## 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 [2–17] 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 [11–17] 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,

With such a general local fringe model and matching strategy, the GRPT has many merits: (1) as both $a\left(x,y\right)$ and $b\left(x,y\right)$ 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,

the GRPT is found to be successful where the fringe is dense or ${\omega}_{\text{TLF}}$ is high, and becomes problematic where the fringe is sparse, or ${\omega}_{\text{TLF}}$ 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 $\sigma =7.5$ and $\lambda =20$ 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.## 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, ${a}_{e}\left(x,y;\epsilon ,\eta \right)$ and ${b}_{e}\left(x,y;\epsilon ,\eta \right)\mathrm{cos}\left[{\phi}_{e}\left(x,y;\epsilon ,\eta \right)\right]$, should be distinguishable from each other so that their respective parameters can converge to their true values. Since ${a}_{e}\left(x,y;\epsilon ,\eta \right)$ is assumed to be a plane, ${b}_{e}\left(x,y;\epsilon ,\eta \right)\mathrm{cos}\left[{\phi}_{e}\left(x,y;\epsilon ,\eta \right)\right]$ should vary sufficiently fast. In other words, the TLF should be high enough. In the Fourier domain, the spectra of ${a}_{e}\left(x,y;\epsilon ,\eta \right)$, $0.5{b}_{e}\left(x,y;\epsilon ,\eta \right)exp\left[j{\phi}_{e}\left(x,y;\epsilon ,\eta \right)\right]$ and $0.5{b}_{e}\left(x,y;\epsilon ,\eta \right)exp\left[-j{\phi}_{e}\left(x,y;\epsilon ,\eta \right)\right]$ (*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 $\sigma $, there is an energy leakage of about $2/\sigma $ for all three terms [21]. To separate these three terms in the Fourier domain at least partially, roughly ${\omega}_{\text{TLF}}=2/\sigma $ 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, $\left[{\phi}_{0}\left(x,y\right),{\omega}_{x}\left(x,y\right),{\omega}_{y}\left(x,y\right)\right]$ and $\left[-{\phi}_{0}\left(x,y\right),-{\omega}_{x}\left(x,y\right),-{\omega}_{y}\left(x,y\right)\right]$, 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 ${\omega}_{\text{TLF}}=2/\sigma $, which is inversely proportional to the window size $\sigma $. Thus we can increase the window size to reduce this requirement. In other words, we can set the window size adaptively as,
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,

$$\sigma =\{\begin{array}{ccc}{\sigma}_{\mathrm{min}}& if& \tilde{\sigma}\le {\sigma}_{\mathrm{min}}\\ \tilde{\sigma}& if& {\sigma}_{\mathrm{min}}<\tilde{\sigma}<{\sigma}_{\mathrm{max}}\\ {\sigma}_{\mathrm{max}}& if& \tilde{\sigma}\ge {\sigma}_{\mathrm{max}}\end{array},$$where ${\sigma}_{\mathrm{min}}$ and ${\sigma}_{\mathrm{max}}$ are the smallest and largest window sizes. Consider that the problem only exists for sparse fringes, we can set ${\sigma}_{\mathrm{min}}$ as in the GRPT, for example, ${\sigma}_{\mathrm{min}}=7.5$, 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 $10\le {\sigma}_{\mathrm{max}}\le 20$. Incorporating these considerations, a simpler window setting also can be used,

$$\sigma =\{\begin{array}{ccc}{\sigma}_{\mathrm{min}}& if& \tilde{\sigma}\le \left({\sigma}_{\mathrm{min}}+{\sigma}_{\mathrm{max}}\right)/2\\ {\sigma}_{\mathrm{max}}& if& \tilde{\sigma}>\left({\sigma}_{\mathrm{min}}+{\sigma}_{\mathrm{max}}\right)/2\end{array}.$$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$$U\left(x,y\right)={\displaystyle \sum _{\left(\epsilon ,\eta \right)\in {N}_{x,y}}(w\left(x,y;\epsilon ,\eta \right)\{{\left[f\left(\epsilon ,\eta \right)-{f}_{e}\left(x,y;\epsilon ,\eta \right)\right]}^{2}}+\left({\lambda}_{a}{R}_{a}+{\lambda}_{b}{R}_{b}+{\lambda}_{\phi}{R}_{\phi}\right)m\left(\epsilon ,\eta \right)\}),$$
with

$${R}_{a}={\left[{a}_{0}\left(\epsilon ,\eta \right)-{a}_{e}\left(x,y;\epsilon ,\eta \right)\right]}^{2},$$$${R}_{b}={\left[{b}_{0}\left(\epsilon ,\eta \right)-{b}_{e}\left(x,y;\epsilon ,\eta \right)\right]}^{2},$$$${R}_{\phi}={\left[{\phi}_{0}\left(\epsilon ,\eta \right)-{\phi}_{e}\left(x,y;\epsilon ,\eta \right)\right]}^{2},$$where${\lambda}_{a}$and ${\lambda}_{b}$are regularizing parameters of the background regularization term and the modulation regularization term, respectively; ${\lambda}_{\phi}$ is the same as $\lambda $ 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 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 $\left({x}_{s},{y}_{s}\right)$ with the lowest NI in the demodulation register; calculate the ${\omega}_{\text{TLF}}$ of this selected pixel by Eq. (8) and set the window size by Eq. (10) or Eq. (11); demodulate the unprocessed pixels adjacent to $\left({x}_{s},{y}_{s}\right)$ 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 ${\sigma}_{\mathrm{min}}=7.5$,${\sigma}_{\mathrm{max}}=15$, ${\lambda}_{a}=0$, ${\lambda}_{b}=0$, ${\lambda}_{\phi}=20$ 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 ${\lambda}_{a}=10$ and ${\lambda}_{b}=10$ 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 ${R}_{a}$ and ${R}_{b}$. We can also compare the NI numbers shown in Fig. 2(e) with those in Fig. 2(b) and find that ${R}_{a}$ and ${R}_{b}$ 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.

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.

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 ${\sigma}_{\mathrm{min}}=3$,${\sigma}_{\mathrm{max}}=12$, ${\lambda}_{a}=10$, ${\lambda}_{b}=10$, ${\lambda}_{\phi}=20$ 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 ${\sigma}_{\mathrm{min}}$ has been set in order to deal with this fringe pattern containing complex fringe details. The calculation time of Fig. 6(a) is 28min.

#### 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.

#### 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 $\lambda =0.15$ 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].

## 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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