Abstract

We present new extensions of the two-step, triangular-pattern phase-shifting method for different numbers of phase-shifting steps to increase measurement accuracy and to analyze the influence of the number of phase-shifting steps and pitch of the projected triangular intensity-profile pattern on the measurement accuracy. Phase-shifting algorithms to generate the intensity ratio, essential for surface reconstruction, were developed for each measurement method. Experiments determined that higher measurement accuracy can be obtained with a greater number of phase-shifting steps and a lower value of pitch, as long as the pitch is appropriately selected to be divisible by the number of phase-shifting steps and not below an optimal value, where intensity-ratio unwrapping failure would occur.

© 2007 Optical Society of America

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  1. K. Creath, "Phase-measurement interferometry techniques," in Progress in Optics, Vol. XXVI, E. Wolf, ed. (Elsevier Science, Amsterdam, 1988), pp. 349-393.
    [CrossRef]
  2. M. Halioua and H. C. Liu, "Optical three-dimensional sensing by phase measuring profilometry," Opt. Lasers Eng. 11, 185-215 (1989).
    [CrossRef]
  3. J. E. Greivenkamp and J. H. Bruning, "Phase shifting interferometry," in Optical Shop Testing, D. Malacara, ed. (Wiley, 1992), pp. 501-598.
  4. X. Y. He, D. Q. Zou, S. Liu, and Y. F. Guo, "Phase-shifting analysis in moiré interferometry and its application in electronic packaging," Opt. Eng. 37, 1410-1419 (1998).
    [CrossRef]
  5. Y. B. Choi and S. W. Kim, "Phase-shifting grating projection moiré topography," Opt. Eng. 37, 1005-1010 (1998).
    [CrossRef]
  6. B. Carrihill and R. Hummel, "Experiments with the intensity ratio depth sensor," in Computer Vision, Graphics and Image Processing (Academic, 1985), pp. 337-358.
  7. T. Miyasaka and K. Araki, "Development of real time 3-D measurement system using intensity ratio method," in Proc. ISPRS Commission III, 34, Part 3B, Photogrammetric Computer vision (PCV02), (Graz, 2002), pp. 181-185.
  8. P. Jia, J. Kofman, and C. English, "Two-step triangular-pattern phase-shifting method for three-dimensional object-shape measurement," Opt. Eng. (to be published).
  9. P. S. Huang, S. Zhang, and F.-P. Chiang, "Trapezoidal phase-shifting method for three-dimensional shape measurement," Opt. Eng. 44, 123601 (2005).
    [CrossRef]
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    [CrossRef] [PubMed]
  12. R. M. Goldstein, H. A. Zebker, and C. L. Werner, "Satellite radar interferometry:two-dimensional phase unwrapping," Radio Sci. 23, 713-720 (1988).
    [CrossRef]
  13. M. Fujigaki and Y. Morimoto, "Accuracy of real-time shape measurement by phase-shifting grid method using correlation," JSME International Journal , Series A 43, 314-320 (2000).
  14. P. Jia, J. Kofman, and C. English, "Comparison of linear and nonlinear calibration methods for phase-measuring profilometry," Opt. Eng. 46, 043601 (2007).
    [CrossRef]
  15. H. Guo, H. He, and M. Chen, "Gamma correction for digital fringe projection profilometry," Appl. Opt. 43, 2906-2914 (2004).
    [CrossRef] [PubMed]
  16. P. Jia, J. Kofman, C. English, and A. Deslauriers, "Two-step triangular phase-shifting method for 3-D object-shape measurement," Proc SPIE 6049, 141-150 (2005).

2007 (1)

P. Jia, J. Kofman, and C. English, "Comparison of linear and nonlinear calibration methods for phase-measuring profilometry," Opt. Eng. 46, 043601 (2007).
[CrossRef]

2005 (2)

P. Jia, J. Kofman, C. English, and A. Deslauriers, "Two-step triangular phase-shifting method for 3-D object-shape measurement," Proc SPIE 6049, 141-150 (2005).

P. S. Huang, S. Zhang, and F.-P. Chiang, "Trapezoidal phase-shifting method for three-dimensional shape measurement," Opt. Eng. 44, 123601 (2005).
[CrossRef]

2004 (1)

2000 (1)

M. Fujigaki and Y. Morimoto, "Accuracy of real-time shape measurement by phase-shifting grid method using correlation," JSME International Journal , Series A 43, 314-320 (2000).

1998 (2)

X. Y. He, D. Q. Zou, S. Liu, and Y. F. Guo, "Phase-shifting analysis in moiré interferometry and its application in electronic packaging," Opt. Eng. 37, 1410-1419 (1998).
[CrossRef]

Y. B. Choi and S. W. Kim, "Phase-shifting grating projection moiré topography," Opt. Eng. 37, 1005-1010 (1998).
[CrossRef]

1997 (2)

1989 (1)

M. Halioua and H. C. Liu, "Optical three-dimensional sensing by phase measuring profilometry," Opt. Lasers Eng. 11, 185-215 (1989).
[CrossRef]

1988 (1)

R. M. Goldstein, H. A. Zebker, and C. L. Werner, "Satellite radar interferometry:two-dimensional phase unwrapping," Radio Sci. 23, 713-720 (1988).
[CrossRef]

Araki, K.

T. Miyasaka and K. Araki, "Development of real time 3-D measurement system using intensity ratio method," in Proc. ISPRS Commission III, 34, Part 3B, Photogrammetric Computer vision (PCV02), (Graz, 2002), pp. 181-185.

Bruning, J. H.

J. E. Greivenkamp and J. H. Bruning, "Phase shifting interferometry," in Optical Shop Testing, D. Malacara, ed. (Wiley, 1992), pp. 501-598.

Carrihill, B.

B. Carrihill and R. Hummel, "Experiments with the intensity ratio depth sensor," in Computer Vision, Graphics and Image Processing (Academic, 1985), pp. 337-358.

Chen, M.

Chiang, F.-P.

P. S. Huang, S. Zhang, and F.-P. Chiang, "Trapezoidal phase-shifting method for three-dimensional shape measurement," Opt. Eng. 44, 123601 (2005).
[CrossRef]

Choi, Y. B.

Y. B. Choi and S. W. Kim, "Phase-shifting grating projection moiré topography," Opt. Eng. 37, 1005-1010 (1998).
[CrossRef]

Creath, K.

K. Creath, "Phase-measurement interferometry techniques," in Progress in Optics, Vol. XXVI, E. Wolf, ed. (Elsevier Science, Amsterdam, 1988), pp. 349-393.
[CrossRef]

Deslauriers, A.

P. Jia, J. Kofman, C. English, and A. Deslauriers, "Two-step triangular phase-shifting method for 3-D object-shape measurement," Proc SPIE 6049, 141-150 (2005).

English, C.

P. Jia, J. Kofman, and C. English, "Comparison of linear and nonlinear calibration methods for phase-measuring profilometry," Opt. Eng. 46, 043601 (2007).
[CrossRef]

P. Jia, J. Kofman, C. English, and A. Deslauriers, "Two-step triangular phase-shifting method for 3-D object-shape measurement," Proc SPIE 6049, 141-150 (2005).

P. Jia, J. Kofman, and C. English, "Two-step triangular-pattern phase-shifting method for three-dimensional object-shape measurement," Opt. Eng. (to be published).

Fang, Q.

Fujigaki, M.

M. Fujigaki and Y. Morimoto, "Accuracy of real-time shape measurement by phase-shifting grid method using correlation," JSME International Journal , Series A 43, 314-320 (2000).

Goldstein, R. M.

R. M. Goldstein, H. A. Zebker, and C. L. Werner, "Satellite radar interferometry:two-dimensional phase unwrapping," Radio Sci. 23, 713-720 (1988).
[CrossRef]

Greivenkamp, J. E.

J. E. Greivenkamp and J. H. Bruning, "Phase shifting interferometry," in Optical Shop Testing, D. Malacara, ed. (Wiley, 1992), pp. 501-598.

Guo, H.

Guo, Y. F.

X. Y. He, D. Q. Zou, S. Liu, and Y. F. Guo, "Phase-shifting analysis in moiré interferometry and its application in electronic packaging," Opt. Eng. 37, 1410-1419 (1998).
[CrossRef]

Halioua, M.

M. Halioua and H. C. Liu, "Optical three-dimensional sensing by phase measuring profilometry," Opt. Lasers Eng. 11, 185-215 (1989).
[CrossRef]

He, H.

He, X. Y.

X. Y. He, D. Q. Zou, S. Liu, and Y. F. Guo, "Phase-shifting analysis in moiré interferometry and its application in electronic packaging," Opt. Eng. 37, 1410-1419 (1998).
[CrossRef]

Huang, P. S.

P. S. Huang, S. Zhang, and F.-P. Chiang, "Trapezoidal phase-shifting method for three-dimensional shape measurement," Opt. Eng. 44, 123601 (2005).
[CrossRef]

Hummel, R.

B. Carrihill and R. Hummel, "Experiments with the intensity ratio depth sensor," in Computer Vision, Graphics and Image Processing (Academic, 1985), pp. 337-358.

Jia, P.

P. Jia, J. Kofman, and C. English, "Comparison of linear and nonlinear calibration methods for phase-measuring profilometry," Opt. Eng. 46, 043601 (2007).
[CrossRef]

P. Jia, J. Kofman, C. English, and A. Deslauriers, "Two-step triangular phase-shifting method for 3-D object-shape measurement," Proc SPIE 6049, 141-150 (2005).

P. Jia, J. Kofman, and C. English, "Two-step triangular-pattern phase-shifting method for three-dimensional object-shape measurement," Opt. Eng. (to be published).

Kim, S. W.

Y. B. Choi and S. W. Kim, "Phase-shifting grating projection moiré topography," Opt. Eng. 37, 1005-1010 (1998).
[CrossRef]

Kofman, J.

P. Jia, J. Kofman, and C. English, "Comparison of linear and nonlinear calibration methods for phase-measuring profilometry," Opt. Eng. 46, 043601 (2007).
[CrossRef]

P. Jia, J. Kofman, C. English, and A. Deslauriers, "Two-step triangular phase-shifting method for 3-D object-shape measurement," Proc SPIE 6049, 141-150 (2005).

P. Jia, J. Kofman, and C. English, "Two-step triangular-pattern phase-shifting method for three-dimensional object-shape measurement," Opt. Eng. (to be published).

Liu, H. C.

M. Halioua and H. C. Liu, "Optical three-dimensional sensing by phase measuring profilometry," Opt. Lasers Eng. 11, 185-215 (1989).
[CrossRef]

Liu, S.

X. Y. He, D. Q. Zou, S. Liu, and Y. F. Guo, "Phase-shifting analysis in moiré interferometry and its application in electronic packaging," Opt. Eng. 37, 1410-1419 (1998).
[CrossRef]

Miyasaka, T.

T. Miyasaka and K. Araki, "Development of real time 3-D measurement system using intensity ratio method," in Proc. ISPRS Commission III, 34, Part 3B, Photogrammetric Computer vision (PCV02), (Graz, 2002), pp. 181-185.

Morimoto, Y.

M. Fujigaki and Y. Morimoto, "Accuracy of real-time shape measurement by phase-shifting grid method using correlation," JSME International Journal , Series A 43, 314-320 (2000).

Werner, C. L.

R. M. Goldstein, H. A. Zebker, and C. L. Werner, "Satellite radar interferometry:two-dimensional phase unwrapping," Radio Sci. 23, 713-720 (1988).
[CrossRef]

Zebker, H. A.

R. M. Goldstein, H. A. Zebker, and C. L. Werner, "Satellite radar interferometry:two-dimensional phase unwrapping," Radio Sci. 23, 713-720 (1988).
[CrossRef]

Zhang, S.

P. S. Huang, S. Zhang, and F.-P. Chiang, "Trapezoidal phase-shifting method for three-dimensional shape measurement," Opt. Eng. 44, 123601 (2005).
[CrossRef]

Zheng, S.

Zou, D. Q.

X. Y. He, D. Q. Zou, S. Liu, and Y. F. Guo, "Phase-shifting analysis in moiré interferometry and its application in electronic packaging," Opt. Eng. 37, 1410-1419 (1998).
[CrossRef]

Appl. Opt. (3)

JSME International Journal (1)

M. Fujigaki and Y. Morimoto, "Accuracy of real-time shape measurement by phase-shifting grid method using correlation," JSME International Journal , Series A 43, 314-320 (2000).

Opt. Eng. (5)

P. Jia, J. Kofman, and C. English, "Comparison of linear and nonlinear calibration methods for phase-measuring profilometry," Opt. Eng. 46, 043601 (2007).
[CrossRef]

X. Y. He, D. Q. Zou, S. Liu, and Y. F. Guo, "Phase-shifting analysis in moiré interferometry and its application in electronic packaging," Opt. Eng. 37, 1410-1419 (1998).
[CrossRef]

Y. B. Choi and S. W. Kim, "Phase-shifting grating projection moiré topography," Opt. Eng. 37, 1005-1010 (1998).
[CrossRef]

P. Jia, J. Kofman, and C. English, "Two-step triangular-pattern phase-shifting method for three-dimensional object-shape measurement," Opt. Eng. (to be published).

P. S. Huang, S. Zhang, and F.-P. Chiang, "Trapezoidal phase-shifting method for three-dimensional shape measurement," Opt. Eng. 44, 123601 (2005).
[CrossRef]

Opt. Lasers Eng. (1)

M. Halioua and H. C. Liu, "Optical three-dimensional sensing by phase measuring profilometry," Opt. Lasers Eng. 11, 185-215 (1989).
[CrossRef]

Proc SPIE (1)

P. Jia, J. Kofman, C. English, and A. Deslauriers, "Two-step triangular phase-shifting method for 3-D object-shape measurement," Proc SPIE 6049, 141-150 (2005).

Radio Sci. (1)

R. M. Goldstein, H. A. Zebker, and C. L. Werner, "Satellite radar interferometry:two-dimensional phase unwrapping," Radio Sci. 23, 713-720 (1988).
[CrossRef]

Other (4)

J. E. Greivenkamp and J. H. Bruning, "Phase shifting interferometry," in Optical Shop Testing, D. Malacara, ed. (Wiley, 1992), pp. 501-598.

B. Carrihill and R. Hummel, "Experiments with the intensity ratio depth sensor," in Computer Vision, Graphics and Image Processing (Academic, 1985), pp. 337-358.

T. Miyasaka and K. Araki, "Development of real time 3-D measurement system using intensity ratio method," in Proc. ISPRS Commission III, 34, Part 3B, Photogrammetric Computer vision (PCV02), (Graz, 2002), pp. 181-185.

K. Creath, "Phase-measurement interferometry techniques," in Progress in Optics, Vol. XXVI, E. Wolf, ed. (Elsevier Science, Amsterdam, 1988), pp. 349-393.
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Two-step triangular-pattern phase-shifting algorithm. (a) Two phase-shifted triangular intensity-profile patterns, (b) intensity ratio as repeated triangles, and (c) intensity-ratio ramp after removal of the separate triangles.

Fig. 2
Fig. 2

Three-step triangular-pattern phase-shifting algorithm. (a) Three phase-shifted triangular intensity-profile patterns, (b) intensity ratio as repeated triangles, and (c) intensity-ratio ramp after removal of the separate triangles.

Fig. 3
Fig. 3

(Color online) Phase-shifted triangular intensity-profile patterns: (a) four-step, (b) five-step, and (c) six-step.

Fig. 4
Fig. 4

(Color online) Schematic diagram of the 3-D shape measurement system based on triangular intensity-profile pattern projection.

Fig. 5
Fig. 5

Depth measurements across the flat plate positioned 25   mm from the reference position using the triangular-pattern phase-shifting method with different number of phase-shifting steps and different values of pitch in the projected triangular patterns. For each group, measurement results are for the two-, three-, four-, five-, and six-step triangular-pattern phase-shifting methods, respectively as follows: (a)–(e) (Group A) for a pitch of 15 pixels, (f)–(j) (Group B) pitch of 16 pixels, (k)–(o) (Group C) pitch of 18 pixels, and (p)–(t) (Group D) pitch of 20 pixels.

Fig. 6
Fig. 6

Measurement errors over the full range of depth using optimal values of pitch of the triangular-pattern phase-shifting methods for different number of phase-shifting steps, determined by measurement of a flat plate. Measurement errors are calculated based on all pixels of the 648 × 494 resolution image.

Fig. 7
Fig. 7

3-D shape measurement of a human-head mask with triangular-pattern phase-shifting methods. (a)–(e) Reconstructed 3-D shaded models with two-, three-, four-, five-, and six-step phase-shifting methods, respectively, using optimal values of pitch of the triangular fringe pattern. The triangular pattern was generated with a pitch of 16 pixels in (a) and (c), 15 pixels in (b) and (d), and 18 pixels in (e).

Fig. 8
Fig. 8

3-D shaded models of a wooden mask reconstructed from measurement using the six-step triangular-pattern phase-shifting method, shown in different orientations in (a)–(e). The triangular pattern projected was generated using a pitch of 18 pixels.

Tables (4)

Tables Icon

Table 1 Impact of Pitch on Measurement Error for Two- and Three-Step Triangular-Pattern Phase-Shifting Methods a

Tables Icon

Table 2 Measurement Errors in Depth Measurement of a Flat Plate Using the Triangular-Pattern Phase-Shifting Method with Different Number of Phase-Shifting Steps and Different Values of Pitch of the Projected Triangular Pattern a

Tables Icon

Table 3 Optimal Values of Pitch for the Triangular-Pattern Phase-Shifting Methods with Different Number of Phase Steps

Tables Icon

Table 4 Measurement Errors in Depth Measurement of a Flat Plate Using the Sinusoidal-Pattern Phase-Shifting Method with Three and Four Phase-Shifting Steps and Different Values of Pitch of the Projected Sinusoidal Pattern a

Equations (12)

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

I 1 ( x , y ) = { 2 I m ( x , y ) T x + I min ( x , y ) + I m ( x , y ) 2 x [ 0 , T 4 ) 2 I m ( x , y ) T x + I min ( x , y ) + 3 I m ( x , y ) 2 x [ T 4 , 3 T 4 ) 2 I m ( x , y ) T x + I min ( x , y ) 3 I m ( x , y ) 2 x [ 3 T 4 , T )
I 2 ( x , y ) =
{ 2 I m ( x , y ) T x + I min ( x , y ) + I m ( x , y ) 2 x [ 0 , T 4 ) 2 I m ( x , y ) T x + I min ( x , y ) I m ( x , y ) 2 x [ T 4 , 3 T 4 ) 2 I m ( x , y ) T x + I min ( x , y ) + 5 I m ( x , y ) 2 x [ 3 T 4 , T )
I m ( x , y ) = I max ( x , y ) I min ( x , y )
r 0 ( x , y ) = | I 1 ( x , y ) I 2 ( x , y ) | I m ( x , y ) .
r ( x , y ) = 2 × round ( R 1 2 ) + ( 1 ) R + 1 r 0 ( x , y ) R = 1 , 2 , 3 , 4
I i ( x , y ) = { 2 I m ( x , y ) T ( x + δ i ) + I min ( x , y ) + I m ( x , y ) 2 x + δ i [ 0 , T 4 ) 2 I m ( x , y ) T ( x + δ i ) + I min ( x , y ) + 3 I m ( x , y ) 2 x + δ i [ T 4 , 3 T 4 ) 2 I m ( x , y ) T ( x + δ i ) + I min ( x , y ) 3 I m ( x , y ) 2 x + δ i [ 3 T 4 , T )
δ i = ( i 1 ) T N i = 1 , 2 ,   …   ,   N ,   N 2
r 0 ( x , y ) = I high ( x , y ) I med ( x , y ) + I low ( x , y ) I min ( x , y ) I m ( x , y )
r 0 ( x , y ) = | | I 1 ( x , y ) I 3 ( x , y ) | | I 2 ( x , y ) I 4 ( x , y ) | | I m ( x , y )
r 0 ( x , y ) = I high ( x , y ) I m e d 1 ( x , y ) + I m e d 2 ( x , y ) I m e d 3 ( x , y ) + I l o w ( x , y ) I min ( x , y ) I m ( x , y )
r 0 ( x , y ) = I high ( x , y ) I m e d 1 ( x , y ) + I m e d 2 ( x , y ) I m e d 3 ( x , y ) + I m e d 4 ( x , y ) I low ( x , y ) I m ( x , y )

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