Abstract

We develop an error-free nonuniform phase-stepping algorithm (nPSA) based on principal component analysis (PCA). PCA-based algorithms typically give phase-demodulation errors when applied to nonuniform phase-shifted interferograms. We present a straightforward way to correct those PCA phase-demodulation errors. We give mathematical formulas to fully analyze PCA-based nPSA (PCA-nPSA). These formulas give a) the PCA-nPSA frequency transfer function (FTF), b) its corrected Lissajous figure, c) the corrected PCA-nPSA formula, d) its harmonic robustness (RH), and e) its signal-to-noise-ratio (SNR). We show that the PCA-nPSA can be seen as a linear quadrature filter and, as consequence, one can find its FTF. Using the FTF, we show why plain PCA often fails to demodulate nonuniform phase-shifted fringes. Previous works on PCA-nPSA (without FTF), give specific numerical/experimental fringe data to “visually demonstrate” that their new nPSA works better than its competitors. This often leads to biased/favorable fringe pattern selections which “visually demonstrate” the superior performance of their new nPSA. This biasing is herein totally avoided because we provide figures-of-merit formulas based on linear systems and stochastic process theories. However, and for illustrative purposes only, we provide specific fringe data phase-demodulation, including comprehensive analysis and comparisons.

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

1. Introduction

Phase-shifting interferometry is a widely used optical metrology technique to demodulate the phase from a set of phase-shifted interferograms [1–7]. The demodulated phase contains the searched measuring information from N phase-shifted fringe patterns. Traditionally phase shifting algorithms (PSAs) require precise phase-shifting steps, however, it is not easy to be absolutely sure that an interferometer have a zero-error phase shifter. Therefore, methods to estimate nonuniform phase-shifting steps and the desired modulating phase from nonlinear phase-shifted data have been investigated [8–23]. The Lissajous ellipse fitting technique is one of the earliest phase demodulation methods for dealing with nonuniform phase-stepped images [24–26]. If the demodulated analytic signal forms a Lissajous circle, one obtains an error-free measuring phase [24–26]. For erroneous phase demodulation, the Lissajous figure of the analytic signal is not a circle; it is an ellipse. The ellipse fitting method converts the Lissajous ellipse into a Lissajous circle, and the phase demodulation error is eliminated [27–32]. The Lissajous technique is a powerful technique to correct phase demodulation errors; it is currently based on least squares fitting of a rotated and origin-shifted ellipse; this has however its own difficulties which are sometimes not trivially solved [27–32].

Another nonuniform phase-steps algorithm (nPSA) is the principal component analysis (PCA) of phase-shifted fringe data [33–38]; we call this the PCA-nPSA technique. This is a subspace technique because it finds two orthogonal signals from N correlated nonuniform phase-shifted fringe images. Plain/unmodified PCA-nPSA demodulates the phase from fringe images without the explicit knowledge about their nonlinear phase shifts. The PCA-nPSA technique has low computational cost, it is linear, non-iterative, and it can deal with spatially varying background illumination and fringe contrast. Therefore, it seems at first glance, that PCA-nPSA would deal with all possible situations of nonuniform/linear phase-shifted phase demodulation [33–38].

In spite of all those good properties, the PCA-nPSA has however some disadvantages which often gives inacceptable phase demodulation errors [37–40]. The PCA-nPSA users may not be aware of the phase-demodulation errors, and may therefore reach erroneous conclusions in phase metrology engineering [37–40]. A well studied PCA-nPSA limitation occurs when less-than-one spatial fringe is present within the fringe pattern [33–40]. But the problem of “few spatial-fringes” is in our view, a pseudo-problem. That is because one can easily introduce as many spatial fringes as desire simply by introducing a large spatial carrier (a large tilt), and the few spatial-fringes “problem” is gone [1]. A more recent attempt, and good review, to improve the PCA-nPSA using the Lissajous figure is given in [40].

In this work we show that the PCA-nPSA can be regarded as a linear quadrature filter applied to nonuniform phase-shifted fringe data. And as any other linear filter, it is possible to find its Fourier spectrum through its frequency transfer function (FTF) [1]. Finding the FTF of the PCA-nPSA one can see the reason why plain PCA often fails to demodulate, error-free, a set of nonuniform phase-shifted data. With the FTF at hand, one can find the signal-to-noise ratio (SNR) and fringe harmonics robustness of the PCA-nPSA from first principles of stochastic process and linear systems theories [1].

2. Nonuniform phase-shifting fringe images

We first describe the continuous-time fringe model as,

I(x,y,t)=a(x,y)+b(x,y)cos[φ(x,y)+ω0t];t.
Where a(x,y), b(x,y), and φ(x,y) are the background, the amplitude, and the phase of the fringes respectively. The parameter ω0 is the angular frequency of the fringes. Without loss of generality we assume ω0 = 1.0 (radians/second). The temporal Fourier transformF[] of the fringe is,
F[I(t)]=aδ(ω)+b2eiφδ(ω+1)+b2eiφδ(ω1).
For all (x,y)L×L, and i=1; see Fig. 1.

 

Fig. 1 Panel (a) shows 9 nonuniform temporal samples of a continuous-time fringe. Panel (b) shows the Fourier spectrum of the continuous-time fringe.

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The temporal continuous fringe in Eq. (1) is sampled at nonuniform times (tn) as,

In=[a+bcos(φ+ω0t)]δ(ttn)dt;n{0,1,...,N1}.
It is usual to label the fringe samples by their nonlinear phase-steps θn=ω0tn as,
In=a+bcos(φ+θn);(x,y)L×L.
Note that ω0=1.0 and tn are not meaningful; the relevant data are the nonuniform phase-steps θn=ω0tn. Figure 1(a) shows 9 nonlinearly-spaced phase-shifted samples (in red), and Fig. 1(b) the Fourier spectrum of the continuous temporal fringe (in blue).

3. PCA-nPSA phase demodulation formula

Principal component analysis (PCA) was invented by Karl Pearson in 1901 [42]. It is a statistical procedure that uses a linear transformation that converts hundreds of correlated observations into a subset of linearly uncorrelated signals called the principal components of the data. In phase-shifting demodulation, the PCA is used to find 2 orthogonal signals (an analytic signal) from few temporal fringe samples. The fact of using a handful (instead of hundreds) of nonuniform phase-shifted data translates into an erroneous phase estimation, making the PCA-nPSA (if not properly corrected) inadequate for precision optical metrology. That is why several works have been published to correct the PCA-nPSA to cope with this residual phase demodulation error [37–41].

We now construct the desired PCA-nPSA formula. We start by modeling N nonuniform phase-shifted samples as,

In(x,y)=a(x,y)+b(x,y)cos[φ(x,y)+θn];n={0,1,...,N1}.
The images have (x,y)L×L pixels. Firstly, we estimate the background of the fringes as,
a^(x,y)=1Nn=0N1In(x,y).
The following step in PCA is to compute the covariance matrix,
[C]m,n=1L2x=0L1y=0L1[Im(x,y)a^(x,y)][In(x,y)a^(x,y)].
We now find the N eigenvalues λn and eigenvectors vn of matrix C as,
Cvn=λnvn;n{0,1,...,N1}.
Assuming that the largest eigenvalues are (λ0,λ1) (Cv0=λ0v0, Cv1=λ1v1), then the PCA-nPSA formula is given by,
A(x,y)=n=0N1([v0]n+i[v1]n)In(x,y).
Here A(x,y) is the demodulated analytic signal, and its phase is given by arg[A(x,y)]. Note that the PCA as herein presented does not need to vectorize back-and-forth the fringe images [33–36]. Equation (9) is the non-corrected (plain) PCA-nPSA formula. This is an interesting result because, however implicit in the original PCA technique [33], it has not been explicitly given as an standard phase-shifting algorithm formula.

The first work on PCA as nPSA was presented as a linear algorithm which could demodulate any set of phase-shifted fringes, almost error-free [33]. Afterwards this was found not to be exact and several attempts have been made to improve plain PCA [35,38–40]. However, plain PCA can be combined with the advanced iterative algorithm (AIA), obtaining a PCA-AIA algorithm which reduces the phase-errors left by plain PCA [35]. The PCA calculation provides a good first phase estimation helping the AIA converge faster [35]. Note that the AIA estimates both, the modulating phase φ(x,y) and the nonuniform phase-steps {θ0,θ1,...,θN1} [35].

4. Correcting the Lissajous ellipse of the PCA-nPSA analytic signal

The Lissajous figure of the PCA-nPSA (Eq. (9)) is obtained by the following parametric plot,

r[φ(x,y)]=Re[A(x,y)]i+Im[A(x,y)]j.
Here i and j are the real and imaginary unit vectors. Since the PCA finds orthogonal eigenvectors, these Lissajous ellipses r[φ] are always non-rotated; they are easily transformed into circles (with zero phase demodulation error) by first calculating the ratio,
ρ=x=0L1y=0L1|Im[A(x,y)]|x=0L1y=0L1|Re[A(x,y)]|;(0<ρ1.0).
Here || denotes the absolute value. Equation (11) is very robust to noise because a single parameter is estimated from an entire image. With ρ, we may transform the Lissajous ellipse into a circle by modifying the plain PCA-nPSA (Eq. (9)) as,
A2(x,y)=n=0N1(ρ[v0]n+i[v1]n)In(x,y).
Now we have a Lissajous circle, i.e. an error-free phase arg[A2(x,y)]. Equation (12) constitutes the corrected PCA-nPSA formula. The Lissajous ellipses are then transformed into circles without explicitly knowing the nonuniform phase-steps {θ0,...,θN1}.

5. Frequency transfer function (FTF) for the PCA-nPSA

The phase-steps {θ0,...,θN1} can be known by combining the PCA and AIA (PCA-AIA) algorithms [35]. Having {θ0,...,θN1} we can obtain the FTF from the PCA-nPSA formula because it may be seen as a convolution product (see [1]),

A(x,y)=n=0N1c¯nIn(x,y)=[h¯(t)I(x,y,t)]t=N1,h(t)=n=0N1cnδ(tθn1.0);cn=ρ[v0]n+i[v1]n;I(x,y,t)=n=0N1I(x,y,θn)δ(tθn1.0).
Where h(t) is the impulse response and I(x,y,t) is the fringe data. The overbar stands for the complex conjugate, and (*) is the linear convolution symbol. The Fourier transform H(ω)=F[h(t)] is the FTF of the corrected PCA-nPSA,
H(ω)=F[n=0N1cnδ(tθn1.0)]=n=0N1cneiθnω.
To obtain a quadrature signal, H(ω)must comply at least, with the following conditions,
H(1)=0;H(0)=0;H(+1)0.
The response H(1)=0 rejects the conjugate signal (b/2)eiφδ(ω+1), and H(0)=0rejects the background aeiφδ(ω). Then one obtains an error-free analytic signal as,
A(x,y)=b(x,y)2H(1)eiφ(x,y).
If the response at H(1) is not zero, then one obtains an erroneous analytic signal given by,
Aerror=b2[eiφH(1)+eiφH(1)]=b2H(1)eiφ[1+H(1)H(1)e2iφ];(x,y)LxL.
A non-zero H(1) generate a detuning-like phase-demodulation error [H(1)/H(1)]e2iφ. This is the typical phase error given by plain PCA-nPSA.

6. Signal-to-noise ratio gain (GSNR) and fringe harmonic robustness

Once the FTF (Eq. (14)) is obtained, one can find the SNR and harmonics robustness of the PCA-nPSA from basic stochastic process and linear systems theories (for more details see pages 48-85 in [1]). Without the FTF people usually rely on particular fringe images (sometimes favorably biased) which may lead to over-optimistic conclusions [9–40].

6.1 Signal-to-noise ratio gain GSNR of the PCA-nPSA formula

The SNR of the analytic signal (Eq. (13)) for nonuniform sampled fringes corrupted by additive white Gaussian noise (AWGN) with power density N(ω)=η/2, ω(π,π) is [1].

SNR=QuadratureSignalPowerFilteredAWGNPower=(b2/4)(η/2)|H(1.0)|2n=0N1|cn|2.
From this formula we define the SNR-gain (GSNR) as [1],
GSNR=|H(1.0)|2n=0N1|cn|2;(0<GSNRN).
The figure of merit GSNR is the SNR-gain of the analytic signal with respect to the SNR of the fringe data. For example GSNR=N means that the analytic signal has N-times higher SNR than the fringe data. The figure of merit GSNR substantially decreases for highly nonuniform phase-step fringes [41].

6.2 Harmonic robustness RH for N-steps PCA-nPSA

Harmonic-distorted phase-shifted fringes may be modeled by,

I(φ,θn)=a+bcos[φ+θn]+k=2bkcos[kφ+kθn];(x,y)LxL.
Then the demodulated analytic signal A(x,y) for harmonic distorted fringes is given by,
A(x,y)=b2eiφH(1)+k=2(bk2)[eikφH(k)+eikφH(k)].
Where H(k) and H(k) is the FTF response to the k-th harmonics k = {2,3,...}. Assuming that the harmonics amplitude decreases as bk = (1/k) we define the harmonic robustness as,
RH=QuadratureSignalPowerTotalFringeHarmonicPower=|H(1.0)|2k=2(1k2)[|H(k)|2+|H(k)|2].
A large RH figure means high fringe harmonics robustness. In contrast, a low RH figure means high harmonics power (low harmonics robustness).

7. Computer simulations

For illustrative purposes we now offer two simulations for plain and corrected PCA-nPSA applied to 3 and 9 nonuniform phase sampled fringe images. These examples are given to show the Fourier spectral response of plain/corrected PCA-nPSA.

7.1 Plain PCA-nPSA applied to 3 nonuniform phase-step fringe images

Let us start with the 3 nonuniform phase-shifted fringe images shown in Fig. 2 and Fig. 3.

 

Fig. 2 Three nonlinear phase-shifted samples taken at θn = {0, 1.49, 5.13} radians.

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Fig. 3 The 3 nonuniform phase-shifted, harmonics free, fringe images.

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The nonlinear phase-shifting are θn={0,1.49,5.13} radians. The fringes are shown in Fig. 3.

Then we apply the PCA to these 3 images as,

[Cn,m]=[3.893.730.163.7317.0213.30.1613.313.46](λ0λ1λ2)=(28.955.420)
Taking the largest eigenvalues Cv0=λ0v0 and Cv1=λ1v1, the PCA-nPSA is,
A(x,y)=n=02([v0]n+i[v1]n)In(x,y).
The demodulated phase arg[A(x,y)] is given in Fig. 4 along with its phase error.

 

Fig. 4 Here we show plain PCA-nPSA demodulated phase, and its phase-error. This figure uses the standard 256 gray-level linear-map from [-π,π) to [0,255). The amplitude of the two phase-error images was multiplied by 2 for illustrative purposes.

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The FTF, H(ω), of the plain PCA-nPSA is,

H(ω)=n=02([v0]n+i[v1]n)eiθnω.
From Fig. 5(a), we see that |H(1)|>0 and the erroneous analytic signal is given by,
A=b2H(1)eiφ[1+H(1)H(1)e2iφ]=b2eiφ[1+0.31e2iφ];(x,y).
As Eq. (26) shows, |H(1)=1.0 and |H(1)=0.31 giving an error of 0.31e2iφ.

 

Fig. 5 Correcting 3-steps plain PCA-nPSA. Panel (a) shows the FTF, analytic signal, and Lissajous ellipse of plain PCA. Panel (b) shows the corrected FTF along its analytic signal and Lissajous circle. We also show the SNR-gain slight degradation due to FTF correction.

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7.2 Lissajous figures for plain and corrected PCA-nPSA with 3 phase steps

Figure 5(a) shows the FTF, analytic signal, and Lissajous ellipse of plain PCA-nPSA.

The (noise-robust) correcting factor ρ for these 3 nonuniform phase-steps in Fig. 2 is,

ρ=x=0L1y=0L1|Im[A(x,y)]|x=0L1y=0L1|Re[A(x,y)]|=0.432.
We then use ρ to correct the PCA-nPSA obtaining the Lissajous circle in Fig. 5(b). From Fig. 5 we see that GSNR reduces from 1.96 to 1.2. On the other hand, the harmonic robustness RH decreases from 1.3 to 0.66. That is, the corrected PCA-nPSA is more sensitive to noise and harmonics than plain PCA-nPSA. However, the phase error of plain PCA-nPSA is intolerable (see Fig. 4).

7.3 Lissajous figures for plain and corrected PCA-nPSA with 9 phase steps

To further illustrate our technique, we now present the figures of merit for plain and corrected 9-steps PCA-nPSA. Figure 6 shows 4 out of 9 noiseless fringe images.

 

Fig. 6 Here we show 4 out of 9 nonlinear phase-shifted, harmonics free, fringe patterns.

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Figure 7 shows the 9 phase-steps θn = {0, 1.13, 2.49, 1.52, 3.55, 3.78, 6.2, 6.42, 8.74},

 

Fig. 7 The red dots plot represents 9 nonlinear/nonuniform phase steps.

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Figure 8 shows the FTF, analytic signal, and Lissajous figures of plain PCA and corrected FTF. Plain PCA-nPSA has a SNR-gain of GSNR = 8.11, and harmonics robustness of RH = 4.316. In contrast, the corrected PCA-nPSA has a SNR-gain of GSNR = 7.12, and harmonics robustness of RH = 3.381. Again, plain PCA-nPSA has better figure of merits than the corrected PCA-nPSA, but the demodulated phase error is intolerable.

 

Fig. 8 Correction of 9-steps plain PCA-nPSA. Panel (a) shows the FTF, analytic signal, and Lissajous ellipse of plain PCA-nPSA applied to Fig. 7 data. Panel (b) shows the corrected FTF, analytic signal, and Lissajous circle. Note that the SNR-gain decreases due to FTF correction.

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Please note that we omit the demodulated phase and phase-error maps for this 9-step example because they are virtually undistinguishable to Fig. 4 (for 3-steps) and do not contribute to a better understanding of the proposed technique.

8. Conclusion

We have presented a very simple way to correct the technique of principal component analysis (PCA) applied to phase-demodulation of nonuniform phase-shifting fringes. We can summarize the contributions of this work as,

  • a) We have presented a PCA-nPSA procedure which does not need to vectorize the two-dimensional fringe images (Eqs. (5)-(9)).
  • b) The Lissajous figures of the PCA-nPSA demodulated analytic signal are always non-rotated ellipses. These non-rotated ellipses are corrected to circles using a single scale factor (Eqs. (11)-(12)).
  • c) Knowing the phase steps {θ0,θ1,...,θN1}, one can find the PCA-nPSA Fourier spectral response H(ω) or frequency transfer function (FTF).
  • d) The FTF of the PCA-nPSA is then used to estimate the SNR-gain (GSNR), for fringes corrupted by additive white Gaussian noise, and the harmonics robustness RH for the plain/corrected PCA-nPSA (Eqs. (19) and (22)).
  • e) We have shown that plain PCA-nPSA has better SNR and harmonics robustness (RH) than corrected PCA-nPSA. But plain PCA-nPSA produces, in general, inacceptable phase demodulation errors.

In brief, we have shown that the cost for correcting plain PCA's demodulated phase are small decreases in SNR and harmonics robustness.

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41. M. Servin, M. Padilla, G. Garnica, and G. Paez, “Design of nonlinearly spaced phase-shifting algorithms using their frequency transfer function,” Appl. Opt. 58(4), 1134–1138 (2019). [CrossRef]   [PubMed]  

42. K. Pearson, “On Lines and Planes of Closest Fit to Systems of Points in Space,” Philos. Mag. 2(11), 559–572 (1901). [CrossRef]  

References

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    [Crossref]
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  29. 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).
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  30. F. Liu, Y. Wu, F. Wu, and W. Song, “Generalized phase shifting interferometry based on Lissajous calibration technology,” Opt. Lasers Eng. 83, 106–115 (2016).
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  31. F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert Huang prefiltering,” J. Opt. 18(10), 105604 (2016).
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  32. B. Kimbrough, “Correction of errors in polarization based dynamic phase shifting interferometers,” Int. J. Optomechatronics 8(4), 304–312 (2014).
    [Crossref]
  33. J. Vargas, J. A. Quiroga, and T. Belenguer, “Phase-shifting interferometry based on principal component analysis,” Opt. Lett. 36(8), 1326–1328 (2011).
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  34. J. Vargas, J. A. Quiroga, and T. Belenguer, “Analysis of the principal component algorithm in phase-shifting interferometry,” Opt. Lett. 36(12), 2215–2217 (2011).
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  35. J. Vargas, C. Sorzano, J. Estrada, and J. Carazo, “Generalization of the principal component analysis algorithm for interferometry,” Opt. Commun. 286, 130–134 (2013).
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  36. J. Vargas and C. Sorzano, “Quadrature component analysis for interferometry,” Opt. Lasers Eng. 51(5), 637–641 (2013).
    [Crossref]
  37. J. Xu, W. Jin, L. Chai, and Q. Xu, “Phase extraction from randomly phase-shifted interferograms by combining principal component analysis and least squares method,” Opt. Express 19(21), 20483–20492 (2011).
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  38. 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).
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  39. K. Yatabe, K. Ishikawa, and Y. Oikawa, “Improving principal component analysis based phase extraction method for phase-shifting interferometry by integrating spatial information,” Opt. Express 24(20), 22881–22891 (2016).
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  40. K. Yatabe, K. Ishikawa, and Y. Oikawa, “Simple, flexible, and accurate phase retrieval method for generalized phase-shifting interferometry,” J. Opt. Soc. Am. A 34(1), 87–96 (2017).
    [Crossref] [PubMed]
  41. M. Servin, M. Padilla, G. Garnica, and G. Paez, “Design of nonlinearly spaced phase-shifting algorithms using their frequency transfer function,” Appl. Opt. 58(4), 1134–1138 (2019).
    [Crossref] [PubMed]
  42. K. Pearson, “On Lines and Planes of Closest Fit to Systems of Points in Space,” Philos. Mag. 2(11), 559–572 (1901).
    [Crossref]

2019 (1)

2017 (1)

2016 (7)

K. Yatabe, K. Ishikawa, and Y. Oikawa, “Improving principal component analysis based phase extraction method for phase-shifting interferometry by integrating spatial information,” Opt. Express 24(20), 22881–22891 (2016).
[Crossref] [PubMed]

K. Ishikawa, K. Yatabe, N. Chitanont, Y. Ikeda, Y. Oikawa, T. Onuma, H. Niwa, and M. Yoshii, “High-speed imaging of sound using parallel phase-shifting interferometry,” Opt. Express 24(12), 12922–12932 (2016).
[Crossref] [PubMed]

J. Deng, D. Wu, K. Wang, and J. Vargas, “Precise phase retrieval under harsh conditions by constructing new connected interferograms,” Sci. Rep. 6(1), 24416 (2016).
[Crossref] [PubMed]

Y. Xu, Y. Wang, Y. Ji, H. Han, and W. Jin, “Three-frame generalized phase-shifting interferometry by a Euclidean matrix norm algorithm,” Opt. Lasers Eng. 84, 89–95 (2016).
[Crossref]

X. Xu, X. Lu, J. Tian, J. Shou, D. Zheng, and L. Zhong, “Random phase-shifting interferometry based on independent component analysis,” Opt. Commun. 370, 75–80 (2016).
[Crossref]

F. Liu, Y. Wu, F. Wu, and W. Song, “Generalized phase shifting interferometry based on Lissajous calibration technology,” Opt. Lasers Eng. 83, 106–115 (2016).
[Crossref]

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert Huang prefiltering,” J. Opt. 18(10), 105604 (2016).
[Crossref]

2015 (4)

2014 (3)

J. Deng, L. Zhong, H. Wang, H. Wang, W. Zhang, F. Zhang, S. Ma, and X. Lu, “1-Norm character of phase shifting interferograms and its application in phase shift extraction,” Opt. Commun. 316, 156–160 (2014).
[Crossref]

A. Albertazzi, A. V. Fantin, D. P. Willemann, and M. E. Benedet, “Phase maps retrieval from sequences of phase shifted images with unknown phase steps using generalized N-dimensional Lissajous figures—principles and applications,” Int. J. Optomechatronics 8(4), 340–356 (2014).
[Crossref]

B. Kimbrough, “Correction of errors in polarization based dynamic phase shifting interferometers,” Int. J. Optomechatronics 8(4), 304–312 (2014).
[Crossref]

2013 (5)

J. Vargas, C. Sorzano, J. Estrada, and J. Carazo, “Generalization of the principal component analysis algorithm for interferometry,” Opt. Commun. 286, 130–134 (2013).
[Crossref]

J. Vargas and C. Sorzano, “Quadrature component analysis for interferometry,” Opt. Lasers Eng. 51(5), 637–641 (2013).
[Crossref]

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] [PubMed]

H. Guo and Z. Zhang, “Phase shift estimation from variances of fringe pattern differences,” Appl. Opt. 52(26), 6572–6578 (2013).
[Crossref] [PubMed]

R. Juarez-Salazar, C. Robledo-Sánchez, C. Meneses-Fabian, F. Guerrero-Sánchez, and L. A. Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least squares method,” Opt. Lasers Eng. 51(5), 626–632 (2013).
[Crossref]

2011 (3)

2009 (1)

2008 (1)

2007 (1)

2005 (1)

A. Patil and P. Rastogi, “Approaches in generalized phase shifting interferometry,” Opt. Lasers Eng. 43(3-5), 475–490 (2005).
[Crossref]

2004 (2)

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]

Y. Awatsuji, M. Sasada, and T. Kubota, “Parallel quasi-phase-shifting digital holography,” Appl. Phys. Lett. 85(6), 1069–1071 (2004).
[Crossref]

1997 (1)

1996 (1)

1995 (1)

I.-B. Kong and S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34(1), 183–188 (1995).
[Crossref]

1994 (1)

C. T. Farrell and M. A. Player, “Phase-step insensitive algorithms for phase-shifting interferometry,” Meas. Sci. Technol. 5(6), 648–654 (1994).
[Crossref]

1992 (1)

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

1991 (2)

G. Lai and T. Yatagai, “Generalized phase-shifting interferometry,” J. Opt. Soc. Am. A 8(5), 822–827 (1991).
[Crossref]

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]

1988 (1)

1985 (1)

1901 (1)

K. Pearson, “On Lines and Planes of Closest Fit to Systems of Points in Space,” Philos. Mag. 2(11), 559–572 (1901).
[Crossref]

Aguilar, L. A.

R. Juarez-Salazar, C. Robledo-Sánchez, C. Meneses-Fabian, F. Guerrero-Sánchez, and L. A. Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least squares method,” Opt. Lasers Eng. 51(5), 626–632 (2013).
[Crossref]

Albertazzi, A.

A. Albertazzi, A. V. Fantin, D. P. Willemann, and M. E. Benedet, “Phase maps retrieval from sequences of phase shifted images with unknown phase steps using generalized N-dimensional Lissajous figures—principles and applications,” Int. J. Optomechatronics 8(4), 340–356 (2014).
[Crossref]

Awatsuji, Y.

Y. Awatsuji, M. Sasada, and T. Kubota, “Parallel quasi-phase-shifting digital holography,” Appl. Phys. Lett. 85(6), 1069–1071 (2004).
[Crossref]

Belenguer, T.

Benedet, M. E.

A. Albertazzi, A. V. Fantin, D. P. Willemann, and M. E. Benedet, “Phase maps retrieval from sequences of phase shifted images with unknown phase steps using generalized N-dimensional Lissajous figures—principles and applications,” Int. J. Optomechatronics 8(4), 340–356 (2014).
[Crossref]

Bokor, J.

Cai, L. Z.

Carazo, J.

J. Vargas, C. Sorzano, J. Estrada, and J. Carazo, “Generalization of the principal component analysis algorithm for interferometry,” Opt. Commun. 286, 130–134 (2013).
[Crossref]

Chai, L.

Chen, M.

Chen, Q.

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert Huang prefiltering,” J. Opt. 18(10), 105604 (2016).
[Crossref]

Chen, W.

Cheng, X. C.

Cheng, Y.-Y.

Chitanont, N.

Deng, J.

J. Deng, D. Wu, K. Wang, and J. Vargas, “Precise phase retrieval under harsh conditions by constructing new connected interferograms,” Sci. Rep. 6(1), 24416 (2016).
[Crossref] [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).
[Crossref] [PubMed]

J. Deng, L. Zhong, H. Wang, H. Wang, W. Zhang, F. Zhang, S. Ma, and X. Lu, “1-Norm character of phase shifting interferograms and its application in phase shift extraction,” Opt. Commun. 316, 156–160 (2014).
[Crossref]

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] [PubMed]

Dong, G. Y.

Estrada, J.

J. Vargas, C. Sorzano, J. Estrada, and J. Carazo, “Generalization of the principal component analysis algorithm for interferometry,” Opt. Commun. 286, 130–134 (2013).
[Crossref]

Fan, J.

Fantin, A. V.

A. Albertazzi, A. V. Fantin, D. P. Willemann, and M. E. Benedet, “Phase maps retrieval from sequences of phase shifted images with unknown phase steps using generalized N-dimensional Lissajous figures—principles and applications,” Int. J. Optomechatronics 8(4), 340–356 (2014).
[Crossref]

Farrell, C. T.

C. T. Farrell and M. A. Player, “Phase-step insensitive algorithms for phase-shifting interferometry,” Meas. Sci. Technol. 5(6), 648–654 (1994).
[Crossref]

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

Gao, P.

Garnica, G.

Geist, E.

Goldberg, K. A.

Guerrero-Sánchez, F.

R. Juarez-Salazar, C. Robledo-Sánchez, C. Meneses-Fabian, F. Guerrero-Sánchez, and L. A. Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least squares method,” Opt. Lasers Eng. 51(5), 626–632 (2013).
[Crossref]

Guo, H.

Han, B.

Han, H.

Y. Xu, Y. Wang, Y. Ji, H. Han, and W. Jin, “Three-frame generalized phase-shifting interferometry by a Euclidean matrix norm algorithm,” Opt. Lasers Eng. 84, 89–95 (2016).
[Crossref]

Hao, J.

Harder, I.

He, J.

Q. Liu, Y. Wang, J. He, and F. Ji, “Phase shift extraction and wavefront retrieval from interferograms with background and contrast fluctuations,” J. Opt. 17(2), 025704 (2015).
[Crossref]

Hou, X.

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert Huang prefiltering,” J. Opt. 18(10), 105604 (2016).
[Crossref]

Ikeda, Y.

Ishikawa, K.

Ji, F.

Q. Liu, Y. Wang, J. He, and F. Ji, “Phase shift extraction and wavefront retrieval from interferograms with background and contrast fluctuations,” J. Opt. 17(2), 025704 (2015).
[Crossref]

Ji, Y.

Y. Xu, Y. Wang, Y. Ji, H. Han, and W. Jin, “Three-frame generalized phase-shifting interferometry by a Euclidean matrix norm algorithm,” Opt. Lasers Eng. 84, 89–95 (2016).
[Crossref]

Jin, W.

Y. Xu, Y. Wang, Y. Ji, H. Han, and W. Jin, “Three-frame generalized phase-shifting interferometry by a Euclidean matrix norm algorithm,” Opt. Lasers Eng. 84, 89–95 (2016).
[Crossref]

J. Xu, W. Jin, L. Chai, and Q. Xu, “Phase extraction from randomly phase-shifted interferograms by combining principal component analysis and least squares method,” Opt. Express 19(21), 20483–20492 (2011).
[Crossref] [PubMed]

Juarez-Salazar, R.

R. Juarez-Salazar, C. Robledo-Sánchez, C. Meneses-Fabian, F. Guerrero-Sánchez, and L. A. Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least squares method,” Opt. Lasers Eng. 51(5), 626–632 (2013).
[Crossref]

Kim, S.-W.

I.-B. Kong and S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34(1), 183–188 (1995).
[Crossref]

Kimbrough, B.

B. Kimbrough, “Correction of errors in polarization based dynamic phase shifting interferometers,” Int. J. Optomechatronics 8(4), 304–312 (2014).
[Crossref]

Kinnstaetter, K.

Kong, I.-B.

I.-B. Kong and S.-W. Kim, “General algorithm of phase-shifting interferometry by iterative least-squares fitting,” Opt. Eng. 34(1), 183–188 (1995).
[Crossref]

Kubota, T.

Y. Awatsuji, M. Sasada, and T. Kubota, “Parallel quasi-phase-shifting digital holography,” Appl. Phys. Lett. 85(6), 1069–1071 (2004).
[Crossref]

Lai, G.

Lara-Cortes, F. A.

Li, C.

Lindlein, N.

Liu, F.

F. Liu, Y. Wu, F. Wu, and W. Song, “Generalized phase shifting interferometry based on Lissajous calibration technology,” Opt. Lasers Eng. 83, 106–115 (2016).
[Crossref]

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert Huang prefiltering,” J. Opt. 18(10), 105604 (2016).
[Crossref]

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).
[Crossref] [PubMed]

Liu, Q.

Q. Liu, Y. Wang, J. He, and F. Ji, “Phase shift extraction and wavefront retrieval from interferograms with background and contrast fluctuations,” J. Opt. 17(2), 025704 (2015).
[Crossref]

Lohmann, A. W.

Lu, X.

X. Xu, X. Lu, J. Tian, J. Shou, D. Zheng, and L. Zhong, “Random phase-shifting interferometry based on independent component analysis,” Opt. Commun. 370, 75–80 (2016).
[Crossref]

J. Deng, L. Zhong, H. Wang, H. Wang, W. Zhang, F. Zhang, S. Ma, and X. Lu, “1-Norm character of phase shifting interferograms and its application in phase shift extraction,” Opt. Commun. 316, 156–160 (2014).
[Crossref]

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] [PubMed]

Lv, X.

Ma, S.

J. Deng, L. Zhong, H. Wang, H. Wang, W. Zhang, F. Zhang, S. Ma, and X. Lu, “1-Norm character of phase shifting interferograms and its application in phase shift extraction,” Opt. Commun. 316, 156–160 (2014).
[Crossref]

Mantel, K.

Medecki, H.

Meneses-Fabian, C.

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).
[Crossref] [PubMed]

R. Juarez-Salazar, C. Robledo-Sánchez, C. Meneses-Fabian, F. Guerrero-Sánchez, and L. A. Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least squares method,” Opt. Lasers Eng. 51(5), 626–632 (2013).
[Crossref]

Meng, X. F.

Niwa, H.

Oikawa, Y.

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]

Onuma, T.

Padilla, M.

Paez, G.

Patil, A.

A. Patil and P. Rastogi, “Approaches in generalized phase shifting interferometry,” Opt. Lasers Eng. 43(3-5), 475–490 (2005).
[Crossref]

Patorski, K.

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert Huang prefiltering,” J. Opt. 18(10), 105604 (2016).
[Crossref]

Pearson, K.

K. Pearson, “On Lines and Planes of Closest Fit to Systems of Points in Space,” Philos. Mag. 2(11), 559–572 (1901).
[Crossref]

Player, M. A.

C. T. Farrell and M. A. Player, “Phase-step insensitive algorithms for phase-shifting interferometry,” Meas. Sci. Technol. 5(6), 648–654 (1994).
[Crossref]

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

Qin, J.

Quiroga, J. A.

Rastogi, P.

A. Patil and P. Rastogi, “Approaches in generalized phase shifting interferometry,” Opt. Lasers Eng. 43(3-5), 475–490 (2005).
[Crossref]

Robledo-Sánchez, C.

R. Juarez-Salazar, C. Robledo-Sánchez, C. Meneses-Fabian, F. Guerrero-Sánchez, and L. A. Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least squares method,” Opt. Lasers Eng. 51(5), 626–632 (2013).
[Crossref]

Sasada, M.

Y. Awatsuji, M. Sasada, and T. Kubota, “Parallel quasi-phase-shifting digital holography,” Appl. Phys. Lett. 85(6), 1069–1071 (2004).
[Crossref]

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]

Schwider, J.

Servin, M.

Shen, X. X.

Shou, J.

X. Xu, X. Lu, J. Tian, J. Shou, D. Zheng, and L. Zhong, “Random phase-shifting interferometry based on independent component analysis,” Opt. Commun. 370, 75–80 (2016).
[Crossref]

Song, W.

F. Liu, Y. Wu, F. Wu, and W. Song, “Generalized phase shifting interferometry based on Lissajous calibration technology,” Opt. Lasers Eng. 83, 106–115 (2016).
[Crossref]

Sorzano, C.

J. Vargas, C. Sorzano, J. Estrada, and J. Carazo, “Generalization of the principal component analysis algorithm for interferometry,” Opt. Commun. 286, 130–134 (2013).
[Crossref]

J. Vargas and C. Sorzano, “Quadrature component analysis for interferometry,” Opt. Lasers Eng. 51(5), 637–641 (2013).
[Crossref]

Streibl, N.

Sun, W. J.

Tejnil, E.

Tian, J.

X. Xu, X. Lu, J. Tian, J. Shou, D. Zheng, and L. Zhong, “Random phase-shifting interferometry based on independent component analysis,” Opt. Commun. 370, 75–80 (2016).
[Crossref]

Trusiak, M.

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert Huang prefiltering,” J. Opt. 18(10), 105604 (2016).
[Crossref]

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.

J. Deng, D. Wu, K. Wang, and J. Vargas, “Precise phase retrieval under harsh conditions by constructing new connected interferograms,” Sci. Rep. 6(1), 24416 (2016).
[Crossref] [PubMed]

J. Vargas and C. Sorzano, “Quadrature component analysis for interferometry,” Opt. Lasers Eng. 51(5), 637–641 (2013).
[Crossref]

J. Vargas, C. Sorzano, J. Estrada, and J. Carazo, “Generalization of the principal component analysis algorithm for interferometry,” Opt. Commun. 286, 130–134 (2013).
[Crossref]

J. Vargas, J. A. Quiroga, and T. Belenguer, “Analysis of the principal component algorithm in phase-shifting interferometry,” Opt. Lett. 36(12), 2215–2217 (2011).
[Crossref] [PubMed]

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

Wan, Y.

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert Huang prefiltering,” J. Opt. 18(10), 105604 (2016).
[Crossref]

Wang, H.

J. Deng, L. Zhong, H. Wang, H. Wang, W. Zhang, F. Zhang, S. Ma, and X. Lu, “1-Norm character of phase shifting interferograms and its application in phase shift extraction,” Opt. Commun. 316, 156–160 (2014).
[Crossref]

J. Deng, L. Zhong, H. Wang, H. Wang, W. Zhang, F. Zhang, S. Ma, and X. Lu, “1-Norm character of phase shifting interferograms and its application in phase shift extraction,” Opt. Commun. 316, 156–160 (2014).
[Crossref]

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] [PubMed]

Wang, J.

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert Huang prefiltering,” J. Opt. 18(10), 105604 (2016).
[Crossref]

Wang, K.

J. Deng, D. Wu, K. Wang, and J. Vargas, “Precise phase retrieval under harsh conditions by constructing new connected interferograms,” Sci. Rep. 6(1), 24416 (2016).
[Crossref] [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).
[Crossref] [PubMed]

Wang, Y.

Y. Xu, Y. Wang, Y. Ji, H. Han, and W. Jin, “Three-frame generalized phase-shifting interferometry by a Euclidean matrix norm algorithm,” Opt. Lasers Eng. 84, 89–95 (2016).
[Crossref]

Q. Liu, Y. Wang, J. He, and F. Ji, “Phase shift extraction and wavefront retrieval from interferograms with background and contrast fluctuations,” J. Opt. 17(2), 025704 (2015).
[Crossref]

Wang, Y. R.

Wang, Z.

Willemann, D. P.

A. Albertazzi, A. V. Fantin, D. P. Willemann, and M. E. Benedet, “Phase maps retrieval from sequences of phase shifted images with unknown phase steps using generalized N-dimensional Lissajous figures—principles and applications,” Int. J. Optomechatronics 8(4), 340–356 (2014).
[Crossref]

Wu, D.

J. Deng, D. Wu, K. Wang, and J. Vargas, “Precise phase retrieval under harsh conditions by constructing new connected interferograms,” Sci. Rep. 6(1), 24416 (2016).
[Crossref] [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).
[Crossref] [PubMed]

Wu, F.

F. Liu, Y. Wu, F. Wu, and W. Song, “Generalized phase shifting interferometry based on Lissajous calibration technology,” Opt. Lasers Eng. 83, 106–115 (2016).
[Crossref]

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert Huang prefiltering,” J. Opt. 18(10), 105604 (2016).
[Crossref]

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).
[Crossref] [PubMed]

Wu, Y.

F. Liu, Y. Wu, F. Wu, and W. Song, “Generalized phase shifting interferometry based on Lissajous calibration technology,” Opt. Lasers Eng. 83, 106–115 (2016).
[Crossref]

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert Huang prefiltering,” J. Opt. 18(10), 105604 (2016).
[Crossref]

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).
[Crossref] [PubMed]

Wyant, J. C.

Xu, J.

Xu, Q.

Xu, X.

X. Xu, X. Lu, J. Tian, J. Shou, D. Zheng, and L. Zhong, “Random phase-shifting interferometry based on independent component analysis,” Opt. Commun. 370, 75–80 (2016).
[Crossref]

Xu, X. F.

Xu, Y.

Y. Xu, Y. Wang, Y. Ji, H. Han, and W. Jin, “Three-frame generalized phase-shifting interferometry by a Euclidean matrix norm algorithm,” Opt. Lasers Eng. 84, 89–95 (2016).
[Crossref]

Yamaguchi, I.

Yao, B.

Yatabe, K.

Yatagai, T.

Yoshii, M.

Yu, Y.

Zhang, D.

Zhang, F.

J. Deng, L. Zhong, H. Wang, H. Wang, W. Zhang, F. Zhang, S. Ma, and X. Lu, “1-Norm character of phase shifting interferograms and its application in phase shift extraction,” Opt. Commun. 316, 156–160 (2014).
[Crossref]

Zhang, H.

Zhang, T.

Zhang, W.

J. Deng, L. Zhong, H. Wang, H. Wang, W. Zhang, F. Zhang, S. Ma, and X. Lu, “1-Norm character of phase shifting interferograms and its application in phase shift extraction,” Opt. Commun. 316, 156–160 (2014).
[Crossref]

Zhang, Z.

Zheng, D.

X. Xu, X. Lu, J. Tian, J. Shou, D. Zheng, and L. Zhong, “Random phase-shifting interferometry based on independent component analysis,” Opt. Commun. 370, 75–80 (2016).
[Crossref]

Zhong, L.

X. Xu, X. Lu, J. Tian, J. Shou, D. Zheng, and L. Zhong, “Random phase-shifting interferometry based on independent component analysis,” Opt. Commun. 370, 75–80 (2016).
[Crossref]

J. Deng, L. Zhong, H. Wang, H. Wang, W. Zhang, F. Zhang, S. Ma, and X. Lu, “1-Norm character of phase shifting interferograms and its application in phase shift extraction,” Opt. Commun. 316, 156–160 (2014).
[Crossref]

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] [PubMed]

Appl. Opt. (4)

Appl. Phys. Lett. (1)

Y. Awatsuji, M. Sasada, and T. Kubota, “Parallel quasi-phase-shifting digital holography,” Appl. Phys. Lett. 85(6), 1069–1071 (2004).
[Crossref]

Int. J. Optomechatronics (2)

A. Albertazzi, A. V. Fantin, D. P. Willemann, and M. E. Benedet, “Phase maps retrieval from sequences of phase shifted images with unknown phase steps using generalized N-dimensional Lissajous figures—principles and applications,” Int. J. Optomechatronics 8(4), 340–356 (2014).
[Crossref]

B. Kimbrough, “Correction of errors in polarization based dynamic phase shifting interferometers,” Int. J. Optomechatronics 8(4), 304–312 (2014).
[Crossref]

J. Opt. (2)

Q. Liu, Y. Wang, J. He, and F. Ji, “Phase shift extraction and wavefront retrieval from interferograms with background and contrast fluctuations,” J. Opt. 17(2), 025704 (2015).
[Crossref]

F. Liu, J. Wang, Y. Wu, F. Wu, M. Trusiak, K. Patorski, Y. Wan, Q. Chen, and X. Hou, “Simultaneous extraction of phase and phase shift from two interferograms using Lissajous figure and ellipse fitting technology with Hilbert Huang prefiltering,” J. Opt. 18(10), 105604 (2016).
[Crossref]

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

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X. Xu, X. Lu, J. Tian, J. Shou, D. Zheng, and L. Zhong, “Random phase-shifting interferometry based on independent component analysis,” Opt. Commun. 370, 75–80 (2016).
[Crossref]

J. Deng, L. Zhong, H. Wang, H. Wang, W. Zhang, F. Zhang, S. Ma, and X. Lu, “1-Norm character of phase shifting interferograms and its application in phase shift extraction,” Opt. Commun. 316, 156–160 (2014).
[Crossref]

J. Vargas, C. Sorzano, J. Estrada, and J. Carazo, “Generalization of the principal component analysis algorithm for interferometry,” Opt. Commun. 286, 130–134 (2013).
[Crossref]

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]

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F. Liu, Y. Wu, F. Wu, and W. Song, “Generalized phase shifting interferometry based on Lissajous calibration technology,” Opt. Lasers Eng. 83, 106–115 (2016).
[Crossref]

J. Vargas and C. Sorzano, “Quadrature component analysis for interferometry,” Opt. Lasers Eng. 51(5), 637–641 (2013).
[Crossref]

A. Patil and P. Rastogi, “Approaches in generalized phase shifting interferometry,” Opt. Lasers Eng. 43(3-5), 475–490 (2005).
[Crossref]

Y. Xu, Y. Wang, Y. Ji, H. Han, and W. Jin, “Three-frame generalized phase-shifting interferometry by a Euclidean matrix norm algorithm,” Opt. Lasers Eng. 84, 89–95 (2016).
[Crossref]

R. Juarez-Salazar, C. Robledo-Sánchez, C. Meneses-Fabian, F. Guerrero-Sánchez, and L. A. Aguilar, “Generalized phase-shifting interferometry by parameter estimation with the least squares method,” Opt. Lasers Eng. 51(5), 626–632 (2013).
[Crossref]

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

J. Vargas, J. A. Quiroga, and T. Belenguer, “Analysis of the principal component algorithm in phase-shifting interferometry,” Opt. Lett. 36(12), 2215–2217 (2011).
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J. Deng, D. Wu, K. Wang, and J. Vargas, “Precise phase retrieval under harsh conditions by constructing new connected interferograms,” Sci. Rep. 6(1), 24416 (2016).
[Crossref] [PubMed]

Other (2)

M. Servin, J. A. Quiroga, and M. Padilla, Fringe Pattern Analysis for Optical Metrology, (WILEY-VCH, 2014).

H. Schreiber and J. H. Bruning, “Phase shifting interferometry,” in Optical Shop Testing (Wiley, 2006), pp. 547–666.

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Figures (8)

Fig. 1
Fig. 1 Panel (a) shows 9 nonuniform temporal samples of a continuous-time fringe. Panel (b) shows the Fourier spectrum of the continuous-time fringe.
Fig. 2
Fig. 2 Three nonlinear phase-shifted samples taken at θn = {0, 1.49, 5.13} radians.
Fig. 3
Fig. 3 The 3 nonuniform phase-shifted, harmonics free, fringe images.
Fig. 4
Fig. 4 Here we show plain PCA-nPSA demodulated phase, and its phase-error. This figure uses the standard 256 gray-level linear-map from [-π,π) to [0,255). The amplitude of the two phase-error images was multiplied by 2 for illustrative purposes.
Fig. 5
Fig. 5 Correcting 3-steps plain PCA-nPSA. Panel (a) shows the FTF, analytic signal, and Lissajous ellipse of plain PCA. Panel (b) shows the corrected FTF along its analytic signal and Lissajous circle. We also show the SNR-gain slight degradation due to FTF correction.
Fig. 6
Fig. 6 Here we show 4 out of 9 nonlinear phase-shifted, harmonics free, fringe patterns.
Fig. 7
Fig. 7 The red dots plot represents 9 nonlinear/nonuniform phase steps.
Fig. 8
Fig. 8 Correction of 9-steps plain PCA-nPSA. Panel (a) shows the FTF, analytic signal, and Lissajous ellipse of plain PCA-nPSA applied to Fig. 7 data. Panel (b) shows the corrected FTF, analytic signal, and Lissajous circle. Note that the SNR-gain decreases due to FTF correction.

Equations (27)

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

I( x,y,t )=a(x,y)+b(x,y)cos[ φ(x,y)+ ω 0 t ];t.
F[ I(t) ]=aδ(ω)+ b 2 e iφ δ( ω+1 )+ b 2 e iφ δ( ω1 ).
I n = [ a+bcos( φ+ ω 0 t ) ]δ(t t n ) dt;n{0,1,...,N1}.
I n =a+bcos( φ+ θ n );(x,y)L×L.
I n ( x,y )=a(x,y)+b(x,y)cos[ φ(x,y)+ θ n ];n={0,1,...,N1}.
a ^ (x,y)= 1 N n=0 N1 I n (x,y) .
[ C ] m,n = 1 L 2 x=0 L1 y=0 L1 [ I m (x,y) a ^ (x,y) ][ I n (x,y) a ^ (x,y) ] .
C v n = λ n v n ;n{0,1,...,N1}.
A(x,y)= n=0 N1 ( [ v 0 ] n +i [ v 1 ] n ) I n (x,y) .
r[ φ(x,y) ]=Re[ A(x,y) ]i+Im[ A(x,y) ]j.
ρ= x=0 L1 y=0 L1 | Im[ A(x,y) ] | x=0 L1 y=0 L1 | Re[ A(x,y) ] | ;( 0<ρ1.0 ).
A 2 (x,y)= n=0 N1 ( ρ [ v 0 ] n +i [ v 1 ] n ) I n (x,y) .
A(x,y)= n=0 N1 c ¯ n I n (x,y) = [ h ¯ (t)I(x,y,t) ] t=N1 , h(t)= n=0 N1 c n δ( t θ n 1.0 ) ; c n =ρ [ v 0 ] n +i [ v 1 ] n ; I(x,y,t)= n=0 N1 I(x,y, θ n )δ( t θ n 1.0 ) .
H(ω)=F[ n=0 N1 c n δ( t θ n 1.0 ) ]= n=0 N1 c n e i θ n ω .
H(1)=0;H(0)=0;H(+1)0.
A(x,y)= b(x,y) 2 H(1) e iφ(x,y) .
A error = b 2 [ e iφ H(1)+ e iφ H(1) ]= b 2 H(1) e iφ [ 1+ H(1) H(1) e 2iφ ];(x,y)LxL.
SNR= QuadratureSignalPower FilteredAWGNPower = ( b 2 /4) (η/2) | H(1.0) | 2 n=0 N1 | c n | 2 .
G SNR = | H(1.0) | 2 n=0 N1 | c n | 2 ;(0< G SNR N).
I(φ, θ n )=a+bcos[ φ+ θ n ]+ k=2 b k cos[ kφ+k θ n ] ;(x,y)LxL.
A(x,y)= b 2 e iφ H(1)+ k=2 ( b k 2 )[ e ikφ H(k)+ e ikφ H(k) ] .
R H = QuadratureSignalPower TotalFringeHarmonicPower = | H(1.0) | 2 k=2 ( 1 k 2 )[ | H(k) | 2 + | H(k) | 2 ] .
[ C n,m ]=[ 3.89 3.73 0.16 3.73 17.02 13.3 0.16 13.3 13.46 ]( λ 0 λ 1 λ 2 )=( 28.95 5.42 0 )
A(x,y)= n=0 2 ( [ v 0 ] n +i [ v 1 ] n ) I n (x,y) .
H(ω)= n=0 2 ( [ v 0 ] n +i [ v 1 ] n ) e i θ n ω .
A= b 2 H(1) e iφ [ 1+ H(1) H(1) e 2iφ ]= b 2 e iφ [ 1+0.31 e 2iφ ];(x,y).
ρ= x=0 L1 y=0 L1 | Im[ A(x,y) ] | x=0 L1 y=0 L1 | Re[ A(x,y) ] | =0.432.

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