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Abstract
Phase unwrapping in regions of abnormal fringes remains an unresolved issue. In this paper, we present an approach that combines an image-inpainting strategy based on an adaptive window which is obtained according to the density and orientation of fringe patterns and a quality-guided algorithm for phase unwrapping. First, a threshold is set to a quality map to detect the target region. Second, the target region is filled with new phase values by the adaptive image-inpainting method. Then, a quality-guided phase unwrapping algorithm is applied to this newly generated wrapped phase map. Finally, postprocessing of the unwrapped result is performed. The method is validated through several simulation and experiments. The results demonstrate that the proposed algorithm is effective in the presence of abnormal fringes.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Optical interferometry [1,2], synthetic aperture radar (SAR) [3,4], and magnetic resonance imaging (MRI) [5] are well-known techniques for measuring the surface deformation of objects. By applying an appropriate measurement strategy, the measured information of the object is converted to a phase map, referred to as the phase fringe pattern, which is a wrapped (modulo 2π) phase. Phase unwrapping is a significant process that is used to obtain absolute phase values from these wrapped phases. However, under real-world conditions, phase unwrapping is difficult because of noise, aliasing, shadows and fractures.
In recent decades, various algorithms have been proposed to perform practical phase unwrapping. As discussed in [6], the phase unwrapping problem can be solved from two extreme points of view. In the first perspective, all efforts are focused on the phase unwrapping algorithm without any special algorithm for filtering the phase fringe patterns. These algorithms can be generally divided into two groups: path-following algorithms [7–11] and minimum-norm algorithms [12–16]. The performance of path-following algorithms largely depends on the reliability of a prechosen integration path. Although it performs well in many cases, this algorithm is unstable if the integration path runs across or around pixels with a problematic phase, which is corrupted by serious noise or phase aliasing. The minimum-norm algorithms treat the unwrapping process as a global optimization problem. The results of these algorithms appear smooth; nevertheless, in some cases, the results are not accurate for each pixel, and the residual unwrapping error is relatively large. In the second point of view, all efforts are concentrated on improving the filter algorithms. These phase unwrapping algorithms feature filtering the phase noise to obtain a sufficiently high-quality wrapped phase map, which makes the unwrapping process easier. When the phase fringe pattern is corrupted by random noise, these filter-based algorithms [17–20] perform rather well under this condition. Whereas, the filter algorithms cannot adequately deal with the wrapped phase maps obtained by our laser interferometer system to measure a gear’s tooth flank [21–24]. The measured tooth flank is a continuous surface, but its surface is complex and includes extremely small pits and convexities at the microlevel. These factors introduced by the manufacturing process can lead to regional abnormal fringes in the wrapped phase that are larger than the optical elements. Filter-based algorithms have no effect under such conditions [25].
To address this issue, a quality-guided phase unwrapping algorithm based on image inpainting (II-QG) was proposed by our group [25]. The algorithm refills the phase in the doubtful region with the phase in the reliable region. After that, a new wrapped phase map without a doubtful region is obtained, and the following phase unwrapping would be easier to proceed. However, as the research progresses, the algorithm shows several problems. On the one hand, a threshold cannot be obtained through the threshold calculation method under some circumstances. On the other hand, the shape and size of the template window during the filling process is exactly the same all the time; the filling process performs worse when the density and curvature of the fringe pattern show great change.
Based on the above discussion, we present a phase unwrapping algorithm based on the adaptive image inpainting of fringe patterns. In this paper, we argue that the target region should be obtained before the filling process. According to a predetermined threshold, the wrapped phase map could be divided into two regions, the doubtful region (the target region) and the reliable region (the source region). Then, a fringe orientation and fringe density map can be defined according to the original wrapped phase map. Afterwards, specify the adaptive template window to fill the doubtful region with the phase data from the source region. For each replacement of the pixel, the algorithm updates the shape of the adaptive window based on the fringe orientation of the point in the fringe orientation map. In the same way, the size of the adaptive window changes with the fringe density of the pixel, depending on the fringe density map. After these preprocessing operations, the doubtful region is refilled with an appropriate phase from the reliable region, and a new wrapped phase map can be achieved with a high degree of quality. Then, the phase unwrapping becomes relatively trivial. This algorithm overcomes the limitation of the II-QG algorithm that utilizes all the same size and shape windows, and it has a wide adaptation range for wrapped phase maps.
The remainder of this paper is organized as follows. The proposed algorithm is elaborated in Section 2. In Section 3, the method is verified through several experiments. Finally, the conclusions are presented in Section 4.
2. Description of the algorithm
The proposed algorithm consists of the following steps: definition of the target region, fill the target region with adaptive window and apply quality-guided algorithm for phase unwrapping.
2.1 Definition of the target region
Our algorithm aims to restore the problematic wrapped phase, so these abnormal fringe pattern regions should be identified first. It is well known that several quality maps are useful in evaluating the goodness of the phase [1,26]. Among these quality maps, the modulation map directly comes from the interferogram, which can be applied to evaluate whether the quality of the wrapped phase is doubtful. The modulation is defined as follows [26]:
When the fringe modulation map is obtained, a threshold can be set to mask out the doubtful region. Nevertheless, the selection of a threshold is a significant step. If the threshold is too low, then too many doubtful pixels are considered reliable pixels that largely limit the contribution of the algorithm. On the other hand, if the threshold is too high, then too many reliable pixels are missing. The phase restoration process may fail because of the lack of adequately matching template information. According to the residue theorem [1,12], the doubtful region should include most of the residues. A threshold selection method from the image histogram is proposed by our group based on this fact, and it can also be applied to mask out the target region. The detail of this threshold selection method is as follows:
- (1) Search all the residue points in the wrapped phase map and meanwhile record their positions in the corresponding quality map. Mark the mean of the gray value of these points as ${G_1}$ and other normal points as ${G_2}$.
- (2) Count the number the residues with different gray values. Then generate the histogram which shows the relationship between gray values and the percentage of residues.
- (4) Verify whether $G(k )$ is fulfilled with the equation:$$G({k + 1} )- G(k )> {{{w_1}{G_1}} \mathord{\left/ {\vphantom {{{w_1}{G_1}} 2}} \right.} 2} + {{{w_2}{G_2}} \mathord{\left/ {\vphantom {{{w_2}{G_2}} 2}} \right.} 2},$$where ${w_1}$ and ${w_2}$ are the weighting coefficients of mean of the gray value of residues and normal points, respectively. If so, $G(k )$ is considered as the threshold ${T_n}$.
Based on the above process, the quality map can be binarized with threshold ${T_n}$. And therefore the abnormal fringe pattern regions need to be restored can be obtained.
2.2 Phase restoration of the target region
In our algorithm, the phase restoration of the target region is based on the image-inpainting technique. Because the phase fringe pattern shows strong texture characteristics, the region-filling algorithm is applied [27]. This algorithm is one of the most successful exemplar-based image inpainting algorithms. It specifies a part of region in the image as an exemplar block which is used for region filling. Therefore the algorithm is more suitable for images with textures. A brief process of the region-filling algorithm is shown in Fig. 1 as follows.
Fig. 1. Diagram of region-filling process: (a) original image with the target region $\Omega $,the contour $\delta \Omega $, and the source region $\Phi $; (b) patch ${\Psi _p}$ centered at point p $({p \in \delta \Omega } )$; (c)most likely candidate ${\Phi _q}$; (d) partial filling of ${\Psi _p}$. Only the target region of ${\Psi _p}$ is filled with the corresponding phase value in ${\Phi _q}$.
Before the region-filling process, the target region $\Omega $ and the source region $\Phi $ must be defined [25], and the adaptive window should also be specified. Then, the algorithm iterates the following three steps until all pixels have been filled.
- (1) Calculation of the patch priorities. Given a patch ${\Psi _p}$ centered at point p, its priority $P(p )$ is defined as the product of two terms: where $C(p )$ is the confidence term and $D(p )$ is the data term, which are defined as follows:$$C(p )= \frac{{\sum\nolimits_{q \in {\Psi _p} \cap \Phi } {C(q )} }}{{|{{\Psi _p}} |}}\;\;\;\;D(p )= \frac{{|{\nabla I_p^ \bot \cdot {n_p}} |}}{\alpha },$$where $|{{\Psi _p}} |$ is the area of ${\Psi _p}$, $\alpha $ is the normalization factor, ${n_p}$ is the unit vector orthogonal to the front $\delta \Omega $ in point p, and $\nabla I_p^ \bot $ is the isophote at point p. Initially, $C(p )$ is set to zero($\forall p \in \Omega $). The term $C(p )$ can be regarded as a measure which indicates the amount of reliable information nearby the pixel p. And $D(p )$ is a kind of strength of isophotes hitting the contour $\delta \Omega $ at each iteration.
- (2) Propagating texture and structure information. As all priorities of the pixels on the border of the target regions are obtained in the last step, we could find the pixel which has the highest priority and mark it as $\Psi_{\hat{p}}$. And meanwhile we also define a patch $\Psi_{\hat{q}}$ which is most similar to $\Psi_{\hat{p}}$. Formally, the patch follows the requirement as$${\Psi _{\hat{q}}} = \arg \mathop {\min }\limits_{{\Psi _q} \in \Phi } d({\Psi _{\hat{q}}},{\Psi _q}),$$where $d({\Psi _{\hat{q}}},{\Psi _q})$ is defined as the sum of squared differences of the already filled pixels in the two patches. Based on that, the propagation of texture and structure information can be achieved from the source region to the target region per patch.
- (3) Updating confidence values. When each pixel in patch ${\Psi _{\hat{p}}}$ is filled with a new pixel value, the confidence $C(p )$ is updated as follows: Gradually, the value of confidence decays with the filling proceed.
2.3 Determination of the adaptive window
As we mentioned before, in order to obtain the appropriate filling phase value, an adaptive window should be specified before the region-filling process. We develop an algorithm to determine the adaptive window that could change its shape and size according to the fringe orientation and fringe density. The fringe orientation and fringe density could be obtained as follows.
2.3.1 Determination of fringe orientations
The fringe orientation can be defined as the direction in which the fringe pattern light intensity direction derivative is zero. So the fringe orientation can be obtained through the gradient method [28]. First of all, define ${\phi _x}$ and ${\phi _y}$ which represent the gradients along the x and y directions of the current point, respectively. That is,
Figure 2(a) shows part of the simulated fringe pattern, and Fig. 2(b) shows its corresponding fringe orientation map. The gray value 0 denotes direction 0, and the gray value 255 denotes direction $\pi$.
Fig. 2. (a) Simulation wrapped phase map, (b) estimation of fringe orientation map of (a), (c) estimation of fringe density map of (a).
2.3.2 Determination of the fringe density
In practice, the phase fringe density often changes greatly. Obviously, the window size at the region-filling process should also be changed with the density. Based on this process, we specify that the window size depends on the distribution of the fringe density map, which can be estimated by the variation in the fringe gray value on the fringe normal direction [30,31].
Assume x denotes the fringe normal direction and y is orthogonal to x. The fringe pattern shows a maximum accumulated difference along direction x, so we can consider that the difference along direction y is introduced by noise. Therefore, we can obtain
where $f({x,y} )$ is the ideal fringe intensity of point $({x,y} )$ without noise, and ${D_x}$, ${D_y}$ represent the accumulated gray difference in directions x and y, respectively. For one exact period of the fringe, B is the amplitude of the fringe; thus, the accumulated difference along direction x of this period of the fringe should be 2B for each row of data in the processing window. Then, the data under the processing window of $L \times L$ can be written as where n is the number of fringes in the processing window. Based on the normal and tangent directions of the fringe, the fringe density can be estimated as follows:2.3.3 Determination of the adaptive window
When we obtain both the estimation of the fringe orientation and density maps, we can specify the adaptive window of an exact point ${p_0}({x,y} )$. Suppose its local fringe orientation is ${\theta _0}$, then its two neighboring points ${p_1}({{x_1},{y_1}} )$ and ${p_{\textrm{ - }1}}({{x_{\textrm{ - }1}},{y_{\textrm{ - }1}}} )$ along this fringe orientation could be obtained, and so on. The points ${p_n}({{x_n},{y_n}} )$ and ${p_{\textrm{ - }n}}({{x_{\textrm{ - }n}},{y_{\textrm{ - }n}}} )$ could be calculated as:
Fig. 3. (a) The propagation of point ${p_0}$, (b) an adaptive window of ${p_0}$, (c) different adaptive window of different fringe density.
As previously mentioned, the size of the region-filling window should be also adapted based on the fringe density. Thus, we define a smaller adaptive window size for dense fringes and vice versa. Based on the discussion above, we develop such a region-filling window that could alter its shape and size adaptively to the local fringe orientation and density, as shown in Fig. 3(c).
2.4 Quality-guided phase unwrapping
After restoration, a new wrapped phase map is generated through the above method. Then the quality-guided phase unwrapping algorithm is applied which the PDV map is used as the quality map. As a general phase unwrapping algorithm which has been discussed in [1,11], so there is no need to describe its detail here.
3. Experimental results and discussion
In this section, in order to verify the performance of the proposed method, several simulations and experiments were carried out. The proposed algorithm is applied to these wrapped phase images and compared with traditional quality-guided algorithm with PDV map (TQGA) [1], weighted least-squares algorithm (WLS) [1] and traditional image-inpainting algorithm (II-QG) [25].
3.1 Simulation validation
In order to verify the performance of the proposed method in different cases, we made a series of simulation of two conditions. One of them is the condition of rapid change of fringe density, another one is the case of discontinuous phase.
The first simulated wrapped phase with dimensions of $220 \times 220$ is shown in Fig. 4(a). It is generated by two inclined planes with different directions. Therefore, the phase in horizontal direction is not continuous phase, there exists dramatic phase changes. After gaussian white noise with a signal-to-noise ratio of 24 dB is added in the phase map, Fig. 4(b) is obtained. And its corresponding real phase map is shown in Fig. 4(c).
Fig. 4. Left: diagram of the percentage of pixels. Right: (a) wrapped phase map; (b) wrapped phase map with noise; (c) the real phase; (d) TQGA; (e) WLS; (f) Proposed.
Then several algorithms are applied to unwrap the phase, and their results are shown in Figs. 4(d)–4(f). They show the results of TQGA, WLS and the proposed algorithm, respectively. The dark pixels in the figures denote the discontinuous pixels. Apparently, the result from TQGA has the most of the discontinuous phase points. And the proposed method has fewer than that. The result from WLS seems perform best which has the least discontinuous phase points among all these algorithms. In fact, this conclusion cannot be reached yet.
Besides, we also define a kind of diagram as shown in the left of Fig. 4. The diagram can present the absolute error of the result. In the figure, the x axis represents the threshold of error, and the y axis presents the percentage of pixels where the errors are below the threshold. For example, when x=2.5, the result of TQGA is 0.914 which means there are 91.4% of pixels have phase unwrapping error below 2.5 rad. Therefore, based on this definition, it can be concluded that the curve which is close to the top left of the diagram has better unwrapping result than the one under it.
According to this figure, the blue curve which represents the result of WLS is far under the other two methods. That means the result is very disappointing. Obviously in Fig. 4(e), the real phase jump in the middle of the map is missing. It reaches a relatively smooth result by distorting the real phase. That’s the reason why it has the least discontinuities. Thus, WLS actually performs the worst in this sense. While from any point of criteria, the proposed method performs better than the others.
The second simulated wrapped phase with dimensions of $200 \times 200$ is shown in Fig. 5(a). It is generated by a peak function in Matlab. In the same way, the gaussian white noise with a signal-to-noise ratio of 24 dB is added in the phase map, then the noised wrapped phase map is obtained, as shown in Fig. 5(b). The real phase map of it is shown in Fig. 5(c). The unwrapped phase maps are presented in Figs. 5(d)–5(f) for the proposed algorithm and another two algorithms.
Fig. 5. Left: diagram of the percentage of pixels. Right: (a) wrapped phase map; (b) wrapped phase map with noise; (c) the real phase; (d) TQGA; (e) WLS; (f) Proposed.
As indicated by these results, the phase discontinuities of the proposed algorithm is considerably lower than the other two algorithms. Followed by WLS, same as the first simulation unwrapping results, TQGA has the most discontinuous phase points.
This time we also draw a diagram of the percentage of pixels as shown in Fig. 5. The left figure of Fig. 5, the diagram has the same definition as shown in Fig. 4. So we can reach the similar conclusion that the TWS algorithm still performs the worst. And the proposed method also performs better than the other two.
According to these results, the unwrapping results come from WLS method performs bad on these abnormal fringe regions, not as expected. There are two reasons for that. On one hand, because the method is essentially a two-dimensional least squares fit, if there are residual points in the wrapped phase data, there will always be distortions in the unwrapped results. On the other hand, the selection of the weight coefficient in the weighted least square method is very important. The traditional method usually uses the quality map as the weight coefficient to calculate the weight, but at many abnormal fringes, there is often exist the case that the traditional quality map cannot correctly reflect the phase quality [16]. And this is also the reason of poor performance by TQGA.
3.2 Experimental test
Besides simulation, several experiments are conducted with interferograms taken from the laser interferometric system for the measurement of gear tooth flank shape deviation.
The first experimental interferogram comes from a helical gear tooth flank, as shown in Fig. 6, which includes $1196 \times 257$ pixels. As shown in Fig. 6, the fringe density changes dramatically, especially in region A and region B marked with a red dotted rectangle. To make a clear comparison, we performed a close examination of these two regions, as shown in Fig. 7. The target region is shown in the dark in Fig. 7(a), and Fig. 7(b) shows the original wrapped phase. The II-QG and proposed algorithms restored results are shown in Fig. 7(c) and Fig. 7(d), respectively. Comparing these two restored results with the original wrapped phase, the fringe texture of the new wrapped phase maps that are generated by the proposed algorithm become clearer, especially in regions m and n marked with red lines.
Fig. 7. (a) Target region; (b) original wrapped phase map; (c) restored wrapped phase map by II-QG; (d) restored wrapped phase map by proposed method.
As previously mentioned, the quality of the given phase data can be evaluated by quality maps. Therefore, the comparison of quality maps before and after restoration is another way to verify the performance of the algorithm. Three kinds of quality maps, Fig. 7(b), Fig. 7(c), and Fig. 7(d), are calculated, as shown in Fig. 8 and Fig. 9. Figure 8(a) and Fig. 9(a) are PDV quality maps, Fig. 8(b) and Fig. 9(b) are PSEU quality maps and Fig. 8(c) and Fig. 9(c) are MPG quality maps. In Fig. 8 and Fig. 9, the higher the intensity of pixels in these quality maps, the higher the quality and therefore the confidence level in the corresponding pixels in the wrapped phase field. As we can see from both Fig. 8 and Fig. 9, the quality maps derived from II-QG help to reduce the low-quality phase points to some degree. However, the proposed method has better results than II-QG, especially in region B, as shown in Fig. 9. Region B has a denser fringe density, which introduces many low-quality phase points, but after restoration through the proposed method, the majority of these phase points become high-quality points, which are easy to unwrap.
Fig. 8. Phase quality maps of the wrapped phase of region A; (a) PDV quality map, (b) PSEU quality map, (c) MPG quality map.
Fig. 9. Phase quality maps of the wrapped phase of region B; (a) PDV quality map, (b) PSEU quality map, (c) MPG quality map.
For a further comparison, the unwrapped results of these two regions are shown in Fig. 10, where the discontinuous phase points are marked with dark pixels. As Fig. 10(b) shows, before the restoration process, there are many dark pixels gathered near the abnormal fringes in both regions A and B. In Fig. 10(c), as the above quality maps illustrate, the II-QG algorithm has some effect in region A, in which a large part of the discontinuous points are removed, but only a small proportion of discontinuous points are removed in region B. And the WLS seems perform better than TQGA and II-QG which has fewer discontinuous points in both regions. In contrast, the unwrapped result of the proposed algorithm is shown in Fig. 10(d), which performs the least number of discontinuous phase points.
Fig. 10. Discontinuity phase results; (a) wrapped phase, (b) unwrapped result of TQGA, (c) unwrapped result of II-QG, (d) unwrapped result of the proposed method, (e) unwrapped result of WLS.
In addition to the above experiment, a second experiment is performed on a spur gear tooth flank, as shown in Fig. 11(a); its corresponding wrapped phase map is shown in Fig. 11(b). This wrapped phase map includes $596 \times 580$ pixels. The phase fringe pattern of this experiment is slightly different from above; the fringes are clear and sparse in the middle of the gear flank but dense and vague in the top and root regions, which are shown in regions C and D in Fig. 11(a), respectively. The phases in regions C and D are relatively difficult for phase unwrapping, so they are selected to compare the performance among the different phase unwrapping algorithms.
Fig. 11. Experimental wrapped phase; (a) the experimental interferogram ($596 \times 580$), (b) wrapped phase map.
The II-QG and the proposed algorithms are applied to restore the phase fringe pattern first, and the results are shown in Fig. 12. In region C, we can see that II-QG does have some effect to reduce some of the abnormal fringe patterns but performs worse than the proposed algorithm, in which most of these abnormal fringes are reduced significantly. Clearly, in region D, the wrapped phase map generated by II-QG performs even worse than the original wrapped phase map. This result means that after phase restoration, the II-QG algorithm brings more abnormal fringe patterns than before. Although it seems there are only slight changes in the result of the proposed algorithm, we can verify its effect through the unwrapped results.
Fig. 12. (a) Original wrapped phase map; (b) restored wrapped phase map by II-QG; (c) restored wrapped phase map by proposed method.
The first row of Fig. 13 and Fig. 14 refer to the unwrapped results obtained by the traditional quality-guided algorithm (TQGA), the II-QG algorithm, the proposed algorithm and WLS algorithm. The dark pixels in second row of Fig. 13 and Fig. 14 denote the discontinuous phase points corresponding to the unwrapped results. The results in regions C and D, as shown in Fig. 13 and Fig. 14, respectively, indicate that the TQGA performs worse than the other three algorithms, which have several discontinuous phase points. The II-QG algorithm does not perform as well; in region C, it reduces fewer discontinuous phase points than the proposed method. The WLS algorithm presents as many discontinuities as the proposed algorithm in this region. In region D, the II-QG algorithm reduces some of the discontinuous phase points but also brings in some discontinuities that did not previously exist. While the WLS algorithm show the most discontinuities this time. Among these results, the proposed algorithm performs as expected and removes most of these discontinuous phase points, especially in region D. To make a clear comparison of these results, we also extract one row of these unwrapped results in the place marked with the red line, as shown in Fig. 11(b). This comparison is shown in Fig. 15. In Fig. 15, the unwrapped row data of the TQGA, the II-QG algorithm, the proposed algorithm and WLS have nearly the same results in most areas. Whereas, it be noticed that in segments o, p and q, there are clearly general rises or falls and deviations of the unwrapped result of the TQGA and II-QG algorithms marked with the green and blue lines, respectively. These changes cause an error greater than π. In comparison with the other algorithms, as we expected, the red line, which represents the proposed algorithm, is a relatively smooth line without sudden changes or deviations. A comparison of the locations where these sudden changes emerge indicates that these sudden changes are the places where a number of discontinuous phase points exist.
Fig. 13. Different phase unwrapping results of region C: (a) TQGA; (b) II-QG; (c) Proposed; (d) WLS.
Fig. 14. Unwrapped results and discontinuity phase results of different methods of region D: (a) TQGA; (b) II-QG; (c) Proposed; (d) WLS.
After the above experiment, we performed another experiment on a spur gear tooth flank with drum-shaped error, as shown in Fig. 16(a). This wrapped phase map includes $632 \times 574$ pixels. This shape of the phase fringe pattern is different than the pattern from the other two experiments, as discussed previously, which performs arc-shaped fringes in some places. This kind of fringe can influence the result of the image-inpainting process. The phases in regions E and F are selected for a close examination in Fig. 16(b), and their corresponding restored new wrapped phase maps are shown in Fig. 16(c) and Fig. 16(d), which are generated by the II-QG and the proposed methods, respectively. First row of Fig. 17 shows the unwrapped results of region E through the TQGA, II-QG, proposed methods and WLS. Second row of Fig. 17 shows the discontinuous phase points of the corresponding unwrapped results, respectively. As we can see from Fig. 17(a) and Fig. 17(d), there are many discontinuous phase points. After phase restoration and phase unwrapping by II-QG, some of them are removed, but some discontinuities emerge where they did not exist before. As for the results from the proposed method in Fig. 17(c), (a) large part of discontinuities are significantly eliminated. The unwrapped results of region F and the corresponding discontinuities are shown in Fig. 18. We can see that many discontinuities are distributed mainly in regions r, s and t, which are marked with white circles in Fig. 18(a). The discontinuities in region r are removed by both II-QG and the proposed algorithm, but in regions s and t, II-QG has only a minimal effect in reducing the discontinuous phase points compared with the proposed algorithm.
Fig. 16. (a) Close view of original wrapped phase map; (b) close view of restored wrapped phase map by II-QG; (c) close view of restored wrapped phase map by proposed method.
Fig. 17. Unwrapped results and discontinuity phase results of different methods of region E: (a) TQGA; (b) II-QG; (c) Proposed; (d) WLS.
Fig. 18. Unwrapped results and discontinuity phase results of different methods of region F: (a) TQGA; (b) II-QG; (c) Proposed; (d) WLS.
The three experimental unwrapped results are evaluated with the number of discontinuous phase points. This result is shown in Table 1. The table illustrates that the proposed method has the least discontinuities in all three experimental tests, which indicates its validity and reliability. The computation time of these algorithms has also been compared, as shown in Table 2. The proposed algorithm performs more slowly than the other three algorithms. In consideration of the application, more attention is paid to the accuracy of the unwrapped result and the low demand of real-time, so a better tradeoff exists between time consumption and the accuracy of the result. Considering the influence of the execution efficiency of the algorithm, the algorithm is more suitable for occasions with lower requirements for measuring real-time performance. In conclusion, the proposed algorithm shows satisfactory performance among all these methods.
Table 1. Number of discontinuities of different methods.
Table 2. Computation time comparison among different methods(s).
4. Conclusion
This paper presents a new algorithm based on adaptive phase restoration strategy for phase unwrapping. First, the proposed algorithm is applied to several simulations to verify the feasibility and robustness. Then, three real experiments are carried out to test the performance of the proposed algorithm and the results were compared with three traditional algorithms. The results demonstrate that the proposed method can restore the phase value in abnormal fringes and most of the discontinuities of the unwrapped results are eliminated, as expected. The proposed method performs better than the other algorithms in all cases.
Funding
National Natural Science Foundation of China (51275397, 61803302, 61805195).
Disclosures
The authors declare no conflicts of interest.
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