Abstract
Phase unwrapping still plays an important role in the optical metrology field. The phase unwrapping process has direct influence on the accuracy of final results. The aim of this article is to evaluate the performance of several well-known phase unwrapping algorithms for noisy phase measurement. The results of the examined algorithms on simulated and real phase data are presented, and the conclusions regarding the performances of each studied algorithm are given.
NOMENCLATURE
| SPs | = |
Singular points |
| Quality-Sort | = |
Algorithm based on sorting by reliability following a non-contiguous path |
| LS-DCT | = |
Least-squares method with discrete cosine transform |
| SSPU | = |
Singularity-spreading phase unwrapping method |
| RC | = |
Rotational compensator method |
| RC + DC | = |
Rotational and direct compensator method |
| LC | = |
Localized compensator method |
| FTP | = |
Fourier transform Profilometry system |
| Ũ | = |
Unwrapped phase value |
| ∇Ũ | = |
The gradients of Ũ is computed by planar fitting function |
| Δ(∇Ũ) | = |
The errors of gradient |
| Ũ0 | = |
The true phase value |
| σ | = |
The root mean square of residual from Ũ |
| N | = |
The one dimensional image size |
| p | = |
The proportional decay related to N |
1. INTRODUCTION
Phase unwrapping is a crucial and challenging step to most data-processing chains based on phase information in many fields of research, such as magnetic resonance imaging,[ Citation 1 ] optical interferometry[ Citation 2 ] and Profilometry.[ Citation 3 ] It has direct influence on the accuracy of final results. This is why many algorithms have been proposed to phase unwrapping. However, there is no agreement between the current phase unwrapping algorithms for different applications owing to the existence of disturbance (inconsistency) in the measured phase data. In the case that there is no disturbance in the phase data, the unwrapped phase can be obtained by integrating the phase gradients over the whole data set which is independent of the integration path. However, there are several sources of inconsistencies. Firstly, phase aliasing occurs when the true phase changes by more than one cycle (2π radian) between samples, which was caused by either long baselines, objects discontinuities or high deformation. The second source is the noise which may be caused by speckle noise, and/or fringe breaks. In the presence of these errors, it is hard to unwrap a phase map correctly. All these sources of inconsistency produce what is called singular points (SPs) in the measured phase maps, which spreads errors throughout the measured domain. Thus, the difficult problem that faces phase unwrapping methods is the basis of how to deal with the SPs, and to process their singularities in the phase data. Most of phase unwrapping algorithms cannot distinguish the SPs that exist owing to real object discontinuities from other SPs which appear because of the noise during the measurement in the phase data. However, the previous study[ Citation 4 ] gives a comparison for quality maps and guided strategies for unwrapping singularities due to object discontinuities and noise.
In order to solve the singularities of SPs which existed due to noise, many phase unwrapping algorithms have been proposed in the past. When we focus on the methods that handle the SPs directly, the phase unwrapping algorithms are classified into two types according to the nature of the unwrapped results path following phase unwrapping methods and least squares based methods. Path following phase unwrapping methods[ Citation5-13 ] are pixel-to-pixel integration techniques which rely on local wrapped phase values along a chosen path to construct a correct true unwrapped phase in the absence of SPs in the wrapped phase map. However, this is not always the case, because the presence of noise or corrupted areas in the wrapped phase map makes the integration path becomes dependent. To avoid this situation, corrupted areas which are SPs must be identified; balanced and isolated by using branch cuts from the rest of the good pixels in the phase map. Once SPs are isolated, phase unwrapping will avoid these branch cuts and take an independent path. Thus, it retrieves correct phase data. Least squares based methods[ Citation14-17 ] are completely different from the path following methods. These methods in general, to a certain degree, minimize the difference between the gradients of the wrapped and the gradients of the unwrapped solution in both x and y directions. This problem is considered being described by a solution of the partial differential equation by appending symmetrical images outside the original image, and by taking the Neumann condition as a boundary condition. However, these methods still indirectly deal with the SP problem because their solution is obtained by integrating over the SPs to minimize the gradient differences. The unwrapped phase maps do not contain any continuous phase gaps, which commonly appeared in the path following methods. The path dependency, which is considered as error or consequence of the inconsistencies, is spread throughout the whole domain in order to avoid any large localized errors. Algorithms based on spreading singularity[ Citation18-21 ] are also classified into the same category of least-square methods since they spread the singularity, like the least-square methods do. In terms of accuracy, the method using localized compensator is superior to the other methods; however, it requires a high computational time cost to produce its unwrapped results.
To find the most accurate phase unwrapping algorithm, criterions have to be set for the evaluation of the quality of the unwrapped results. In this article, we investigate the performance of the eight existing phase unwrapping methods for noisy phase data obtained from different applications. A brief explanation regarding each studied algorithm is given in Section 2. Furthermore, the evaluation of the studied methods is carried out based on a group of simulated and real phase data obtained from different applications. Two criterions are set to study the performance of the compared algorithms. First criterion is to examine the accuracy of the studied unwrapping algorithms; second one is aimed to analyze the time cost required to produce the unwrapped phase results of each unwrapping algorithm, as explained in Section 3. Some conclusions are given in Section 4 after the error analysis and the performance evaluation of the examined phase unwrapping methods.
2. PHASE UNWRAPPING ALGORITHMS
This article presents a comprehensive evaluation of eight phase unwrapping algorithms, which have been categorized into two main groups: path-following methods and least-squares methods. Path-following algorithms perform unwrapping pixel based on its neighborhood, whereas least-squares methods see all the entire pixels which will be unwrapped. The studied algorithms are selected because they may give acceptable unwrapped phase results in such applications. The following algorithms are examined in this paper: Goldstein et al.'s path-following method,[ Citation 6 ] Flynn's minimum weighted discontinuity method,[ Citation 10 ] algorithm based on sorting by reliability following a non-contiguous path (Quality-Sort),[ Citation 12 ] the least-squares method with discrete cosine transform (LS-DCT),[ Citation 17 ] singularity-spreading phase unwrapping method (SSPU),[ Citation 19 ] rotational compensator method (RC),[ Citation 20 ] rotational and direct compensator method (RC + DC),[ Citation 22 ] and localized compensator method (LC).[ Citation 21 ]
A short description of each studied method will be illustrated, with the emphasis on the basic ideas. For complete descriptions and formulas regarding each algorithm we referred to the original papers.
2.1. Goldstein et al.'s Path-Following Method
The Goldstein et al. method is one of the earliest branch cut methods.[ Citation 6 ] This method creates trees that connect a number of nearest neighbor SPs where the net charge of every tree should be zero. Therefore, when this method produces a tree which is not neutral and is closer to the border than any neutralizing SP, this tree is neutralized by connecting it to the nearest border pixel. This method is very fast but it tends to isolate areas with dense SPs because branch cuts in such areas often close on themselves. The weakness of this method is the lack of a weighting factor. Furthermore, choosing wrong single branch cut causes errors propagate than over the whole image.
2.2. Flynn's Minimum Weighted Discontinuity Method
Flynn method suggests that the discontinuities in the measured phase data, which are SPs, must be restricted to the areas of noise and true discontinuity in the profile.[ Citation 10 ] These discontinuity areas can be often identified owing to their low quality. The elementary operation of this algorithm is to partition the phase image into two connected regions, to raise the unwrapped phase by 2π in one of the regions, and to reduce the minimal weighted sum of discontinuities. This is done repeatedly until no suitable partitions exist. The major disadvantage of Flynn method is that it required considerable high computational time cost since it use iteration process during finding its unwrapped results by minimizing the weighted sum of discontinuity magnitudes. Furthermore, if many of the weights are high, this method may make poor SP pairing owing low weight paths, therefore, the paths connecting the SPs are not available.
2.3. Sorting by Reliability Following a Non-Contiguous Path Method (Quality-Sort)
The Quality-Sort method uses quality guided map to unwrap the phase data.[ Citation 12 ] The main issues of this method are the choice of the reliability function and the design of the unwrapping path. It begins with calculating the reliability values of pixels which depends on computing the second-derivative for pixels. Then, it calculates reliability values of edges by summing the reliabilities of two pixels of which the edge links. After that this method sorts all edges by value; those edges with higher reliability are unwrapped first. However, this method has a minimum signal-to-noise ratio for the wrapped measured phase data in which this method starts to fail to obtain an accurate unwrapped phase result.
2.4. Least-Squares Method with Discrete Cosine Transforms (LS-DCT)
LS-DCT method tries to minimize the difference between the gradients of the wrapped measured data and those of the unwrapped solution by solving the Poisson equation under the Neumann boundary condition.[ Citation 17 ] The nature of LS-DCT method is to spread the singularity of SPs to the whole domain of the measured phase data including regular regions that have no SPs. It means that phase error which occurred owing to SPs is also propagated to the regular region. Therefore the unwrapped result obtained by LS-DCT method has phase errors with unique density.
2.5. Singularity-Spreading Phase Unwrapping Method (SSPU)
The SSPU algorithm is based on compensating the phase singularities to cancel their effect.[ Citation 19 ] This method first defines SPs distribution in the phase map. Next, the compensators are not only added to the pixel values at the SPs but also to those around the SPs; this method repeats the same process over the whole image. However, the SSPU method spreads the singularities to the entire domain of the image and this is considered as a disadvantage of the SSPU method. Another disadvantage found in this method is the requirement of large computational time to obtain a converged result. When the maximum residue value after the spreading process is small, but not a negligible value, it needs to repeat the process many times.
2.6. Rotational Compensator Method (RC)
The RC method uses local phase information to compensate the singularity parts of the phase map caused by existence of SPs.[ Citation 20 ] It can compute the compensators through superposing the effect of each SP by adding an integral of isotropic singular function along any loops. After that, the unwrapped phase at a point is retrieved by summing the wrapped phase difference and compensators at this point with the unwrapped phase of the previous point. However, the RC method has a drawback of undesired phase error because the RC should be applied to the regular region with no SPs, so as to the singular region. In addition, the RC method required high computational time cost when the measured phase data contains many SPs.
2.7. Rotational and Direct Compensator Method (RC + DC)
The RC + DC method is based on coupling the RC and the direct compensator (DC) techniques to compute the compensators, and the choice of technique depends on the SPs' locations.[ Citation 22 ] This method identifies SPs' locations, then creates SP pairs and defines adjoining and non-adjoining pairs. The RC + DC method uses DC as the compensators for the adjoining SP pairs, and uses RC to compute the compensators of non-adjoining pairs. The adjoining pair is a dipole that consists of two SPs with opposite signs separated by one pixel horizontally or vertically. The DC value is just 2π. The RC + DC method is fast, however, its accuracy is not guaranteed, since the accuracy depends on reducing the number of times of using the RC technique that increases the phase distortion in the unwrapped results.
2.8. Localized Compensator Method (LC)
The LC method regularizes the inconsistencies in local areas, which are clusters, around the SPs by integrating the solution of Poisson's equation for each cluster to evaluate the compensators.[ Citation 21 ] In other words, firstly, the LC method needs to determine cluster groups, and then it computes the compensators depending on the solution of Poisson's equation for each cluster. In terms of accuracy, the LC method is superior to the other methods. Despite this, the LC method has the disadvantage of computational cost since this method requires a long time cost to compute the compensators. The LC method has the same merit of the RC + DC method regarding the phase errors' spread owing to the singularity effect of the adjoining SP dipole pairs on the regular regions that contains no SPs. In addition, the LC method confines the phase error for other SPs in local regions which are clusters.
3. RESULTS AND DISCUSSIONS
3.1. Unwrapping for Simulated Wrapped Data of Known Phase Map
To evaluate the accuracy of the compared phase unwrapping algorithms,[ Citation 6 , Citation 10 , Citation 12 , Citation 17 , Citation19-22 ] we provide known phase maps that are not wrapped. The simulated phase data are two cases of phase maps with the same gradient (0.1, −0.1) cycle/pixel, and the image area 100 × 100 pixels. However, these cases of the simulated phase data contain a set of noise with normal distributions but with different standard deviations (0.15 and 0.2 cycle). In order to compare the number of fringe lines in the wrapped phase data and the unwrapped phase results easily, the unit of phase in this article is represented by cycle which is equivalent to 2π rad. The induced noise caused disturbances, which are SPs, in the phase map. The first simulated noisy phase map with the standard deviation 0.15 cycle is shown as the original and wrapped phase in Figures (a) and (b), respectively. This phase data has 453 positive SPs and 456 negative SPs, as shown in Figure (c); the sum of them exceeds 9% of the number of all pixels. The unwrapped and rewrapped phase results obtained by each examined algorithm are shown in Figure . From the figure, it can be observed that there are many discontinuities in the unwrapped results of Goldstein[ Citation 6 ] and Quality-Sort methods. This means that these methods provide inaccurate unwrapped results, as shown in Figures (a) and (c), respectively. Meanwhile, the unwrapped phase maps obtained by the other studied methods have no phase gap. To evaluate the accuracy of these methods, we count the number of stripes in the wrapped phase data and compare them with their numbers in the rewrapped results. It can be observed that the unwrapped results obtained by Flynn[ Citation 10 ] and LC methods have same numbers of lines as the wrapped phase data does, as shown in Figures (b) and (h), respectively. This observation can be also confirmed by Table , which provides a quantitative comparison to examine the accuracy between the evaluated methods for the two cases of the simulated phase data. In the table, the second column shows the gradients of the unwrapped phase results (Ũ), which are computed by planar fitting function. In the meantime, the third column shows the errors of gradient; the σ is the root mean square of residual for the phase results. The σ of the original phase data is not equal to zero since the original data contains noise with the given standard deviations. In the case of noise with 0.15 cycle, Table reveals that both Flynn and LC algorithms exhibited the best accuracy between the studied algorithms, because they give the smallest error in terms of Δ(∇Ũ).
Table 1. Accuracy comparison of the studied unwrapping algorithms for the simulated phase data cases
Figure 1 Simulated phase data contains noise with standard deviation 0.15 cycle: (a) the original phase data, (b) the wrapped data, and (c) the distribution map of SPs.
Figure 2 Comparison accuracy of unwrapped and rewrapped phase maps of simulated phase data for noisy case with standard deviation 0.15 cycle: in each sub-figure, the left-hand side figure shows unwrapped phase map where phase increases with increasing of brightness, and the right-hand side figure shows rewrapped phase map. (a) Goldstein et al. method, (b) Flynn method, (c) Quality-Sort method, (d) LS-DCT method, (e) SSPU method, (f) RC method, (g) RC + DC method, and (h) LC method.
Figure shows a comparison of required computational time of the studied methods for various image sizes; with the same data component in terms of the gradients and the standard deviation of the noise contained in the data. In the figure, the horizontal axis N denotes one-dimensional area size in pixels. The computational time for each phase unwrapping algorithm is measured using a PC with Intel Core 2 DUO CPU installed, with 2.13 GHz clock in a single CPU operation mode. The computing language used to implement the compared phase unwrapping algorithms is C language. From the figure, it is shown that Flynn[ Citation 10 ] method has the highest computational time cost and Goldstein[ Citation 6 ] method has the lowest one. Furthermore, Table shows the required time cost of N = 500 pixel for each algorithm, and it shows the proportional decay related to this one dimensional image size, for more clarification regarding the computational time cost. Although both Flynn[ Citation 10 ] and LC methods show the most accurate unwrapped results among all algorithms, it is also observed that Flynn has the highest computational time cost. This implies that the LC method has the best performance among all examined algorithms in terms of accuracy and time cost of unwrapped results.
Figure 3 Required computational time of each studied algorithm for simulated phase data has 0.15 cycle standard deviation of noise with various image sizes.
Table 2. Execution time required for the compared algorithms to obtain their unwrapped results for simulated phase data with 0.15 cycle standard deviation of noise case when the image size is 500 × 500 pixels
For the second simulated phase data with 0.2 cycle standard deviation of noise, the comparison is given only to Flynn[ Citation 10 ] and LC methods to explore their performance in the case of high noise since they showed the best accuracy for 0.15 cycle noise. This phase data has 1033 positive SPs and 1031 negative SPs; the sum of them exceeds 20% of the number of all pixels. The unwrapped and rewrapped results obtained by the Flynn[ Citation 10 ] and LC methods are shown in Figures (a) and (b), respectively. It can be seen that the unwrapped results of both methods are smooth and have no phase jumps; however, the rewrapped phase result of the LC method has a large distortion as shown in Figure (b). These distorted regions correspond to the regions around the large clusters size. It indicates that the gradients for the unwrapped result of LC method are not accurate enough. Also, this is clear in Table , as shown for the case of 0.2 cycle standard deviation of noise, that Flynn method is better in terms of the accuracy even though it has a higher computational time cost to produce its unwrapped results, which is 136.3 sec, than the LC does, which is 44.75 sec. However, the advantage of computational cost of LC method becomes smaller than in the case of noise with 0.15 cycle which is caused by the nature of clustering in the LC method. When the noise is large, such as the 0.2 cycle case, the cluster size becomes bigger and the singularity cannot be confined into narrow area.
Figure 4 Comparison accuracy of the unwrapped phase results for simulated phase data has noise with standard deviation 0.2 cycle: (a) the unwrapped and rewrapped results obtained by Flynn method, and (b) the unwrapped and rewrapped results obtained by the LC method. The left side in (a) and (b) the phase increases with the increases of brightness.
3.2. Phase Unwrapping for Experimental Phase Data
We present two actual phase data examples obtained by experiments to demonstrate the applicability of the compared algorithms for noisy phase measurements.
3.2.1. Experimental data obtained by interferometer
The first experimental example is gained from the analysis of fringe pattern obtained by interferometer. The object of this experiment is the temperature measurement of heated gas around a candle flame through measuring the phase shift caused by the flame. The fringe pattern and its wrapped phase map obtained by Fourier domain method[ Citation23-25 ] are shown in Figures (a) and (b), respectively. The wrapped phase map includes 2532 SPs within the image area of 256 × 170 pixels, as shown in Figure (c). Furthermore, the unwrapped results and their rewrapped results obtained by each compared algorithm are given in Figure . It is observed that the unwrapped results of Goldstein[ Citation 6 ] and Quality-Sort methods have many phase jumps. This indicates that these methods provide inaccurate unwrapped results, as shown in Figures (a) and (c), respectively. On the other hand, the other unwrapping methods produce smooth unwrapped results, as shown in Figure . By comparing the number of stripe lines in the wrapped phase data and their number in the rewrapped results of these algorithms from the mid bottom point of each figure, it is observed that the unwrapped results obtained by the Flynn method and LC method have identical numbers of stripe lines to the wrapped phase data, as shown in Figures (b) and (h), respectively. However, the execution time, 449.60 sec, required for the Flynn algorithm to obtain its unwrapped result is much higher compared to that of the LC algorithm, 52.44 sec, as shown in Table . Therefore, it can be explained that LC algorithm gives accurate unwrapped phase results with low time cost.
Figure 5 Experimental phase data obtained by interferometer for candle flame: (a) The observed fringe pattern with enhancement of contrast; (b) wrapped phase map obtained by Fourier domain method; and (c) SPs distributions, (positive and negative SPs are represented by white and black dots, respectively).
Figure 6 Comparison accuracy for the examined phase unwrapping algorithms of interferometric phase data for candle flame; in each sub-figure, the left-hand side figure shows unwrapped phase map where phase increases with increasing of brightness, and the right-hand side figure shows rewrapped phase map. (a) Goldstein et al. method, (b) Flynn method, (c) Quality-Sort method, (d) LS-DCT method, (e) SSPU method, (f) RC method, (g) RC + DC method, and (h) LC method.
Table 3. Execution time required for the compared algorithms to obtain their unwrapped results for each case of real measured phase data, the time is measured by second
3.2.2. Experimental data obtained by FTP
The second example of real phase data is for a 3-D object surface, which is measured by using Fourier transform Profilometry system (FTP).[ Citation 3 , Citation 26 ] The measured object is a ping-pong ball which is roughly 40 mm in diameter. The deformed fringe pattern, the wrapped phase image and its corresponding SP distribution map are shown in Figure . The wrapped phase map includes 580 SPs within the image area of 408 × 312 pixels, most of which are found in the background and around the boundary of the object shown in Figure (c) as white and black pixels. Figure shows the accuracy comparison of the studied algorithms in regard to the unwrapped and rewrapped phase shift results of the object. From the unwrapped results of Goldstein[ Citation 6 ] and Quality-Sort methods, we can see that these methods provide inaccurate results since these results have many phase jumps, as shown in Figures (a) and (c), respectively. Furthermore, it can be observed that there are phase discontinues around the object in the rewrapped result of Flynn[ Citation 10 ] method, as well around the top edge in the rewrapped result of LC method, as shown in Figures (b) and (h), respectively. This indicates that Flynn and LC methods produce inaccurate results. In contrast, it seems that the other methods give good results. For the reason of evaluating the accuracy of the other studied methods, the cross-sectional profiles of the ball height for three different y-positions of each algorithm are shown in Figure ; where y = 0 mm is corresponding to the cross-section which goes through the center of the ball. It can be figured out from these results that the SSPU method has an error in the symmetry of height around the center more than the other methods do, as illustrated in Figure (e). In comparison to this the LS-DCT, RC, and RC + DC methods can correctly compute the height point of the ball which is related to the ball radius (R = 20 mm), as shown in Figures (d), (f), and (g), respectively. In addition, the execution time needed to obtain the unwrapped phase results for the studied algorithms is illustrated in Table . From this table, it can be inferred that both LS-DCT and RC + DC methods successfully unwrap the FTP experimental data with reasonable time cost and satisfactory results.
Figure 7 Experimental measured data obtained by FTP technique: (a) deformed grating image, (b) wrapped phase image, and (c) map of distribution of SPs.
Figure 8 Comparison accuracy for the studied phase unwrapping algorithms FTP measured phase data; in each sub-figure, the left-hand side figure shows unwrapped phase map where phase increases with increasing of brightness, and the right-hand side figure shows rewrapped phase map. (a) Goldstein et al. method, (b) Flynn method, (c) Quality-Sort method, (d) LS-DCT method, (e) SSPU method, (f) RC method, (g) RC + DC method, and (h) LC method.
Figure 9 Comparison accuracy of the cross-sectional profile of the object's height for the FTP data obtained by each examined algorithm: (a) Goldstein et al. method, (b) Flynn method, (c) Quality-Sort method, (d) LS-DCT method, (e) SSPU method, (f) RC method, (g) RC + DC method, and (h) LC method.
4. CONCLUSIONS
This article shows a comprehensive evaluation for eight phase unwrapping algorithms for noisy phase measurements. After applying the studied algorithms to noisy simulated and real phase data in which the ratio of SPs number in these data ranges from 0.5% to 20%, it is clear that the performance of the tested algorithms varies significantly. Goldstein[ Citation 6 ] and Quality-Sort methods are very fast methods but generally yield unreliable results in moderate and high levels of noise. These methods cause many phase jumps in the unwrapped results. The Flynn[ Citation 10 ] method performs better, but its results require high computational time cost. The least-square methods generally perform well for noisy phase data. However, the LS-DCT, SSPU, RC and RC + DC methods spread the singularities to the entire domain of the image. Those spreads cause averaging of the unwrapped phases. Meanwhile, the LC method can localize the averaged areas, and the phase distortion is suppressed. However, when the cluster size is large in the LC method, it has errors around the edges. Despite this, it can be concluded that the LC algorithm gives accurate unwrapped phase results with low time cost for noisy phase measurements.
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