Abstract

Based on Fourier domain estimation, a novel self-calibration phase-shifting algorithm, named Mid-Band Spatial Spectrum Matching (MSSM), is proposed to achieve phase retrieval from a small amount of phase-shifting interferograms containing very few fringes (defined as ultra-sparse fringe pattern(USFP)), which is still a difficult problem for optical interferometry. Both simulation and experimental results demonstrate that the proposed MSSM algorithm can accurately and rapidly achieve the phase distribution encoded in USFP while other current self-calibration algorithms fail, and this will supply a powerful tool to extend the application of phase-shifting interferometry.

© 2017 Optical Society of America

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References

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  1. T. C. Poon, Digital holography and three-dimensional display (Springer US, 2006).
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    [PubMed]
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    [PubMed]
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    [PubMed]
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    [PubMed]
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    [PubMed]
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    [PubMed]
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    [PubMed]
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2017 (4)

2016 (2)

M. Trusiak, Ł. Służewski, and K. Patorski, “Single shot fringe pattern phase demodulation using Hilbert-Huang transform aided by the principal component analysis,” Opt. Express 24(4), 4221–4238 (2016).
[PubMed]

L. Fengwei, W. Jing, W. Yongqian, W. Fan, T. Maciej, P. Krzysztof, W. Yongjian, C. Qiang, and H. Xi, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert–Huang prefiltering,” J. Opt. 18, 105604 (2016).

2015 (4)

2014 (1)

2013 (1)

2011 (2)

2009 (3)

2008 (1)

2005 (2)

2004 (1)

1998 (1)

D. Paganin and K. A. Nugent, “Noninterferometric Phase Imaging with Partially Coherent Light,” Phys. Rev. Lett. 80, 2586–2589 (1998).

1997 (1)

1995 (1)

I.-B. Kong and S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34, 183–188 (1995).

1992 (2)

C. T. Farrell and M. A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3, 953 (1992).

K. G. Larkin and B. F. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9, 1740–1748 (1992).

1982 (2)

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72, 156–160 (1982).

C. J. Morgan, “Least-squares estimation in phase-measurement interferometry,” Opt. Lett. 7(8), 368–370 (1982).
[PubMed]

1942 (1)

F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects,” Physica 9, 686–698 (1942).

Abdulhalim, I.

Asundi, A.

Belenguer, T.

Cai, L. Z.

Chen, W.

Cheng, X. C.

Choo, C. O.

Colomb, T.

Cuche, E.

Deng, J.

Depeursinge, C.

Ding, H.

Dong, G. Y.

Emery, Y.

Fan, J.

Fan, W.

L. Fengwei, W. Jing, W. Yongqian, W. Fan, T. Maciej, P. Krzysztof, W. Yongjian, C. Qiang, and H. Xi, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert–Huang prefiltering,” J. Opt. 18, 105604 (2016).

Farrell, C. T.

C. T. Farrell and M. A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3, 953 (1992).

Fengwei, L.

L. Fengwei, W. Jing, W. Yongqian, W. Fan, T. Maciej, P. Krzysztof, W. Yongjian, C. Qiang, and H. Xi, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert–Huang prefiltering,” J. Opt. 18, 105604 (2016).

Gao, P.

Geist, E.

Gillette, M. U.

Han, B.

Hao, J.

Harder, I.

Hariharan, P.

Ina, H.

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72, 156–160 (1982).

Ishikawa, K.

Jing, W.

L. Fengwei, W. Jing, W. Yongqian, W. Fan, T. Maciej, P. Krzysztof, W. Yongjian, C. Qiang, and H. Xi, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert–Huang prefiltering,” J. Opt. 18, 105604 (2016).

Kim, S.-W.

I.-B. Kong and S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34, 183–188 (1995).

Kobayashi, S.

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72, 156–160 (1982).

Kong, I.-B.

I.-B. Kong and S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34, 183–188 (1995).

Krzysztof, P.

L. Fengwei, W. Jing, W. Yongqian, W. Fan, T. Maciej, P. Krzysztof, W. Yongjian, C. Qiang, and H. Xi, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert–Huang prefiltering,” J. Opt. 18, 105604 (2016).

Lara-Cortes, F. A.

Larkin, K. G.

Li, B.

Li, C.

Lindlein, N.

Liu, F.

Liu, S.

Lu, X.

Lv, X.

Maciej, T.

L. Fengwei, W. Jing, W. Yongqian, W. Fan, T. Maciej, P. Krzysztof, W. Yongjian, C. Qiang, and H. Xi, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert–Huang prefiltering,” J. Opt. 18, 105604 (2016).

Magistretti, P. J.

Mantel, K.

Marquet, P.

Meneses-Fabian, C.

Meng, X. F.

Millet, L.

Mir, M.

Morgan, C. J.

Nugent, K. A.

D. Paganin and K. A. Nugent, “Noninterferometric Phase Imaging with Partially Coherent Light,” Phys. Rev. Lett. 80, 2586–2589 (1998).

Oikawa, Y.

Oreb, B. F.

Osten, W.

Paganin, D.

D. Paganin and K. A. Nugent, “Noninterferometric Phase Imaging with Partially Coherent Light,” Phys. Rev. Lett. 80, 2586–2589 (1998).

Patorski, K.

Pedrini, G.

Player, M. A.

C. T. Farrell and M. A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3, 953 (1992).

Popescu, G.

Qiang, C.

L. Fengwei, W. Jing, W. Yongqian, W. Fan, T. Maciej, P. Krzysztof, W. Yongjian, C. Qiang, and H. Xi, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert–Huang prefiltering,” J. Opt. 18, 105604 (2016).

Qin, J.

Qiu, X.

Quiroga, J. A.

Rappaz, B.

Rogers, J.

Roy, M.

Safrani, A.

Saide, D.

Schmit, J.

Shen, X. X.

Sluzewski, L.

Sun, W. J.

Takeda, M.

M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. A 72, 156–160 (1982).

Tian, J.

Trusiak, M.

Unarunotai, S.

Vargas, J.

Wang, H.

Wang, K.

Wang, Y.

Wang, Y. R.

Wang, Z.

Weijuan, Q.

Wu, D.

Wu, F.

Wu, Y.

Xi, H.

L. Fengwei, W. Jing, W. Yongqian, W. Fan, T. Maciej, P. Krzysztof, W. Yongjian, C. Qiang, and H. Xi, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert–Huang prefiltering,” J. Opt. 18, 105604 (2016).

Xiong, J.

Xu, X. F.

Yamaguchi, I.

Yao, B.

Yatabe, K.

Yingjie, Y.

Yongjian, W.

L. Fengwei, W. Jing, W. Yongqian, W. Fan, T. Maciej, P. Krzysztof, W. Yongjian, C. Qiang, and H. Xi, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert–Huang prefiltering,” J. Opt. 18, 105604 (2016).

Yongqian, W.

L. Fengwei, W. Jing, W. Yongqian, W. Fan, T. Maciej, P. Krzysztof, W. Yongjian, C. Qiang, and H. Xi, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert–Huang prefiltering,” J. Opt. 18, 105604 (2016).

Zernike, F.

F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects,” Physica 9, 686–698 (1942).

Zhang, D.

Zhang, H.

Zhang, T.

Zhang, Y.

Zhong, L.

Zhou, Y.

Appl. Opt. (1)

J. Opt. (1)

L. Fengwei, W. Jing, W. Yongqian, W. Fan, T. Maciej, P. Krzysztof, W. Yongjian, C. Qiang, and H. Xi, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert–Huang prefiltering,” J. Opt. 18, 105604 (2016).

J. Opt. Soc. Am. A (3)

Meas. Sci. Technol. (1)

C. T. Farrell and M. A. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3, 953 (1992).

Opt. Eng. (1)

I.-B. Kong and S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34, 183–188 (1995).

Opt. Express (9)

Z. Wang, L. Millet, M. Mir, H. Ding, S. Unarunotai, J. Rogers, M. U. Gillette, and G. Popescu, “Spatial light interference microscopy (SLIM),” Opt. Express 19(2), 1016–1026 (2011).
[PubMed]

Y. Wang, B. Li, L. Zhong, J. Tian, and X. Lu, “Spatial dual-orthogonal (SDO) phase-shifting algorithm by pre-recomposing the interference fringe,” Opt. Express 25(15), 17446–17456 (2017).
[PubMed]

Y. Wang, X. Qiu, J. Xiong, B. Li, L. Zhong, S. Liu, Y. Zhou, J. Tian, and X. Lu, “General spatial phase-shifting interferometry by optimizing the signal retrieving function,” Opt. Express 25(7), 7170–7180 (2017).
[PubMed]

M. Roy, J. Schmit, and P. Hariharan, “White-light interference microscopy: minimization of spurious diffraction effects by geometric phase-shifting,” Opt. Express 17(6), 4495–4499 (2009).
[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).
[PubMed]

F. Liu, Y. Wu, and F. Wu, “Correction of phase extraction error in phase-shifting interferometry based on Lissajous figure and ellipse fitting technology,” Opt. Express 23(8), 10794–10807 (2015).
[PubMed]

J. Deng, K. Wang, D. Wu, X. Lv, C. Li, J. Hao, J. Qin, and W. Chen, “Advanced principal component analysis method for phase reconstruction,” Opt. Express 23(9), 12222–12231 (2015).
[PubMed]

C. Meneses-Fabian and F. A. Lara-Cortes, “Phase retrieval by Euclidean distance in self-calibrating generalized phase-shifting interferometry of three steps,” Opt. Express 23(10), 13589–13604 (2015).
[PubMed]

M. Trusiak, Ł. Służewski, and K. Patorski, “Single shot fringe pattern phase demodulation using Hilbert-Huang transform aided by the principal component analysis,” Opt. Express 24(4), 4221–4238 (2016).
[PubMed]

Opt. Lett. (11)

Q. Weijuan, Y. Yingjie, C. O. Choo, and A. Asundi, “Digital holographic microscopy with physical phase compensation,” Opt. Lett. 34(8), 1276–1278 (2009).
[PubMed]

P. Gao, B. Yao, N. Lindlein, K. Mantel, I. Harder, and E. Geist, “Phase-shift extraction for generalized phase-shifting interferometry,” Opt. Lett. 34(22), 3553–3555 (2009).
[PubMed]

J. Vargas, J. A. Quiroga, and T. Belenguer, “Phase-shifting interferometry based on principal component analysis,” Opt. Lett. 36(8), 1326–1328 (2011).
[PubMed]

J. Deng, H. Wang, D. Zhang, L. Zhong, J. Fan, and X. Lu, “Phase shift extraction algorithm based on Euclidean matrix norm,” Opt. Lett. 38(9), 1506–1508 (2013).
[PubMed]

A. Safrani and I. Abdulhalim, “Real-time phase shift interference microscopy,” Opt. Lett. 39(17), 5220–5223 (2014).
[PubMed]

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

P. Marquet, B. Rappaz, P. J. Magistretti, E. Cuche, Y. Emery, T. Colomb, and C. Depeursinge, “Digital holographic microscopy: a noninvasive contrast imaging technique allowing quantitative visualization of living cells with subwavelength axial accuracy,” Opt. Lett. 30(5), 468–470 (2005).
[PubMed]

G. Pedrini, W. Osten, and Y. Zhang, “Wave-front reconstruction from a sequence of interferograms recorded at different planes,” Opt. Lett. 30(8), 833–835 (2005).
[PubMed]

X. F. Xu, L. Z. Cai, Y. R. Wang, X. F. Meng, W. J. Sun, H. Zhang, X. C. Cheng, G. Y. Dong, and X. X. Shen, “Simple direct extraction of unknown phase shift and wavefront reconstruction in generalized phase-shifting interferometry: algorithm and experiments,” Opt. Lett. 33(8), 776–778 (2008).
[PubMed]

I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22(16), 1268–1270 (1997).
[PubMed]

C. J. Morgan, “Least-squares estimation in phase-measurement interferometry,” Opt. Lett. 7(8), 368–370 (1982).
[PubMed]

Phys. Rev. Lett. (1)

D. Paganin and K. A. Nugent, “Noninterferometric Phase Imaging with Partially Coherent Light,” Phys. Rev. Lett. 80, 2586–2589 (1998).

Physica (1)

F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects,” Physica 9, 686–698 (1942).

Other (2)

M. Servín, J. A. Quiroga, and J. M. Padilla, Fringe Pattern Analysis for Optical Metrology: Theory, Algorithms, and Applications (Wiley-VCH, 2014).

T. C. Poon, Digital holography and three-dimensional display (Springer US, 2006).

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Figures (8)

Fig. 1
Fig. 1 Schematic of the spectral division, in which the background spectrum A( f x ), noise spectrum o( f x )and interference term T 1 ( f x ), T 2 ( f x )are located in the low-frequency band, high-frequency band and mid-frequency band, respectively.
Fig. 2
Fig. 2 (a1)(a2)Two-frame simulated interference patterns encoded with phase ϕ 1 and ϕ 2 , respectively; (b1)(b2) the theoretical phase distributions of ϕ 1 and ϕ 2 , respectively; (c1) (e1) (g1) the retrieved phases ϕ 1 with MSSM, PCA and AIA algorithms, respectively; (d1) (f1) (h1) the differences between (b1) and (c1), (e1), (g1), respectively; (c2) (e2) (g2) the retrieved phase ϕ 2 with MSSM, PCA and AIA algorithms, respectively; (d2) (f2) (h2) the differences between (b2) and (c2), (e2), (g2), respectively.
Fig. 3
Fig. 3 Variation curves of RMSE with different parameters (a) t ; (b) k; (c) SNR; the 2-dimentional distribution variation of RMSEs with δ 1 , δ 2 achieved by different algorithms (d) MSSM; (e) PCA; (f) AIA.
Fig. 4
Fig. 4 Variation curves of (a) RMSE and (b) the error level (l1 and l2) with the filtering window σ.
Fig. 5
Fig. 5 Experimental result of fringe-pattern with many fringes (a) one-frame fringe-pattern; (b) the intercepted calculation area from (a); (c) the reference phase distribution of (b); the retrieved phases and the corresponding differences between the reference phase and the achieved phases with different algorithms (d) (g) MSSM; (e) (h) PCA; (f) (i)AIA, respectively.
Fig. 6
Fig. 6 Experimental results of USFP (a) One-frame interferogram; (b) the intercepted USFP from (a); (c) the reference phase map of (b); the retrieved phases and the corresponding differences between the reference phase and the achieved phases with different algorithms (d) (g) MSSM; (e) (h) PCA; (f) (i)AIA, respectively.
Fig. 7
Fig. 7 Experimental setup of on-axis phase-shifting digital holographic microscopy system. MO1,MO2: microscope objective; PZT: piezoelectric ceramic transducer; M1, M2: mirror; BS1, BS2: beam splitter.
Fig. 8
Fig. 8 Experimental results of a T-lymphocyte cell (a) three-frame interferograms; (b) the reference phase of (a) achieved by temporal Fourier transform algorithm; the unwrapped phase distributions and the corresponding differences between the reference phase and the wrapped phases achieved with different algorithms (c)(f) PCA; (d)(g) MSSM; (e)(h) AIA; (i) the cross-section curves of the 170th row in (f), (g) and (h), respectively.

Tables (3)

Tables Icon

Table 1 RMSE, PVE and calculation time of phase retrieval with different algorithms in Fig. 5

Tables Icon

Table 2 RMSE, PVE and calculation time of phase retrieval with different algorithms in Fig. 6

Tables Icon

Table 3 RMSE, PVE and calculation time of phase retrieval with different algorithms in Fig. 8

Equations (14)

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

I n (x,y)=A(x,y)+B(x,y)cos[ϕ(x,y)+ δ n ]+ o n (x,y)n=1......N.
I 1 ( f x )=A( f x )+ζ[B(x)cosϕ(x)]+ o 1 ( f x ) =A( f x )+ T 1 ( f x )+ o 1 ( f x ) .
I 2 ( f x )=A( f x )+ζ{B(x)cos[ϕ(x)+ δ 1 ]}+ o 2 ( f x ) =A( f x )+cos δ 1 T 1 ( f x )sin δ 1 T 2 ( f x )+ o 2 ( f x ) .
I 3 ( f x )=A( f x )+ζ{B(x)cos[ϕ(x)+ δ 2 ]}+ o 3 ( f x ) =A( f x )+cos δ 2 T 1 ( f x )sin δ 2 T 2 ( f x )+ o 3 ( f x ) .
I ˜ 1 ( f x ) T ˜ 1 ( f x ).
I ˜ 2 ( f x ) T ˜ 1 ( f x )cos δ 1 T ˜ 2 ( f x )sin δ 1 .
I ˜ 3 ( f x ) T ˜ 1 ( f x )cos δ 2 T ˜ 2 ( f x )sin δ 2 .
C 1 I ˜ 2 ( f x )+ C 2 I ˜ 3 ( f x )= I ˜ 1 ( f x ).
C 1 cos δ 1 + C 2 cos δ 2 =1.
C 1 sin δ 1 = C 2 sin δ 2 .
( A(x,y) B(x,y)cosϕ(x,y) B(x,y)sinϕ(x,y) )= ( 1 cos δ 1 sin δ 1 1 cos δ 2 sin δ 2 1 cos δ 3 sin δ 3 ) 1 ( I 1 I 2 I 3 ).
ϕ(x,y)=arctan[ B(x,y)sinϕ(x,y) B(x,y)cosϕ(x,y) ].
l 1 = f x , f y abs[ A ˜ ( f x , f y )] f x , f y abs[ T ˜ 1 ( f x , f y )] .
l 2 = f x , f y abs[ A ˜ ( f x , f y )] f x , f y abs[ T ˜ 2 ( f x , f y )] .

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