. Li, L. G. Hassebrook, and C. Guan, “Optimized two-frequency phase-measuring-profilometry light-sensor temporal-noise sensitivity,” J. Opt. Soc. Am. A 20, 106-115 (2003).
    [CrossRef]
  8. A. Patil, R. Langoju, P. Rastogi, and S. Ramani, “Statistical study and experimental verification of high-resolution methods in phase-shifting interferometry,” J. Opt. Soc. Am. A 24, 794-813 (2007).
    [CrossRef]
  9. G. Nico, “Noise-residue filtering of interferometric phase images,” J. Opt. Soc. Am. A 17, 1962-1974 (2000).
    [CrossRef]
  10. C. A. Poynton, “'Gamma' and its disguises: the nonlinear mappings of intensity in perception, CRTs, film and video,” SMPTE J. 102, 1099-1108 (1993).
    [CrossRef]
  11. H. Farid, “Blind inverse gamma correction,” IEEE Trans. Image Process. 10, 1428-1433 (2001).
    [CrossRef]
  12. A. Siebert, “Retrieval of gamma corrected images,” Pattern Recogn. Lett. 22, 249-256 (2001).
    [CrossRef]
  13. K. A. Stetson and W. R. Brohinsky, “Electrooptic holography and its application to hologram interferometry,” Appl. Opt. 24, 3631-3637 (1985).
    [CrossRef] [PubMed]
  14. H. Guo and M. Chen, “Fourier analysis of the sampling characteristics of the phase-shifting algorithm,” Proc. SPIE 5180, 437-444 (2003).
    [CrossRef]
  15. P. S. Huang, C. Zhang, and F. Chiang, “High-speed 3D shape measurement based on digital fringe projection,” Opt. Eng. 42, 163-168 (2003).
    [CrossRef]
  16. S. Zhang and S. Yau, “Generic nonsinusoidal phase error correction for three-dimensional shape measurement using a digital video projector,” Appl. Opt. 46, 36-43 (2007).
    [CrossRef]
  17. X. Su, W. Zhou, G. von Baly, and D. Vukicevic, “Automated phase-measuring profilometry using defocused projection of a ronchi grating,” Opt. Commun. 94, 561-573 (1992).
    [CrossRef]
  18. M. J. Baker, J. Xi, and J. F. Chicharo, “Elimination of γ non-linear luminance effects for digital video projection phase measuring profilometers,” in 4th IEEE International Symposium on Electronic Design, Test and Applications (IEEE Computer Society, 2008), pp. 496-501.
  19. K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, “Phase shifting for nonsinusoidal waveforms with phase-shift errors,” Opt. Eng. 12, 761-768 (1995).
  20. P. S. Huang, Q. J. Hu, and F. Chiang, “Double three-step phase-shifting algorithm,” Appl. Opt. 41, 4503-4509 (2002).
    [CrossRef] [PubMed]
  21. Y. Hu, J. Xi, J. F. Chicharo, and Z. Yang, “Improved three-step phase shifting profilometry using digital fringe pattern projection,” in Proceedings of the International Conference on Computer Graphics, Imaging and Visualization (ACS, 2006), pp. 161-167.
  22. Y. Hu, J. Xi, E. Li, J. Chicharo, and Z. Yang, “Three-dimensional profilometry based on shift estimation of projected fringe patterns,” Appl. Opt. 45, 678-687 (2006).
    [CrossRef] [PubMed]
  23. Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35, 51-60 (1996).
    [CrossRef] [PubMed]
  24. M. J. Baker, J. Xi, and J. F. Chicharo, “Neural network digital fringe calibration technique for structured light profilometers,” Appl. Opt. 46, 1233-1243 (2007).
    [CrossRef] [PubMed]
  25. S. Zhang and P. S. Huang, “Phase error compensation for a 3-D shape measurement system based on the phase-shifting method,” Opt. Eng. 46, 063601 (2007).
    [CrossRef]
  26. P. Jia, J. Kofman, and C. English, “Intensity-ratio error compensation for triangular-pattern phase-shifting profilometry,” J. Opt. Soc. Am. A 24, 3150-3158 (2007).
    [CrossRef]
  27. H. Guo, H. He, and M. Chen, “Gamma correction for digital fringe projection profilometry,” Appl. Opt. 43, 2906-2914 (2004).
    [CrossRef] [PubMed]
  28. B. Pan, L. H. Q. Kemao, and A. Asundi, “Phase error analysis and compensation for nonsinusoidal waveforms in phase-shifting digital fringe projection profilometry,” Opt. Lett. 34, 416-418 (2009).
    [CrossRef] [PubMed]

2009 (1)

2007 (5)

2006 (1)

2004 (2)

H. Guo, H. He, and M. Chen, “Gamma correction for digital fringe projection profilometry,” Appl. Opt. 43, 2906-2914 (2004).
[CrossRef] [PubMed]

J. Salvi, J. Pages, and J. Batlle, “Pattern codification strategies in structured light systems,” Pattern Recogn. 37, 827-849 (2004).
[CrossRef]

2003 (4)

H. Guo and M. Chen, “Fourier analysis of the sampling characteristics of the phase-shifting algorithm,” Proc. SPIE 5180, 437-444 (2003).
[CrossRef]

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

J. Li, L. G. Hassebrook, and C. Guan, “Optimized two-frequency phase-measuring-profilometry light-sensor temporal-noise sensitivity,” J. Opt. Soc. Am. A 20, 106-115 (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).
[CrossRef] [PubMed]

2002 (1)

2001 (2)

H. Farid, “Blind inverse gamma correction,” IEEE Trans. Image Process. 10, 1428-1433 (2001).
[CrossRef]

A. Siebert, “Retrieval of gamma corrected images,” Pattern Recogn. Lett. 22, 249-256 (2001).
[CrossRef]

2000 (1)

1996 (1)

1995 (2)

C. Rathjen, “Statistical properties of phase-shift algorithms,” J. Opt. Soc. Am. A 12, 1997-2008 (1995).
[CrossRef]

K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, “Phase shifting for nonsinusoidal waveforms with phase-shift errors,” Opt. Eng. 12, 761-768 (1995).

1993 (2)

X. Su, G. von Bally, and D. Vukicevic, “Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun. 98, 141-150 (1993).
[CrossRef]

C. A. Poynton, “'Gamma' and its disguises: the nonlinear mappings of intensity in perception, CRTs, film and video,” SMPTE J. 102, 1099-1108 (1993).
[CrossRef]

1992 (1)

X. Su, W. Zhou, G. von Baly, and D. Vukicevic, “Automated phase-measuring profilometry using defocused projection of a ronchi grating,” Opt. Commun. 94, 561-573 (1992).
[CrossRef]

1985 (1)

1984 (1)

Asundi, A.

Baker, M. J.

M. J. Baker, J. Xi, and J. F. Chicharo, “Neural network digital fringe calibration technique for structured light profilometers,” Appl. Opt. 46, 1233-1243 (2007).
[CrossRef] [PubMed]

M. J. Baker, J. Xi, and J. F. Chicharo, “Elimination of γ non-linear luminance effects for digital video projection phase measuring profilometers,” in 4th IEEE International Symposium on Electronic Design, Test and Applications (IEEE Computer Society, 2008), pp. 496-501.

Batlle, J.

J. Salvi, J. Pages, and J. Batlle, “Pattern codification strategies in structured light systems,” Pattern Recogn. 37, 827-849 (2004).
[CrossRef]

Brohinsky, W. R.

Chen, M.

H. Guo, H. He, and M. Chen, “Gamma correction for digital fringe projection profilometry,” Appl. Opt. 43, 2906-2914 (2004).
[CrossRef] [PubMed]

H. Guo and M. Chen, “Fourier analysis of the sampling characteristics of the phase-shifting algorithm,” Proc. SPIE 5180, 437-444 (2003).
[CrossRef]

Chiang, F.

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

P. S. Huang, Q. J. Hu, and F. Chiang, “Double three-step phase-shifting algorithm,” Appl. Opt. 41, 4503-4509 (2002).
[CrossRef] [PubMed]

Chicharo, J.

Chicharo, J. F.

M. J. Baker, J. Xi, and J. F. Chicharo, “Neural network digital fringe calibration technique for structured light profilometers,” Appl. Opt. 46, 1233-1243 (2007).
[CrossRef] [PubMed]

M. J. Baker, J. Xi, and J. F. Chicharo, “Elimination of γ non-linear luminance effects for digital video projection phase measuring profilometers,” in 4th IEEE International Symposium on Electronic Design, Test and Applications (IEEE Computer Society, 2008), pp. 496-501.

Y. Hu, J. Xi, J. F. Chicharo, and Z. Yang, “Improved three-step phase shifting profilometry using digital fringe pattern projection,” in Proceedings of the International Conference on Computer Graphics, Imaging and Visualization (ACS, 2006), pp. 161-167.

English, C.

Farid, H.

H. Farid, “Blind inverse gamma correction,” IEEE Trans. Image Process. 10, 1428-1433 (2001).
[CrossRef]

Farrant, D. I.

K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, “Phase shifting for nonsinusoidal waveforms with phase-shift errors,” Opt. Eng. 12, 761-768 (1995).

Guan, C.

Guo, H.

H. Guo, H. He, and M. Chen, “Gamma correction for digital fringe projection profilometry,” Appl. Opt. 43, 2906-2914 (2004).
[CrossRef] [PubMed]

H. Guo and M. Chen, “Fourier analysis of the sampling characteristics of the phase-shifting algorithm,” Proc. SPIE 5180, 437-444 (2003).
[CrossRef]

Halioua, M.

Hassebrook, L. G.

He, H.

Hibino, K.

K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, “Phase shifting for nonsinusoidal waveforms with phase-shift errors,” Opt. Eng. 12, 761-768 (1995).

Hu, Q. J.

Hu, Y.

Y. Hu, J. Xi, E. Li, J. Chicharo, and Z. Yang, “Three-dimensional profilometry based on shift estimation of projected fringe patterns,” Appl. Opt. 45, 678-687 (2006).
[CrossRef] [PubMed]

Y. Hu, J. Xi, J. F. Chicharo, and Z. Yang, “Improved three-step phase shifting profilometry using digital fringe pattern projection,” in Proceedings of the International Conference on Computer Graphics, Imaging and Visualization (ACS, 2006), pp. 161-167.

Huang, P. S.

S. Zhang and P. S. Huang, “Phase error compensation for a 3-D shape measurement system based on the phase-shifting method,” Opt. Eng. 46, 063601 (2007).
[CrossRef]

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

P. S. Huang, Q. J. Hu, and F. Chiang, “Double three-step phase-shifting algorithm,” Appl. Opt. 41, 4503-4509 (2002).
[CrossRef] [PubMed]

Jia, P.

Kemao, L. H. Q.

Kofman, J.

Langoju, R.

Larkin, K. G.

K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, “Phase shifting for nonsinusoidal waveforms with phase-shift errors,” Opt. Eng. 12, 761-768 (1995).

Lau, D. L.

Li, E.

Li, J.

Liu, H. C.

Nico, G.

Oreb, B. F.

K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, “Phase shifting for nonsinusoidal waveforms with phase-shift errors,” Opt. Eng. 12, 761-768 (1995).

Pages, J.

J. Salvi, J. Pages, and J. Batlle, “Pattern codification strategies in structured light systems,” Pattern Recogn. 37, 827-849 (2004).
[CrossRef]

Pan, B.

Patil, A.

Poynton, C. A.

C. A. Poynton, “'Gamma' and its disguises: the nonlinear mappings of intensity in perception, CRTs, film and video,” SMPTE J. 102, 1099-1108 (1993).
[CrossRef]

Ramani, S.

Rastogi, P.

Rathjen, C.

Royer, D.

S. Zhang, D. Royer, and S. T. Yau, “High-resolution, real-time-geometry video acquisition system,” in International Conference on Computer Graphics and Interactive Techniques. ACM SIGGRAPH 2006 (ACM, 2006), Emerging Technologies, article 14.

Salvi, J.

J. Salvi, J. Pages, and J. Batlle, “Pattern codification strategies in structured light systems,” Pattern Recogn. 37, 827-849 (2004).
[CrossRef]

Siebert, A.

A. Siebert, “Retrieval of gamma corrected images,” Pattern Recogn. Lett. 22, 249-256 (2001).
[CrossRef]

Srinivasan, V.

Stetson, K. A.

Su, X.

X. Su, G. von Bally, and D. Vukicevic, “Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun. 98, 141-150 (1993).
[CrossRef]

X. Su, W. Zhou, G. von Baly, and D. Vukicevic, “Automated phase-measuring profilometry using defocused projection of a ronchi grating,” Opt. Commun. 94, 561-573 (1992).
[CrossRef]

Surrel, Y.

von Bally, G.

X. Su, G. von Bally, and D. Vukicevic, “Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun. 98, 141-150 (1993).
[CrossRef]

von Baly, G.

X. Su, W. Zhou, G. von Baly, and D. Vukicevic, “Automated phase-measuring profilometry using defocused projection of a ronchi grating,” Opt. Commun. 94, 561-573 (1992).
[CrossRef]

Vukicevic, D.

X. Su, G. von Bally, and D. Vukicevic, “Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun. 98, 141-150 (1993).
[CrossRef]

X. Su, W. Zhou, G. von Baly, and D. Vukicevic, “Automated phase-measuring profilometry using defocused projection of a ronchi grating,” Opt. Commun. 94, 561-573 (1992).
[CrossRef]

Xi, J.

M. J. Baker, J. Xi, and J. F. Chicharo, “Neural network digital fringe calibration technique for structured light profilometers,” Appl. Opt. 46, 1233-1243 (2007).
[CrossRef] [PubMed]

Y. Hu, J. Xi, E. Li, J. Chicharo, and Z. Yang, “Three-dimensional profilometry based on shift estimation of projected fringe patterns,” Appl. Opt. 45, 678-687 (2006).
[CrossRef] [PubMed]

M. J. Baker, J. Xi, and J. F. Chicharo, “Elimination of γ non-linear luminance effects for digital video projection phase measuring profilometers,” in 4th IEEE International Symposium on Electronic Design, Test and Applications (IEEE Computer Society, 2008), pp. 496-501.

Y. Hu, J. Xi, J. F. Chicharo, and Z. Yang, “Improved three-step phase shifting profilometry using digital fringe pattern projection,” in Proceedings of the International Conference on Computer Graphics, Imaging and Visualization (ACS, 2006), pp. 161-167.

Yang, Z.

Y. Hu, J. Xi, E. Li, J. Chicharo, and Z. Yang, “Three-dimensional profilometry based on shift estimation of projected fringe patterns,” Appl. Opt. 45, 678-687 (2006).
[CrossRef] [PubMed]

Y. Hu, J. Xi, J. F. Chicharo, and Z. Yang, “Improved three-step phase shifting profilometry using digital fringe pattern projection,” in Proceedings of the International Conference on Computer Graphics, Imaging and Visualization (ACS, 2006), pp. 161-167.

Yau, S.

Yau, S. T.

S. Zhang, D. Royer, and S. T. Yau, “High-resolution, real-time-geometry video acquisition system,” in International Conference on Computer Graphics and Interactive Techniques. ACM SIGGRAPH 2006 (ACM, 2006), Emerging Technologies, article 14.

Zhang, C.

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

Zhang, S.

S. Zhang and P. S. Huang, “Phase error compensation for a 3-D shape measurement system based on the phase-shifting method,” Opt. Eng. 46, 063601 (2007).
[CrossRef]

S. Zhang and S. Yau, “Generic nonsinusoidal phase error correction for three-dimensional shape measurement using a digital video projector,” Appl. Opt. 46, 36-43 (2007).
[CrossRef]

S. Zhang, D. Royer, and S. T. Yau, “High-resolution, real-time-geometry video acquisition system,” in International Conference on Computer Graphics and Interactive Techniques. ACM SIGGRAPH 2006 (ACM, 2006), Emerging Technologies, article 14.

Zhou, W.

X. Su, W. Zhou, G. von Baly, and D. Vukicevic, “Automated phase-measuring profilometry using defocused projection of a ronchi grating,” Opt. Commun. 94, 561-573 (1992).
[CrossRef]

Appl. Opt. (8)

IEEE Trans. Image Process. (1)

H. Farid, “Blind inverse gamma correction,” IEEE Trans. Image Process. 10, 1428-1433 (2001).
[CrossRef]

J. Opt. Soc. Am. A (5)

Opt. Commun. (2)

X. Su, G. von Bally, and D. Vukicevic, “Phase-stepping grating profilometry: utilization of intensity modulation analysis in complex objects evaluation,” Opt. Commun. 98, 141-150 (1993).
[CrossRef]

X. Su, W. Zhou, G. von Baly, and D. Vukicevic, “Automated phase-measuring profilometry using defocused projection of a ronchi grating,” Opt. Commun. 94, 561-573 (1992).
[CrossRef]

Opt. Eng. (3)

K. Hibino, B. F. Oreb, D. I. Farrant, and K. G. Larkin, “Phase shifting for nonsinusoidal waveforms with phase-shift errors,” Opt. Eng. 12, 761-768 (1995).

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

S. Zhang and P. S. Huang, “Phase error compensation for a 3-D shape measurement system based on the phase-shifting method,” Opt. Eng. 46, 063601 (2007).
[CrossRef]

Opt. Express (1)

Opt. Lett. (1)

Pattern Recogn. (1)

J. Salvi, J. Pages, and J. Batlle, “Pattern codification strategies in structured light systems,” Pattern Recogn. 37, 827-849 (2004).
[CrossRef]

Pattern Recogn. Lett. (1)

A. Siebert, “Retrieval of gamma corrected images,” Pattern Recogn. Lett. 22, 249-256 (2001).
[CrossRef]

Proc. SPIE (1)

H. Guo and M. Chen, “Fourier analysis of the sampling characteristics of the phase-shifting algorithm,” Proc. SPIE 5180, 437-444 (2003).
[CrossRef]

SMPTE J. (1)

C. A. Poynton, “'Gamma' and its disguises: the nonlinear mappings of intensity in perception, CRTs, film and video,” SMPTE J. 102, 1099-1108 (1993).
[CrossRef]

Other (3)

Y. Hu, J. Xi, J. F. Chicharo, and Z. Yang, “Improved three-step phase shifting profilometry using digital fringe pattern projection,” in Proceedings of the International Conference on Computer Graphics, Imaging and Visualization (ACS, 2006), pp. 161-167.

M. J. Baker, J. Xi, and J. F. Chicharo, “Elimination of γ non-linear luminance effects for digital video projection phase measuring profilometers,” in 4th IEEE International Symposium on Electronic Design, Test and Applications (IEEE Computer Society, 2008), pp. 496-501.

S. Zhang, D. Royer, and S. T. Yau, “High-resolution, real-time-geometry video acquisition system,” in International Conference on Computer Graphics and Interactive Techniques. ACM SIGGRAPH 2006 (ACM, 2006), Emerging Technologies, article 14.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (12)

Fig. 1
Fig. 1

Ideal and distorted ( γ = 2.2 ) sine wave (top), and the magnitude of the distorted sine wave in the frequency domain (bottom).

Fig. 2
Fig. 2

Coefficients B k (top), ratios of B k + 1 B k (center), and | B k | at γ = 3.18 in log scale (bottom).

Fig. 3
Fig. 3

Phase error (top), RMS of phase error for different N at γ = 3.18 in log scale (center), and RMS of phase error for different N = 3 , 4, 5, 6 with γ [ 1 , 5 ] in log scale (bottom).

Fig. 4
Fig. 4

Gamma computation versus the number of patterns, N, showing the RMS of error in log scale.

Fig. 5
Fig. 5

Distorted (solid curves) and corrected (dotted–dashed lines) phase error using constrained phase optimization without gamma calibration for γ = 3.18 and N = 3 (top), γ = 6.36 and N = 4 (center), and γ = 10.36 and N = 5 (bottom).

Fig. 6
Fig. 6

Photograph of the scanned plastic fish with varying texture.

Fig. 7
Fig. 7

Best-fit curve for a pixel with a measured gamma value of 2.153 (top) and the histogram of the calibrated gamma over the entire image, where the mean is 2.2123 with variance 2.67 × 10 4 (bottom).

Fig. 8
Fig. 8

3D reconstructed depth (solid curves) before and after (dashed curves) gamma-corrected phase of the textureless foam board for N = 3 with gamma calibration (top), N = 3 without gamma calibration (second from top), N = 4 without gamma calibration (second from bottom), and N = 5 without gamma calibration (bottom).

Fig. 9
Fig. 9

3D reconstructions of the textured plastic fish viewed from the front and from above for (left) the uncorrected and (right) corrected phase with gamma calibration.

Fig. 10
Fig. 10

3D reconstructions of the textured plastic fish viewed from the front and from above for (left) the uncorrected and (right) corrected phase with N = 3 constrained gamma and phase optimization.

Fig. 11
Fig. 11

3D reconstructions of the textured plastic fish viewed from the front and from above for (left) the uncorrected and (right) corrected phase with N = 4 constrained gamma and phase optimization.

Fig. 12
Fig. 12

3D reconstructions of the textured plastic fish viewed from the front and from above for (left) the uncorrected and (right) corrected phase with N = 5 constrained gamma and phase optimization.

Tables (2)

Tables Icon

Table 1 RMS of Phase Error a

Tables Icon

Table 2 Phase Error Comparison for Different Algorithms

Equations (48)

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

I n p = A p + B p cos ( 2 π f y p 2 π n N ) ,
I n c = A c + B c cos ( ϕ 2 π n N ) .
A c = 1 N n = 0 N 1 I n c .
B c = 2 N { [ n = 0 N 1 I n c sin ( 2 π n N ) ] 2 + [ n = 0 N 1 I n c cos ( 2 π n N ) ] 2 } 0.5 .
ϕ = arctan [ n = 0 N 1 I n c sin ( 2 π n N ) n = 0 N 1 I n c cos ( 2 π n N ) ] ,
I n p = α p [ 0.5 + 0.5 cos ( 2 π f y p 2 π n N ) ] + β p ,
I n c = α α p [ 0.5 + 0.5 cos ( ϕ 2 π n N ) ] γ ,
I n c = 0.5 γ α α p m = 0 [ ( γ m ) cos m ( ϕ 2 π n N ) ] .
I n c = A + k = 1 { B k cos [ k ( ϕ 2 π n N ) ] } ,
A = 0.5 B 0 ,
B k = 0.5 γ 1 α α p m = 0 ( b k , m )
b k , m = 0.5 2 m + k ( γ 2 m + k ) ( 2 m + k m )
( γ k ) b k , m 2 ( k + 1 ) b k + 1 , m = 2 m b k , m + 2 m b k + 1 , m .
( γ k ) B ̂ k 2 ( k + 1 ) B ̂ k + 1 = ( γ k 1 ) B ̂ k + 1 .
B k + 1 B k = B ̂ k + 1 B ̂ k = γ k γ + k + 1
A ̃ c = 0.5 B 0 + m = 1 [ B m N cos ( m N ϕ ) ] ,
Δ A c = 0.5 α α p [ 1 0.5 γ 1 m = 0 ( b 0 , m ) ] m = 1 [ B m N cos ( m N ϕ ) ] ,
S N = 2 N n = 0 N 1 [ I n c sin ( 2 π n N ) ] = B 1 sin ( ϕ ) + k = 1 { B k N + 1 sin [ ( k N + 1 ) ϕ ] } k = 1 { B k N 1 sin [ ( k N 1 ) ϕ ] } ,
C N = 2 N n = 0 N 1 [ I n c cos ( 2 π n N ) ] = B 1 cos ( ϕ ) + k = 1 { B k N + 1 cos [ ( k N + 1 ) ϕ ] } + k = 1 { B k N 1 cos [ ( k N 1 ) ϕ ] } ,
B ̃ c = [ ( S N ) 2 + ( C N ) 2 ] 0.5 = B 1 D 0.5 ,
D = { k = 1 [ b k N sin ( k N ϕ ) ] } 2 + { 1 + k = 1 [ b k N + cos ( k N ϕ ) ] } 2 ,
b k N = G k N 1 ( γ ) G k N + 1 ( γ ) ,
b k N + = G k N 1 ( γ ) + G k N + 1 ( γ ) ,
G m ( γ ) = B m B 1 = n = 2 m ( γ n + 1 γ + n ) .
Δ B c = 0.5 α α p [ 1 0.5 γ 2 D 0.5 m = 0 ( b 1 , m ) ] ,
ϕ ̃ = arctan ( S N C N ) = arctan [ sin ( ϕ ) + H s ( γ , ϕ ) cos ( ϕ ) + H c ( γ , ϕ ) ] ,
H s ( γ , ϕ ) = k = 1 G k N + 1 ( γ ) sin [ ( k N + 1 ) ϕ ] k = 1 G k N 1 ( γ ) sin [ ( k N 1 ) ϕ ] ,
H c ( γ , ϕ ) = k = 1 G k N + 1 ( γ ) cos [ ( k N + 1 ) ϕ ] + k = 1 G k N 1 ( γ ) cos [ ( k N 1 ) ϕ ] .
Δ ϕ = arctan { k = 1 [ b k N sin ( k N ϕ ) ] 1 + k = 1 [ b k N + cos ( k N ϕ ) ] } .
S m , N = 2 N n = 0 N 1 [ I n c sin ( m 2 π n N ) ] ,
C m , N = 2 N n = 0 N 1 [ I n c cos ( m 2 π n N ) ] ,
S m , N = B m sin ( m ϕ ) + k = 1 { B k N + m sin [ ( k N + m ) ϕ ] } k = 1 { B k N m sin [ ( k N m ) ϕ ] } ,
C m , N = B m cos ( m ϕ ) + k = 1 { B k N + m cos [ ( k N + m ) ϕ ] } + k = 1 { B k N m cos [ ( k N m ) ϕ ] } ,
B m 2 N { [ n = 0 N 1 I n c sin ( m 2 π n N ) ] 2 + [ n = 0 N 1 I n c cos ( m 2 π n N ) ] 2 } 0.5
B cos = 2 k = 0 B N 2 + k N cos [ ( 0.5 N + k N ) ϕ ] ,
ϕ m = arctan ( S m , N C m , N ) ,
f 1 ( γ , ϕ ) = arctan [ sin ( ϕ ) + H s ( γ , ϕ ) cos ( ϕ ) + H c ( γ , ϕ ) ] ϕ ̃ ,
γ = B 1 + 2 B 2 B 1 B 2 ,
f 2 ( ϕ ) = arctan [ sin ( ϕ ) + H s ( γ , ϕ ) cos ( ϕ ) + H c ( γ , ϕ ) ] ϕ ̃
f 3 ( ϕ ) = arctan [ sin ( ϕ ) + H s ( γ ¯ , ϕ ) cos ( ϕ ) + H c ( γ ¯ , ϕ ) ] ϕ ̃ .
f c 1 ( γ , ϕ ) = f n c 1 ( γ , ϕ ) f d c 1 ( γ , ϕ ) A ̃ c B ̃ c ,
f n c 1 ( γ , ϕ ) = 0.5 G 0 ( γ ) + k = 1 [ G 3 k ( γ ) cos ( 3 k ϕ ) ] ,
f d c 1 ( γ , ϕ ) = { [ sin ( ϕ ) + H s ( γ , ϕ ) ] 2 + [ cos ( ϕ ) + H c ( γ , ϕ ) ] 2 } 0.5 ,
f c 2 ( γ , ϕ ) = f n c 2 ( γ , ϕ ) f d c 1 ( γ , ϕ ) B ̃ cos B ̃ c ,
f n c 2 ( γ , ϕ ) = 2 m = 0 { G 2 + 4 m ( γ ) cos [ ( 2 + 4 m ) ϕ ] } ,
f c 3 ( γ , ϕ ) = arctan [ sin ( 2 ϕ ) + H s 2 ( γ , ϕ ) cos ( 2 ϕ ) + H c 2 ( γ , ϕ ) ] ϕ ̃ 2 ,
H s 2 ( γ , ϕ ) = k = 1 G k N + 2 ( γ ) sin [ ( k N + 2 ) ϕ ] k = 1 G k N 2 ( γ ) sin [ ( k N 2 ) ϕ ] ,
H c 2 ( γ , ϕ ) = k = 1 G k N + 2 ( γ ) cos [ ( k N + 2 ) ϕ ] + k = 1 G k N 2 ( γ ) cos [ ( k N 2 ) ϕ ] ,

Metrics