Abstract

Hilbert transform (HT) has been employed to compensate phase error arising from the nonlinear effect in phase shifting profilometry (PSP). However, in most common situations, pure HT may lead to a significant system error, which has a negative impact on subsequent phase error compensation. In this paper, system error from HT of non-stationary and non-continuous fringe is analyzed, and then a novel phase error suppression approach is presented. The cosine fringe without direct current (DC) component is reconstructed to eliminate the influence of non-smooth reflectivity, and the fractional periods at both ends of the reconstructed fringe are extended to generate fringe with integer number of periods. And then the HT is applied to the reconstructed and extended fringe. Finally, a revised phase-shifting algorithm is employed to calculate the phase with the fringe after HT. The proposed approach is suitable for PSP of the surface with non-smooth reflectivity (e.g. texture of complex colors), which is demonstrated in a series of experiments.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (4)

S. Feng, C. Zuo, T. Tao, Y. Hu, M. Zhang, Q. Chen, and G. Gu, “Robust dynamic 3-D measurements with motion-compensated phase-shifting profilometry,” Opt. Lasers Eng. 103, 127–138 (2018).
[Crossref]

P. Zhao, N. Gao, Z. Zhang, F. Gao, and X. Jiang, “Performance analysis and evaluation of direct phase measuring deflectometry,” Opt. Lasers Eng. 103, 24–33 (2018).
[Crossref]

C. Chen, Y. Wan, and Y. Cao, “Instability of projection light source and real-time phase error correction method for phase-shifting profilometry,” Opt. Express 26(4), 4258–4270 (2018).
[Crossref] [PubMed]

A. Kamagara, X. Wang, and S. Li, “Nonlinear gamma correction via normed bicoherence minimization in optical fringe projection metrology,” Opt. Eng. 57(3), 034107 (2018).
[Crossref]

2017 (4)

2015 (4)

2012 (1)

Z. Zhang, “Review of single-shot 3D shape measurement by phase calculation-based fringe projection techniques,” Opt. Lasers Eng. 50(8), 1097–1106 (2012).
[Crossref]

2011 (1)

2010 (2)

2009 (1)

2008 (1)

J. Tian, X. Peng, and X. Zhao, “A generalized temporal phase unwrapping algorithm for three-dimensional profilometry,” Opt. Lasers Eng. 46(4), 336–342 (2008).
[Crossref]

2007 (1)

2006 (1)

2004 (1)

2003 (2)

P. S. Huang, C. Zhang, and F.-P. Chiang, “High-speed 3-D shape measurement based on digital fringe projection,” Opt. Eng. 42(1), 163–169 (2003).
[Crossref]

G. H. Notni and G. Notni, “Digital fringe projection in 3D shape measurement: an error analysis,” Proc. SPIE 5144, 372–380 (2003).
[Crossref]

1999 (2)

C. Coggrave and J. M. Huntley, “High-speed surface profilometer based on a spatial light modulator and pipeline image processor,” Opt. Eng. 38(9), 1573–1582 (1999).
[Crossref]

S. Kakunai, T. Sakamoto, and K. Iwata, “Profile measurement taken with liquid-crystal gratings,” Appl. Opt. 38(13), 2824–2828 (1999).
[Crossref] [PubMed]

1996 (1)

S. L. Hahn, “Comments on “A tabulation of Hilbert transforms for electrical engineers”,” IEEE Trans. Commun. 44(7), 768 (1996).
[Crossref]

An, Y.

Asundi, A.

Cai, Z.

Z. Cai, X. Liu, Q. Tang, X. Peng, and Y. Yin, “Comparison of active, passive and adaptive phase error compensation methods using a universal phase error model,” Proc. SPIE 10250, 102502Z (2017).
[Crossref]

Z. Cai, X. Liu, H. Jiang, D. He, X. Peng, S. Huang, and Z. Zhang, “Flexible phase error compensation based on Hilbert transform in phase shifting profilometry,” Opt. Express 23(19), 25171–25181 (2015).
[Crossref] [PubMed]

Cao, Y.

Chen, C.

Chen, M.

Chen, Q.

S. Feng, C. Zuo, T. Tao, Y. Hu, M. Zhang, Q. Chen, and G. Gu, “Robust dynamic 3-D measurements with motion-compensated phase-shifting profilometry,” Opt. Lasers Eng. 103, 127–138 (2018).
[Crossref]

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(1), 163–169 (2003).
[Crossref]

Coggrave, C.

C. Coggrave and J. M. Huntley, “High-speed surface profilometer based on a spatial light modulator and pipeline image processor,” Opt. Eng. 38(9), 1573–1582 (1999).
[Crossref]

Da, F.

Feng, S.

S. Feng, C. Zuo, T. Tao, Y. Hu, M. Zhang, Q. Chen, and G. Gu, “Robust dynamic 3-D measurements with motion-compensated phase-shifting profilometry,” Opt. Lasers Eng. 103, 127–138 (2018).
[Crossref]

Gao, B. Z.

Gao, F.

P. Zhao, N. Gao, Z. Zhang, F. Gao, and X. Jiang, “Performance analysis and evaluation of direct phase measuring deflectometry,” Opt. Lasers Eng. 103, 24–33 (2018).
[Crossref]

Gao, N.

P. Zhao, N. Gao, Z. Zhang, F. Gao, and X. Jiang, “Performance analysis and evaluation of direct phase measuring deflectometry,” Opt. Lasers Eng. 103, 24–33 (2018).
[Crossref]

Gu, G.

S. Feng, C. Zuo, T. Tao, Y. Hu, M. Zhang, Q. Chen, and G. Gu, “Robust dynamic 3-D measurements with motion-compensated phase-shifting profilometry,” Opt. Lasers Eng. 103, 127–138 (2018).
[Crossref]

Guo, H.

Hahn, S. L.

S. L. Hahn, “Comments on “A tabulation of Hilbert transforms for electrical engineers”,” IEEE Trans. Commun. 44(7), 768 (1996).
[Crossref]

Hao, Q.

Hassebrook, L. G.

He, D.

He, H.

Heist, S.

Hoang, T.

Hu, Y.

S. Feng, C. Zuo, T. Tao, Y. Hu, M. Zhang, Q. Chen, and G. Gu, “Robust dynamic 3-D measurements with motion-compensated phase-shifting profilometry,” Opt. Lasers Eng. 103, 127–138 (2018).
[Crossref]

Huang, L.

Huang, P. S.

P. S. Huang, C. Zhang, and F.-P. Chiang, “High-speed 3-D shape measurement based on digital fringe projection,” Opt. Eng. 42(1), 163–169 (2003).
[Crossref]

Huang, S.

Huntley, J. M.

C. Coggrave and J. M. Huntley, “High-speed surface profilometer based on a spatial light modulator and pipeline image processor,” Opt. Eng. 38(9), 1573–1582 (1999).
[Crossref]

Iwata, K.

Jiang, H.

Jiang, X.

P. Zhao, N. Gao, Z. Zhang, F. Gao, and X. Jiang, “Performance analysis and evaluation of direct phase measuring deflectometry,” Opt. Lasers Eng. 103, 24–33 (2018).
[Crossref]

Kakunai, S.

Kamagara, A.

A. Kamagara, X. Wang, and S. Li, “Nonlinear gamma correction via normed bicoherence minimization in optical fringe projection metrology,” Opt. Eng. 57(3), 034107 (2018).
[Crossref]

Kemao, Q.

Kühmstedt, P.

Lau, D. L.

Li, S.

A. Kamagara, X. Wang, and S. Li, “Nonlinear gamma correction via normed bicoherence minimization in optical fringe projection metrology,” Opt. Eng. 57(3), 034107 (2018).
[Crossref]

Li, Y.

Li, Z.

Liu, K.

Liu, X.

Nguyen, D.

Notni, G.

Notni, G. H.

G. H. Notni and G. Notni, “Digital fringe projection in 3D shape measurement: an error analysis,” Proc. SPIE 5144, 372–380 (2003).
[Crossref]

Pan, B.

Peng, X.

Z. Cai, X. Liu, Q. Tang, X. Peng, and Y. Yin, “Comparison of active, passive and adaptive phase error compensation methods using a universal phase error model,” Proc. SPIE 10250, 102502Z (2017).
[Crossref]

Y. Yin, M. Wang, B. Z. Gao, X. Liu, and X. Peng, “Fringe projection 3D microscopy with the general imaging model,” Opt. Express 23(5), 6846–6857 (2015).
[Crossref] [PubMed]

Z. Cai, X. Liu, H. Jiang, D. He, X. Peng, S. Huang, and Z. Zhang, “Flexible phase error compensation based on Hilbert transform in phase shifting profilometry,” Opt. Express 23(19), 25171–25181 (2015).
[Crossref] [PubMed]

J. Tian, X. Peng, and X. Zhao, “A generalized temporal phase unwrapping algorithm for three-dimensional profilometry,” Opt. Lasers Eng. 46(4), 336–342 (2008).
[Crossref]

Sakamoto, T.

Seah, H. S.

Shi, Y.

Tang, H.

Tang, Q.

Z. Cai, X. Liu, Q. Tang, X. Peng, and Y. Yin, “Comparison of active, passive and adaptive phase error compensation methods using a universal phase error model,” Proc. SPIE 10250, 102502Z (2017).
[Crossref]

Tao, T.

S. Feng, C. Zuo, T. Tao, Y. Hu, M. Zhang, Q. Chen, and G. Gu, “Robust dynamic 3-D measurements with motion-compensated phase-shifting profilometry,” Opt. Lasers Eng. 103, 127–138 (2018).
[Crossref]

Tian, J.

J. Tian, X. Peng, and X. Zhao, “A generalized temporal phase unwrapping algorithm for three-dimensional profilometry,” Opt. Lasers Eng. 46(4), 336–342 (2008).
[Crossref]

Towers, C. E.

Towers, D. P.

Tünnermann, A.

Wan, Y.

Wang, C.

Wang, M.

Wang, X.

A. Kamagara, X. Wang, and S. Li, “Nonlinear gamma correction via normed bicoherence minimization in optical fringe projection metrology,” Opt. Eng. 57(3), 034107 (2018).
[Crossref]

Wang, Y.

Wang, Z.

Yau, S.-T.

Yin, Y.

Z. Cai, X. Liu, Q. Tang, X. Peng, and Y. Yin, “Comparison of active, passive and adaptive phase error compensation methods using a universal phase error model,” Proc. SPIE 10250, 102502Z (2017).
[Crossref]

Y. Yin, M. Wang, B. Z. Gao, X. Liu, and X. Peng, “Fringe projection 3D microscopy with the general imaging model,” Opt. Express 23(5), 6846–6857 (2015).
[Crossref] [PubMed]

Zhan, G.

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(1), 163–169 (2003).
[Crossref]

Zhang, M.

S. Feng, C. Zuo, T. Tao, Y. Hu, M. Zhang, Q. Chen, and G. Gu, “Robust dynamic 3-D measurements with motion-compensated phase-shifting profilometry,” Opt. Lasers Eng. 103, 127–138 (2018).
[Crossref]

Zhang, S.

Zhang, Z.

P. Zhao, N. Gao, Z. Zhang, F. Gao, and X. Jiang, “Performance analysis and evaluation of direct phase measuring deflectometry,” Opt. Lasers Eng. 103, 24–33 (2018).
[Crossref]

Z. Cai, X. Liu, H. Jiang, D. He, X. Peng, S. Huang, and Z. Zhang, “Flexible phase error compensation based on Hilbert transform in phase shifting profilometry,” Opt. Express 23(19), 25171–25181 (2015).
[Crossref] [PubMed]

Z. Zhang, “Review of single-shot 3D shape measurement by phase calculation-based fringe projection techniques,” Opt. Lasers Eng. 50(8), 1097–1106 (2012).
[Crossref]

Z. Zhang, C. E. Towers, and D. P. Towers, “Time efficient color fringe projection system for 3D shape and color using optimum 3-frequency Selection,” Opt. Express 14(14), 6444–6455 (2006).
[Crossref] [PubMed]

Zhao, P.

P. Zhao, N. Gao, Z. Zhang, F. Gao, and X. Jiang, “Performance analysis and evaluation of direct phase measuring deflectometry,” Opt. Lasers Eng. 103, 24–33 (2018).
[Crossref]

Zhao, X.

J. Tian, X. Peng, and X. Zhao, “A generalized temporal phase unwrapping algorithm for three-dimensional profilometry,” Opt. Lasers Eng. 46(4), 336–342 (2008).
[Crossref]

Zheng, D.

Zhong, K.

Zuo, C.

S. Feng, C. Zuo, T. Tao, Y. Hu, M. Zhang, Q. Chen, and G. Gu, “Robust dynamic 3-D measurements with motion-compensated phase-shifting profilometry,” Opt. Lasers Eng. 103, 127–138 (2018).
[Crossref]

Appl. Opt. (6)

IEEE Trans. Commun. (1)

S. L. Hahn, “Comments on “A tabulation of Hilbert transforms for electrical engineers”,” IEEE Trans. Commun. 44(7), 768 (1996).
[Crossref]

J. Opt. Soc. Am. A (1)

Opt. Eng. (3)

A. Kamagara, X. Wang, and S. Li, “Nonlinear gamma correction via normed bicoherence minimization in optical fringe projection metrology,” Opt. Eng. 57(3), 034107 (2018).
[Crossref]

C. Coggrave and J. M. Huntley, “High-speed surface profilometer based on a spatial light modulator and pipeline image processor,” Opt. Eng. 38(9), 1573–1582 (1999).
[Crossref]

P. S. Huang, C. Zhang, and F.-P. Chiang, “High-speed 3-D shape measurement based on digital fringe projection,” Opt. Eng. 42(1), 163–169 (2003).
[Crossref]

Opt. Express (6)

Opt. Lasers Eng. (4)

J. Tian, X. Peng, and X. Zhao, “A generalized temporal phase unwrapping algorithm for three-dimensional profilometry,” Opt. Lasers Eng. 46(4), 336–342 (2008).
[Crossref]

S. Feng, C. Zuo, T. Tao, Y. Hu, M. Zhang, Q. Chen, and G. Gu, “Robust dynamic 3-D measurements with motion-compensated phase-shifting profilometry,” Opt. Lasers Eng. 103, 127–138 (2018).
[Crossref]

P. Zhao, N. Gao, Z. Zhang, F. Gao, and X. Jiang, “Performance analysis and evaluation of direct phase measuring deflectometry,” Opt. Lasers Eng. 103, 24–33 (2018).
[Crossref]

Z. Zhang, “Review of single-shot 3D shape measurement by phase calculation-based fringe projection techniques,” Opt. Lasers Eng. 50(8), 1097–1106 (2012).
[Crossref]

Opt. Lett. (3)

Proc. SPIE (2)

Z. Cai, X. Liu, Q. Tang, X. Peng, and Y. Yin, “Comparison of active, passive and adaptive phase error compensation methods using a universal phase error model,” Proc. SPIE 10250, 102502Z (2017).
[Crossref]

G. H. Notni and G. Notni, “Digital fringe projection in 3D shape measurement: an error analysis,” Proc. SPIE 5144, 372–380 (2003).
[Crossref]

Other (2)

M. Klingspor, Hilbert Transform: Mathematical Theory and Applications to Signal processing, Master thesis, (Linköping University, 2015).

F. R. Kschischang, The hilbert transform, (University of Toronto, 2006).

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Figures (12)

Fig. 1
Fig. 1 Simulation of spectra aliasing due to non-smooth reflectivity. (a) Non-stationary fringe whose amplitude is modulated by a step function. (b) Spectra of the step function and cosine fringe contained in (a). (c) Phase errors of 3-step phase-shifting, with fringes before and after HT respectively.
Fig. 2
Fig. 2 Circular convolution in DHT
Fig. 3
Fig. 3 Simulation of spectrum leakage due to fractional period. (a) Fringe containing a fractional period. (b) The spectrum of fringe in (a). (c) Phase errors of 3-step phase-shifting, with fringes before and after HT respectively.
Fig. 4
Fig. 4 Simulation of phase error distribution in3-step phase-shifting (γ = 2.5).
Fig. 5
Fig. 5 Fringe extension for 3-step phase-shifting. (a) Source section search based on wrapped phase. (b)-(d) Fractional period extension at both ends, for the 3 fringe images of 3-step phase-shifting respectively. (e) Phase errors before and after fringe extension respectively.
Fig. 6
Fig. 6 Proposed approach of phase error suppression based on HT
Fig. 7
Fig. 7 The maximum phase errors in simulations for different values of γ and step numbers of phase-shifting
Fig. 8
Fig. 8 Experiments on a colored plate. (a) Partial display of color photography. (b) Partial display of the fringe image. (c) Intensity curve corresponding to the red dashed line in (b). (d) Phase error distributions of 3-step phase-shifting.
Fig. 9
Fig. 9 Phase error suppression with the proposed approach. (a) Wrapped phase calculated with captured fringe images. (b) Reconstrued fringe with phase in (a). (c) Distributions of different phase errors.
Fig. 10
Fig. 10 Experiment on a colorful craft. (a) Color photo. (b) Captured fringe image.
Fig. 11
Fig. 11 Reconstructed 3D surface. (a) 3D surface without phase error suppression. (b) 3D surface with proposed approach of phase error suppression. (c), (d) Enlarged partial views of areas marked with the red rectangle in (a) and (b) respectively.
Fig. 12
Fig. 12 Profiles of the 3D surface on the position marked with the green line in Fig. 9(a)

Tables (1)

Tables Icon

Table 1 Statistics of simulated phase errors in 3-step phase-shifting.

Equations (26)

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

I n p ( x , y ) = α α p { 0.5 + 0.5 cos [ ϕ ( x , y ) + δ n ] } , n = 1 , 2 , , N
I n c = α α p ( 0.5 + 0.5 cos ϕ n ) γ
Δ ϕ = ϕ c ϕ = arc tan { m = 1 [ ( G m N + 1 G m N 1 ) sin ( m N ϕ ) ] 1 + m = 1 [ ( G m N + 1 + G m N 1 ) cos ( m N ϕ ) ] }
G s = i = 2 s γ i + 1 γ + i
ϕ = arc tan [ G N 1 sin ( N ϕ ) 1 + G N 1 cos ( N ϕ ) ]
u H ( t ) = H [ u ( t ) ] = b sin ( t )
I n c = A + 0.5 γ 1 α α p k = 1 [ B k cos ( k ϕ n ) ]
u H [ n ] = h [ n ] u [ n ] = i = 0 N 1 h ( i ) u N ( n i )
I ˜ n c = cos ϕ n c = cos ( ϕ n + Δ ϕ ) , n = 0 , 1 , ... , N 1
Δ ϕ ˜ H = arc tan G N 1 2 sin ( 2 N ϕ ) 4 G N 1 2 4 G N 1 cos ( N ϕ )
tan Δ ϕ ˜ H tan Δ ϕ = G N 1 2 sin ( 2 N ϕ ) 4 G N 1 2 4 G N 1 cos ( N ϕ ) / G N 1 sin ( N ϕ ) 1 + G N 1 cos ( N ϕ ) = 2 | G N 1 | cos ( N ϕ ) 1 + G N 1 cos ( N ϕ ) 4 G N 1 2 4 G N 1 cos ( N ϕ ) | G N 1 | 2 cos ( N ϕ )
Δ ϕ ˜ H Δ ϕ | max = tan Δ ϕ ˜ H tan Δ ϕ | max = | G N 1 | 2
L s = min { i | s . t . | ϕ i ϕ i 1 | > T H , i = 2 , 3 , 4... }
L e = min { i | s . t . ( ϕ i ϕ s ) ( ϕ i + 1 ϕ s ) 0 , i = L s , L s + 1 , ... }
I n ( x , y ) = a + b cos [ ϕ ( x , y ) + δ n ] , n = 1 , 2 , , N
ϕ = arc tan [ n = 1 N I n sin δ n n = 1 N I n cos δ n ]
I n H ( x , y ) = b sin [ ϕ ( x , y ) + δ n ] , n = 1 , 2 , , N
ϕ H = arc tan [ n = 1 N I n H cos δ n n = 1 N I n H sin δ n ]
ϕ = arc tan [ G N 1 sin ( N ϕ ) 1 + G N 1 cos ( N ϕ ) ] arc tan [ G N 1 sin ( N ϕ ) ] G N 1 sin ( N ϕ )
cos Δ ϕ = cos [ G N 1 sin ( N ϕ ) ] 1 1 2 G N 1 2 sin 2 ( N ϕ ) sin Δ ϕ = sin [ G N 1 sin ( N ϕ ) ] G N 1 sin ( N ϕ )
I ˜ n c = cos ϕ n cos Δ ϕ sin ϕ n sin Δ ϕ = cos ϕ n { 1 1 2 G N 1 2 sin 2 ( N ϕ ) } + sin ϕ n { G N 1 sin ( N ϕ ) } = ( 1 1 4 G N 1 2 ) cos ϕ n + 1 4 G N 1 2 cos 2 N ϕ cos ϕ n + G N 1 sin ( N ϕ ) sin ϕ n
I ˜ n c H = H [ I ˜ n c ] = ( 1 1 4 G N 1 2 ) sin ϕ n + 1 4 G N 1 2 sin ( 2 N ϕ ) cos ϕ n G N 1 cos ( N ϕ ) sin ϕ n = [ 1 1 4 G N 1 2 G N 1 cos ( N ϕ ) ] sin ϕ n + 1 4 G N 1 2 sin ( 2 N ϕ ) cos ϕ n
ϕ ˜ c H = arc tan [ n = 1 N I ˜ n c H cos δ n n = 1 N I ˜ n c H sin δ n ]
Δ ϕ ˜ H = ϕ ˜ c H ϕ = arc tan [ tan ( ϕ ˜ c H ϕ ) ] = arc tan ( tan ϕ ˜ c H tan ϕ 1 + tan ϕ ˜ c H tan ϕ ) = arc tan n = 1 N I ˜ n c H cos ϕ n n = 1 N I ˜ n c H sin ϕ n
Δ ϕ ˜ H = arc tan [ 1 1 4 G N 1 2 G N 1 cos ( N ϕ ) ] n = 1 N sin ϕ n cos ϕ n + 1 4 G N 1 2 sin ( 2 N ϕ ) n = 1 N cos ϕ n cos ϕ n [ 1 1 4 G N 1 2 G N 1 cos ( N ϕ ) ] n = 1 N sin ϕ n sin ϕ n + 1 4 G N 1 2 sin ( 2 N ϕ ) n = 1 N sin ϕ n cos ϕ n = arc tan 1 2 [ 1 1 4 G N 1 2 G N 1 cos ( N ϕ ) ] n = 1 N sin 2 ϕ n + 1 8 G N 1 2 sin ( 2 N ϕ ) [ N + n = 1 N cos 2 ϕ n ] 1 2 [ 1 1 4 G N 1 2 G N 1 cos ( N ϕ ) ] [ N n = 1 N cos 2 ϕ n ] + 1 8 G N 1 2 sin ( 2 N ϕ ) n = 1 N sin 2 ϕ n = arc tan [ 4 G N 1 2 4 G N 1 cos ( N ϕ ) ] n = 1 N sin 2 ϕ n + G N 1 2 sin ( 2 N ϕ ) [ N + n = 1 N cos 2 ϕ n ] [ 4 G N 1 2 4 G N 1 cos ( N ϕ ) ] [ N n = 1 N cos 2 ϕ n ] + G N 1 2 sin ( 2 N ϕ ) n = 1 N sin 2 ϕ n
Δ ϕ ˜ H = arc tan N G N 1 2 sin ( 2 N ϕ ) N [ 4 G N 1 2 4 G N 1 cos ( N ϕ ) ] = arc tan G N 1 2 sin ( 2 N ϕ ) 4 G N 1 2 4 G N 1 cos ( N ϕ )

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