Abstract

A generalized regularized phase tracker (GRPT) for demodulation of a single fringe pattern was recently proposed. It is very successful for many fringe patterns. However, the GRPT has poor performance in the area where the fringe pattern is sparse. An improved GRPT (iGRPT) with two novel improvements is proposed to overcome the problem. First, the fixed window used in the GRPT is replaced by a spatially adaptive window. Second, a background regularization term and a modulation regularization term are incorporated in the cost function. With these two improvements, the proposed iGRPT can successfully demodulate sparse fringes and thus improves the demodulation capability of the GRPT. Simulation and experimental results are presented to verify the performance of the iGRPT.

© 2013 Optical Society of America

1. Introduction

In optical metrology, phase demodulation from a single fringe pattern is of great interest in applications where the use of phase shifting and carrier methods [1] is difficult, for example, when transient mechanical processes are measured or the environment is hostile. Various methods have been proposed [217] to deal with this problem, such as the adaptive quadrature filters, the spiral phase quadrature transform, the correlation technique and the regularized phase tracker (RPT). The RPT [1117] is one of the most powerful approaches and has been well recognized, but it has two disadvantages: the necessity of a normalized fringe pattern as input and the sensitivity to critical points. To avoid these two drawbacks, a generalized regularized phase tracker (GRPT) [18] was recently proposed. The GRPT can successfully deal with many fringe patterns. However, it has poor performance in the area where the fringe pattern is sparse. In this paper, an improved GRPT (iGRPT) with two novel improvements is proposed to overcome the problem. First, the fixed window used in the GRPT is replaced by a spatially adaptive window. Second, a background regularization term and a modulation regularization term are incorporated in the cost function. The proposed iGRPT can successfully demodulate sparse fringes and thus improves the demodulation capability of the GRPT.

The rest of the paper is organized as follows. The GRPT will be briefly reviewed in Sec. 2. The proposed iGRPT will then be presented in Sec. 3. Results and discussions will be given in Sec. 4. A conclusion will be drawn in Sec. 5.

2. Generalized regularized phase tracker

A typical fringe pattern is written as,

f(x,y)=a(x,y)+b(x,y)cos[φ(x,y)]+n(x,y),
where (x,y) is the pixel coordinate; f(x,y) denotes the recorded intensity of the fringe pattern; a(x,y) is the background; b(x,y) is the modulation; φ(x,y) is the phase term related to the physical quantity being measured; n(x,y) is the noise term. The common objective of fringe pattern analysis is to recover the phase φ(x,y) from the intensityf(x,y). The basic idea of the GRPT is to locally match the fringe pattern with a local fringe model by minimizing a cost function defined as [18],
U(x,y)=(ε,η)Nx,y(w(x,y;ε,η){[f(ε,η)fe(x,y;ε,η)]2+λ[φ0(ε,η)φe(x,y;ε,η)]2m(ε,η)}),
with
w(x,y;ε,η)=exp{[(xε)2+(yη)2]/(2σ2)},
fe(x,y;ε,η)=ae(x,y;ε,η)+be(x,y;ε,η)cos[φe(x,y;ε,η)],
ae(x,y;ε,η)=a0(x,y)+ax(x,y)(εx)+ay(x,y)(ηy),
be(x,y;ε,η)=b0(x,y)+bx(x,y)(εx)+by(x,y)(ηy),
φe(x,y;ε,η)=φ0(x,y)+ωx(x,y)(εx)+ωy(x,y)(ηy)+12cxx(x,y)(εx)2+12cyy(x,y)(ηy)2+cxy(x,y)(εx)(ηy),
where w(x,y;ε,η) is a Gaussian window with a standard deviation σ;Nx,y is a square neighborhood region centering at (x,y) and consisting of (4σ+1)×(4σ+1) pixels. The coordinates of these pixels are (ε,η); fe(x,y;ε,η) denotes the local fringe model in Nx,y; ae(x,y;ε,η) is the local background model in Nx,y; a0(x,y) is the background estimation; ax(x,y) and ay(x,y) are the first order derivatives of the background; similarly, be(x,y;ε,η) is the local modulation model in Nx,y; b0(x,y) is the modulation estimation; bx(x,y) and by(x,y) are the first order derivatives of the modulation; φe(x,y;ε,η) is the local phase model in Nx,y; φ0(x,y) is the phase estimation; ωx(x,y), ωy(x,y), cxx(x,y), cyy(x,y) and cxy(x,y) are the first and second order phase derivatives; m(ε,η) is an indicator that equals one if the pixel (ε,η) has already been demodulated and equals zero otherwise; λ is a regularizing parameter that controls the smoothness of the phase estimation φ0(x,y). By minimizing the cost function in Eq. (2) on a pixel-by-pixel basis with respect to the twelve parameters, a0(x,y), ax(x,y), ay(x,y), b0(x,y), bx(x,y), by(x,y), φ0(x,y), ωx(x,y), ωy(x,y), cxx(x,y), cyy(x,y) and cxy(x,y), the given fringe pattern can be demodulated. To avoid the problem of the sensitivity to critical points, the number of iterations (NI) in the optimization process is adopted as the quality measure to guide the demodulation path.

With such a general local fringe model and matching strategy, the GRPT has many merits: (1) as both a(x,y) and b(x,y) are included in the fringe model, no fringe normalization is needed; (2) as the fringe model reflects the real fringe pattern well, high demodulation accuracy can be achieved; (3) as the fringe model is reasonable in a large window, after the window-based matching, the noise is implicitly and sufficiently suppressed and no post-denoising is needed; (4) with a robust NI-guided demodulation path, critical points can be successfully circumvented. With these merits, the GRPT can indeed successfully demodulate many fringe patterns [18], but unfortunately not all. Using the following total local frequency (TLF) to indicate the fringe density,

ωTLF=ωx2+ωy2,
the GRPT is found to be successful where the fringe is dense or ωTLF is high, and becomes problematic where the fringe is sparse, or ωTLF is low. For example, in Fig. 1(a), the fringe pattern is dense in the inner part and sparse in the outer region. The GRPT with σ=7.5 and λ=20 applying to Fig. 1(a) gives the result in Fig. 1(b), which shows that the GRPT is only partially successful. The seed pixel to start the GRPT is selected at (130, 100), which is located in the dense area near the central part of the fringe pattern. Figure 1(c) shows the NI map where a higher value means the algorithm is more difficult to converge.

 figure: Fig. 1

Fig. 1 Demodulation of a simulated fringe pattern by the GRPT. (a) simulated fringe pattern, (b) phase result obtained by the GRPT with σ=7.5 and λ=20, (c) NI quality map of (b).

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3. Improved Generalized Regularized phase tracker

3.1 Analysis of the problem in the GRPT

The reason of the undesired outcome of the GRPT exposed in Fig. 1 is explained as follows. In the local fringe model in Eq. (4), two terms, ae(x,y;ε,η) and be(x,y;ε,η)cos[φe(x,y;ε,η)], should be distinguishable from each other so that their respective parameters can converge to their true values. Since ae(x,y;ε,η) is assumed to be a plane, be(x,y;ε,η)cos[φe(x,y;ε,η)] should vary sufficiently fast. In other words, the TLF should be high enough. In the Fourier domain, the spectra of ae(x,y;ε,η), 0.5be(x,y;ε,η)exp[jφe(x,y;ε,η)] and 0.5be(x,y;ε,η)exp[jφe(x,y;ε,η)] (j is the imaginary unit) should be separated sufficiently [19, 20]. Knowing that when we process a fringe pattern locally with a window size of σ, there is an energy leakage of about 2/σ for all three terms [21]. To separate these three terms in the Fourier domain at least partially, roughly ωTLF=2/σ is required. Otherwise these terms will mix up and the parameters are less uniquely determined, which may affect the correct convergence in the GRPT optimization. Obviously this limitation only affects the fringe pattern where the TLF is low.

It is interesting to note that the above problem is much less significant in the RPT [11, 12] because there are only two minima, [φ0(x,y),ωx(x,y),ωy(x,y)] and [φ0(x,y),ωx(x,y),ωy(x,y)], in the RPT energy function for noiseless fringe patterns and they are easily distinguishable. However, the problem becomes more significant for noisy fringe patterns. We will return to this point in Section 4.3.

3.2 The improved GRPT (iGRPT)

To solve the spectrum separation problem in the GRPT, the following two strategies are adopted into the proposed iGRPT:

  • (1) The requirement for the TLF is roughly ωTLF=2/σ, which is inversely proportional to the window size σ. Thus we can increase the window size to reduce this requirement. In other words, we can set the window size adaptively as,
    σ˜=2/ωTLF,

    Consider the window size should not be too large so that our local fringe model given in Eqs. (4-7) is suitable, and not be too small so that the GRPT is insensitive to noise, the following window size setting can be adopted,

    σ={σminifσ˜σminσ˜ifσmin<σ˜<σmaxσmaxifσ˜σmax,

    where σmin and σmax are the smallest and largest window sizes. Consider that the problem only exists for sparse fringes, we can set σmin as in the GRPT, for example, σmin=7.5, unless the fringe is locally too complicated and a smaller window size has to be used. To avoid the window size becomes too large, we can empirically set 10σmax20. Incorporating these considerations, a simpler window setting also can be used,

    σ={σminifσ˜(σmin+σmax)/2σmaxifσ˜>(σmin+σmax)/2.

    The adaptive size selection stragegy implicietly helps the iGRPT optimization to converge correctly.

  • (2) When the fringe is sparse, the twelve parameters in the fringe model tend to be less unique and mix with each other. In order to prevent this trend, we can explicitly prevent the parameters from slipping away too much from their already demodulated values in the neighboring areas. To do so, a background regularization term and a modulation regularization term are adopted in the cost function Eq. (2). The cost function becomes
    U(x,y)=(ε,η)Nx,y(w(x,y;ε,η){[f(ε,η)fe(x,y;ε,η)]2+(λaRa+λbRb+λφRφ)m(ε,η)}),

    with

    Ra=[a0(ε,η)ae(x,y;ε,η)]2,
    Rb=[b0(ε,η)be(x,y;ε,η)]2,
    Rφ=[φ0(ε,η)φe(x,y;ε,η)]2,

    whereλaand λbare regularizing parameters of the background regularization term and the modulation regularization term, respectively; λφ is the same as λ used in Eq. (2). The modulation regularization was first adopted in Ref [13], and the background regularization term is first adopted in this paper. The background and modulation regulization strategy explicietly helps the iGRPT optimization to converge correctly.

3.3 The implementation of the iGRPT

The implementation of the iGRPT is similar to the GRPT and is not very complicated. It is outlined as follows.

  • Step 1: Select a pixel with high signal-to-noise ratio as the seed pixel (x0,y0); estimate the ωTLF of this pixel and set the window size according to Eq. (10) or Eq. (11);
  • Step 2: Demodulate the seed pixel by minimizing the cost function in Eq. (12); push the seed pixel into a demodulation register. Only one register is needed for the iGRPT;
  • Step 3: Select the pixel (xs,ys) with the lowest NI in the demodulation register; calculate the ωTLF of this selected pixel by Eq. (8) and set the window size by Eq. (10) or Eq. (11); demodulate the unprocessed pixels adjacent to (xs,ys) by minimizing the cost function in Eq. (12); remove the selected pixel from the register;
  • Step 4: Push the processed pixels into the register and sort the pixels according to their NIs;
  • Step 5: Repeat step 3 and step 4 until the register is empty.

4. Results and discussions

4.1 Verification of the iGRPT

The example in Fig. 1(a) mentioned earlier is a simulated fringe pattern (256 × 256 pixels) spoiled by Gaussian noise. It is now demodulated by the iGRPT with the same seed pixel used in Fig. 1. Figure 2(a) shows the phase result obtained by the iGRPT with σmin=7.5,σmax=15, λa=0, λb=0, λφ=20 and the widow size is determined by Eq. (10). The corresponding NI quality map is shown in Fig. 2(b) and the window size is shown in Fig. 2(c). The phase result is obviously improved by using the spatially adaptive window size. Figure 2(d) shows the phase result obtained by the iGRPT with the same parameters as Fig. 2(a) except that λa=10 and λb=10 are set. The corresponding NI quality map and the window size are shown in Fig. 2(e) and Fig. 2(f), respectively. The phase result is further improved by using the additional explicit regularizations Ra and Rb. We can also compare the NI numbers shown in Fig. 2(e) with those in Fig. 2(b) and find that Ra and Rb make the optimization converge more easily. Figure 2(g) shows the phase result obtained by the iGRPT with the same parameters as Fig. 2(d) except that the window size is determined by Eq. (11). The corresponding NI quality map and the window size are shown in Fig. 2(h) and Fig. 2(i), respectively. The phase result is also acceptable. The mean absolute errors (MAEs), i.e., the mean value of the absolute difference between the extracted phase and the true phase, are calculated to be 0.485rad, 0.038rad and 0.043rad for Figs. 2(a), 2(d) and 2(g), respectively. The peak absolute errors (PAEs) of these three phase results are 2.129rad, 0.743rad and 0.826rad, respectively. All calculations are carried out on a personal Pentium Dual E8400 computer with 3.0GHz main frequency by MATLAB programming. The calculation time of Figs. 2(a), 2(d) and 2(g) are 85min, 40min and 38min, respectively.

 figure: Fig. 2

Fig. 2 Demodulation of a simulated fringe pattern by the iGRPT. From left to right, demodulatd phase, NI quality map and the window size are shown. The first row:σmin=7.5,σmax=15,λa=0, λb=0, λφ=20 and the window size is determined by Eq. (10); the second row: same parameters as in the first row, except that λa=10 and λb=10 are set; the third row: same parameters as in the second row except that the window size is determined by Eq. (11).

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An experimental fringe pattern (256 × 256 pixels) captured from electronic speckle pattern interferometry (ESPI) is shown in Figs. 3(a). The demodulation in Fig. 1 is repeated on Fig. 3(a) with the seed pixel of (130, 150) near the central part of the fringe pattern. The obtained results are shown in Figs. 3(b) and 3(c). The phase result is only partly successful. The various demodulations in Fig. 2 are then repeated on Fig. 3(a) and the obtained results are shown in Fig. 4, which verify the effectiveness of the proposed iGRPT. The calculation time of Figs. 4(a), 4(d) and 4(g) are 63min, 25min and 24min, respectively.

 figure: Fig. 3

Fig. 3 Demodulation of an experimental fringe pattern by the GRPT. (a) experimental fringe pattern, (b) phase result obtained by the GRPT with σ=7.5 and λ=20, (c) NI quality map of (b).

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 figure: Fig. 4

Fig. 4 Demodulation of Fig. 3(a) by the iGRPT. From left to right, demodulatd phase, NI quality map and the window size are shown. The first row:σmin=7.5,σmax=15,λa=0, λb=0, λφ=20 and the window size is determined by Eq. (10); the second row: same parameters as in the first row, except that λa=10 and λb=10 are set; the third row: same parameters as in the second row except that the window size is determined by Eq. (11).

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Another experimental fringe pattern (256 × 256 pixels) captured from electronic speckle pattern interferometry (ESPI) is shown in Figs. 5(a). This fringe pattern is complex and its demodulation is challenging. The demodulation in Fig. 1 is repeated on Fig. 5(a) with the seed pixel of (138, 22) located in the left side of the fringe pattern. The obtained results are shown in Figs. 5(b) and 5(c). The demodulation is failed. Figure 6(a) shows the phase result obtained by the iGRPT with σmin=3,σmax=12, λa=10, λb=10, λφ=20 and the window size is determined by Eq. (10). The corresponding NI quality map is shown in Fig. 6(b) and the window size is shown in Fig. 6(c). The phase result is satisfactory except for the lower left corner of the fringe pattern which is near the image boarder. A smaller σmin has been set in order to deal with this fringe pattern containing complex fringe details. The calculation time of Fig. 6(a) is 28min.

 figure: Fig. 5

Fig. 5 Demodulation of an experimental fringe pattern by the GRPT. (a) experimental fringe pattern, (b) phase result obtained by the GRPT with σ=7.5 and λ=20, (c) NI quality map of (b).

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 figure: Fig. 6

Fig. 6 Demodulation of Fig. 5(a) by the iGRPT with σmin=3,σmax=12,λa=10, λb=10, λφ=20 and the window size is determined by Eq. (10). (a) phase result obtained by the iGRPT, (b) NI quality map of (a), (c) window size.

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4.2 The iGRPT with the pre-removal of the background

We can also remove the background before demodulation so that the iGRPT only estimates nine instead of twelve parameters. To test the feasibility and effectiveness of this idea, the backgrounds of Fig. 1(a), Fig. 3(a) and Fig. 5(a) are removed by a high-pass filter [11, 12] and the iGRPT without background estimation is then applied to them. The parameter settings used here are the same as those used for Fig. 2(d), Fig. 4(d) and Fig. 6(a), respectively, except that their backgrounds are not estimated. The demodulation results, including the phase results, the NI quality maps and the window sizes are shown in Figs. 7(a)-7(c), Figs. 7(d)-7(f) and Figs. 7(g)-7(i), respectively. The phase result shown in Fig. 7(a) is better than the phase result shown in Fig. 2(d) because the background can be ideally removed in this simulated fringe pattern. The MAEs and PAEs of this phase result are 0.025rad and 0.563rad, respectively, which are lower than those in Fig. 2(d). However, the phase results shown in Figs. 7(d) and 7(g) are worse than those shown in Figs. 4(d) and 6(a) because the background removal becomes more challenging in these experimental fringe patterns. These results reveal that the iGRPT without background estimation is preferred when the background of a fringe pattern can be correctly removed. It is notable that the iGRPT then seems similar to the work in [13], but the iGRPT differentiate itself from the work in [13] by many significant ingredients such as quadratic phase model, larger and spatially adaptive window size, and NI guidance, and consequently the iGRPT is more accurate and robust. The calculation time of Figs. 7(a), 7(d) and 7(g) are 38min, 22min and 25min, respectively.

 figure: Fig. 7

Fig. 7 Demodulation of fringe patterns by the iGRPT without background estimation. From left to right, demodulatd phase, NI quality map and the window size are shown. The first row: demodulation results obtained from the simulated fringe pattern shown in Fig. 1(a); the second row: demodulation results obtained from the experimental fringe pattern shown in Fig. 3(a); the third row: demodulation results obtained from the experimental fringe pattern shown in Fig. 5(a).

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4.3 The comparison between the iGRPT and the RPT

As analyzed in Section 3.1, it is very interesting to find that the RPT [11, 12] is able to demodulate a fringe pattern with very low local frequencies. However, we also notice that with the presence of noise, the RPT also requires a certain amount of local frequency, roughly estimated as 1/(half window size). Furthermore, as highlighted in [12], the RPT has the problems with the noise disturbance and critical points. To compare the iGRPT with the RPT [12], all three fringe patterns used above, Fig. 1(a), Fig. 3(a) and Fig. 5(a), are tested. A rectangle window and λ=0.15 are used in the RPT. For Fig. 1(a), the RPT with different window sizes are performed without noise filtering and the results are shown in Fig. 8. The small window sizes (5 × 5 and 9 × 9) are sensitive to noise, as shown in Figs. 8(b) and 8(c). A big window size of 21 × 21 can obtain a smooth phase result as shown in Fig. 8(f), but it fails at the central part of the fringe pattern because in such a big window the linear phase model used in the RPT [12] is no longer suitable. The other two window sizes (13 × 13 and 17 × 17) produce the most successful results. The MAEs of these two phase results are 0.065rad and 0.058rad, respectively. The PAEs of these two phase results are 1.676rad and 1.713rad, respectively. Both MAEs and PAEs are higher than those in Figs. 2(d) and 2(g) by the iGRPT. The calculation time of Figs. 8(b)-8(f) are 1.5min, 2.0min, 3.0min, 3.9min and 4.6min, respectively. For Fig. 3(a), windowed Fourier filtering [21] is used to suppress the noise, and the 2D Hilbert transform [22] is used to normalize the fringe pattern. The normalized result is shown in Fig. 9(a). The demodulation results obtained from this normalized fringe pattern by the RPT with different window sizes are shown in Figs. 9(b)-9(f). All demodulations are successful. This is because the noise has been suppressed and the phase curvature for this example is not large. The calculation time of Figs. 9(b)-9(f) are 0.7min, 1.3min, 1.9min, 3.0min and 4.4min, respectively. For Fig. 5(a), the normalized fringe pattern obtained by the windowed Fourier filtering and 2D Hilbert transform is shown in Fig. 10(a). The demodulation results obtained from this normalized fringe pattern by the RPT with different window sizes are shown in Figs. 10(b)-10(f). The phase results shown in Figs. 10(b) and 10(c) obtained with two small window sizes (5 × 5 and 9 × 9) are quite successful but not yet satisfactory, because small windows are sensitive to intensity disturbance. The other window sizes fail the demodulation in the upper right part, because the large phase curvature there cannot be well approximated by the linear phase model in large windows. The calculation time of Figs. 10(b)-10(f) are 0.8min, 1.2min, 1.9min, 3.0min and 4.5min, respectively. From these examples, the iGRPT is found to be more time-consuming but more robust and accurate than the RPT [12].

 figure: Fig. 8

Fig. 8 Demodulation of the simulated fringe pattern by the RPT [12] with different window sizes. (a) the same simulated fringe pattern as shown in Fig. 1(a), (b)-(f) the phase results obtained by the RPT with window sizes 5 × 5, 9 × 9, 13 × 13, 17 × 17 and 21 × 21, respectively.

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 figure: Fig. 9

Fig. 9 Demodulation of the experimental fringe pattern shwon in Fig. 3(a) by the RPT [12] with different window sizes. (a) the normalized fringe pattern of Fig. 3(a), (b)-(f) the phase results obtained by the RPT from the normalized fringe pattern (a) with window sizes 5 × 5, 9 × 9, 13 × 13, 17 × 17 and 21 × 21, respectively.

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 figure: Fig. 10

Fig. 10 Demodulation of the experimental fringe pattern shwon in Fig. 5(a) by the RPT [12] with different window sizes. (a) the normalized fringe pattern of Fig. 5(a), (b)-(f) the phase results obtained by the RPT from the normalized fringe pattern (a) with window sizes 5 × 5, 9 × 9, 13 × 13, 17 × 17 and 21 × 21, respectively.

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5. Conclusion

An improved GRPT (iGRPT) with two novel improvements for demodulation of a single closed fringe pattern is proposed. First, the fixed window used in the GRPT is replaced by a spatially adaptive window. Second, a background regularization term and a modulation regularization term are incorporated in the cost function. With these two improvements, the proposed iGRPT can successfully demodulate sparse fringes and thus improves the demodulation capability of the GRPT. Simulation and experimental results are presented to verify the performance of the iGRPT. We mention that, although the iGRPT greatly increases the demodulation capability, it still has the problem when the fringe becomes even sparser, which is a natural consequence of the ill-posed problem of fringe demodulation [1]. If the background is easy to remove, we can apply the iGRPT to a fringe pattern without the background in order to process very sparse fringe patterns. The iGRPT is currently time-consuming and its acceleration is under consideration. A possible way to accelerate the demodulation speed of the iGRPT is to use parallel computing. One can divide a fringe pattern into a number of blocks, and start the same number of iGRPT processes by a multicore computer [23], each process for a block, to demodulate the fringe pattern simultaneously. The phase results obtained by this method need to be corrected because the signs of the phase results among the blocks could be ambiguous. The phase ambiguity problem can be solved by performing a sign propagation process among the blocks [24].

Acknowledgements

We thank Dr Qifeng Yu and Dr Yu Fu for providing us Fig. 3(a) and Fig. 5(a), respectively. This work was partially supported by the Singapore MOE Academic Research Fund Tier 1 (RG11/10) and the National Natural Science Foundation of China (NSFC) (No.11332005).

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18. L. Kai and Q. Kemao, “A generalized regularized phase tracker for demodulation of a single fringe pattern,” Opt. Express 20(11), 12579–12592 (2012). [CrossRef]   [PubMed]  

19. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72(1), 156–160 (1982). [CrossRef]  

20. B. V. Dorrío and J. L. Fernández, “Phase-evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Technol. 10(3), R33–R55 (1999). [CrossRef]  

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

22. J. A. Quiroga and M. Servin, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun. 224(4-6), 221–227 (2003). [CrossRef]  

23. W. J. Gao, Q. Kemao, H. X. Wang, F. Lin, and S. H. Soon, “Parallel computing for fringe pattern processing: A multicore CPU approach in MATLAB® environment,” Opt. Lasers Eng. 47(11), 1286–1292 (2009). [CrossRef]  

24. L. Kai and Q. Kemao, “Fast frequency-guided sequential demodulation of a single fringe pattern,” Opt. Lett. 35(22), 3718–3720 (2010). [CrossRef]   [PubMed]  

References

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  • |

  1. D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, 1998).
  2. J. L. Marroquin, M. Servin, and R. Rodriguez-Vera, “Adaptive quadrature filters and the recovery of phase from fringe pattern images,” J. Opt. Soc. Am. A 14(8), 1742–1753 (1997).
    [Crossref]
  3. J. L. Marroquin, R. Rodriguez-Vera, and M. Servin, “Local phase from local orientation by solution of a sequence of linear systems,” J. Opt. Soc. Am. A 15(6), 1536–1544 (1998).
    [Crossref]
  4. K. G. Larkin, D. J. Bone, and M. A. Oldfield, “Natural demodulation of two-dimensional fringe patterns. I. general background of the spiral phase quadrature transform,” J. Opt. Soc. Am. A 18(8), 1862–1870 (2001).
    [Crossref] [PubMed]
  5. M. Servin, J. A. Quiroga, and J. L. Marroquin, “General n-dimensional quadrature transform and its application to interferogram demodulation,” J. Opt. Soc. Am. A 20(5), 925–934 (2003).
    [Crossref] [PubMed]
  6. E. Robin, V. Valle, and F. Brémand, “Phase demodulation method from a single fringe pattern based on correlation with a polynomial form,” Appl. Opt. 44(34), 7261–7269 (2005).
    [Crossref] [PubMed]
  7. M. Rivera, “Robust phase demodulation of interferograms with open or closed fringes,” J. Opt. Soc. Am. A 22(6), 1170–1175 (2005).
    [Crossref] [PubMed]
  8. J. C. Estrada, M. Servin, and J. L. Marroquín, “Local adaptable quadrature filters to demodulate single fringe patterns with closed fringes,” Opt. Express 15(5), 2288–2298 (2007).
    [Crossref] [PubMed]
  9. Q. Kemao and S. Hock Soon, “Sequential demodulation of a single fringe pattern guided by local frequencies,” Opt. Lett. 32(2), 127–129 (2007).
    [Crossref] [PubMed]
  10. O. Dalmau-Cedeño, M. Rivera, and R. Legarda-Saenz, “Fast phase recovery from a single close-fringe pattern,” J. Opt. Soc. Am. A 25(6), 1361–1370 (2008).
    [Crossref]
  11. M. Servin, J. L. Marroquin, and F. J. Cuevas, “Demodulation of a single interferogram by use of a two-dimensional regularized phase-tracking technique,” Appl. Opt. 36(19), 4540–4548 (1997).
    [Crossref] [PubMed]
  12. M. Servin, J. L. Marroquin, and F. J. Cuevas, “Fringe-follower regularized phase tracker for demodulation of closed-fringe interferograms,” J. Opt. Soc. Am. A 18(3), 689–695 (2001).
    [Crossref]
  13. R. Legarda-Sáenz, W. Osten, and W. Jüptner, “Improvement of the Regularized Phase Tracking Technique for the Processing of Nonnormalized Fringe Patterns,” Appl. Opt. 41(26), 5519–5526 (2002).
    [Crossref] [PubMed]
  14. R. Legarda-Saenz and M. Rivera, “Fast half-quadratic regularized phase tracking for nonnormalized fringe patterns,” J. Opt. Soc. Am. A 23(11), 2724–2731 (2006).
    [Crossref] [PubMed]
  15. H. Wang and Q. Kemao, “Frequency guided methods for demodulation of a single fringe pattern,” Opt. Express 17(17), 15118–15127 (2009).
    [Crossref] [PubMed]
  16. C. Tian, Y. Y. Yang, D. Liu, Y. J. Luo, and Y. M. Zhuo, “Demodulation of a single complex fringe interferogram with a path-independent regularized phase-tracking technique,” Appl. Opt. 49(2), 170–179 (2010).
    [Crossref] [PubMed]
  17. H. Wang, K. Li, and Q. Kemao, “Frequency guided methods for demodulation of a single fringe pattern with quadratic phase matching,” Opt. Lasers Eng. 49(4), 564–569 (2011).
    [Crossref]
  18. L. Kai and Q. Kemao, “A generalized regularized phase tracker for demodulation of a single fringe pattern,” Opt. Express 20(11), 12579–12592 (2012).
    [Crossref] [PubMed]
  19. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72(1), 156–160 (1982).
    [Crossref]
  20. B. V. Dorrío and J. L. Fernández, “Phase-evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Technol. 10(3), R33–R55 (1999).
    [Crossref]
  21. Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementations,” Opt. Lasers Eng. 45(2), 304–317 (2007).
    [Crossref]
  22. J. A. Quiroga and M. Servin, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun. 224(4-6), 221–227 (2003).
    [Crossref]
  23. W. J. Gao, Q. Kemao, H. X. Wang, F. Lin, and S. H. Soon, “Parallel computing for fringe pattern processing: A multicore CPU approach in MATLAB® environment,” Opt. Lasers Eng. 47(11), 1286–1292 (2009).
    [Crossref]
  24. L. Kai and Q. Kemao, “Fast frequency-guided sequential demodulation of a single fringe pattern,” Opt. Lett. 35(22), 3718–3720 (2010).
    [Crossref] [PubMed]

2012 (1)

2011 (1)

H. Wang, K. Li, and Q. Kemao, “Frequency guided methods for demodulation of a single fringe pattern with quadratic phase matching,” Opt. Lasers Eng. 49(4), 564–569 (2011).
[Crossref]

2010 (2)

2009 (2)

W. J. Gao, Q. Kemao, H. X. Wang, F. Lin, and S. H. Soon, “Parallel computing for fringe pattern processing: A multicore CPU approach in MATLAB® environment,” Opt. Lasers Eng. 47(11), 1286–1292 (2009).
[Crossref]

H. Wang and Q. Kemao, “Frequency guided methods for demodulation of a single fringe pattern,” Opt. Express 17(17), 15118–15127 (2009).
[Crossref] [PubMed]

2008 (1)

2007 (3)

2006 (1)

2005 (2)

2003 (2)

2002 (1)

2001 (2)

1999 (1)

B. V. Dorrío and J. L. Fernández, “Phase-evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Technol. 10(3), R33–R55 (1999).
[Crossref]

1998 (1)

1997 (2)

1982 (1)

Bone, D. J.

Brémand, F.

Cuevas, F. J.

Dalmau-Cedeño, O.

Dorrío, B. V.

B. V. Dorrío and J. L. Fernández, “Phase-evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Technol. 10(3), R33–R55 (1999).
[Crossref]

Estrada, J. C.

Fernández, J. L.

B. V. Dorrío and J. L. Fernández, “Phase-evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Technol. 10(3), R33–R55 (1999).
[Crossref]

Gao, W. J.

W. J. Gao, Q. Kemao, H. X. Wang, F. Lin, and S. H. Soon, “Parallel computing for fringe pattern processing: A multicore CPU approach in MATLAB® environment,” Opt. Lasers Eng. 47(11), 1286–1292 (2009).
[Crossref]

Hock Soon, S.

Ina, H.

Jüptner, W.

Kai, L.

Kemao, Q.

L. Kai and Q. Kemao, “A generalized regularized phase tracker for demodulation of a single fringe pattern,” Opt. Express 20(11), 12579–12592 (2012).
[Crossref] [PubMed]

H. Wang, K. Li, and Q. Kemao, “Frequency guided methods for demodulation of a single fringe pattern with quadratic phase matching,” Opt. Lasers Eng. 49(4), 564–569 (2011).
[Crossref]

L. Kai and Q. Kemao, “Fast frequency-guided sequential demodulation of a single fringe pattern,” Opt. Lett. 35(22), 3718–3720 (2010).
[Crossref] [PubMed]

H. Wang and Q. Kemao, “Frequency guided methods for demodulation of a single fringe pattern,” Opt. Express 17(17), 15118–15127 (2009).
[Crossref] [PubMed]

W. J. Gao, Q. Kemao, H. X. Wang, F. Lin, and S. H. Soon, “Parallel computing for fringe pattern processing: A multicore CPU approach in MATLAB® environment,” Opt. Lasers Eng. 47(11), 1286–1292 (2009).
[Crossref]

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

Q. Kemao and S. Hock Soon, “Sequential demodulation of a single fringe pattern guided by local frequencies,” Opt. Lett. 32(2), 127–129 (2007).
[Crossref] [PubMed]

Kobayashi, S.

Larkin, K. G.

Legarda-Saenz, R.

Legarda-Sáenz, R.

Li, K.

H. Wang, K. Li, and Q. Kemao, “Frequency guided methods for demodulation of a single fringe pattern with quadratic phase matching,” Opt. Lasers Eng. 49(4), 564–569 (2011).
[Crossref]

Lin, F.

W. J. Gao, Q. Kemao, H. X. Wang, F. Lin, and S. H. Soon, “Parallel computing for fringe pattern processing: A multicore CPU approach in MATLAB® environment,” Opt. Lasers Eng. 47(11), 1286–1292 (2009).
[Crossref]

Liu, D.

Luo, Y. J.

Marroquin, J. L.

Marroquín, J. L.

Oldfield, M. A.

Osten, W.

Quiroga, J. A.

Rivera, M.

Robin, E.

Rodriguez-Vera, R.

Servin, M.

Soon, S. H.

W. J. Gao, Q. Kemao, H. X. Wang, F. Lin, and S. H. Soon, “Parallel computing for fringe pattern processing: A multicore CPU approach in MATLAB® environment,” Opt. Lasers Eng. 47(11), 1286–1292 (2009).
[Crossref]

Takeda, M.

Tian, C.

Valle, V.

Wang, H.

H. Wang, K. Li, and Q. Kemao, “Frequency guided methods for demodulation of a single fringe pattern with quadratic phase matching,” Opt. Lasers Eng. 49(4), 564–569 (2011).
[Crossref]

H. Wang and Q. Kemao, “Frequency guided methods for demodulation of a single fringe pattern,” Opt. Express 17(17), 15118–15127 (2009).
[Crossref] [PubMed]

Wang, H. X.

W. J. Gao, Q. Kemao, H. X. Wang, F. Lin, and S. H. Soon, “Parallel computing for fringe pattern processing: A multicore CPU approach in MATLAB® environment,” Opt. Lasers Eng. 47(11), 1286–1292 (2009).
[Crossref]

Yang, Y. Y.

Zhuo, Y. M.

Appl. Opt. (4)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (8)

R. Legarda-Saenz and M. Rivera, “Fast half-quadratic regularized phase tracking for nonnormalized fringe patterns,” J. Opt. Soc. Am. A 23(11), 2724–2731 (2006).
[Crossref] [PubMed]

M. Servin, J. L. Marroquin, and F. J. Cuevas, “Fringe-follower regularized phase tracker for demodulation of closed-fringe interferograms,” J. Opt. Soc. Am. A 18(3), 689–695 (2001).
[Crossref]

O. Dalmau-Cedeño, M. Rivera, and R. Legarda-Saenz, “Fast phase recovery from a single close-fringe pattern,” J. Opt. Soc. Am. A 25(6), 1361–1370 (2008).
[Crossref]

M. Rivera, “Robust phase demodulation of interferograms with open or closed fringes,” J. Opt. Soc. Am. A 22(6), 1170–1175 (2005).
[Crossref] [PubMed]

J. L. Marroquin, M. Servin, and R. Rodriguez-Vera, “Adaptive quadrature filters and the recovery of phase from fringe pattern images,” J. Opt. Soc. Am. A 14(8), 1742–1753 (1997).
[Crossref]

J. L. Marroquin, R. Rodriguez-Vera, and M. Servin, “Local phase from local orientation by solution of a sequence of linear systems,” J. Opt. Soc. Am. A 15(6), 1536–1544 (1998).
[Crossref]

K. G. Larkin, D. J. Bone, and M. A. Oldfield, “Natural demodulation of two-dimensional fringe patterns. I. general background of the spiral phase quadrature transform,” J. Opt. Soc. Am. A 18(8), 1862–1870 (2001).
[Crossref] [PubMed]

M. Servin, J. A. Quiroga, and J. L. Marroquin, “General n-dimensional quadrature transform and its application to interferogram demodulation,” J. Opt. Soc. Am. A 20(5), 925–934 (2003).
[Crossref] [PubMed]

Meas. Sci. Technol. (1)

B. V. Dorrío and J. L. Fernández, “Phase-evaluation methods in whole-field optical measurement techniques,” Meas. Sci. Technol. 10(3), R33–R55 (1999).
[Crossref]

Opt. Commun. (1)

J. A. Quiroga and M. Servin, “Isotropic n-dimensional fringe pattern normalization,” Opt. Commun. 224(4-6), 221–227 (2003).
[Crossref]

Opt. Express (3)

Opt. Lasers Eng. (3)

H. Wang, K. Li, and Q. Kemao, “Frequency guided methods for demodulation of a single fringe pattern with quadratic phase matching,” Opt. Lasers Eng. 49(4), 564–569 (2011).
[Crossref]

W. J. Gao, Q. Kemao, H. X. Wang, F. Lin, and S. H. Soon, “Parallel computing for fringe pattern processing: A multicore CPU approach in MATLAB® environment,” Opt. Lasers Eng. 47(11), 1286–1292 (2009).
[Crossref]

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

Opt. Lett. (2)

Other (1)

D. Malacara, M. Servín, and Z. Malacara, Interferogram Analysis for Optical Testing (Marcel Dekker, 1998).

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Figures (10)

Fig. 1
Fig. 1 Demodulation of a simulated fringe pattern by the GRPT. (a) simulated fringe pattern, (b) phase result obtained by the GRPT with σ=7.5 and λ=20 , (c) NI quality map of (b).
Fig. 2
Fig. 2 Demodulation of a simulated fringe pattern by the iGRPT. From left to right, demodulatd phase, NI quality map and the window size are shown. The first row: σ min =7.5 , σ max =15 , λ a =0 , λ b =0 , λ φ =20 and the window size is determined by Eq. (10); the second row: same parameters as in the first row, except that λ a =10 and λ b =10 are set; the third row: same parameters as in the second row except that the window size is determined by Eq. (11).
Fig. 3
Fig. 3 Demodulation of an experimental fringe pattern by the GRPT. (a) experimental fringe pattern, (b) phase result obtained by the GRPT with σ=7.5 and λ=20 , (c) NI quality map of (b).
Fig. 4
Fig. 4 Demodulation of Fig. 3(a) by the iGRPT. From left to right, demodulatd phase, NI quality map and the window size are shown. The first row: σ min =7.5 , σ max =15 , λ a =0 , λ b =0 , λ φ =20 and the window size is determined by Eq. (10); the second row: same parameters as in the first row, except that λ a =10 and λ b =10 are set; the third row: same parameters as in the second row except that the window size is determined by Eq. (11).
Fig. 5
Fig. 5 Demodulation of an experimental fringe pattern by the GRPT. (a) experimental fringe pattern, (b) phase result obtained by the GRPT with σ=7.5 and λ=20 , (c) NI quality map of (b).
Fig. 6
Fig. 6 Demodulation of Fig. 5(a) by the iGRPT with σ min =3 , σ max =12 , λ a =10 , λ b =10 , λ φ =20 and the window size is determined by Eq. (10). (a) phase result obtained by the iGRPT, (b) NI quality map of (a), (c) window size.
Fig. 7
Fig. 7 Demodulation of fringe patterns by the iGRPT without background estimation. From left to right, demodulatd phase, NI quality map and the window size are shown. The first row: demodulation results obtained from the simulated fringe pattern shown in Fig. 1(a); the second row: demodulation results obtained from the experimental fringe pattern shown in Fig. 3(a); the third row: demodulation results obtained from the experimental fringe pattern shown in Fig. 5(a).
Fig. 8
Fig. 8 Demodulation of the simulated fringe pattern by the RPT [12] with different window sizes. (a) the same simulated fringe pattern as shown in Fig. 1(a), (b)-(f) the phase results obtained by the RPT with window sizes 5 × 5, 9 × 9, 13 × 13, 17 × 17 and 21 × 21, respectively.
Fig. 9
Fig. 9 Demodulation of the experimental fringe pattern shwon in Fig. 3(a) by the RPT [12] with different window sizes. (a) the normalized fringe pattern of Fig. 3(a), (b)-(f) the phase results obtained by the RPT from the normalized fringe pattern (a) with window sizes 5 × 5, 9 × 9, 13 × 13, 17 × 17 and 21 × 21, respectively.
Fig. 10
Fig. 10 Demodulation of the experimental fringe pattern shwon in Fig. 5(a) by the RPT [12] with different window sizes. (a) the normalized fringe pattern of Fig. 5(a), (b)-(f) the phase results obtained by the RPT from the normalized fringe pattern (a) with window sizes 5 × 5, 9 × 9, 13 × 13, 17 × 17 and 21 × 21, respectively.

Equations (15)

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f( x,y )=a( x,y )+b( x,y )cos[ φ( x,y ) ]+n( x,y ),
U( x,y )= ( ε,η ) N x,y ( w( x,y;ε,η ){ [ f( ε,η ) f e ( x,y;ε,η ) ] 2 +λ [ φ 0 ( ε,η ) φ e ( x,y;ε,η ) ] 2 m( ε,η ) } ) ,
w( x,y;ε,η )=exp{ [ ( xε ) 2 + ( yη ) 2 ] / ( 2 σ 2 ) },
f e ( x,y;ε,η )= a e ( x,y;ε,η )+ b e ( x,y;ε,η )cos[ φ e ( x,y;ε,η ) ],
a e ( x,y;ε,η )= a 0 ( x,y )+ a x ( x,y )( εx )+ a y ( x,y )( ηy ),
b e ( x,y;ε,η )= b 0 ( x,y )+ b x ( x,y )( εx )+ b y ( x,y )( ηy ),
φ e ( x,y;ε,η )= φ 0 ( x,y )+ ω x ( x,y )( εx )+ ω y ( x,y )( ηy )+ 1 2 c xx ( x,y ) ( εx ) 2 + 1 2 c yy ( x,y ) ( ηy ) 2 + c xy ( x,y )( εx )( ηy ),
ω TLF = ω x 2 + ω y 2 ,
σ ˜ =2/ ω TLF ,
σ={ σ min if σ ˜ σ min σ ˜ if σ min < σ ˜ < σ max σ max if σ ˜ σ max ,
σ={ σ min if σ ˜ ( σ min + σ max )/2 σ max if σ ˜ >( σ min + σ max )/2 .
U( x,y )= ( ε,η ) N x,y ( w( x,y;ε,η ){ [ f( ε,η ) f e ( x,y;ε,η ) ] 2 + ( λ a R a + λ b R b + λ φ R φ )m( ε,η ) } ),
R a = [ a 0 ( ε,η ) a e ( x,y;ε,η ) ] 2 ,
R b = [ b 0 ( ε,η ) b e ( x,y;ε,η ) ] 2 ,
R φ = [ φ 0 ( ε,η ) φ e ( x,y;ε,η ) ] 2 ,

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