Abstract

Phase-shifting techniques are extremely important in modern optical metrology. The advanced iterative algorithm (AIA) is an elegant, flexible and effective phase-shifting algorithm that can extract phase from fringe patterns with arbitrary unknown phase-shifts. However, comparing it with traditional phase-shifting algorithms, AIA has not been sufficiently investigated on (i) its applicability to different types of fringe patterns; (ii) its performance with respect to different phase-shifts, frame numbers and noise levels and thus the possibility of further improvement; and (iii) the predictability of its accuracy. To solve these problems, a series of innovations are proposed in this paper. First, condition numbers are introduced to characterize the least squares matrices used in AIA, and subsequently a fringe density requirement is suggested for the success of AIA. Second, the performance of AIA regarding different phase-shifts, frame numbers and noise levels is thoroughly evaluated by simulations, based on which, an overall phase error model is established. With such understanding, three individual improvements of AIA, i.e., controlling phase-shifts, controlling frame numbers and suppressing noise, are proposed for better performance of AIA. Third, practical methods for estimating the overall phase errors are developed to make the AIA performance predictable even before AIA is executed. We then integrate all these three innovations into an enhanced AIA (eAIA), which solves all the problems we mentioned earlier. The significant contributions of eAIA include the insurability of the convergence, the controllability of the performance, and achievability of a desired accuracy. An experiment is carried out to demonstrate the effectiveness of eAIA.

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

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References

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2013 (1)

2012 (1)

2011 (1)

2010 (1)

Q. Kemao, H. Wang, W. Gao, L. Feng, and S. H. Soon, “Phase extraction from arbitrary phase-shifted fringe patterns with noise suppression,” Opt. Laser. Eng. 48(6), 684–689 (2010).
[Crossref]

2009 (1)

2007 (1)

Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementations,” Opt. Laser. Eng. 45(2), 304–317 (2007).
[Crossref]

2004 (1)

1997 (1)

1996 (2)

Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35(1), 51–60 (1996).
[Crossref]

J. Immerkaer, “Fast noise variance estimation,” Comput. Vis. Image. Und. 64(2), 300–302 (1996).
[Crossref]

1994 (1)

G. E. Healey and R. Kondepudy, “Radiometric CCD camera calibration and noise estimation,” IEEE Trans. Pattern Anal. Machine Intell. 16(3), 267–276 (1994).
[Crossref]

1993 (1)

1992 (3)

K. Larkin and B. Oreb, “Design and assessment of symmetrical phase-shifting algorithms,” J. Opt. Soc. Am. A 9(10), 1740–1748 (1992).
[Crossref]

C. Farrell and M. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3(10), 953–958 (1992).
[Crossref]

R. Jozwicki, M. Kujawinska, and L. A. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing,” Opt. Eng. 31(3), 422–434 (1992).
[Crossref]

1990 (1)

1988 (1)

K. Creath, “Phase-measurement interferometry techniques,” Prog. Opt. 26(26), 349–393 (1988).
[Crossref]

1987 (1)

1984 (1)

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23(4), 234350 (1984).
[Crossref]

1974 (1)

1966 (1)

P. Carré, “Installation et utilisation du comparateur photoélectrique et interférentiel du Bureau International des Poids et Mesures,” Metrologia 2(1), 13–23 (1966).
[Crossref]

Belenguer, T.

Belsley, D. A.

D. A. Belsley, E. Kuh, and R. E. Welsch, Regression diagnostics: Identifying influential data and sources of collinearity (John Wiley & Sons, 2005).

Brangaccio, D.

Brophy, C. P.

Bruning, J. H.

Carazo, J.

Carré, P.

P. Carré, “Installation et utilisation du comparateur photoélectrique et interférentiel du Bureau International des Poids et Mesures,” Metrologia 2(1), 13–23 (1966).
[Crossref]

Creath, K.

K. Creath, “Phase-measurement interferometry techniques,” Prog. Opt. 26(26), 349–393 (1988).
[Crossref]

Deng, J.

Dewar, J.

D. Zill and J. Dewar, Algebra and Trigonometry (Jones & Bartlett Publishers, 2011).

Eiju, T.

Estrada, J.

Fan, J.

Farrell, C.

C. Farrell and M. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3(10), 953–958 (1992).
[Crossref]

Feng, L.

Q. Kemao, H. Wang, W. Gao, L. Feng, and S. H. Soon, “Phase extraction from arbitrary phase-shifted fringe patterns with noise suppression,” Opt. Laser. Eng. 48(6), 684–689 (2010).
[Crossref]

Gallagher, J.

Gao, W.

Q. Kemao, H. Wang, W. Gao, L. Feng, and S. H. Soon, “Phase extraction from arbitrary phase-shifted fringe patterns with noise suppression,” Opt. Laser. Eng. 48(6), 684–689 (2010).
[Crossref]

Greivenkamp, J. E.

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23(4), 234350 (1984).
[Crossref]

Han, B.

Hariharan, P.

Healey, G. E.

G. E. Healey and R. Kondepudy, “Radiometric CCD camera calibration and noise estimation,” IEEE Trans. Pattern Anal. Machine Intell. 16(3), 267–276 (1994).
[Crossref]

Herriott, D. R.

Immerkaer, J.

J. Immerkaer, “Fast noise variance estimation,” Comput. Vis. Image. Und. 64(2), 300–302 (1996).
[Crossref]

Jozwicki, R.

R. Jozwicki, M. Kujawinska, and L. A. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing,” Opt. Eng. 31(3), 422–434 (1992).
[Crossref]

Kemao, Q.

Q. Kemao, H. Wang, W. Gao, L. Feng, and S. H. Soon, “Phase extraction from arbitrary phase-shifted fringe patterns with noise suppression,” Opt. Laser. Eng. 48(6), 684–689 (2010).
[Crossref]

Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementations,” Opt. Laser. Eng. 45(2), 304–317 (2007).
[Crossref]

Q. Kemao, Windowed fringe pattern analysis (SPIE Press Bellingham, Wash, USA, 2013).

Kondepudy, R.

G. E. Healey and R. Kondepudy, “Radiometric CCD camera calibration and noise estimation,” IEEE Trans. Pattern Anal. Machine Intell. 16(3), 267–276 (1994).
[Crossref]

Kuh, E.

D. A. Belsley, E. Kuh, and R. E. Welsch, Regression diagnostics: Identifying influential data and sources of collinearity (John Wiley & Sons, 2005).

Kujawinska, M.

R. Jozwicki, M. Kujawinska, and L. A. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing,” Opt. Eng. 31(3), 422–434 (1992).
[Crossref]

Larkin, K.

Lu, X.

Malacara, D.

D. Malacara, Optical shop testing (John Wiley & Sons, 2007).

Nakamura, J.

J. Nakamura, Image sensors and signal processing for digital still cameras (CRC, 2016).

Oreb, B.

Parlett, B. N.

B. N. Parlett, The symmetric eigenvalue problem (siam, 1998).

Player, M.

C. Farrell and M. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3(10), 953–958 (1992).
[Crossref]

Quiroga, J. A.

Rosenfeld, D.

Salbut, L. A.

R. Jozwicki, M. Kujawinska, and L. A. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing,” Opt. Eng. 31(3), 422–434 (1992).
[Crossref]

Servin, M.

Soon, S. H.

Q. Kemao, H. Wang, W. Gao, L. Feng, and S. H. Soon, “Phase extraction from arbitrary phase-shifted fringe patterns with noise suppression,” Opt. Laser. Eng. 48(6), 684–689 (2010).
[Crossref]

Sorzano, C.

Surrel, Y.

Vargas, J.

Wang, H.

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).
[Crossref]

Q. Kemao, H. Wang, W. Gao, L. Feng, and S. H. Soon, “Phase extraction from arbitrary phase-shifted fringe patterns with noise suppression,” Opt. Laser. Eng. 48(6), 684–689 (2010).
[Crossref]

Wang, Z.

Welsch, R. E.

D. A. Belsley, E. Kuh, and R. E. Welsch, Regression diagnostics: Identifying influential data and sources of collinearity (John Wiley & Sons, 2005).

White, A.

Zhang, D.

Zhong, L.

Zill, D.

D. Zill and J. Dewar, Algebra and Trigonometry (Jones & Bartlett Publishers, 2011).

Appl. Opt. (5)

Comput. Vis. Image. Und. (1)

J. Immerkaer, “Fast noise variance estimation,” Comput. Vis. Image. Und. 64(2), 300–302 (1996).
[Crossref]

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

G. E. Healey and R. Kondepudy, “Radiometric CCD camera calibration and noise estimation,” IEEE Trans. Pattern Anal. Machine Intell. 16(3), 267–276 (1994).
[Crossref]

J. Opt. Soc. Am. A (2)

Meas. Sci. Technol. (1)

C. Farrell and M. Player, “Phase step measurement and variable step algorithms in phase-shifting interferometry,” Meas. Sci. Technol. 3(10), 953–958 (1992).
[Crossref]

Metrologia (1)

P. Carré, “Installation et utilisation du comparateur photoélectrique et interférentiel du Bureau International des Poids et Mesures,” Metrologia 2(1), 13–23 (1966).
[Crossref]

Opt. Eng. (2)

J. E. Greivenkamp, “Generalized data reduction for heterodyne interferometry,” Opt. Eng. 23(4), 234350 (1984).
[Crossref]

R. Jozwicki, M. Kujawinska, and L. A. Salbut, “New contra old wavefront measurement concepts for interferometric optical testing,” Opt. Eng. 31(3), 422–434 (1992).
[Crossref]

Opt. Express (1)

Opt. Laser. Eng. (2)

Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementations,” Opt. Laser. Eng. 45(2), 304–317 (2007).
[Crossref]

Q. Kemao, H. Wang, W. Gao, L. Feng, and S. H. Soon, “Phase extraction from arbitrary phase-shifted fringe patterns with noise suppression,” Opt. Laser. Eng. 48(6), 684–689 (2010).
[Crossref]

Opt. Lett. (4)

Prog. Opt. (1)

K. Creath, “Phase-measurement interferometry techniques,” Prog. Opt. 26(26), 349–393 (1988).
[Crossref]

Other (6)

Q. Kemao, Windowed fringe pattern analysis (SPIE Press Bellingham, Wash, USA, 2013).

D. Malacara, Optical shop testing (John Wiley & Sons, 2007).

D. A. Belsley, E. Kuh, and R. E. Welsch, Regression diagnostics: Identifying influential data and sources of collinearity (John Wiley & Sons, 2005).

D. Zill and J. Dewar, Algebra and Trigonometry (Jones & Bartlett Publishers, 2011).

B. N. Parlett, The symmetric eigenvalue problem (siam, 1998).

J. Nakamura, Image sensors and signal processing for digital still cameras (CRC, 2016).

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Figures (15)

Fig. 1.
Fig. 1. Typical fringe patterns used for simulation. (a) A noiseless straight fringe pattern with Np = 1. (b) A circular fringe pattern with Np= 2 and additive noise with σ/B = 10%. (c) A complex fringe pattern with Np= 3 and ωc = 0 and additive white noise with σ/B = 20%
Fig. 2.
Fig. 2. Fringe density requirement. (a) κ(Aps) of φ1, φ2 and φ3 w.r.t Np. (b) RMSEs of phases from noiseless fringe pattern with φ1, φ2 and φ3 w.r.t Np. (c) RMSEs of phases from noisy fringe pattern with φ1, φ2 and φ3 w.r.t Np.
Fig. 3.
Fig. 3. A complex fringe pattern with φ3 (Np = 10 and ωc = 0.25) as phase and additive white noise with σ/B = 10%
Fig. 4.
Fig. 4. Influence of phase-shifts. (a) RMSEs of phases w.r.t κ(Ap)1/2. (b) Mean RMSEs of phases w.r.t κ(Ap)1/2.
Fig. 5.
Fig. 5. Influence of frame number. (a) RMSEs of phases w.r.t M. (b) RMSEs of phases w.r.t M−1/2.
Fig. 6.
Fig. 6. RMSE of phases w.r.t σ/B.
Fig. 7.
Fig. 7. Error model validation. (a) RMSEs of phases w.r.t EAIA with different M. (b) RMSEs of phases w.r.t EAIA with different σ/B.
Fig. 8.
Fig. 8. RMSEs of phases of controlled phase-shifts group and random phase-shifts group.
Fig. 9.
Fig. 9. Improvement by frame number control. (a) Mean values of κ(Ap) w.r.t M.(b) Mean values of RMSEs of phases w.r.t M.
Fig. 10.
Fig. 10. RMSEs of phases from unfiltered fringe patterns, MF filtered fringe patterns and WFF filtered fringe patterns w.r.t σ/B.
Fig. 11.
Fig. 11. Validation of AIA error prediction. (a) RMSEs of phases w.r.t ÊAIA with different M. (b) RMSEs of phases w.r.t ÊAIA with different σ.
Fig. 12.
Fig. 12. Flowchart of eAIA.
Fig. 13.
Fig. 13. Fringe patterns and ground true phase. (a) One of acquired fringe patterns. (b) The ground true phase.
Fig. 14.
Fig. 14. Performance of eAIA. (a) RMSE of phases and ÊAIA w.r.t frame number. (b) Estimated phase-shifts of the 7 frames.
Fig. 15.
Fig. 15. Phase calculated with different number of frames. (a) Phase calculated with 3 frames. (b) Phase calculated with 7 frames.

Equations (41)

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I i j t = A i j + B i j cos ( φ j + δ i ) , i = 1 , 2 , , M ; j = 1 , 2 , , N ,
I i j = I i j t + n i j ,
I i j t = a j + b j cos δ i + c j sin δ i ,
S j = i = 1 M ( I i j t I i j ) 2 = i = 1 M ( a j + b j cos δ i + c j sin δ i I i j ) 2 .
A p X p , j = B p , j ,
A p = [ M i = 1 M cos δ i i = 1 M sin δ i i = 1 M cos δ i i = 1 M cos 2 δ i i = 1 M cos δ i sin δ i i = 1 M sin δ i i = 1 M cos δ i sin δ i i = 1 M sin 2 δ i ] ,
X p , j = A p 1 B p , j ,
φ j = tan 1 ( c j b j ) .
I i j t = a i + b i cos φ j + c i sin φ j ,
S i = j = 1 N ( I i j t I i j ) 2 = j = 1 N ( a i + b i cos φ j + c i sin φ j I i j ) 2 .
A ps X ps , i = B ps , i ,
A ps = [ 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 j j = 1 N cos φ j sin φ j j = 1 N sin φ j j = 1 N cos φ j sin φ j j = 1 N sin 2 φ j ] ,
X ps , i = A ps 1 B ps , i ,
δ i = tan 1 ( c i b i ) .
| ( δ i k δ 1 k ) ( δ i k 1 δ 1 k 1 ) | < ε ,
κ ( H ) = σ max ( H ) σ min ( H ) ,
A ps x , x = x T A ps x = j = 1 N ( x 1 + cos φ j x 2 + sin φ j x 3 ) 2 0 ,
σ min ( A ps ) x T A ps x x T x σ max ( A ps ) .
σ max ( A ps ) N ,
σ min ( A ps ) min ( j = 1 N cos 2 φ j , j = 1 N sin 2 φ j ) .
κ ( A ps ) N min ( j = 1 N cos 2 φ j , j = 1 N sin 2 φ j ) .
cos 2 φ + sin 2 φ = 1 ,
min ( j = 1 N cos 2 φ j , j = 1 N sin 2 φ j ) N 2 .
κ ( A ps ) 2 ,
φ 1 ( x , y ) = 2 N p π N x x ,
φ 2 ( x , y ) = 2 N p π ( N x 2 ) 2 + ( N y 2 ) 2 r 2 ,
φ 3 ( x , y ) = N p × peaks s ( N x , N y ) + ω c x
E AIA = 0.42 ( κ ( A p ) + 2 ) × σ / B M
I i j t = A i j + C i j + C i j ,
C i j = 1 2 B i j exp [ j ( φ j + δ i ) ] = 1 2 B i j exp ( j δ i ) exp ( j φ j ) ,
C i j = C 1 j exp ( j δ i ) ,
F [ I ( ξ x , ξ y ; i ) ] = F [ A ] + F [ C ( ξ x , ξ y ; i ) ] + F [ C ( ξ x , ξ y ; i ) ] ,
( u , v ) = a r g max ξ x , ξ y i = 1 M | F [ C ( ξ x , ξ y ; i ) ] | .
δ ^ i = F [ C ( u , v ; i ) ] F [ C ( u , v ; 1 ) ] .
B ^ j = b j 2 + c j 2 ,
B ^ = 1 N j = 1 N B ^ j .
σ ^ i = 2 π 1 6 ( N x 2 ) ( N y 2 ) I i d | I i d N | ,
N = [ 1 2 1 2 4 2 1 2 1 ] ,
σ ^ = 1 M i = 1 M σ ^ i
E ^ AIA = 0.42 ( κ ( A ^ p ) + 2 ) × σ ^ / B ^ M .
M = ceil [ 2 ( σ ^ / B ^ τ ) 2 ] ,

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