Phase demodulation from a single fringe pattern is a challenging task but of interest. A frequency-guided regularized phase tracker and a frequency-guided sequential demodulation method with Levenberg-Marquardt optimization are proposed to demodulate a single fringe pattern. Demodulation path guided by the local frequency from the highest to the lowest is applied in both methods. Since critical points have low local frequency values, they are processed last so that the spurious sign problem caused by these points is avoided. These two methods can be considered as alternatives to the effective fringe follower regularized phase tracker. Demodulation results from one computer-simulated and two experimental fringe patterns using the proposed methods will be demonstrated.
©2009 Optical Society of America
Various optical interferometric techniques produce fringe patterns. The useful information is often encoded in the phase of fringe patterns. A fringe pattern is typically in the form of1]. The three unknowns, a(x, y), b(x, y) and ϕ(x, y), make the retrieval of phase information difficult. Phase shifting technique  is often used for phase measurement, which requires several phase-shifted fringe patterns. Carrier method with Fourier transform is also frequently used  by introducing a carrier frequency into a fringe pattern. However, there are always cases where multiple fringe patterns or carrier frequencies are hard to obtain, and consequently phase demodulation of a single fringe pattern is important. Usually a high-pass filter is used to remove the background intensity [2, 3], followed by clipping [2, 3] or Hilbert transform  to normalize the fringe pattern so that the fringe amplitude is one. Before these steps, noise needs to be removed to increase the reliability of the normalization result, for which windowed Fourier filtering  and coherence enhancing diffusion [6, 7], among others, can be applied. After all these preprocessing, the resulting fringe pattern can be described as
The inherent sigh ambiguity that results from the cosinusoidal function brings the difficulty of retrieving phase from Eq. (2). To demodulate a single fringe pattern, various methods have been proposed [2, 3, 8–15], Servin et al. proposed a popular demodulation method for closed fringe patterns, called the regularized phase tracker (RPT) . It is later improved by introducing the fringe follower, a scanning strategy, to avoid critical points . Meanwhile, Marroquin et al. proposed an adaptive quadrature filter based on Bayesian estimation theory and complex-valued Markov random-field prior models [8, 9]. The minimization of the energy function is complex and computational expensive. A local adaptive quadrature filter  is then proposed by Estrada et al. for closed fringe patterns, which has interesting similarity with windowed Fourier ridges algorithm . Rivera proposed a phase-propagation algorithm that is more robust to noise than RPT . In the other way, Larkin et al. proposed a two dimensional Hilbert transform for fringe patterns which turns the demodulation problem into orientation estimation problem . It was later generalized to arbitrary dimensions by Servin et al. . A frequency-guided sequential demodulation (FSD) is proposed by Kemao and Soon , in which an expensive exhaustive search is used. Recently, Dalmau-Cedeno et al. proposed a demodulation method by dividing the fringe patterns into tiles which are analyzed by an open-fringe-analysis algorithm. The phase sign is propagated and determined tile by tile so that the algorithm is more robust and faster .
Among all these algorithms, the fringe follower regularized phase tracker (FFRPT)  and the frequency-guided sequential demodulation (FSD)  can achieve high effectiveness with simple implementation. Both methods work on a normalized fringe pattern. In both methods, the phase is assumed to be locally linear and consequently a planar phase is searched to match the local fringe data by minimizing similar energy functions. Further, both methods adopt scanning strategies to avoid critical points. Due to the effectiveness, simplicity and similarity of these two methods, they are of interest and considered in this paper.
Though FFRPT is effective, there are cases that the fringe follower scanning strategy does not work. We will give an example in Sec. 2.1. We thus propose to adopt the frequency guided scanning strategy into RPT (we abbreviate it as FGRPT), as an alternative to FFRPT. As for FSD, in its original version , it suffers from low computation speed as an exhaustive search algorithm is used. We propose to use Levenberg-Marquardt algorithm  for optimization so that the computation speed is greatly improved (we abbreviate it as FSD-LM). As a fact, FSD-LM is faster than FFRPT and FGRPT, which makes it useful in practice.
The left of the paper is organized as follows. FFRPT and FSD will be introduced in Sec. 2.1 and Sec. 2.2, respectively. Two proposed methods, FGRPT and FSD-LM, will then be presented in Sec. 3.1 and 3.2, respectively. Results and discussions will be given in Sec. 4. A conclusion will be drawn in Sec. 5.
2. Fringe-follower regularized phase tracker and frequency-guided sequential demodulation
2.1 Fringe-follower regularized phase tracker
The basic idea of regularized phase tracker (RPT) is that the phase is assumed to be locally linear, which is usually true in a fringe pattern. For a concerned pixel and its neighbors, an energy function is defined as2, 3], while other optimization methods can also be used. Initial estimation of ϕ0(x, y), ωx(x, y) and ωy(x, y) are needed for optimization.
In RPT approach, critical points might introduce spurious phase signs  and sign errors will propagate. To solve this problem, a fringe follower scanning strategy is used , by which critical points are processed later than their surrounding pixels. It uses the fringe intensity to guide the RPT demodulation process. Low intensity pixels are defined as difficult points. Therefore, high intensity pixels of the interferogram fringe patterns are scanned first to avoid the difficult points where fringe follower regularized phase tracker (FFRPT) may go wrong. All pixels in the fringe pattern are classified by binarizing them into two categories, bright and dark, based on their intensity values. An empty bright register and an empty dark register are initialized to store pixels in bright and dark categories, respectively. After a pixel is demodulated, its eight neighboring pixels are pushed into these two registers according to their binarized values. The bright register is always processed first in a “first in first out” order. Only when the bright register is empty, the pixels in the dark register will be processed also in a “first in first out” order. Alternatively, high intensity pixels can be defined as difficult points and thus low intensity pixels are scanned first.
FFRPT is a robust and automatic method to demodulate a single fringe pattern. The algorithm can directly estimate smooth phase. The scanning strategy is very effective and efficient. However, one potential problem is that, critical points may not be classified into a single intensity category, i.e., critical points with high intensity values are pushed to the bright register while the other critical points are pushed to the dark registers. Thus some critical points in the bright register might be demodulated earlier than their surrounding pixels. An example is given below to show this problem.
A computer-simulated noisy fringe pattern is shown in Fig. 1(a) and its true phase is shown in Fig. 1(b). Both the background and the amplitude of the fringe are assigned to 1 and speckle noise is simulated. Windowed Fourier filtering  is used to suppress the noise, while clipping [2, 3] and Hilbert transform  are used to normalize the fringe pattern. The normalization result is shown in Fig. 1(c). The normalized fringe pattern is then binarized, as shown in Fig. 1(d). It can be seen that the critical points of the fringe pattern cannot be classified into a single bright or dark category. The phase demodulated by FFRPT is shown in Fig. 1(e), where low intensity pixels are treated as difficult points. A demodulation error region can be noticed, as indicated by the red ellipse in Fig. 1(e). Similarly, Fig. 1(f) is the phase obtained by FFRPT if high intensity pixels are treated as difficult points. Two error regions are found as indicated by the red ellipses in Fig. 1(f). This potential problem inspires us to adopt the frequency guided scanning strategy which will be introduced in subsection 3.1.
2.2 Frequency-guided sequential demodulation
Frequency-guided sequencial demodulation (FSD) is another useful method to demodulate a single fringe pattern. The first step of FSD is to directly retrieve the phase from Eq. (2),
The second step is to estimate the local frequency information which will be used to solve the phase ambiguity problem. The frequency ωx(x, y) and ωy(x, y) are calculated by minimizing the following cost function:Eq. (3). Since ϕ0(x, y) has already been calculated by Eq. (5), regularization term in Eq. (3) is not used in this method; only ωx(x, y) and ωy(x, y) need to be estimated. In the original FSD, ωx(x, y) and ωy(x, y) are obtained by exhaustive search within intervals ωxl(x, y) ≤ ωx(x, y) ≤ ωxh(x, y) and ωyl(x, y) ≤ ωy(x, y) ≤ ωyh(x, y), respectively. Candidates of ωx(x, y) are uniformly sampled from ωxl (lower bound) to ωxh (upper bound) with a sampling interval ωxi. Candidates of ωy(x, y) are generated similarly. Once ωx(x, y) and ωy(x, y) have been obtained, the local frequency density is calculated as
The third step is to remove the phase ambiguity using the local frequency information. Since critical points have low local frequencies approaching zero, naturally the local frequency information instead of the intensity of a fringe pattern can be used to differentiate the critical points from other pixels in the fringe pattern and guide the demodulation process. With this scanning strategy and by applying a constraint that local frequencies ωx(x, y) and ωy(x, y) are spatially continuous, the sign of the phase can be determined. Details can be found in . The phase unwrapping can be carried out simultaneously.
Since only two parameters, ωx(x, y) and ωy(x, y), need to be optimized, it is feasible to use exhaustive search algorithm which guarantees a global minimum. However it increases the computation cost. A faster algorithm is nevertheless expected if possible, which inspires us to adopt Levenberg-Marquardt optimization into FSD and is introduced in subsection 3.2.
3. Proposed methods
3.1 Frequency-guided regularized phase tracker
As stated on Section 2.1, FFRPT has a potential problem about its scanning strategy, especially when the critical points in a fringe pattern cannot be classified into a single intensity category. Using local frequency ω(x, y) to guide the demodulation path is naturally a suitable alternative to the fringe follower. A frequency-guided regularized phase tracker (FGRPT) is proposed. As critical points have low local frequencies, high frequency points are processed first so that the critical points will be guaranteed to be processed last and therefore reduce the possibility of error propagating from critical points.
The FGRPT algorithm is outlined as follows.
- 1) Randomly choose a pixel as a seed pixel;
- 2) As no initial values are provided for the seed pixel, FSD is used to demodulate the seed pixel: obtain the phase from Eq. (5), then obtain ωx(x, y) and ωy(x, y) from Eq. (6) by exhaustive search. Compute ω(x, y) as in Eq. (7). If it is smaller than a pre-defined threshold, go back to step 1; otherwise, push the seed pixel into a demodulation register. Only one register is needed for FGRPT.
- 3) Select the pixel with the highest local frequency ω(x, y) in the demodulation register; demodulate its four adjacent pixels by estimating ϕ0(x, y), ωx(x, y) and ωy(x, y) that minimize energy function in Eq. (3). Levenberg-Marquardt instead of gradient descent is used in this paper. Details on the Levenberg-Marquardt optimization method and its initial value setting will be discussed later.
- 4) Push the processed pixels into the register and sort the pixels according to their local frequencies ω(x, y).
- 5) Repeat step 3 and 4 until all pixels are processed.
The frequency guided scanning strategy needs additional sorting of the frequencies. A quality-guided path following method  is used, taking the frequency values as its quality map. A quick sorting algorithm  is adopted to increase the processing speed.
In order to minimize Eq. (3), Levenberg-Marquardt optimization method is used in this paper . It is a damped Gauss-Newton method, interpolating between Gauss-Newton method and gradient descent method. The gradient descent method has good performance in the initial stage but takes long time to converge. Meanwhile, the Gauss-Newton method is very efficient in the later part of convergence. The Levenberg-Marquardt optimization has advantages of both Gauss-Newton method and gradient descent method. It has higher convergence speed and is robust that in some cases it can find a solution even the starting point is far away. Levenberg-Marquardt is adopted for both FFRPT and FGRPT for fair comparison, though gradient descent method is originally adopted in [2, 3].
For optimization methods, the initial values affect the optimization speed and final results. Well-chosen initial values will reduce the possibility of falling into local minimum and computation time. In this paper, the initial value for each pixel is chosen as the weighted sum of its eight neighbors, which is defined asEq. (3) indicating whether this pixel has been processed; U(ε, η) is the energy of the pixel defined by Eq. (3). Low energy value of a pixel embodies accurate and reliable estimation of demodulation parameters. Thus low energy can be treated as high quality of demodulation and thus a high weight is assigned to this pixel. The same idea is applied to ωx(x, y) and ωy(x, y). The denominator in Eq. (8) is used for normalization purpose.
After the fringe pattern is demodulated, a further refinement can be processed when necessary. Since the energy value can be used as a quality measure, the refinement needs only to be carried out in areas with low quality, i.e. high energy. Since all the neighboring information is known at the refinement stage, i.e., m(x, y) = 1 for all pixels, a better estimation of initial condition can be made, which may lead to better optimization results. The partial refinement is applicable to both FFRPT and FGRPT. However, in all our examples presented in Section 4, the refinement is not applied.
3.2 Frequency-guided sequential demodulation - Levenberg Marquardt
By directly calculating the phase using Eq. (5), FSD reduces the three variables ϕ0(x, y), ωx(x, y) and ωy(x, y) to two for optimization. The regularization term is not included in the energy function in Eq. (6). Thus the optimization problem is simpler. As mentioned earlier, an exhaustive search is used in original FSD. In order to increase the optimization speed, the success of using the Levenberg-Marquardt optimization in FGRPT in Sec. 3.1 suggests that the Levenberg-Marquardt optimization can also be adopted for FSD to replace the exhaustive search, which becomes the proposed frequency-guided sequential demodulation - Levenberg Marquardt (FSD-LM) method.
FSD-LM is guided by local frequencies and shares similar outline as FGRPT in Section 3.1. The differences are the energy function used and initial values for the Levenberg-Marquardt optimization. FSD-LM uses Eq. (6) as its energy function. For a pixel to be demodulated, its initial values of ωx(x, y) and ωy(x, y) are set as a weighted sum of its already-demodulated neighbors, as discussed in Sec. 3.1. As for ϕ0(x, y), it is obtained by Eq. (5). As mentioned in Sec. 2.2, there is a sign ambiguity problem for ϕ0(x, y) which needs to be settled before the optimization of ωx(x, y) and ωy(x, y). Since the energy value using ϕ0(x, y) with a wrong sign will be higher than that using ϕ0(x, y) with a correct sign, the correct sign of ϕ0(x, y) can thus be determined. Then the optimization process for the concerned pixel starts. Note that ϕ0(x, y) remains unchanged and only ωx(x, y) and ωy(x, y) are optimized. The phase ϕ0(x, y) is unwrapped in this stage. Therefore, the resulting phase is continuous. It is interesting to note that the above initial value setting for the sign-determined phase ϕ0(x, y) are also applicable to FGRPT.
Since phase in FSD-LM is directly calculated by Eq. (5), its quality depends on the preprocessing and normalization results. Therefore, the unwrapped phase may need extra post-processing, where windowed Fourier filtering can be used . FSD-LM can also be seen as an alternative method for FFRPT. It is faster than FFRPT and FGRPT. When normalization is well-done, FSD-LM can produce reasonably good results.
4. Results and discussions
In this section, the results of FGRPT and FSD-LM will be presented on a computer simulated fringe pattern and two experimental fringe patterns. All the phase results obtained are continuous and unwrapped. These phases are rewrapped for comparison purpose. A 9×9 neighborhood for Nxy and a value of 0.3 for λ are used for all the three examples.
Figure 1(a) is used again to validate FGRPT and FSD-LM. Figure 2(a) shows the local frequency ω(x, y) estimated using FGRPT, which is used to guide the demodulation process. The frequency is estimated along with phase demodulation, as indicated in the algorithm outline given in Subsection 3.1. Figures 2(b) and 2(c) are two snapshots showing the sequences of the FGRPT demodulation. From the snapshots, the critical points are well avoided during the processing and guaranteed to be processed later than their surroundings. Figure 2(d) is the phase obtained by FGRPT. Figure 2(e) is the estimated frequency using FSD-LM method. Figure 2(f) is the phase result of FSD-LM. The demodulation sequence for FSD-LM is almost the same as Fig. 2(b) and 2(c). Both methods guarantee that the critical points are processed last and they successfully demodulate the fringe pattern. Thus they can be alternative methods to FFRPT. FGRPT consumes similar computation time as FFRPT while FSD-LM is faster than FFRPT and FGRPT. In detail, for the image of Fig. 1(a) of size 256×256 processed in Intel® Xeon® quad-core CPUs of 2.5GHz main frequency using C++ programming, FSD-LM needs about 13s while FSD needs 276s. Addition 7s is needed if windowed Fourier filtering is applied to post-process the obtained phase for both FSD and FSD-LM methods. FFRPT needs about 32s and FGRPT needs about 34s. The mean square errors for the cosine function of the retrieved phase are 0.0125, 0.0097, and 0.0095 for FGRPT, FSD-LM and FSD, respectively. The phase result of FSD is very similar to FSD-LM therefore not shown here. Though FSD-LM does not use exhaustive search, it yields very similar results as FSD and performs 21 times faster.
Figure 3(a) shows an experimental fringe pattern from speckle shearography. The figure is of size 259×208. Coherence enhancing diffusion [6,7] is used for denoising the fringe pattern. The average value of the fringe pattern is then deducted in order to remove the background intensity. The Hilbert transform  is used for normalization. The normalized result is shown in Fig. 3(b). Equation (8) is used for initial value setting for FGRPT. Figure 3(c) shows the local frequency ω(x, y) estimated by FGRPT which is used to guide the demodulation process. The phase result is shown in Fig. 3(d). Figure 3(e) is the cosine value of Fig. 3(d). Figure 3(f) shows the frequency value obtained by FSD-LM. Figure 3(g) shows the phase result using FSD-LM method and Fig. 3(h) is the cosine value of Fig. 3(g). The results by both FGRPT and FSD-LM are satisfactory. The quantitative data are presented in Table 1 .
Figure 4(a) shows another experimental fringe pattern from electronic speckle pattern interferometry. The figure size is 259×208. Coherence enhancing diffusion [6, 7] is used for denoising the fringe pattern. The average value of the fringe pattern is deducted to remove the background intensity. The clipping [2, 3] is used for normalization. The normalized result is shown in Fig. 4(b). FGRPT uses same initial value setting as FSD-LM discussed in Sec. 3.2 for this fringe pattern. Figure 4(c) shows the local frequency ω(x, y) estimated by FGRPT. The phase result is shown in Fig. 4(d). Figure 4(e) is the cosine value of Fig. 4(d). Figure 4(f) shows the frequency value obtained by FSD-LM applied to Fig. 4(b). Figure 4(g) shows the phase result using FSD-LM method and Fig. 4(h) is the cosine value of Fig. 4(g). Again, the results by both methods are satisfactory. The quantitative data are also presented in Table 1.
Table 1 quantitatively presents the results of the above three figures. The accuracy is measured as the mean square errors between the cosine value of the retrieved phase and cosine value of the ground true phase for the first example. Since the true phases of the experimental examples are unknown, and the normalization shows good results, the accuracies of example 2 and example 3 are measured as the mean square errors between the cosine value of the retrieved phase and the normalized fringe patterns. FGRPT is able to demodulate all the examples even when FFRPT fails. FSD-LM obtains similar error rates as FSD but with much less time required. It is interesting to note that FSD-LM sometimes gives even lower error rates than FSD, which is possibly due to that we use the normalized fringe patterns as ground-truth which is not very suitable. It inspires us to look for better comparison criteria in the future. Nevertheless, the shown error rates well measure the similarity between the demodulated results and original fringe patterns.
Two frequency guided method, the frequency guided regularized phase tracker (FGRPT) and sequential demodulation with Levenberg-Marquardt optimization (FSD-LM), are proposed to demodulate a single fringe pattern. Since critical points have low frequency values, frequency information can be used to guarantee that they are performed last to avoid the propagation of spurious sign on these points. These two methods can be considered as alternatives to the fringe-follower regularized phase tracker (FFRPT) technique. When critical points cannot be classified into a single register, FFRPT may go wrong. FGRPT and FSD-LM are not affected by this problem. When fringe pattern is well normalized, FGRPT consumes similar execution time as FFRPT, while FSD-LM is faster than FFRPT. The demodulation results of computer simulated and experimental fringe patterns show that the two proposed algorithms are effective and efficient.
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