## Abstract

This paper proposes a robust phase unwrapping algorithm (RPUA) for phase unwrapping in the presence of noise and segmented phase. The RPUA method presents a new model of phase derivatives combined with error-correction iterations to achieve an anti-noise effect. Moreover, it bridges the phase islands in the spatial domain using numerical carrier frequency and fringe extrapolation thus eliminating height faults to enable solving segmented phase unwrapping. Numerical simulation and comparison with three conventional methods were performed, proving the high robustness and efficiency of the RPUA. Further, three experiments demonstrated that the RPUA can obtain the unwrapped phase under different noise accurately and possesses the capability to process segmented phases, indicating reliable practicality.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Phase unwrapping is the process of retrieving the true unwrapped phase by removing the ambiguity in the wrapped phase, which is typically generated by the arctangent function [1]. It is an essential procedure for many interferometric measurement applications, such as interferometry, digital holographic [2], synthetic aperture radar imaging (SAR) [3,4], magnetic resonance imaging (MRI) [5], and profilometry [6]. However, in practical applications, phase unwrapping is difficult to realize in the presence of noise or isolated regions. In the last few decades, many phase unwrapping approaches have been developed. Typically, these methods can be classified as path-following methods [7–19], minimum-norm methods [20–31], and other methods [32–39]. The path-following methods utilize the phase residues or phase quality maps to search for a suitable path, and subsequently perform line integration of the modulo 2π mapped wrapped phase differences along the selected path to avoid the accumulation of errors [1]. Based on this principle, many phase unwrapping methods with different path selection strategies have been proposed, such as the branch-cut algorithm [7–12], quality-guided algorithm [13–18], and minimum weighted discontinuity algorithm [19]. These methods can obtain accurate solutions, but they are susceptible to phase noise or phase residues as severe noise in the wrapped phase affects its reliability. In contrast, the minimum-norm algorithms construct an evaluation function of the difference between the wrapped phase gradients and the unwrapped phase gradients, and obtain a robust unwrapped phase by optimizing the function. Many methods have been proposed to accomplish this task, including the least square (LS)-based algorithms [20–26], transport of intensity equation (TIE) or fast Fourier transform (FFT)-based algorithms [27–31]. The minimum-norm algorithms are efficient and are not affected by phase noise to a certain extent because they are independent of the path. Typically, they can obtain a smooth unwrapped phase from the noisy wrapped phase and have been extensively used. Moreover, minimum-norm algorithms are based on the assumption that the absolute value of the phase derivative between adjacent pixels is less than π, which implies that the wrapped phase difference is the true gradient field [1]. However, severe phase noise, such as electronic speckle noise, may render this assumption null as high noise may cause a phase difference exceeding π between adjacent pixels outside the 2π jump area of the wrapped phase, which introduces a large unwrapping error. To improve the performance of phase unwrapping in the presence of high noise, spatial filtering has been proposed to process the wrapped phase to reduce noise [32–35]. However, filtering methods may erase some phase jumps or useful information, and many are time-consuming.

The above methods described are for the solutions wherein the phase is continuous as a whole, but in practical applications, portions of the mirror are expected to be occurred obscured or broken, which divides the phase into segmented phase, also called “phase islands.” For example, in an assembled telescope system, multiple primary or secondary mirrors are often partially covered by the “spider” support structure [40]. In addition to the phase noise, the phase unwrapping is also affected by these phase islands, which causes large errors. Though these phase islands are part of the complete phase, but the wrapped phase cannot reflect that. Thus, when unwrapping the entire measurement area, discontinuity among the phase islands is a challenge. Further, it may cause not only errors inside phase islands, but also height (piston) faults among them. Moreover, when multiple average measurements are required, the problem becomes more acute. Phase unwrapping may adjust a particular phase island up in one measurement, but perhaps down in the next, and when these measurements are averaged, the results are ambiguous and cannot be resolved automatically or manually. The method currently employed involves filling in the slope data of the missing area to obtain a continuous local slope, which is accomplished by calculating the difference between the two directions of the surface figure [40].

This paper proposes a robust phase unwrapping algorithm (RPUA) to solve the global phase unwrapping problem with a noisy or segmented phase. The robustness of the proposed method involves two main aspects. One is to construct a new model of phase derivative combined with the least square method and error-correction iteration to achieve the anti-noise phase unwrapping. Another is bridging the phase islands in the spatial domain utilizing numerical carrier frequency and fringe extrapolation to eliminate the height faults among phase islands for segmented phase unwrapping. The remainder of this paper is organized as follows. The proposed method is described in Section 2. Section 3 presents the numerical analysis and simulation, along with comparisons with several representative methods. Section 4 discusses the practical application of the proposed method in laser interferometry, speckle interferometry, and segmented phase measurement. Finally, the conclusions are drawn in Section 5.

## 2. Proposed robust phase unwrapping algorithm

#### 2.1 New model of phase derivatives

The relationship between the unwrapped phase $\varphi (x,y)$ and wrapped phase $\phi (x,y)$,which is modulo 2π and in the range π to -π, can be written as

where $k(x,y)$ is an integer to be determined. Assuming that the phase derivatives of the wrapped phase are the identical to those of the true phase, the unwrapped phase can be calculated by establishing the following Poisson equation:The above calculation is performed without noise. The presence of noise generates errors between noisy and noise-free phase derivatives, which complicates the phase unwrapping. The key to achieving a robust unwrapped phase is to make the phase derivatives of the noisy phase close to that of the noise-free phase. Here, we propose a new model of phase derivatives to realize anti-noise phase unwrapping:

where ${\nabla ^2}$ is the Laplacian operator, $\textrm{sin}({\cdot} )$ is the sine function, and $(x,y)$ is omitted for convenience of description.Without loss of generality, the analysis starts with a noise-free phase. When not affected by noise, the wrapped phase is continuous (except for 2π jump points), which implies that the phase derivatives are small. For these noise-free points, the first-order approximation can be applied

Because the values of the phase derivatives are small, typically smaller than π, the conventional model can also be used to obtain the similar results, except for 2π jump points: where*W*is the wrapped operator. Equation (4) and (5) demonstrate that the proposed model is approximately equivalent to the conventional model for noise-free points The minute approximation error can be eliminated by iteration, as discussed below.

When the phase is disturbed by severe noise, the wrapped phase becomes discontinuous. The phase derivatives of the noisy point are much greater than those of the noise-free point. In the conventional model, these noisy points are regarded as phase jumps in the phase derivatives and generate large errors in the phase unwrapping. However, for the proposed model, we can easily obtain:

It can be found that the phase derivatives of all noisy points whose values are larger than noise-free points are set in the range of -1 to 1. Compared to the conventional model, we have and it is not hard to find where $\sigma $ is the standard deviation.The above analysis shows that the proposed model has phase derivatives similar to those of the conventional model when not affected by noise, while for the severe noise phase, it can make the noisy phase derivatives more close to those of the noise-free phase. To illustrate the difference between the two models, Fig. 1 shows the plots of two different phase derivatives extracted from the middle row in the *x*-direction of Fig. 6(a1), section 3. Figure 1(a) shows that the two plots coincide to a great extent in the absence of noise. When severe phase noise is introduced, as shown in Fig. 1(b), the conventional model contains a large number of high phase jumps that arise for the large value of the noisy phase derivatives. In contrast, the proposed method reduces the noise phase jumps. The standard deviations of the conventional and proposed methods are 1.249 rad and 0.514 rad, respectively, corresponding to Eq. (9).

#### 2.2 Iteration for approximation error correction

The discrete form of the Poisson equation input in Eq. (3) is

*x*and

*y*directions of the wrapped phase, respectively, defined as

*M*and

*N*are the number of grid points with respect to the

*x*and

*y*indices, respectively. The discrete cosine transform (DCT) method is used to calculate the unwrapped phase, and the unwrapped phase can be obtained as

Iterations can be applied to correct the first-order approximation error and smoothing effect. The *n*th corrected phase can be obtained from the following equation:

*n*th iteration, and $< \cdot > $ is the mean operation. Equation (14b) can correct the piston difference between the unwrapped and wrapped phases.

The phase error of the *n*th iteration is defined as

When there is no unwrapping errors in ${\delta _n}(x,y)$, ${\varphi _n}(x,y)$ can be regarded as completely unwrapped and equal to the true phase. The convergence condition is set as

*N*are the predetermined convergence threshold and number of iterations, respectively. The iterative process flow of the proposed RPUA is shown in Fig. 2.

_{it}#### 2.3 Segmented phase unwrapping

Figure 3(a) illustrates the simultaneous phase-shifting interferograms of a primary mirror in a telescope system, which shows that a complete mirror is divided into multiple phase islands by a “spider” support structure. The unwrapped phase of the mirror calculated with conventional phase unwrapping algorithm is shown in Fig. 3(b), which shows that a random height error occurs among phase islands that should be at the same height, resulting in the failure of the phase unwrapping [21].

We propose to solve the phase unwrapping of the segmented phase by bridging the segmented phase in the spatial domain by utilizing numerical carrier frequency and fringe extrapolation, as shown in Fig. 4.

Numerical carrier frequency is added to the wrapped phase

where $T(x,y)$ is the numerical carrier frequency and can be written as where α, β and γ are carrier-frequency coefficients. Regenerating interferogram ${I_T}$ with ${\phi _T}(x,y)$ we obtain The numerical carrier frequency is added to increase the fringe number in the interferogram to make the structural characteristics of the edges of the phase island more obvious, which is helpful in improving the accuracy of the following fringe extrapolation. The added numerical carrier frequency should also satisfy the Nyquist sampling theorem.The difficulty of the segmented phase unwrapping is the absence of any connection among the phase islands. The RPUA utilizes the data of phase islands to extrapolate the fringes to obscured portions. Performing the fringe extrapolation to the regenerated interferogram we obtain

where $Ext\{{\cdot} \}$ is the fringe extrapolation operator. The extrapolated fringes may not be completely in accordance with the phase distribution in the obscured area, but it is not critical. Through this step the connection of the segmented phase is established. The extrapolated interferogram has data in all grid points, where the single-frame Fourier transform method can be applied to retrieve the phaseThereafter, the added numerical carrier frequency is subtracted from the retrieved phase to obtain the complete phase

However, ${\varphi _{COM}}(x,y)$ is not the final true phase because small errors may still exist in the phase islands due to Fourier transform, which will be illustrated in the simulation. Using the calculated phase ${\varphi _{COM}}(x,y)$ to obtain the corrected integer## 3. Numerical simulation and analysis

#### 3.1 Correction of approximation error

The RPUA method produces an approximation error on noise-free points, specifically those with large phase gradients, when suppressing the noise. The application of the iteration reduces the effect of noise as well as the approximation error. During the iterative process, the approximation error decreases as the phase error ${\delta _n}(x,y)$ converges to the threshold, which makes the unwrapped phase close to the noise-free phase.

Numerical simulations were performed to verify the feasibility of RPUA. The initial phase was generated using MATLAB built-in function peaks. To show the difference directly, the plots for the middle row of the phase derivatives in the *x*-direction of the noise-free wrapped phase for the conventional model and RPUA are shown in Fig. 5(a1)–(a6). Here, we used the coefficient of peaks to show the change in the peak to valley of the initial phase (PV* _{init}*). As PV

*increased, the approximation error of the RPUA on points with a large phase gradient also increased. When the coefficient of the peaks was set to 30 (PV*

_{init}*= 439.175 rad), the two-phase derivatives were very different. Figure 5(b1) –(b6) shows the iterative results of the two models with the same iteration convergence conditions. It is evident that the conventional model (blue line) can obtain the unwrapped phase accurately and efficiently, and the RPUA can also achieve the same result with one or two more iterations, which can be negligible. However, when the initial phase has a large dynamic range, the conventional model is not reliable and it is difficult to obtain the correct unwrapped phase. In contrast, the RPUA method remains consistent with the small PV*

_{init}*in terms of both the accuracy and efficiency of the calculation. The results demonstrate that the RPUA method can solve the approximation error via iterations and have a larger dynamic range of the PV*

_{init}*compared to the conventional model.*

_{init}#### 3.2 Phase unwrapping in the presence of noise

To demonstrate the anti-noise performance of the RPUA, different levels of noise were added to the phase and compared with several common phase unwrapping algorithms, which include iterative least square phase unwrapping (ILS) [21], phase unwrapping based on sorting by reliability following a noncontinuous path (SRNCP) [13], and fast PU based on Fourier transformations (FPU) [28].

The convergence condition of the iteration was set to $\varepsilon = {10^{ - 5}},{N_{it}} = 500$. The noise was Gaussian noise, and the standard deviation of the noise (*σ _{noise}*) was set from low to high. Subsequently, different methods were used to unwrap the noisy wrapped phase, and the residuals of each method were calculated, as shown in Fig. 6. It is evident that the accuracy of the phase unwrapping of the four methods was high under low noise; however, as the noise level increased, the accuracy of both the ILS and SRNCP decreased. In particular, under severe noise, obvious unwrapping errors occur in their results. Compared with the FPU, as shown in Fig. 6(e3) and (e4), the RPUA had better PU results when under severe noise. For Fig. 6(g3) and (g4), the standard deviations of the residuals are 0.340 and 0.270 rad, respectively.

Based on this, the noise was kept constant, while the PV* _{init}* was increased. For low noise, the change in PV

*had no significant effect on the unwrapping accuracy of the four methods. Therefore, we only present the results when the noise level was medium and above, as shown in Figs. 7 and 8. Figure 7 shows that when the standard derivative of noise was set to 0.6 rad (medium noise), the increase in PV*

_{init}*causes a significant decrease in the unwrapping accuracy, especially for the ILS and FPU, which have large unwrapping errors in certain local areas. Though the change in PV*

_{init}*had small effect on the SRNCP, with the standard deviations of the residuals 0.025, 0.048, and 0.205 rad, respectively, the accuracy of SRNCP on noisy phase unwrapping is not satisfactory, as shown in Fig. 7(b2)-7(g2). For the RPUA, as shown in Fig. 7(b4)-7(g4), the standard deviations of the residuals were ${10^{ - 15}}$, ${10^{ - 12}}$ and ${10^{ - \textrm{12}}}$ rad, respectively, which demonstrates that the change in PV*

_{init}*has negligible effect on the unwrapping accuracy of the RPUA under medium noise level.*

_{init}Further, when severe noise was introduced, as shown in Fig. 8, the ILS, SRNCP, and FPU methods are not sufficient for obtaining a satisfactory unwrapped phase as large unwrapping errors occur in their residuals, even with a small PV* _{init}*. However, the RPUA is still able to complete the phase unwrapping successfully and obtain a good unwrapped phase, as shown in Fig. 8(b4)-7(g4), and the standard deviations of the residuals were 0.255, 0.275, and 0.304 rad, respectively.

To compare the different approaches more comprehensively, we quantified the phase residuals of each method. If the calculated standard deviations of the residuals were less than ${10^{ - 5}}$ rad, it would be set to 0, which indicates the unwrapped phase was quite similar to the theoretical true phase. The quantitative results are summarized in Table 1, where *M _{noise}* is the maximum noise value.

To observe the phase residuals more directly, the plots of the standard deviations of the residuals versus different noise levels are shown in Fig. 9(a), with coefficient of the peaks set as 9 (PV* _{init}* = 131.914 rad). The plots indicate that the ILS and SRNCP methods are susceptible to noise, and the increase in noise level causes a rapid decrease in the unwrapping performance. The FPU method is not largely affected by the variation of noise, but the unwrapped accuracy is low because the PV

*is high, which is evident from Fig. 7 (e3), 7(g3), and 8 (e3) and 8(g3). Figure 9(b) shows the plots of the calculation accuracy of the different methods versus the PV*

_{init}*with the standard derivative of noise set as 0.6 rad. It is evident that the increase in the PV of the initial phase has a large impact on the other three approaches, while the unwrapped accuracy of the RPUA method shows minimal change. Further, all the quantitative results verify that the RPUA has a high robustness on anti-noise phase unwrapping when the initial phase had severe noise, large PV, or both.*

_{init}#### 3.3 Computation time

In addition to the unwrapping accuracy, the computation cost is an important metric for evaluating the PU algorithm. The computations were performed with MATLAB on a laptop equipped with an Intel Core i7-7700HQ CPU @2.80-GHz and 16GB RAM. Table 2 presents the computation times of the different approaches with different noise levels when the coefficient of the peaks was set to 9 (PV* _{init}* = 131.914 rad).

To compare them more directly, the computation times are shown in Fig. 10. From Table 2 and Fig. 10, it is evident that the time consumption of the ILS is the shortest, and the computational efficiency of the SRNCP and FPU are relatively low. Further, in the presence of low and medium noise levels, the consumption time of the RPUA was very close to that of the ILS. However, when the phase was occupied by severe noise, the ILS could not obtain a reliable unwrapped phase, while The RPUA still had a good efficiency while ensuring unwrapping accuracy. Therefore, the RPUA has a satisfactory computational efficiency under high noise, which improves its practicality.3.4 Phase unwrapping under complex noise

#### 3.4 Phase unwrapping under complex noise

Speckle noise was added to the initial phase to further verify the performance of RPUA on the anti-noise phase unwrapping. The speckle noise is generated with an average size of one pixel using a linear model, as detailed in Ref. [41]. The coefficient of the peaks was set to 10 (PV* _{init}* = 146.572 rad). The wrapped phase was first generated containing a speckle noise, and thereafter, additive Gaussian noise with its standard derivative set as 0.9 rad was added. The unwrapped results for the four methods are shown in Fig. 11. It is evident from the residuals and histograms of the residual that only the RPUA can obtain a continuous unwrapped phase under high complex noise, while the other three methods show large areas of unwrapping errors in their results. The corresponding standard derivatives of the four methods were 7.364, 10.771, 1.477, and 0.569 rad, respectively, which demonstrates the robustness of the RPUA on phase unwrapping in the presence of complex noise.

#### 3.5 Segmented phase unwrapping

To demonstrate the advantage of the RPUA on the segmented phase unwrapping, we simulated the phase of a conventional primary mirror in the telescope system using Zernike coefficients, as shown in Table 3, and manually added a “spider” mask. The wrapped phase is shown in Fig. 12(a). The unwrapped phases of the four methods are shown in Fig. 12(b1) –12(b4), and it can be found that the ILS, SRNCP, and RPUA have the correct unwrapped phase inside each phase island, except the FPUA with an obvious error. Moreover, the residuals shown in Fig. 12 (c1) –12(c4) demonstrate that the ILS and SRNCP introduce a height difference among the phase islands. The unwrapped phase restored by the RPUA is continuous and consistent with the theoretical true phase.

Figure 13 shows the calculation process of the RPUA, where Fig. 13(a), 13(b), and 13(c) show the initial interferogram, regenerated interferogram, and extrapolated interferogram, respectively. The wrapped phase retrieved by the single-frame Fourier transform is illustrated in Fig. 13(d), and Fig. 13(e) shows the unwrapped phase. It is evident that the segmented phases are connected as a whole phase, and the originally occurring areas are filled. However, the result calculated by Fourier transform cannot be regarded as the final phase. As evident from the residual in Fig. 13(f), there are errors at the edge of the phase islands, which should be corrected using Eq. (22), to obtain the true phase.

Subsequently, we added Gaussian noise with its standard derivative set as 0.7 rad to the segmented phase and made the mask more complex to increase the number of phase islands. The unwrapping results of the different approaches are presented in Fig. 14. It is evident that the presence of high noise and increase of the number and area of phase islands caused large phase unwrapping errors inside the phase islands for the ILS, SRNCP, and FPU. The height faults among the phase islands were also observed to be more severe. In contrast, the proposed RPUA method is not affected by the variation in the number and area of the phase islands and can still realize an anti-noise phase unwrapping.

## 4. Experimental results

#### 4.1 RPUA for experimental phase with noise

The proposed method was further tested with an experimental wrapped phase. In the first experiment, a spherical mirror was measured, and to avoid environmental disturbance during the test, a dynamic interferometric scheme was utilized, according to Ref. [42]. The obtained interferogram is shown in Fig. 15(a), and the wrapped phase using the dual-pass phase demodulation method is illustrated in Fig. 15(b). It is evident that the wrapped phase has low noise, and the four unwrapped phases are shown in Fig. 15(c) –(f). All completed the phase unwrapping successfully, and the PV and root mean square (RMS) of the unwrapped phase were the same (2.9579 λ and 0.5614 λ, respectively). Further, the computation times of the ILS, SRNCP, FPU and RPUA were 1.829, 13.309, 3.147, and 1.683 s, respectively.

The second experiment measured the out-of-plane deformation of a rough planar surface with speckle interferometry, according to Ref. [43]. The obtained interferogram and calculated wrapped phase, which was corrupted by severe speckle noise, is shown in Fig. 16(a) and 16(b). The unwrapped phases are illustrated in Fig. 16(c) –(f), wherein the ILS, SRNCP, and FPU still contain large unwrapping errors when reaching the convergence condition, and increasing the number of iterations improves the results minimally. In contrast, the proposed RPUA obtains a good unwrapped phase, which is continuous and has few phase unwrapping errors.

#### 4.2 RPUA for segmented phase measurement

In the third experiment, a spherical mirror with a spider-type mask was added to the mirror. The experiments utilized a synchronous phase-shifting dynamic interferometry according to the Ref. [44], and the phase-shifting interferogram and wrapped phase are shown in Fig. 17(a) and (b). The above four methods were used to perform phase unwrapping, as shown in Fig. 17(c1) –(f1). It is evident that an unexpected height difference occurs in the results of the ILS, SRNCP, and FPU, which cannot be ignored. In contrast, the unwrapping phase of the RPUA has a continuous excess among phase islands, and the phase appears more complete. To observe this more directly, we draw the phase plots for the 300^{th} row (red dotted line) and column (blue dotted line) of the unwrapped phase for each method, as shown in Fig. 17(c2) –(f2). This shows that the blue and red lines of the other three methods all have obvious discontinuities, and the height difference is of the same order of magnitude as the PV of the measured phase, indicating that their results have large errors. In contrast, the two plots of the RPUA are smooth, and there is no height difference among them, which is more consistent with the entire measured surface.

## 5. Conclusion

We proposed and demonstrated a robust phase-unwrapping algorithm for noisy and segmented phase measurements. A new model of phase derivatives was constructed to guarantee the phase unwrapping accuracy and efficiency when the phase was disturbed by different noises. Moreover, the spatial bridging of the segmented phase with numerical carrier frequency and fringe extrapolation ensured successful phase unwrapping when multiple phase islands exist in the wrapped phase. The simulation and experiments both indicate the high robustness of the proposed method on noisy and segmented phase unwrapping. Comparisons with three other conventional algorithms have also been reported. The results reveal the good performance of the RPUA in complex phase unwrapping, providing a reliable alternative for practical applications.

## Funding

National Natural Science Foundation of China (U2031131, 61975079); Key Laboratory Foundation of Equipment Advanced Research (6142604200511).

## Disclosures

The authors declare no conflicts of interest.

## Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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