Abstract

In a recent published work we proposed a technique to recover the absolute phase maps of two fringe patterns with different spatial frequencies. It is demonstrated that a number of selected frequency pairs can be used for the proposed approach, but the published work did not provide a guideline for frequency selection. In addition, the performance of the proposed technique in terms of its anti-noise capability is not addressed. In this paper, the rules for selecting the two frequencies are presented based on theoretical analysis of the proposed technique. Also, when the two frequencies are given, the anti-noise capability of technique is formulated and evaluated. These theoretical conclusions are verified by experimental results.

© 2012 OSA

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References

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2012 (1)

Y. Wang, K. Liu, Q. Hao, X. Wang, D. L. Lau, and L. G. Hassebrook, “Robust active stereo vision using Kullback-Leibler divergence,” IEEE Trans. Pattern Anal. Mach. Intell. 34(3), 548–563 (2012).
[CrossRef] [PubMed]

2011 (1)

2010 (2)

2009 (3)

S. Zhang, “Digital multiple-wavelength phase-shifting algorithm,” Proc. SPIE 7432, 74320N, 74320N-11 (2009).
[CrossRef]

S. Zhang, “Phase unwrapping error reduction framework for a multiple-wavelength phase-shifting algorithm,” Opt. Eng. 48(10), 105601 (2009).
[CrossRef] [PubMed]

K. Houairi and F. Cassaing, “Two-wavelength interferometry: extended range and accurate optical path difference analytical estimator,” J. Opt. Soc. Am. A 26(12), 2503–2511 (2009).
[CrossRef] [PubMed]

2007 (2)

2005 (1)

C. E. Towers, D. P. Towers, and J. D. C. Jones, “Absolute fringe order calculation using optimisied multi-frequency selection in full-filed profilometry,” Opt. Lasers Eng. 43(7), 788–800 (2005).
[CrossRef]

2003 (1)

1997 (3)

1994 (1)

1993 (1)

1987 (1)

1984 (1)

1973 (1)

1971 (1)

Cassaing, F.

Chen, H. J.

Chen, W.

Cheng, Y.-Y.

Chicharo, J.

Creath, K.

Ding, Y.

Fang, J.

Guan, C.

Hao, Q.

Hassebrook, L. G.

Houairi, K.

Huntley, J. M.

Jones, J. D. C.

C. E. Towers, D. P. Towers, and J. D. C. Jones, “Absolute fringe order calculation using optimisied multi-frequency selection in full-filed profilometry,” Opt. Lasers Eng. 43(7), 788–800 (2005).
[CrossRef]

Lau, D. L.

Li, J.

Li, J. L.

Li, X.

Liu, K.

Lv, D. J.

Polhemus, C.

Saldner, H. O.

Su, H. J.

Su, X. Y.

Tan, Y.

Towers, C. E.

C. E. Towers, D. P. Towers, and J. D. C. Jones, “Absolute fringe order calculation using optimisied multi-frequency selection in full-filed profilometry,” Opt. Lasers Eng. 43(7), 788–800 (2005).
[CrossRef]

Towers, D. P.

C. E. Towers, D. P. Towers, and J. D. C. Jones, “Absolute fringe order calculation using optimisied multi-frequency selection in full-filed profilometry,” Opt. Lasers Eng. 43(7), 788–800 (2005).
[CrossRef]

Wang, X.

Y. Wang, K. Liu, Q. Hao, X. Wang, D. L. Lau, and L. G. Hassebrook, “Robust active stereo vision using Kullback-Leibler divergence,” IEEE Trans. Pattern Anal. Mach. Intell. 34(3), 548–563 (2012).
[CrossRef] [PubMed]

Wang, Y.

Wyant, J. C.

Xi, J.

Yau, S. T.

Yu, Y.

Zhang, J.

Zhang, S.

S. Zhang, “Digital multiple-wavelength phase-shifting algorithm,” Proc. SPIE 7432, 74320N, 74320N-11 (2009).
[CrossRef]

S. Zhang, “Phase unwrapping error reduction framework for a multiple-wavelength phase-shifting algorithm,” Opt. Eng. 48(10), 105601 (2009).
[CrossRef] [PubMed]

S. Zhang, X. Li, and S. T. Yau, “Multilevel quality-guided phase unwrapping algorithm for real-time three-dimensional shape reconstruction,” Appl. Opt. 46(1), 50–57 (2007).
[CrossRef] [PubMed]

Zhao, H.

Appl. Opt. (9)

IEEE Trans. Pattern Anal. Mach. Intell. (1)

Y. Wang, K. Liu, Q. Hao, X. Wang, D. L. Lau, and L. G. Hassebrook, “Robust active stereo vision using Kullback-Leibler divergence,” IEEE Trans. Pattern Anal. Mach. Intell. 34(3), 548–563 (2012).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (4)

Opt. Eng. (1)

S. Zhang, “Phase unwrapping error reduction framework for a multiple-wavelength phase-shifting algorithm,” Opt. Eng. 48(10), 105601 (2009).
[CrossRef] [PubMed]

Opt. Express (2)

Opt. Lasers Eng. (1)

C. E. Towers, D. P. Towers, and J. D. C. Jones, “Absolute fringe order calculation using optimisied multi-frequency selection in full-filed profilometry,” Opt. Lasers Eng. 43(7), 788–800 (2005).
[CrossRef]

Opt. Lett. (1)

Proc. SPIE (1)

S. Zhang, “Digital multiple-wavelength phase-shifting algorithm,” Proc. SPIE 7432, 74320N, 74320N-11 (2009).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Absolute phases on reference pattern image.

Fig. 2
Fig. 2

Corresponding relationship between wrapped phase maps and

Fig. 3
Fig. 3

Experiment results whenand. (a) and (b) are the deformed fringe patterns; (c) and (d) are the wrapped phase maps obtained by six-step PSP; (e) and (f) are the recovered absolute phase maps of (c) and (d); (g) and (h) are the wrapped phase maps obtained by three-step PSP; (i) and (j) are the recovered phase of (g) and (h); (k) and (l) are the three dimensional reconstruction results obtained from (e) and (f) respectively.

Fig. 4
Fig. 4

Recovered absolute phases on section y = 800, the section across the palm model. (a) and (b) are the recovered absolute phases on section y = 800 for Fig. 3(e) and 3(f) by six-step PSP; (c) and (d) are the recovered phases on section y = 800 for Fig. 3(i) and 3(j) by three-step PSP.

Fig. 5
Fig. 5

Experiment on the plane with step. (a) and (b) are deformed fringe patterns; (c) and (d) are the wrapped phase maps by six-step PSP (white part are areas covered by object shadows); (e) and (f) are the absolute phase maps recovered by (c) and (d).

Fig. 6
Fig. 6

Experiment by the existing algorithm. (a) and (b) are deformed fringe patterns of f 1 =1 and f 2 =15 ; (c) and (d) are the wrapped phase maps of f 1 =1 and f 2 =15 by six-step PSP; (e) is the absolute phase maps recovered by (c) and (d) based on the existing algorithm. (f) is the three dimensional reconstruction cloud.

Fig. 7
Fig. 7

Recovered absolute phase on section y = 800 of f 2 =15 based on existing algorithm in [6].

Equations (31)

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{ Φ 1 (x)=2π m 1 (x)+ ϕ 1 (x) Φ 2 (x)=2π m 2 (x)+ ϕ 2 (x)
m 2 (x) f 1 m 1 (x) f 2 =Ψ(x)
Φ 1 (x)= f 1 Φ 0 (x) , Φ 2 (x)= f 2 Φ 0 (x)
m 1 (x)={ f 1 /2 [ f 1 ( f 1 mod2+1)]π/ f 1 Φ 0 (x)<π ... ... 1 π/ f 1 Φ 0 (x)<3π/ f 1 0 π/ f 1 < Φ 0 (x)<π/ f 1 1 3π/ f 1 < Φ 0 (x)π/ f 1 ... ... f 1 /2 π < 1 Φ 0 (x)[ f 1 ( f 1 mod2+1)]π/ f 1
m 2 (x)={ f 2 /2 [ f 2 ( f 2 mod2+1)]π/ f 2 Φ 0 (x)<π ... ... 1 π/ f 2 Φ 0 (x)<3π/ f 2 0 π/ f 2 < Φ 0 (x)<π/ f 2 1 3π/ f 2 < Φ 0 (x)π/ f 2 ... ... f 2 /2 π< Φ 0 (x)[ f 2 ( f 2 mod2+1)]π/ f 2
Min m 1 (x), m 2 (x) { | m 2 (x) f 1 m 1 (x) f 2 Φ(x) | }
N= N 1 + N 2 1=2 f 1 /2 +2 f 2 /2 +1
f 2 ϕ 1 ( x a ) f 1 ϕ 2 ( x a ) 2π f 2 ϕ 1 ( x b ) f 1 ϕ 2 ( x b ) 2π or m 2 ( x a ) f 1 m 1 ( x a ) f 2 m 2 ( x b ) f 1 m 1 ( x b ) f 2
m 2 ( x a ) f 1 m 1 ( x a ) f 2 = m 2 ( x b ) f 1 m 1 ( x b ) f 2
m 1 ( x a ) m 1 ( x b ) m 2 ( x a ) m 2 ( x b ) = f 1 f 2
m 1 ( x a ) m 1 ( x b )=k f 1 and m 2 ( x a ) m 2 ( x b )=k f 2
2 f 1 2 m 1 ( x a ) m 1 ( x b )2 f 1 2 and 2 f 2 2 m 2 ( x a ) m 2 ( x b )2 f 2 2
m 1 ( x a ) m 1 ( x b )=± f 1 , m 2 ( x a ) m 2 ( x b )=± f 2
f 1 2 = f 1 2 and f 2 2 = f 2 2
m 2 ( x a ) f 1 m 1 ( x a ) f 2 m 2 ( x b ) f 1 m 1 ( x b ) f 2
m 2max (x)= Φ 0max (x)π/ f 2 2π/ f 2 Φ 0max (x)π/ f 2 2π/ f 2 +1
m 2 (x) f 1 m 1 (x) f 2 m 2max (x) f 1 m 1 (x) f 2
m 2max (x) f 1 m 1 (x) f 2 ( Φ 0max (x)π/ f 2 2π/ f 2 ) f 1 m 1 (x) f 2 = f 1 + f 2 2
m 2 (x) f 1 m 1 (x) f 2 f 1 + f 2 2
m 2min (x)= Φ 0min (x)π/ f 2 2π/ f 2 Φ 0min (x)π/ f 2 2π/ f 2
m 2 (x) f 1 m 1 (x) f 2 f 1 m 2min (x) f 2 m 1 (x)
f 1 m 2min (x) f 2 m 1 (x)( Φ 0min (x)π/ f 2 2π/ f 2 ) f 1 m 1 (x) f 2 = f 1 + f 2 2
m 2 (x) f 1 m 1 (x) f 2 f 1 + f 2 2
f 1 + f 2 2 m 2 (x) f 1 f 2 m 1 (x) f 1 + f 2 2
f 1 + f 2 2 f 2 ϕ 1 (x) f 1 ϕ 2 (x) 2π f 1 + f 2 2
N=2 f 1 /2 +2 f 2 /2 +1
N1=2 f 1 /2 +2 f 2 /2
G= f 1 + f 2 2 f 1 /2 +2 f 2 /2 <2
| f 2 Δ ϕ 1 (x) f 1 Δ ϕ 2 (x) 2π |<0.5
0Δ ϕ max < π f 1 + f 2
f 1 + f 2 < π Δ ϕ max

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