Abstract

This paper analyzes the phase error for a three-dimensional (3D) shape measurement system that utilizes our recently proposed projector defocusing technique. This technique generates seemingly sinusoidal structured patterns by defocusing binary structured patterns and then uses these patterns to perform 3D shape measurement by fringe analysis. However, significant errors may still exist if an object is within a certain depth range, where the defocused fringe patterns retain binary structure. In this research, we experimentally studied a large depth range of defocused fringe patterns, from near-binary to near-sinusoidal, and analyzed the associated phase errors. We established a mathematical phase error function in terms of the wrapped phase and the depth z. Finally, we calibrated and used the mathematical function to compensate for the phase error at arbitrary depth ranges within the calibration volume. Experimental results will be presented to demonstrate the success of this proposed technique.

© 2011 Optical Society of America

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References

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    [CrossRef]
  2. S. Zhang, “Recent progresses on real-time 3-D shape measurement using digital fringe projection techniques,” Opt. Lasers Eng. 48, 149–158 (2010).
    [CrossRef]
  3. S. Lei and S. Zhang, “Digital sinusoidal fringe generation: defocusing binary patterns VS focusing sinusoidal patterns,” Opt. Lasers Eng. 48, 561–569 (2010).
    [CrossRef]
  4. S. Lei and S. Zhang, “Flexible 3-D shape measurement using projector defocusing,” Opt. Lett. 34, 3080–3082 (2009).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  7. Y. Gong and S. Zhang, “Ultrafast 3-D shape measurement with an off-the-shelf DLP projector,” Opt. Express 18, 19743–19754 (2010).
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  18. S. Zhang and S.-T. Yau, “Generic nonsinusoidal phase error correction for three-dimensional shape measurement using a digital video projector,” Appl. Opt. 46, 36–43 (2007).
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2011

Y. Gong and S. Zhang, “High-speed, high-resolution three-dimensional shape measurement using projector defocusing,” Opt. Eng. 50, 023603 (2011).
[CrossRef]

2010

S. Gorthi and P. Rastogi, “Fringe projection techniques: whither we are?,” Opt. Lasers Eng. 48, 133–140 (2010).
[CrossRef]

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

S. Lei and S. Zhang, “Digital sinusoidal fringe generation: defocusing binary patterns VS focusing sinusoidal patterns,” Opt. Lasers Eng. 48, 561–569 (2010).
[CrossRef]

S. Zhang, “Flexible 3-D shape measurement using projector defocusing: Extended measurement range,” Opt. Lett. 35, 931–933 (2010).
[CrossRef]

S. Zhang, D. van der Weide, and J. Olvier, “Superfast phase-shifting method for 3-D shape measurement,” Opt. Express 18, 9684–9689 (2010).
[CrossRef] [PubMed]

Y. Gong and S. Zhang, “Ultrafast 3-D shape measurement with an off-the-shelf DLP projector,” Opt. Express 18, 19743–19754 (2010).
[CrossRef] [PubMed]

2009

2007

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

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

2004

2003

P. S. Huang, C. Zhang, and F.-P. Chiang, “High-speed 3-D shape measurement based on digital fringe projection,” Opt. Eng. 42, l63–168 (2003).
[CrossRef]

1999

1997

L. J. Hornbeck, “Digital light processing for high-brightness, high-resolution applications,” Proc. SPIE 3013, 27–40 (1997).
[CrossRef]

1992

X. Y. Su, W. S. Zhou, G. Von Bally, and D. Vukicevic, “Automated phase-measuring profilometry using defocused projection of a Ronchi grating,” Opt. Commun. 94, 561–573(1992).
[CrossRef]

1983

Asundi, A.

Chen, M.

Chiang, F.-P.

P. S. Huang, C. Zhang, and F.-P. Chiang, “High-speed 3-D shape measurement based on digital fringe projection,” Opt. Eng. 42, l63–168 (2003).
[CrossRef]

Ghiglia, D. C.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).

Gong, Y.

Y. Gong and S. Zhang, “High-speed, high-resolution three-dimensional shape measurement using projector defocusing,” Opt. Eng. 50, 023603 (2011).
[CrossRef]

Y. Gong and S. Zhang, “Ultrafast 3-D shape measurement with an off-the-shelf DLP projector,” Opt. Express 18, 19743–19754 (2010).
[CrossRef] [PubMed]

Gorthi, S.

S. Gorthi and P. Rastogi, “Fringe projection techniques: whither we are?,” Opt. Lasers Eng. 48, 133–140 (2010).
[CrossRef]

Guo, H.

He, H.

Hornbeck, L. J.

L. J. Hornbeck, “Digital light processing for high-brightness, high-resolution applications,” Proc. SPIE 3013, 27–40 (1997).
[CrossRef]

Huang, L.

Huang, P. S.

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

P. S. Huang, C. Zhang, and F.-P. Chiang, “High-speed 3-D shape measurement based on digital fringe projection,” Opt. Eng. 42, l63–168 (2003).
[CrossRef]

Iwata, K.

Kakunai, S.

Kemao, Q.

Lei, S.

S. Lei and S. Zhang, “Digital sinusoidal fringe generation: defocusing binary patterns VS focusing sinusoidal patterns,” Opt. Lasers Eng. 48, 561–569 (2010).
[CrossRef]

S. Lei and S. Zhang, “Flexible 3-D shape measurement using projector defocusing,” Opt. Lett. 34, 3080–3082 (2009).
[CrossRef] [PubMed]

Mutoh, K.

Olvier, J.

Pan, B.

Pritt, M. D.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).

Rastogi, P.

S. Gorthi and P. Rastogi, “Fringe projection techniques: whither we are?,” Opt. Lasers Eng. 48, 133–140 (2010).
[CrossRef]

Sakamoto, T.

Su, X. Y.

X. Y. Su, W. S. Zhou, G. Von Bally, and D. Vukicevic, “Automated phase-measuring profilometry using defocused projection of a Ronchi grating,” Opt. Commun. 94, 561–573(1992).
[CrossRef]

Takeda, M.

van der Weide, D.

Von Bally, G.

X. Y. Su, W. S. Zhou, G. Von Bally, and D. Vukicevic, “Automated phase-measuring profilometry using defocused projection of a Ronchi grating,” Opt. Commun. 94, 561–573(1992).
[CrossRef]

Vukicevic, D.

X. Y. Su, W. S. Zhou, G. Von Bally, and D. Vukicevic, “Automated phase-measuring profilometry using defocused projection of a Ronchi grating,” Opt. Commun. 94, 561–573(1992).
[CrossRef]

Yau, S.-T.

Zhang, C.

P. S. Huang, C. Zhang, and F.-P. Chiang, “High-speed 3-D shape measurement based on digital fringe projection,” Opt. Eng. 42, l63–168 (2003).
[CrossRef]

Zhang, S.

Y. Gong and S. Zhang, “High-speed, high-resolution three-dimensional shape measurement using projector defocusing,” Opt. Eng. 50, 023603 (2011).
[CrossRef]

S. Zhang, D. van der Weide, and J. Olvier, “Superfast phase-shifting method for 3-D shape measurement,” Opt. Express 18, 9684–9689 (2010).
[CrossRef] [PubMed]

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

S. Lei and S. Zhang, “Digital sinusoidal fringe generation: defocusing binary patterns VS focusing sinusoidal patterns,” Opt. Lasers Eng. 48, 561–569 (2010).
[CrossRef]

Y. Gong and S. Zhang, “Ultrafast 3-D shape measurement with an off-the-shelf DLP projector,” Opt. Express 18, 19743–19754 (2010).
[CrossRef] [PubMed]

S. Zhang, “Flexible 3-D shape measurement using projector defocusing: Extended measurement range,” Opt. Lett. 35, 931–933 (2010).
[CrossRef]

S. Lei and S. Zhang, “Flexible 3-D shape measurement using projector defocusing,” Opt. Lett. 34, 3080–3082 (2009).
[CrossRef] [PubMed]

S. Zhang, “Digital multiple-wavelength phase-shifting algorithm,” Proc. SPIE 7432, 74320N (2009).
[CrossRef]

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

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

Zhou, W. S.

X. Y. Su, W. S. Zhou, G. Von Bally, and D. Vukicevic, “Automated phase-measuring profilometry using defocused projection of a Ronchi grating,” Opt. Commun. 94, 561–573(1992).
[CrossRef]

Appl. Opt.

Opt. Commun.

X. Y. Su, W. S. Zhou, G. Von Bally, and D. Vukicevic, “Automated phase-measuring profilometry using defocused projection of a Ronchi grating,” Opt. Commun. 94, 561–573(1992).
[CrossRef]

Opt. Eng.

Y. Gong and S. Zhang, “High-speed, high-resolution three-dimensional shape measurement using projector defocusing,” Opt. Eng. 50, 023603 (2011).
[CrossRef]

P. S. Huang, C. Zhang, and F.-P. Chiang, “High-speed 3-D shape measurement based on digital fringe projection,” Opt. Eng. 42, l63–168 (2003).
[CrossRef]

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

Opt. Express

Opt. Lasers Eng.

S. Gorthi and P. Rastogi, “Fringe projection techniques: whither we are?,” Opt. Lasers Eng. 48, 133–140 (2010).
[CrossRef]

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

S. Lei and S. Zhang, “Digital sinusoidal fringe generation: defocusing binary patterns VS focusing sinusoidal patterns,” Opt. Lasers Eng. 48, 561–569 (2010).
[CrossRef]

Opt. Lett.

Proc. SPIE

L. J. Hornbeck, “Digital light processing for high-brightness, high-resolution applications,” Proc. SPIE 3013, 27–40 (1997).
[CrossRef]

S. Zhang, “Digital multiple-wavelength phase-shifting algorithm,” Proc. SPIE 7432, 74320N (2009).
[CrossRef]

Other

D.Malacara, ed., Optical Shop Testing, 3rd ed. (Wiley, 2007).
[CrossRef]

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithms, and Software (Wiley, 1998).

Supplementary Material (2)

» Media 1: MOV (121 KB)     
» Media 2: MPG (2372 KB)     

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Figures (11)

Fig. 1
Fig. 1

Projector defocusing with increasing depth. While region B develops high-quality sinusoidal fringe patterns, the patterns of region A retain binary structure, which results in phase errors. The signal-to-noise ratio of region C renders it unusable.

Fig. 2
Fig. 2

Example of sinusoidal fringe generation by defocusing binary structured patterns (Media 1). (a) Result when the projector is in focus. (b)–(e) The gradual results when the projector’s focal length increases. (f)–(j) Illustrate the 240th row cross sections of the corresponding above fringe images.

Fig. 3
Fig. 3

Schematic diagram of phase-to-height conversion.

Fig. 4
Fig. 4

Photograph of the experimental system setup.

Fig. 5
Fig. 5

Phase errors computed by finding the difference between the phase obtained from the sinusoidal fringe patterns and that obtained from the defocused binary patterns. (a) One of the three phase-shifted fringe patterns used in the sinusoidal method. (b) One of the three phase-shifted fringe patterns used in the binary method. (c) Cross sections of the absolute phase maps. (d) Phase error.

Fig. 6
Fig. 6

Phase error maps for different depth planes. (a) Depth z = 0 mm , (b) depth z = 40 mm , (c) depth z = 80 mm , and (d) depth z = 120 mm .

Fig. 7
Fig. 7

Phase error compensation using an LUT and its associated polynomial fitting. (a) The LUT in terms of wrapped phase value (Media 2). (b) The polynomial-fitted curve. (c) The residual error after compensation using the LUT shown in (a) (RMS: 0.041 rad ). (d) The residual error after compensation using the fitted polynomial (RMS: 0.051 rad ).

Fig. 8
Fig. 8

Error compensation using the DSP method. (a) The original error (RMS: 0.129 rad ). (b) The original LUT and the polynomial fitting. (c) Phase error after compensation (RMS: 0.045 rad ).

Fig. 9
Fig. 9

Error compensation using the PLUT method. (a) The original error (RMS: 0.129 rad ). (b) The original LUT and the calculated LUT. (c) Phase error after compensation (RMS: 0.037 rad ).

Fig. 10
Fig. 10

Experimental results for a step-height object. (a) 3D result before compensation, (b) 3D result after applying the DSP method, (c) 3D result after applying the PLUT method, (d) 480th row cross sections of original data and the result after applying the DSP method, and (e) 480th row cross sections of the original data and the result after applying the PLUT method.

Fig. 11
Fig. 11

Experimental results for a complex 3D object. (a) Photo of object, (b) 3D result before compensation, (c) 3D result after applying the DSP method, and (d) 3D result after applying the PLUT method.

Tables (1)

Tables Icon

Table 1 Phase Error Before and After Error Compensation with Both Methods for Different Depth z Values

Equations (18)

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I 1 ( x , y ) = I ( x , y ) + I ( x , y ) cos [ ϕ ( x , y ) 2 π / 3 ] ,
I 2 ( x , y ) = I ( x , y ) + I ( x , y ) cos [ ϕ ( x , y ) ] ,
I 3 ( x , y ) = I ( x , y ) + I ( x , y ) cos [ ϕ ( x , y ) + 2 π / 3 ] .
ϕ ( x , y ) = tan 1 [ 3 ( I 1 I 3 ) / ( 2 I 2 I 1 I 3 ) ] .
Φ ( x , y ) = 2 π × m ( x , y ) + ϕ ( x , y ) .
Φ 1 ( x , y ) = ϕ 1 ( x , y ) .
Φ 2 ( x , y ) = 2 π m 2 ( x , y ) + ϕ 2 ( x , y ) .
Φ 2 = 2 Φ 1 .
m 2 ( x , y ) = Round [ Φ 1 ( x , y ) π ϕ 2 ( x , y ) 2 π ] .
m k ( x , y ) = Round [ Φ k 1 ( x , y ) π ϕ k ( x , y ) 2 π ] ,
Φ k ( x , y ) = 2 π m k ( x , y ) + ϕ k ( x , y ) .
Δ Φ D A = Φ D Φ A r = Φ C r Φ A r = Δ Φ A C r .
Δ z ( x , y ) = D B ¯ = A C ¯ · l d + A C ¯ l d A C ¯ Δ Φ A C r = Φ D Φ A r ,
Δ z Δ Φ ( x , y ) = Φ ( x , y ) Φ r ( x , y ) .
z ( x , y ) = z 0 + c 0 × [ Φ ( x , y ) Φ r ( x , y ) ] ,
Δ Φ ( x , y ) = Φ b ( x , y ) Φ s ( x , y ) .
Δ Φ ( x , y ; z ) = Φ b ( x , y ; z ) Φ s ( x , y ; z ) = f ( ϕ ; z ) .
Δ Φ ( x , y ; z ) = k = 0 N ( l = 0 M c k , l z l ) ϕ k .

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