Generalized phase unwrapping method that avoids jump errors for fringe projection profilometry

Zhoujie Wu; Wenbo Guo; Lilian Lu; Qican Zhang

doi:10.1364/OE.436116

1. Introduction

Phase unwrapping is a key issue for three-dimensional (3D) shape measurement based on fringe projection profilometry (FPP) [1–3]. In FPP, the retrieved phase is wrapped between -π and π due to the inverse trigonometric operation, which is ambiguous for further phase-to-height mapping. To eliminate phase ambiguity, spatial and temporal phase unwrapping methods have been developed. The spatial phase unwrapping method detects and removes the 2π phase jumps by judging the phase difference between adjacent pixels, while the temporal phase unwrapping (TPU) method eliminates phase ambiguity by projecting additional patterns to uniquely label each period of the wrapped phase. Because of the pixel-by-pixel phase unwrapping, the TPU method is more suitable to be applied in complex and isolated scenes.

TPU methods can be further classified into the phase-assisted method and the intensity-assisted method depending on the different ways to modulate and retrieve the information of the fringe order. The former mainly includes multiple-frequency or multiple-wavelength method [2] and phase-coding method [4], and the latter mainly includes De Bruijn coding method [5], Grey-level coding method [6] and Gray code coding method [7]. All the TPU methods eliminate phase ambiguity by matching the wrapped phase and the corresponding fringe order. But in actual measurement, TPU methods suffer two types of phase unwrapping errors, which seriously affect the final reconstructed 3D result. The first one is the phase unwrapping error caused by noise, which is uniformly distributed and can be easily eliminated by simple filtering. The second one is the error caused by arctangent calculation and misalignment between the wrapped phase and fringe order. This kind of errors is periodically distributed and gather on the discontinuities of the wrapped phase, so it is named the jump errors. Jump errors are in zonal distribution and the width of the error region further increases in dynamic measurement due to the motion of the object and the active defocusing of the projector [8]. Therefore, it is more challenging to detect and remove the jump errors compared with the noise errors. It should be noted that jump errors do not occur in multiple-frequency methods or multiple-wavelength methods because solving phase order in these methods do not only use the assisted low-frequency wrapped phase but utilize the high-frequency wrapped phase which will be unwrapped. So, the unstable jump of the phase order in the discontinuity is well-matched with the jump of wrapped phase. Therefore, multiple-frequency methods or multiple-wavelength methods are not included in the scope of this work.

The existing methods to overcome this challenge can be classified into post-corrected methods and pre-avoided methods. Among these methods, spatial phase unwrapping [9], median filtering [10], monotonicity detection [11–13], and robust principal component analysis [14] are the post-corrected methods to detect and correct jump errors. These kinds of methods perform well in static scenes. But it is hard to apply them in dynamic measurement because the width of the error region is larger in dynamic scenes, causing difficulty to distinguish the jump errors from the abrupt change of the object. Pre-avoided methods directly avoid the jump errors by staggering the error regions of wrapped phase and the fringe order. Complementary Gray-coded (CGC) method [15] obtain two staggered fringe orders to overcome the jump errors by projecting an additional pattern, which decreases the measuring efficiency. Cyclic complementary Gray-coded (CCGC) and shifting Gray-coded (SGC) methods [16,17] achieved robust phase unwrapping for complex dynamic scenes by utilizing Gray codes in adjacent projected sequences, which is not suitable for static scenes. Tripartite phase unwrapping (Tri-PU) method [18] solves the problem of jump errors by changing the built-up sequence of sinusoidal patterns to obtain staggered wrapped phases and calculating the wrapped phase to divide the fringe order into three parts, which must satisfy the precondition of three-step phase shifting and obtaining reference phase.

To sum up, post-corrected methods have better adaptability and can be applied in all phase unwrapping methods in FPP, but not suitable for dynamic scenes. Pre-avoided methods fundamentally avoid jump errors and can be applied in different measurement scenes but they are not generalized approaches for FPP regardless of phase-shifting steps and phase unwrapping methods. Therefore, an adaptive and generalized pre-avoided phase unwrapping method is required to solve the problem of jump errors in FPP.

In this paper, a phase unwrapping method that avoids jump errors is proposed for FPP. By building two other staggered wrapped phases from the original wrapped phase and dividing each period of fringe order into three parts, the proposed generalized Tri-PU method can be used to avoid rather than compensatorily correct the jump errors. No additional information is required to assist to divide region and no restrictions on the way of obtaining the wrapped phase and fringe order. So, this method is generalized for phase unwrapping, no matter which method is used to obtain wrapped phase and its corresponding fringe order.

2. Principle

2.1 Typical phase unwrapping errors in TPU methods

In TPU methods, sinusoidal patterns are projected to modulate the height information of a measured object, and the modulated wrapped phase can be extracted using different methods. But all the wrapped phase are wrapped in (-π, π] due to arctangent function. To eliminate the phase ambiguity, fringe order is required and can be solved from coding patterns.

In actual measurement, two types of phase unwrapping errors easily occur due to different reasons as shown in Fig. 1. The first one is the error caused by noise, which is uniformly distributed and discrete. So, it can be easily detected and eliminated by the reviewed post-eliminated methods. The second one is the jump errors caused by two main reasons. Firstly, arctangent calculation in phase extraction leads to the unstable jump near the discontinuity of the wrapped phase, therefore the jump edge of the wrapped phase is unreliable for phase unwrapping. In addition, the motion of the object and the defocusing of the projector cause the misalignment between the wrapped phase and fringe order, which leads to the obvious errors in phase unwrapping. The jump errors are periodically distributed and in zonal distribution, so it is hard for the post-eliminated methods to distinguish the jump errors from the abrupt edges of the object when the width of error region is large. Therefore, the jump errors are more difficult to be eliminated compared with the noise-caused errors in FPP. So, in this work, we focus on solving the problem of the latter error (jump errors) and finding a generalized method which can be adapted to different measuring system (focusing and defocusing measuring system) and widely applied in different methods (coding methods and phase extraction methods) in FPP.

Fig. 1. Illustration of typical phase unwrapping errors in FPP.

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2.2 Proposed generalized phase unwrapping method

In our recent work, the pre-avoided Tri-PU method [18] is proposed to avoid jump errors in Gray-code-based phase unwrapping, in which three parts of divided fringe orders are used to unwrap three corresponding staggered wrapped phases. But the precondition of three-step phase shifting and reference wrapped phase in Tri-PU method dramatically limits its generality in FPP.

In this work, a generalized Tri-PU phase unwrapping method that avoided jump errors is proposed for FPP. The framework of the proposed method is shown in Fig. 2. In TPU methods, sinusoidal patterns are projected to modulate the height information of a measured object, and the modulated wrapped phase can be extracted using different methods such as phase-shifting profilometry (PSP) [19], Fourier transform profilometry (FTP) [20], window Fourier transform profilometry (WFTP) [21] and so on. To eliminate the phase ambiguity, fringe order is required and can be calculated by phase coding [4], De Bruijn coding [5], Gray-level coding [6], Gray code coding [7] and so on. In traditional TPU methods, the discontinuities of the wrapped phase and the fringe order should be well aligned to avoid the jump errors, which is hard and even impossible in the actual measurement. In the proposed method, the discontinuity of wrapped phase and fringe order are pre-staggered by building other two staggered wrapped phases from the original wrapped phase and dividing each period of fringe order into three parts. Three parts of the fringe order are used to unwrap the middle part of the corresponding three staggered wrapped phases which has no discontinuity, so the proposed method can be used to avoid rather than compensatorily correct the jump errors.

Fig. 2. Framework of the proposed generalized phase unwrapping method.

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Taking the phase shifting method as an example to illustrate the principle of the proposed method, the i_th deformed fringes can be described as in N-step PSP:

(1)$${I_i}(x,y) = a(x,y) + b(x,y)\cos [\Phi (x,y) - {\delta _i}],i = 1,2,3\ldots N$$

where, a(x,y) and b(x,y) are the average and modulation of patterns; Φ(x,y) is the modulated continuous phase; δ_i=2π(i-1)/N is the phase-shift amount. And the phase shifting algorithm is used to calculate the wrapped phase ϕ(x,y):

(2)$$\phi (x,y) = \textrm{artan} [\frac{{\sum {_{i = 1}^N(I_i^{}(x,y)\sin {\delta _i})} }}{{\sum {_{i = 1}^N(I_i^{}(x,y)\cos {\delta _i})} }}].$$

The periodic natures of the sinusoidal patterns and arctangent function introduce phase ambiguity, so the fringe (phase) order k(x,y) is required to eliminate phase ambiguity.

In actual measurement, jump errors easily occur on the discontinuity as shown in Fig. 3(a). The essential cause of edges errors is the misaligment of jump edge between wrapped phase and fringe order, so phase unwrapping in the edge region of wrapped phase is unreliable. But in the middle region of wrapped phase, both fringe order and wrapped phase are continuous so phase unwrapping on this region is realiable. Based on this point, a generalized phase unwrapping method is proposed to avoid jump errors.

Fig. 3. Principle of the proposed generalized Tri-PU phase unwrapping method. (a) Schematic diagram of region division. (b) Process of tripartite phase unwrapping.

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Firstly, each period of fringe order A(i) is divided into three parts. The middle part of A(i) is judged by realiable phase value of wrapped phase ϕ(x,y):

(3)$${k_m}(x,y) = k(x,y),\textrm{ }where\textrm{ }|{\phi (x,y)} |< \pi /3.$$

And the left and right parts of A(i) are distinguished by the minmum point of wrapped phase:

(4)$${k_l}(x,y) = k(x,y),\textrm{ }where\textrm{ }y < {y_{\left\langle {\min |{\phi (x)} |} \right\rangle }},\textrm{ }(x,y) \in A(i)\textrm{ }and\textrm{ }(x,y) \notin {k_m}$$

(5)$${k_r}(x,y) = k(x,y),\textrm{ }where\textrm{ }y > {y_{\left\langle {\min |{\phi (x)} |} \right\rangle }},\textrm{ }(x,y) \in A(i)\textrm{ }and\textrm{ }(x,y) \notin {k_m}$$

in which, y denotes the direction perpendicular to fringe and <min|ϕ(x)|> denotes the position of the minmum wrpped phase in each row.

After region division of k(x,y), three staggered wrapped phases are calculated using following equation:

(6)$$\left\{ {\begin{array}{l} {{\phi_1}(x,y)\textrm{ = }\boldsymbol{wrap}[\phi (x,y) + 2\pi /3]}\\ {{\phi_2}(x,y) = \phi (x,y)\textrm{ }}\\ {{\phi_3}(x,y)\textrm{ = }\boldsymbol{wrap}[\phi (x,y) - 2\pi /3]} \end{array}} \right.,$$

in which,

(7)$$\boldsymbol{wrap}[\phi ] = \left\{ {\begin{array}{l} {\phi - 2\pi ,\textrm{ }\phi > \pi }\\ {\phi + 2\pi ,\textrm{ }\phi \le - \pi } \end{array}} \right..$$

Finally, three parts of fringe order k_l(x,y), k_m(x,y) and k_r(x,y) can be used to unwrap the middle part of the corresponding wrapped phases ϕ₁(x,y), ϕ₂(x,y) and ϕ₃(x,y) as follows:

(8)$$\Phi (x,y) = \left\{ {\begin{array}{l} {{\phi_1}(x,y) + 2\pi {k_l}(x,y) - 2\pi /3}\\ {{\phi_2}(x,y) + 2\pi {k_m}(x,y)\textrm{ }}\\ {{\phi_3}(x,y) + 2\pi {k_r}(x,y) + 2\pi /3} \end{array}} \right..$$

As shown in Fig. 3(b), the discontinuity of wrapped phase and fringe order are staggered in each subregion of k(x,y), so the jump errors are pre-avoided in the proposed phase unwrapping method rather than post-corrected in the existing methods.

Compared with our recently proposed Tri-PU method [18], this generalized Tri-PU method doesn’t need additional reference plane for region division of fringe order and has no limit for the way of phase extraction. The proposed method is effective only if the wrapped phase and fringe order are properly obtained using any approach. Therefore, it is a generalized phase unwrapping method in FPP.

To further illustrate the proposed method, the algorithm flow chart and the MATLAB codes are shown in Fig. 4. The symbols and acquirement of key arguments are described as follows:

[m,n] is the size of the image and n is the direction perpendicular to fringe direction;
k is the number of fringe period;
KK is the fringe order, which can be calculated by different coding methods;
Mo is the binary modulation template, which can be solved from the sinusoidal fringe pattern [22,23];
Mo_s and Mo_ss are binary templates of each class of fringe order before and after background removing;
phase_H is the original wrapped phase, which can be calculated by different phase analysis methods and phase_H_part is the prat of wrapped phase in each class of fringe order;
phase_H1 and phase_H2 are the staggered wrapped phase using Eq. (6);
thre denotes the position of the minmum wrpped phase in each row;
MMo_l, MMo_m, and MMo_r are the binary templates for the left, middle and right parts of the fringe order.

Fig. 4. Algorithm flow chart and the MATLAB codes of generalized Tri-PU method.

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3. Experiments and results

3.1 Experiments on a 1-bit defocusing projecting measuring system

To verify the effectiveness of the proposed method. We firstly developed a 1-bit binary defocusing projecting measuring system which easily suffers jump errors due to the defocusing of the projector. This system includes a DMD projector (LightCrafter4500) and a Photron FASTCAM Mini UX100 camera with a 16-mm imaging lens. The resolutions of the projector and the camera are 912×1140 pixels (896×1140 pixels are used to generate fringes) and 1280×800 pixels, respectively. This system is developed for high-speed measurement by projecting 1-bit patterns at 2000 fps.

In the first experiment, a complex honeycomb metal specimen is tested using the traditonal Gray coded method and the proposed generalized Tri-PU method. Three phase-shifting sinsoidal patterns with 32 fringe periods are projected for phase modulation and one of these is shown in Fig. 5(a). In addition, 5 Gray code pattens are projected to label the fringe order. Three-step phase-shifitng algorithm is used to obtain the traditional wrapped phase ϕ₂, as shown in Fig. 5(b). And other staggered wrapped phases ϕ₁ and ϕ₃ shown in Fig. 5(c) and 5(d) are calculated using Eq. (6) and Eq. (7). Then, the decoding fringe order is divided into three parts k_l, k_m, k_r using Eqs. (3) - (5) and shown in Figs. 5(e) and 5(h). Finally, k_l, k_m and k_r are used to unwrap the middle part of the corresponding wrapped phase ϕ₁, ϕ₂ and ϕ₃ to avoid the jump errors. The reconstructed results using the traditional and this proposed method are shown in Figs. 5(f) and 5(g) and the enlarged detailed information of results are shown in Figs. 5(i) and 5(j). Experimental results indicate the proposed method can well avoid the jump errors during the complex object 3D measuring using binary defocusing projecting system.

Fig. 5. Experiments on a 1-bit defocusing projecting measuring system. (a) One of the captured deformed fringe pattern. (b) Original wrapped phase. (c)-(d) Staggered wrapped phases. (e) Fringe order. (f) Reconstructed absolute phase using the traditional phase unwrapping method. (g) Reconstructed absolute phase using the proposed generalized Tri-PU method. (h) Divided tripartite regions of fringe order. (i) Enlarged subimage in (f). (j) Enlarged subimage in (g).

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To compare the performance the proposed pre-avoided method with the traditional post-corrected method in the complex measuring scenes, the measuring result of the honeycomb metal specimen with jump errors as shown in Fig. 6(a) are corrected using the post-corrected method. Different from the proposed pre-avoided method, the post-corrected method directly detects and corrects the jump errors using different algorithms, in which median filtering is the typical one. So, in this experiment, median filtering is applied to smooth jump errors and jump errors is detected and corrected by substracting two unwrapping phases before and after filtering. The size of the filtering window increases from 3×3 to 9×9 and the corresponding filtering results and corrected results are shown in Figs. 6(b)–6(d) and Figs. 6(f)–6(h), respectively.

Fig. 6. Comparative experiments using pre-avoided and post-corrected methods. (a) Reconstructed result using the traditional phase unwrapping method. (b)-(d) Filtering results using a filtering window with increasing size. (e) Reconstructed result using the generalized Tri-PU method. (f)-(h) Corresponding corrected results using post-corrected method.

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Results indicate that jump errors are gradually eliminated with the increasing size of the filtering window, but the jump errors region with large width still exist even using relatively large filtering window (9×9) as shown in Fig. 6(h). In addition, sharp edges of the measured object will be also eliminated after median filtering with a large window as shown in Fig. 6(d), and it causes the emerging errors as shown in Fig. 6(h). Therefore, it is hard to distinguish sharp edges from the phase jump errors and might suffer undesired emerging errors due to the strategy of dectecting and correcting in the post-corrected methods. But for the proposed method, it can directly avoid the jump errors while well perserving the sharp edges in the measured scene as shown in Fig. 6(e). The comparative experiments demonstrate that the proposed method is more adaptive and robust compared with the traditional post-corrected methods, so the proposed method can be applied in the complex measuring scene.

3.2 Experiments on an 8-bit focusing projecting measuring system

To verify the generality of the proposed method. We also developed a 8-bit focusing projecting measuring system including a DMD projector (LightCrafter4500) and a Baumer HXC40 camera with a 16-mm imaging lens, which is more common to be applied in FPP. The resolutions of the projector and camera are the same as that of the first experiment. This system is developed for high-accuracy measurement by projecting high-quality 8-bit patterns at 60 fps.

In the second experimrnt, a complex scene including three isolated objects are measured using different phase calculation and unwrapping methods with different fringe frequency. PSP and FTP are two representative phase analysis methods. In this experiment, PSP and π-shift FTP [24] are used to retrive wrapped phase and several typical fringe coding methods are used to calculate the fringe order to verify the generality of the proposed method. Firstly, phase coding method is used to unwrap the PSP-calculated wrapped phase (f=8, 4 patterns for phase retrieval and 3 for fringe order calculation); Grey-level coding method is used to unwrap the PSP-calculated wrapped phase (f=16, 4 patterns for phase retrieval and 2 for fringe order calculation); Gray coding method is used to unwrap the PSP-calculated wrapped phase (f=32, 4 patterns for phase retrieval and 5 for fringe order calculation). In addition, Gray coding method is also used to unwrap the π-shift FTP-calculated wrapped phase (f=64, 2 patterns for phase retrieval and 6 for fringe order calculation). The captured coding patterns of four combining methods for solving fringe order are shown in Fig. 7(a) and one of the sinusoidal patterns in four combining methods is given in Fig. 7(b). Figure 7(c) shows the extracted wrapped phase with different frequency. And the reconstructed results using the traditional phase unwrapping method and the proposed method are shown in Figs. 7(d) and 7(e), respectively. And Fig. 7(f) shows the details of the enlargred subimages Figs. 7(d) and 7(e). The experimental results indicate the proposed generalized Tri-PU method can well avoid the jump errors for different phase-calculated methods and different coding methods with different fringe frequency. And all of them verify the generality of this proposed method.

Fig. 7. Experimental results by different phase calculation and unwrapping methods using a 8-bit focusing projecting measuring system. (a) Captured coding patterns. (b) Captured fringe patterns. (c) Wrapped phase. (d) Reconstructed height using the traditional phase unwrapping method. (e) Reconstructed height using the proposed generalized Tri-PU method. (f) Enlarged subimages of the details in (d) and (e).

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4. Discussion

Our proposed method has the following advantages compared with the existing correction method for jump errors.

Pre-avoided method. The core idea of traditional post-corrected methods such as spatial phase unwrapping [9], median filtering [10], monotonicity detection [11–13], and robust principal component analysis [14] is detecting the jump errors and compensating the unwrapping phase. This kind of method is well suitable for the static and slowly varying scene with narrow error region, but it is hard to correct the wide error region caused by the defocusing of the projector and motion of objects. When the measuring scene has abrupt change or discontinuous surface, it is diffcult to distinguish sharp edges from the phase jump errors and unwrapping phase might suffer undesired emerging errors using the post-corrected methods. In addition, the parameters of the post-corrected methods should be carefully chosen to stimultaneously ensure accurate detection of the jump errors and preserve abrupt change of measuring scene. But for the proposed method in this work, the jump errors are pre-avoided by staggering the discontinuity of the wrapped phase and fringe order, hence it is simple and superior to measure complex scenes with abrupt change.
Generalized phase unwrapping method. The previous pre-avoided methods including CGC [15], CCGC [16], SGC [17] and Tri-PU [18] have respective limitation in measuring methods. CGC, CCGC and SGC methods only work for Gray-coded method. CGC method is suitable for statics scenes by projecting one additional pattern, and CCGC and SGC methods are only suitable for dynamic scenes using Gray codes in the adjacent projecting sequences. The Tri-PU method is limited to phase extration using three-step phase shifting algorithm and additional reference wrapped phase is required to divide region of the fringe order. Compared with the mentioned pre-avoided methods, the proposed method is suitable for the different phase unwrapping methods assisted by fringe order in FPP, once the wrapped phase vlaue and the fringe order information is obtained. Therefore, this method can be widely applied to avoid the jump error in FPP as a generalized method.

Of course the proposed method also has a critical condition to ensure the correct phase unwrapping, because the proposed generalized Tri-PU method has a threshold 1/3 fringe period (2π∕3 phase difference). When the mismatch between the wrapped phase and the phase order is no more than 1/3 fringe period, the proposed method works well.

To better illustrate this question, a schematic diagram of the critical value is shown in Fig. 8. Both the shift of the wrapped phase and phase order will widen the mismatch between them. Motion of the object makes phase order have a direct shift of Δ_order and also causes an indirect phase error Δϕ in wrapped phase as shown in Fig. 8. The motion-induced phase error also leads to a shift of wrapped phase’s discontinuity and the function D(·) is used to denote this mapping relationship from phase error to shifting distance.

Fig. 8. Critical condition of generalized Tri-PU method to ensure successful phase unwrapping.

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It can be seen from Fig. 8 that when the mismatch between the wrapped phase and phase order exceeds 1/3 fringe period T, the middle part of phase order k_m will be used to unwrap the discontinuity of wrapped phase, which will lead to the failure of the proposed method. Therefore, the critical condition of the proposed method to ensure the correct phase unwrapping is concluded in Eq. (9).

(9)$$\max [\frac{{{\Delta _{order}} + D(\Delta \phi )}}{T}] < \frac{1}{3}$$

5. Conclusion

In conclusion, a generalized phase unwrapping method that avoids jump errors is proposed for FPP in this work. By building other two staggered wrapped phases from the original wrapped phase and dividing each period of fringe order into three parts, the proposed generalized Tri-PU method can be used to avoid the jump errors in FPP. Because no additional information is required except for basic wrapped phase and fringe order, this method is suitable for the phase unwrapping method assisted by fringe order in FPP, no matter which method is used to recover the wrapped phase vlaue and the fringe order information.

The experimental results demonstrated the effectiveness and generality of this proposed method. In addition, it is worth noting that the proposed method is the pre-avoid rather than post-corrected method, therefore it is superior to measure complex objects with sharp edges.

Funding

National Natural Science Foundation of China (62075143); National Postdoctoral Program for Innovative Talents (BX2021199).

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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Generalized phase unwrapping method that avoids jump errors for fringe projection profilometry

Abstract

1. Introduction

2. Principle

2.1 Typical phase unwrapping errors in TPU methods

2.2 Proposed generalized phase unwrapping method

3. Experiments and results

3.1 Experiments on a 1-bit defocusing projecting measuring system

3.2 Experiments on an 8-bit focusing projecting measuring system

4. Discussion

5. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (8)

Equations (9)

Optics Express