Abstract

Gram-Schmidt (GS) orthogonal normalization is a fast and efficient two-frame fringe phase demodulation method. However, the precision of the GS method is limited due to the residual background terms and noise, as well as several approximation operations in the GS method. To obtain a phase map with higher accuracy, we propose an algorithm combining GS orthogonal normalization and least squares iterative (LSI) phase shift algorithm (GS&LSI). In our method, the phase was first obtained using GS method, and then a refinement operation using LSI was adopted to get the final wrapped phase map. Because of the LSI process, the demodulation result is greatly improved in many cases. Simulation and experimental result are presented to validate the potential of the proposed method.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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    [Crossref] [PubMed]
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2019 (2)

J. Muñoz-Maciel, V. M. Duran-Ramírez, M. Mora-Gonzalez, F. J. Casillas-Rodriguez, and F. G. Peña-Lecona, “Demodulation of a single closed-fringe interferogram with symmetric wavefront and tilt,” Opt. Commun. 436, 168–173 (2019).
[Crossref]

S. Cao, Y. Wang, X. Lu, and L. Zhong, “Advanced spatial spectrum fitting algorithm for significantly improving the noise resistance ability of self-calibration phase shifting interferometry,” Opt. Lasers Eng. 112, 170–181 (2019).
[Crossref]

2018 (4)

Y. Zhang, X. Tian, and R. Liang, “Random two-step phase shifting interferometry based on Lissajous ellipse fitting and least squares technologies,” Opt. Express 26(12), 15059–15071 (2018).
[Crossref] [PubMed]

H. Lei, Y. Yao, H. Liu, Y. Tian, Y. Yang, and Y. Gu, “Accurate phase extraction algorithm based on Gram–Schmidt orthonormalization and least square ellipse fitting method,” J. Mod. Opt. 65(10), 1199–1209 (2018).
[Crossref]

H. Du, S. Zhang, Z. Zhao, J. Yu, and H. Liu, “Development of two-frame shadow moiré profilometry by iterative self-tuning technique,” Opt. Eng. 57(11), 114101 (2018).
[Crossref]

Y. Zhang, X. Tian, and R. Liang, “Three-step random phase retrieval approach based on difference map normalization and diamond diagonal vector normalization,” Opt. Express 26(22), 29170–29182 (2018).
[Crossref] [PubMed]

2017 (2)

2016 (1)

2015 (1)

2014 (2)

H. Wang, C. Luo, L. Zhong, S. Ma, and X. Lu, “Phase retrieval approach based on the normalized difference maps induced by three interferograms with unknown phase shifts,” Opt. Express 22(5), 5147–5154 (2014).
[Crossref] [PubMed]

J. Ma, Z. Wang, and T. Pan, “Two-dimensional continuous wavelet transform algorithm for phase extraction of two-step arbitrarily phase-shifted interferograms,” Opt. Lasers Eng. 55(4), 205–211 (2014).
[Crossref]

2013 (2)

2012 (2)

2011 (3)

2009 (1)

2008 (1)

2007 (1)

N. A. Ochoa and A. A. Silva-Moreno, “Normalization and noise-reduction algorithm for fringe patterns,” Opt. Commun. 270(2), 161–168 (2007).
[Crossref]

2004 (1)

1997 (1)

1991 (1)

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84(3–4), 118–124 (1991).
[Crossref]

Belenguer, T.

Cao, S.

S. Cao, Y. Wang, X. Lu, and L. Zhong, “Advanced spatial spectrum fitting algorithm for significantly improving the noise resistance ability of self-calibration phase shifting interferometry,” Opt. Lasers Eng. 112, 170–181 (2019).
[Crossref]

Carazo, J. M.

Casillas-Rodriguez, F. J.

J. Muñoz-Maciel, V. M. Duran-Ramírez, M. Mora-Gonzalez, F. J. Casillas-Rodriguez, and F. G. Peña-Lecona, “Demodulation of a single closed-fringe interferogram with symmetric wavefront and tilt,” Opt. Commun. 436, 168–173 (2019).
[Crossref]

Chai, L.

Chen, Y. C.

Cuevas, F. J.

Deck, L. L.

Deng, J.

Du, H.

H. Du, S. Zhang, Z. Zhao, J. Yu, and H. Liu, “Development of two-frame shadow moiré profilometry by iterative self-tuning technique,” Opt. Eng. 57(11), 114101 (2018).
[Crossref]

Duran-Ramírez, V. M.

J. Muñoz-Maciel, V. M. Duran-Ramírez, M. Mora-Gonzalez, F. J. Casillas-Rodriguez, and F. G. Peña-Lecona, “Demodulation of a single closed-fringe interferogram with symmetric wavefront and tilt,” Opt. Commun. 436, 168–173 (2019).
[Crossref]

Estrada, J. C.

Gu, Y.

H. Lei, Y. Yao, H. Liu, Y. Tian, Y. Yang, and Y. Gu, “Accurate phase extraction algorithm based on Gram–Schmidt orthonormalization and least square ellipse fitting method,” J. Mod. Opt. 65(10), 1199–1209 (2018).
[Crossref]

Han, B.

Lee, C. M.

Lei, H.

H. Lei, Y. Yao, H. Liu, Y. Tian, Y. Yang, and Y. Gu, “Accurate phase extraction algorithm based on Gram–Schmidt orthonormalization and least square ellipse fitting method,” J. Mod. Opt. 65(10), 1199–1209 (2018).
[Crossref]

Liang, C. W.

Liang, R.

Lin, P. C.

Ling, T.

Liu, H.

H. Lei, Y. Yao, H. Liu, Y. Tian, Y. Yang, and Y. Gu, “Accurate phase extraction algorithm based on Gram–Schmidt orthonormalization and least square ellipse fitting method,” J. Mod. Opt. 65(10), 1199–1209 (2018).
[Crossref]

H. Du, S. Zhang, Z. Zhao, J. Yu, and H. Liu, “Development of two-frame shadow moiré profilometry by iterative self-tuning technique,” Opt. Eng. 57(11), 114101 (2018).
[Crossref]

Liu, S.

Lu, X.

Luo, C.

Ma, J.

J. Ma, Z. Wang, and T. Pan, “Two-dimensional continuous wavelet transform algorithm for phase extraction of two-step arbitrarily phase-shifted interferograms,” Opt. Lasers Eng. 55(4), 205–211 (2014).
[Crossref]

Ma, S.

Marroquin, J. L.

Mora-Gonzalez, M.

J. Muñoz-Maciel, V. M. Duran-Ramírez, M. Mora-Gonzalez, F. J. Casillas-Rodriguez, and F. G. Peña-Lecona, “Demodulation of a single closed-fringe interferogram with symmetric wavefront and tilt,” Opt. Commun. 436, 168–173 (2019).
[Crossref]

Muñoz-Maciel, J.

J. Muñoz-Maciel, V. M. Duran-Ramírez, M. Mora-Gonzalez, F. J. Casillas-Rodriguez, and F. G. Peña-Lecona, “Demodulation of a single closed-fringe interferogram with symmetric wavefront and tilt,” Opt. Commun. 436, 168–173 (2019).
[Crossref]

Ochoa, N. A.

N. A. Ochoa and A. A. Silva-Moreno, “Normalization and noise-reduction algorithm for fringe patterns,” Opt. Commun. 270(2), 161–168 (2007).
[Crossref]

Okada, K.

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84(3–4), 118–124 (1991).
[Crossref]

Pan, T.

J. Ma, Z. Wang, and T. Pan, “Two-dimensional continuous wavelet transform algorithm for phase extraction of two-step arbitrarily phase-shifted interferograms,” Opt. Lasers Eng. 55(4), 205–211 (2014).
[Crossref]

Patorski, K.

Peña-Lecona, F. G.

J. Muñoz-Maciel, V. M. Duran-Ramírez, M. Mora-Gonzalez, F. J. Casillas-Rodriguez, and F. G. Peña-Lecona, “Demodulation of a single closed-fringe interferogram with symmetric wavefront and tilt,” Opt. Commun. 436, 168–173 (2019).
[Crossref]

Pokorski, K.

Quiroga, J. A.

Saide, D.

Sato, A.

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84(3–4), 118–124 (1991).
[Crossref]

Servin, M.

Servín, M.

Silva-Moreno, A. A.

N. A. Ochoa and A. A. Silva-Moreno, “Normalization and noise-reduction algorithm for fringe patterns,” Opt. Commun. 270(2), 161–168 (2007).
[Crossref]

Sorzano, C. O.

Sorzano, C. O. S.

Tian, C.

Tian, X.

Tian, Y.

H. Lei, Y. Yao, H. Liu, Y. Tian, Y. Yang, and Y. Gu, “Accurate phase extraction algorithm based on Gram–Schmidt orthonormalization and least square ellipse fitting method,” J. Mod. Opt. 65(10), 1199–1209 (2018).
[Crossref]

Trusiak, M.

Tsujiuchi, J.

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84(3–4), 118–124 (1991).
[Crossref]

Vargas, J.

Wang, H.

Wang, Y.

S. Cao, Y. Wang, X. Lu, and L. Zhong, “Advanced spatial spectrum fitting algorithm for significantly improving the noise resistance ability of self-calibration phase shifting interferometry,” Opt. Lasers Eng. 112, 170–181 (2019).
[Crossref]

Wang, Z.

J. Ma, Z. Wang, and T. Pan, “Two-dimensional continuous wavelet transform algorithm for phase extraction of two-step arbitrarily phase-shifted interferograms,” Opt. Lasers Eng. 55(4), 205–211 (2014).
[Crossref]

Z. Wang and B. Han, “Advanced iterative algorithm for phase extraction of randomly phase-shifted interferograms,” Opt. Lett. 29(14), 1671–1673 (2004).
[Crossref] [PubMed]

Wei, T.

Xu, J.

Xu, Q.

Yang, Y.

H. Lei, Y. Yao, H. Liu, Y. Tian, Y. Yang, and Y. Gu, “Accurate phase extraction algorithm based on Gram–Schmidt orthonormalization and least square ellipse fitting method,” J. Mod. Opt. 65(10), 1199–1209 (2018).
[Crossref]

C. Tian, Y. Yang, T. Wei, T. Ling, and Y. Zhuo, “Demodulation of a single-image interferogram using a Zernike-polynomial-based phase-fitting technique with a differential evolution algorithm,” Opt. Lett. 36(12), 2318–2320 (2011).
[Crossref] [PubMed]

Yao, Y.

H. Lei, Y. Yao, H. Liu, Y. Tian, Y. Yang, and Y. Gu, “Accurate phase extraction algorithm based on Gram–Schmidt orthonormalization and least square ellipse fitting method,” J. Mod. Opt. 65(10), 1199–1209 (2018).
[Crossref]

Yu, J.

H. Du, S. Zhang, Z. Zhao, J. Yu, and H. Liu, “Development of two-frame shadow moiré profilometry by iterative self-tuning technique,” Opt. Eng. 57(11), 114101 (2018).
[Crossref]

Zhang, D.

Zhang, F.

Zhang, S.

H. Du, S. Zhang, Z. Zhao, J. Yu, and H. Liu, “Development of two-frame shadow moiré profilometry by iterative self-tuning technique,” Opt. Eng. 57(11), 114101 (2018).
[Crossref]

Zhang, Y.

Zhao, Z.

H. Du, S. Zhang, Z. Zhao, J. Yu, and H. Liu, “Development of two-frame shadow moiré profilometry by iterative self-tuning technique,” Opt. Eng. 57(11), 114101 (2018).
[Crossref]

Zhong, L.

Zhuo, Y.

Appl. Opt. (5)

J. Mod. Opt. (1)

H. Lei, Y. Yao, H. Liu, Y. Tian, Y. Yang, and Y. Gu, “Accurate phase extraction algorithm based on Gram–Schmidt orthonormalization and least square ellipse fitting method,” J. Mod. Opt. 65(10), 1199–1209 (2018).
[Crossref]

Opt. Commun. (3)

K. Okada, A. Sato, and J. Tsujiuchi, “Simultaneous calculation of phase distribution and scanning phase shift in phase shifting interferometry,” Opt. Commun. 84(3–4), 118–124 (1991).
[Crossref]

N. A. Ochoa and A. A. Silva-Moreno, “Normalization and noise-reduction algorithm for fringe patterns,” Opt. Commun. 270(2), 161–168 (2007).
[Crossref]

J. Muñoz-Maciel, V. M. Duran-Ramírez, M. Mora-Gonzalez, F. J. Casillas-Rodriguez, and F. G. Peña-Lecona, “Demodulation of a single closed-fringe interferogram with symmetric wavefront and tilt,” Opt. Commun. 436, 168–173 (2019).
[Crossref]

Opt. Eng. (1)

H. Du, S. Zhang, Z. Zhao, J. Yu, and H. Liu, “Development of two-frame shadow moiré profilometry by iterative self-tuning technique,” Opt. Eng. 57(11), 114101 (2018).
[Crossref]

Opt. Express (8)

M. Trusiak, K. Patorski, and K. Pokorski, “Hilbert-Huang processing for single-exposure two-dimensional grating interferometry,” Opt. Express 21(23), 28359–28379 (2013).
[Crossref] [PubMed]

H. Wang, C. Luo, L. Zhong, S. Ma, and X. Lu, “Phase retrieval approach based on the normalized difference maps induced by three interferograms with unknown phase shifts,” Opt. Express 22(5), 5147–5154 (2014).
[Crossref] [PubMed]

M. Trusiak and K. Patorski, “Two-shot fringe pattern phase-amplitude demodulation using Gram-Schmidt orthonormalization with Hilbert-Huang pre-filtering,” Opt. Express 23(4), 4672–4690 (2015).
[Crossref] [PubMed]

C. Tian and S. Liu, “Two-frame phase-shifting interferometry for testing optical surfaces,” Opt. Express 24(16), 18695–18708 (2016).
[Crossref] [PubMed]

J. Vargas, J. A. Quiroga, T. Belenguer, M. Servín, and J. C. Estrada, “Two-step self-tuning phase-shifting interferometry,” Opt. Express 19(2), 638–648 (2011).
[Crossref] [PubMed]

C. Tian and S. Liu, “Phase retrieval in two-shot phase-shifting interferometry based on phase shift estimation in a local mask,” Opt. Express 25(18), 21673–21683 (2017).
[Crossref] [PubMed]

Y. Zhang, X. Tian, and R. Liang, “Random two-step phase shifting interferometry based on Lissajous ellipse fitting and least squares technologies,” Opt. Express 26(12), 15059–15071 (2018).
[Crossref] [PubMed]

Y. Zhang, X. Tian, and R. Liang, “Three-step random phase retrieval approach based on difference map normalization and diamond diagonal vector normalization,” Opt. Express 26(22), 29170–29182 (2018).
[Crossref] [PubMed]

Opt. Lasers Eng. (2)

S. Cao, Y. Wang, X. Lu, and L. Zhong, “Advanced spatial spectrum fitting algorithm for significantly improving the noise resistance ability of self-calibration phase shifting interferometry,” Opt. Lasers Eng. 112, 170–181 (2019).
[Crossref]

J. Ma, Z. Wang, and T. Pan, “Two-dimensional continuous wavelet transform algorithm for phase extraction of two-step arbitrarily phase-shifted interferograms,” Opt. Lasers Eng. 55(4), 205–211 (2014).
[Crossref]

Opt. Lett. (5)

Other (2)

G. Strang, Introduction to Linear Algebra, 5th ed. (Wellesley Cambridge University, 2016).

D. Malacara, Optical Shop Testing, 3rd ed. (John Wiley & Sons, Inc.,2007).

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Figures (10)

Fig. 1
Fig. 1 Flow chart of GS&LSI phase-shifting algorithms.
Fig. 2
Fig. 2 Simulated phase distribution and the two phase-shifting interferograms. (a) The reference phase, (b) the first interferogram, and (c) the second interferogram.
Fig. 3
Fig. 3 The residual maps obtained by the GS method in the uniform background residuals with different size: (a) a = 0;(b) a = 0.02;(c) a = 0.05;(d) a = 0.10; and the residual maps obtained by the GS&LSI method in the uniform background residuals with different size:(e) a = 0; (f) a = 0.02; (g) a = 0.05; (h) a = 0.10.
Fig. 4
Fig. 4 The residual maps obtained by the GS method in the non-uniform background residuals with different Gaussian parameter: (a) σ = 30;(b) σ = 20;(c) σ = 8;(d) σ = 4; and the residual maps obtained by the GS&LSI method in the non-uniform background residuals with different Gaussian parameter: (e) σ = 30; (f) σ = 20; (g) σ = 8; (h) σ = 4.
Fig. 5
Fig. 5 The iterative demodulation RMS error curve (rad) for GS&LSI in the uniform background residuals with different size: (a) a = 0;(b) a = 0.02;(c) a = 0.05;(d) a = 0.10; and the non-uniform background residuals with different Gaussian parameter: (e) σ = 30;(f) σ = 20;(g) σ = 8;(h) σ = 4.
Fig. 6
Fig. 6 The effect of the amount of phase shift between frames on the measurement results of the two algorithms under uniform background residuals with different sizes: (a) a = 0;(b) a = 0.02;(c) a = 0.05; (d) a = 0.10; and under non-uniform background residuals with different Gaussian parameter: (e) σ = 30;(f) σ = 20;(g) σ = 8;(h) σ = 4.
Fig. 7
Fig. 7 The phase demodulation errors (RMS) of GS and GS&LSI for (a) different fringe number and (b) different random noise.
Fig. 8
Fig. 8 The real open fringes (a) first interferogram; (b) second interferogram; the tested phase by different algorithms (c) ASSF; (d) GS; (e) GS&LSI; and the real closed fringes (f) first interferogram; (g) second interferogram; the tested phase by different algorithms (h) ASSF; (i) GS; (j) GS&LSI.
Fig. 9
Fig. 9 The phase shift value with iteration (a) the open fringes;(b) the closed fringes; and the RMS error with iteration (c) the open fringes; (d) the closed fringes.
Fig. 10
Fig. 10 The complex fringes (a) first interferogram; (b) second interferogram; the tested phase by different algorithms (c) ASSF; (d) GS; (e) GS&LSI.

Tables (4)

Tables Icon

Table 1 The demodulation errors and consumed time for GS and GS&LSI in different background residuals

Tables Icon

Table 2 The phase shift errors of GS&LSI in the different background residuals

Tables Icon

Table 3 The demodulation RMS errors of GS and GS&LSI for two sets of real fringes

Tables Icon

Table 4 The demodulation RMS errors of GS and GS&LSI for complex fringes

Equations (17)

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

I n (x,y)= A n (x,y)+ B n (x,y)cos[φ(x,y)+ δ n ](n=1,2)
I n (x,y) B n (x,y)cos[φ(x,y)+ δ n ](n=1,2)
I 1 * = I 1 / I 1 , I 1 = I 1 / I 1
I ^ 2 = I 2 I 2 , I 1 * I 1 *
I 2 * = I ^ 2 / I ^ 2 , I ^ 2 = I ^ 2 / I ^ 2
| x=1 N x y=1 N y cos 2 (φ)cos(δ) || x=1 N x y=1 N y cos(φ)sin(φ)sinδ | B 1 / x=1 N x y=1 N y ( B 1 cos(φ)) 2 B 2 / x=1 N x y=1 N y ( B 2 sin(φ)) 2
I 1 * = B ' cos(φ) I 2 * = B ' sin(φ)
φ=arctan( I 1 * I 2 * )
I ij = a ij + b ij cos[ φ j + δ i ]
I ij = a i ' + b i ' cos φ j + c i ' sin φ j
S i = j=1 N ( I ij t I ij ) 2 = j=1 N ( a i ' + b i ' cos φ j + c i ' sin φ j I ij ) 2
S i / a i ' =0, S i / b i ' =0, S i / c i ' =0
{ X i }= [ S i ] 1 { R i }
[ a i ' b i ' c i ' ]= [ N j=1 N cos φ j j=1 N sin φ j j=1 N cos φ j j=1 N cos 2 φ i j=1 N sin φ j cos φ j j=1 N sin φ j j=1 N sin φ j cos φ j j=1 N sin 2 φ j ] 1 [ j=1 N I ij j=1 N I ij cos φ j j=1 N I ij sin φ j ]
δ i = tan 1 ( c i ' / b i ' )
cosφ= I 1 a 1 b 1 sinφ= ( b 2 I 1 b 1 cosδ I 2 )( b 2 b 1 a 1 cosδ a 2 ) b 2 sinδ
φ= tan 1 ( ( b 2 I 1 cosδ b 1 I 2 )( b 2 a 1 cosδ b 1 a 2 ) ( I 1 a 1 ) b 2 sinδ )

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