Two-plus-two fringe projection profilometry based on phase-shifted coding

Abstract

In fringe projection profilometry based on temporal phase unwrapping, determining a fringe order map commonly requires a large number of fringes. To reduce the fringe number, this paper proposes a concise absolute phase retrieval algorithm just by projecting four fringes. The first two orthogonal fringes with relatively large frequency can collect reliable height information. The second two fringes are designed the same as the first two, but the only difference is that each 2π-phase of them is shifted by a unique amount, which can robustly label a large number of fringe orders. For decoding the fringes, we develop an average intensity one-time extraction algorithm, which allows for the rapid acquisition of the two pairs of alternating current components. From this, the wrapped phase containing height information and the stair-coded phase providing fringe orders can be directly extracted by arctangent operation in a point-to-point manner. Furthermore, we also develop a universal fringe order correction algorithm that can simultaneously correct the common errors and the misalignment between the wrapped phase and fringe orders. Experiment results demonstrate that this method achieves comparable accuracy and adaptability to the phase-coding method, while utilizing two fewer fringes.

Introduction

Fringe projection profilometry (FPP) as the low-cost and easy-to-operate optical three-dimension(3D) sensing technology can recover complex scenes with full-field of view and pixel-level accuracy^1–3. Therefore, many scholars have been constantly promoting FPP, driving its developments towards high-speed, high-accuracy, large-depth-range, etc.

In FPP, wrapped phase extraction and phase unwrapping are two crucial steps⁴. Wrapped phase extraction can be divided into three categories: phase-shifting method (PSP)^5–7, Fourier transform method (FTP)⁸, and Moiré profilometry (MP)^9,10. Among them, PSP has the outstanding advantages of pixel-wise accuracy, strong robustness, uneven-reflectivity resistance, and ambient-light resistance. However, in order to ensure the above advantages, PSP requires at least three sinusoidal fringes with time-varying phase shifts, which may sacrifice measuring speed to some extent. Recently, Yin et al.¹¹ proposed a novel phase-shifting algorithm. The method excellently reduces the minimum number of fringes in PSP to two, and utilizes a line-circle model to avoid the ambient light disturb and refine the phase information. Concurrently, benefiting from the rapid development of digital sensors and the occurrence of binary fringes^12,13, the potential of PSP for high-speed measurement is increasing. Nevertheless, in this step, the retrieved phase is commonly wrapped within [− π,π] due to the arctangent operation, which makes the periods ambiguous.

Fortunately, the phase unwrapping step can detect and eliminate the 2π discontinuities of the wrapped phase and finally obtain the continuous phase. Phase unwrapping has two main branches: spatial phase unwrapping (SPU)^14,15 and temporal phase unwrapping (TPU)^16–19. SPU only requires a single wrapped phase map, thereby relaxing the number of fringes to some extent. However, it may face challenges when reconstructing multi-isolated objects or encountering low signal-to-noise (SNR) pixels. TPU performs more robustly in the above cases. By introducing extra fringes, TPU generates auxiliary phase maps which can provide the point-to-point fringe order information for the wrapped phase. Phase-coding method (PC) as the high-robust TPU algorithm is first proposed by Wang²⁰. It introduces a stair pattern whose changes are completely aligned with the discontinuous of the wrapped phase, and encodes the stair pattern into the phase domain of three-step phase-shifting fringes. Therefore, the fringe order map extracted by the PSP algorithm has excellent robustness.

Nevertheless, many scholars have been making efforts to further enhance the performance of PC. To utilize high-frequency fringes, Zheng required two sets of extra fringes to ensure the identification of a large number of stairs, but this operation inevitably cause fringe number to increase²¹. Wang generated a multi-period phase-coding map with re-arranged the sequence of stairs to increase the fringe orders, while the method loses point-to-point feature²². Fu utilized the arranged and segmented coded phase to obtain the fringe order map with large codewords²³, yet facing the same problem as the previous method. Besides, reducing the fringe number is also an important work in PC. Zhou encoded sinusoidal fringes and extra fringes into red and blue channels respectively, reducing the fringe number to four, but the color coupling issue may disturb²⁴. An proposed the 3 + 1 phase-coding method and determined fringe orders from only one extra fringe, but the period-by-period comparison incurs high computation costs²⁵. He also used the rotated phase shifts in three phase-shifted fringes as the cue of fringe orders, thereby no extra fringes are needed, but the fringes with destroyed continuity are difficult to collect reliable height information²⁶. Lv proposed an N + 2 phase coding method and uniquely labeled the fringe orders by four adjacent periods of the coded phase²⁷. Liu utilized De-Brujin sequence code and phase gradient code to mark the fringe orders, requiring a total of four fringes²⁸. However, the above two methods may lead to the misinterpretation of the fringe orders when some pixels are invalid. Besides, to enhance the accuracy of the fringe order map, Xing solved the problem of phase errors introduced by the nonlinearity²⁹.

Inspired by the phase-coding method and phase shifting profilometry³⁰, this paper contributes a novel absolute phase retrieval algorithm, which requires only four fringes and does not need complex calculations or the color camera. The first two continuous sinusoidal fringes with orthogonal phases can collect reliable height information, while in the other two orthogonal fringes, the unique phase shifts in each 2π phase can provide the robust cue of fringe order information. To decode the above fringes, a concise and fast algorithm for absolute phase extraction is proposed by tactfully utilizing the phase orthogonality, enabling both wrapped phase extraction and phase unwrapping to be achieved in a point-to-point manner. Besides, a fringe order correction algorithm is also proposed, which can solve the jump errors and the misalignment between the wrapped phase and the fringe order map simultaneously.

Principle

Fringe coding

In fringe coding, two sinusoidal fringes with orthogonal phases are encoded first:

$$\left\{\begin{array}{c}{i}_1(x^p,y^p)=a(x^p,y^p)+b(x^p,y^p)\text{cos}(2\pi f{x}^p)\\ i_2(x^p,y^p)=a(x^p,y^p)+b(x^p,y^p)\text{sin}(2\pi f{x}^p)\end{array}\right)$$ 1

where $a(x^p,y^p)$ and $b(x^p,y^p)$ are the average intensity and the designed modulation respectively, and they are commonly designed as 255/2 grayscale to fully utilize the grayscale of the projector. $f$ presents the frequency, which is designed by 20 Hz in this paper. Notably, the above two fringes are continuous without any break, which can promise reliable height information collection.

Secondly, to provide robust fringe orders for Eq. (1), the other two sinusoidal fringes $i_1^{\prime }(x^p,y^p)$ and $i_2^{\prime }(x^p,y^p)$ are encoded. Due to the phase domain being more robust than intensity in carrying information, each 2π phase of $i_1^{\prime }(x^p,y^p)$ and $i_2^{\prime }(x^p,y^p)$ is shifted by a unique amount to label the fringe orders. The above phase shifts can generate a stair-like phase map which is expressed as Eq. (2):

$${\varphi }_s(x^p,y^p)=\text{Floor}[x^p/T](2\pi /N)-\pi ,$$ 2

where Floor[·] keeps the integer part of a number. T is the number of pixels per period. N is the number of periods and designed the same as $f$. From this, $i_1^{\prime }(x^p,y^p)$ and $i_2^{\prime }(x^p,y^p)$, which can provide a large number of fringe orders and share the same fringe parameters as Eq. (1), can be given as:

$$\left\{\begin{array}{c}{i}_1^{\prime }(x^p,y^p)=a(x^p,y^p)+b(x^p,y^p)\text{cos}[2\pi f{x}^p+{\varphi }_s(x^p,y^p)]\\ i_2^{\prime }(x^p,y^p)=a(x^p,y^p)+b(x^p,y^p)\text{sin}[2\pi f{x}^p+{\varphi }_s(x^p,y^p)]\end{array}\right)$$ 3

where the two alternating current(AC) components are still designed to be orthogonal.

Deformed fringe decoding

The two pairs of encoded fringes are projected onto the objects` surfaces sequentially. Then, two pairs of the deformed fringes as expressed in Eqs. (4, 5) can be collected:

$$\left\{\begin{array}{c}{I}_1(x,y)=A(x,y)+B(x,y)\text{cos}[\varphi (x,y)]\\ I_2(x,y)=A(x,y)+B(x,y)\text{sin}[\varphi (x,y)]\end{array}\right),$$ 4

$$\left\{\begin{array}{c}{I}_1^{\prime }(x,y)=A(x,y)+B(x,y)\text{cos}[\varphi (x,y)+{\varphi }_s(x,y)]\\ I_2^{\prime }(x,y)=A(x,y)+B(x,y)\text{sin}[\varphi (x,y)+{\varphi }_s(x,y)]\end{array}\right)$$ 5

where $A(x,y)$ is the background light intensity, $B(x,y)$ denotes modulation, $\varphi (x,y)$ contains the height information, and ${\varphi }_s(x,y)$ includes the fringe order information. For simple expression, $(x,y)$ is omitted in the remainder of this paper. In order to obtain the phase information, it is crucial to accurately extract the two pairs of AC components from Eqs. (4) and (5). To this end, obtaining and eliminating the background light is the top priority. Therefore, an algorithm for achieving one-time acquisition of the background light intensity is proposed, in which the expressions as follows should be given first:

$$\left\{\begin{array}{c}{[Bcos(\varphi )]}^2+{[Bsin(\varphi )]}^2=B^2\begin{array}{ccc}& & \end{array}\\ {[Bcos(\varphi +{\varphi }_s)]}^2+{[Bsin(\varphi +{\varphi }_s)]}^2=B^2\end{array}\right),$$ 6

where $Bcos(\varphi )$ and $B\text{sin}(\varphi )$ are the AC components of Eq. (4), $Bcos(\varphi +{\varphi }_s)$ and $B\text{sin}(\varphi +{\varphi }_s)$ are the AC components of Eq. (5), and moreover, the expressions are equal to each other. The validity of Eq. (6) is exactly based on the phase orthogonality in each pair of fringes. According to this, substituting Eqs. (4, 5) into Eq. (6), we have:

$${(I_1-A)}^2+{(I_2-A)}^2={(I_1^{\prime }-A)}^2+{(I_2^{\prime }-A)}^2$$ 7

In Eq. (7), A is the only unknown parameter. Therefore, after rewriting Eq. (7), the background light intensity can be obtained by:

$$A=\frac{{(I_1^{\prime })}^2+{(I_2^{\prime })}^2-{(I_1)}^2-{(I_2)}^2}{2(I_1^{\prime }+I_2^{\prime }-I_1-I_2)}$$ 8

So far, in summary, after collecting the four deformed fringes, just by one concise calculation as expressed in Eq. (8), the background intensity can be extracted point-to-point at one time.

Then, the AC components can be easily obtained by subtracting the background intensity from the four deformed fringes, and the expressions can be written as:

$$\left\{\begin{array}{c}{I}_{1AC}=I_1-A=B\text{cos}\varphi \\ I_{2AC}=I_2-A=B\text{sin}\varphi \end{array}\right),$$ 9

$$\left\{\begin{array}{c}{I}_{1AC}^{\prime }=I_1^{\prime }-A=B\text{cos(}\varphi +{\varphi }_s)\\ I_{2AC}^{\prime }=I_2^{\prime }-A=B\text{sin(}\varphi +{\varphi }_s)\end{array}\right),$$ 10

Leveraging the phase orthogonality again, the wrapped phase $\varphi$ containing the height information and the coded phase ${\varphi }_s$ containing the fringe orders can be solved by the arctangent operation:

$$\left\{\begin{array}{c}\varphi ={\text{tan}}^{-1}(I_{2AC}/I_{1AC})\begin{array}{cc}& \end{array}\\ {\varphi }_s={\text{tan}}^{-1}(I_{2AC}^{\prime }/I_{1AC}^{\prime })-\varphi \end{array}\right),$$ 11

Besides, according to the division in Eq. (11), the uneven-reflectivity distribution can also be eliminated. As shown in Fig. 1a, due to arctangent operation, the extracted ${\varphi }_s$ is still unavailable, thereby it needs to be reshaped by the constraint calculation as expressed in Eq. (12):

$${\varphi }_{\mathit{rs}}=\left\{\begin{array}{c}{\varphi }_s+2\pi if{\varphi }_s<0\\ {\varphi }_s\text{else}\end{array}\right),$$ 12

where ${\varphi }_{\mathit{rs}}$ as shown in Fig. 1b is the final coded phase with a large number of stairs.

Fig. 1.

The reshaping process of ${\varphi }_s$. (a) Unreshaped phase, (b) reshaped stair-coded phase.

Then, the fringe order k can be determined from the reshaped stair-coded phase by Eq. (13):

$$k=\text{Round}[[N{\varphi }_{\mathit{rs}}/2\pi ]$$ 13

where Round[·] can obtain the nearest integer. Finally, by the robust guidance of the fringe order map, the wrapped phase $\varphi$ can be mapped to the absolute phase $\mathrm{\Phi }$ in a point-to-point manner:

$$\mathrm{\Phi }=\varphi +2\pi k$$ 14

To illustrate more intuitively, the flow chart of the proposed method is shown in Fig. 2:

Fig. 2.

The flow chart of the proposed method.

Fringe order correction

Due to noise interference, low signal-to-noise ratio (SNR) regions, etc., a fringe order map commonly suffers two types of errors as shown in Fig. 3. One is the misalignment between the stair-changed locations of the fringe order map and the 2π discontinuous locations of the wrapped phase, and the other is the jump error which tends to distribute around the stair-changed positions.

Fig. 3.

The diagram of the fringe order errors.

For solving the problems, researchers have developed various algorithms. One typical method is to use the jumps of the wrapped phase to guide the correction of error fringe orders³¹. Another classic method is to utilize the adaptive filters for accurately eliminating pulse-like jump errors and avoiding polluting height with sharp changes³². Here, we propose a novel correction algorithm based on the frequency of the fringe order occurrence to simultaneously solve the two problems. The algorithm will be elaborated with the correction process of a simulated fringe order map with both jump errors and misalignment. The steps are below:

Step 1: Perform the differential calculation on the wrapped phase $\varphi$(Blue line) in order to obtain the 2π discontinuous location map as shown in the green line of Fig. 4a. Notably, the points at the discontinuous locations $x_1,x_2,x_3,...x_N$ (N = 7 in Fig. 4a) show slight fluctuations around 2π and are contaminated by noise. Therefore, designing a threshold Th (Th = 0.78) using the OTSU algorithm³³ can effectively purify them.

Fig. 4.

The fringe order correction flow chart. (a) Determination of 2π discontinuous location, (b) re-segmentation of uncorrected fringe order map, (c) corrected fringe order.

Step 2: According to the locations $x_1,x_2,x_3,...x_N$, N-1 intervals $(x_1,x_2],(x_2,x_3],(x_3,x_4],...(x_{N-1},x_N]$ can be generated. These intervals precisely align with the wrapped phase, thereby creating refined periods for the uncorrected fringe order map. For example, in Fig. 4b, the intervals illustrated by the black dashed line, segment the original fringe order map into seven periods. Notably, the misalignments between the fringe order map and the intervals, as well as jump errors, exist in some regions, revealing the areas where corrections are necessary.

Step 3: In each period, identify the fringe order that appears with the highest frequency and take it as the true value to correct error fringe orders. For instance, in Period4, two fringe orders k = 3,4 appear, yet it is the most frequently occurring fringe order, k = 4, that is utilized to correct other errors. Finally, the successfully corrected fringe order map is shown in Fig. 4c.

It should be emphasized that the proposed algorithm is not limited to correcting slight errors, but can also address the fringe orders with more significant noise. Besides, it can be extended to other phase-coding methods or gray-coding methods, demonstrating strong universality.

Experiments

Noise tolerance verification

Simulations enable the evaluation of noise tolerance by emulating various noise levels. From this, three different noise levels, 1%, 2%, and a severe case of 5%, are added to the four encoded fringes, respectively. A simulated flat board is utilized as a measured object. Figure 5a–c shows the error distributions under these noise levels. The results reveal that the amplitudes of the three results fluctuate roughly within the range of [− 0.05, 0.05]. For quantitative comparison, the MAE and RMSE of each result are calculated and shown in Table 1.

Fig. 5.

Error distributions under different noise levels.

Table 1.

The accuracy under different noise levels.

Noise level	1%	2%	5%
MAE	0.008	0.014	0.029
RMSE	0.016	0.022	0.040

The results in Table 1 reveal that all three experimental results perform good accuracy under different noise levels, even with 5% noise, the absolute phase accuracy still reaches RMSE = 0.04. which verifies the strong noise tolerance ability of the proposed method.

Feasibility verification

In practical experiments, we built a measuring system which mainly includes the camera (Baumer HXC40) with 2048 × 2048 pixels and the projector (LightCrafter 4500SL02 projector) with 1280 × 800 pixels. The focal length of the camera lens is 8 mm. The computer with an AMD Ryzen 7 5800H processor running at 3.2 GHz is utilized to process the captured images with 2048 × 2048 pixels. Because the projector has slight nonlinearity, we have pre-distorted it before measuring.

Firstly, to verify the feasibility of the proposed method, we measured two isolated gypsum models with few textures at the same time. One of the captured deformed fringes is presented in Fig. 6a. Then, by the proposed algorithm in Eq. (8), the shared background light pattern as shown in Fig. 6b can be extracted at one time. However, the enlarged section in Fig. 6b reveals that there are jumps at a few pixels, which are caused by the denominator approaching zero in Eq. (8). Fortunately, the error jumps can be effectively eliminated by the eight-neighborhood mean algorithm³⁴. From this, the corrected average intensity pattern and its enlarged section as shown in Fig. 6c exhibit the smooth surfaces. Then by the subtractions in Eqs. (9) and (10), two pairs of AC components can be obtained easily and one of them is shown in Fig. 6d. After performing the arctangent operation, the wrapped phase and the coded phase are extracted, and they are shown in Fig. 6e and f respectively. Here, the coded phase still needs to be reshaped by Eq. (12) and determined by Eq. (13), so that the preliminary fringe order map as shown in Fig. 6g can be obtained. To correct the fringe order map, the differential operation is applied to the wrapped phase, yielding the discontinuous location map as shown in Fig. 6h, based on threshold Th = 1.16. Guided by the discontinuous locations, the interval map is generated, as shown in Fig. 6i, where each period is marked using different colors. The final corrected fringe order shown in Fig. 6j is then obtained by identifying the most frequently occurring value in each period of the interval map. Figure 6l presents the cross section of the uncorrected fringe order map at the 1000th line. It can be seen that the slight misalignment and error jump problems inevitably occur. At this point, by utilizing the proposed fringe order correction algorithm, the corrected fringe order map as shown in Fig. 6m is perfectly aligned with the wrapped phase, and jump errors are also suppressed effectively. Then, with the robust guidance of the corrected fringe order map, the unwrapped phase can be obtained as shown in Fig. 6k. Finally, as shown in Fig. 6n, the 3D reconstruction models at different angles fully display the measured objects` morphology. The above results prove the feasibility of the proposed method.

Fig. 6.

Reconstruction results. (a) Captured fringe, (b) average intensity without correction, (c) average intensity after correction, (d) AC component, (e) wrapped phase, (f) unreshaped stair-coded phase, (g) fringe order without correction, (h) discontinuous point map, (i) period map, (j) fringe order after correction, (k) unwrapped phase, (l) cross sections of (e) and (g), (m) cross sections of (e) and (j), (n) reconstructed 3D models at different angles.

Accuracy verification

To evaluate the accuracy of our proposed method, we select two representative methods for comparison: a typical phase coding method requiring six fringes, and a novel 3 + 1 absolute phase retrieval method²⁴ using the same number of fringes as our method. All three methods are employed to measure a stair with a step height of 10 mm under the same measuring environments.

Figure 7a–c shows the three sets of captured fringes. Figure 7d–f shows the reconstructed 3D models without fringe order correction, in which jump errors of 2π integer multiples inevitably exist due to incorrect fringe orders. To quantitatively compare them, the error points of each reconstructed model are counted and shown in Table 2. The results reveal that 3 + 1 method yields the highest number of error points. This is due to its approach of determining the fringe order by comparing the wrapped and the coded phases period-by-period. Noise thus interferes with distinguishing adjacent periods, resulting in misjudgment of the fringe orders. In contrast, our proposed method significantly reduces the number of error points and exhibits performance comparable to the PC method.

Fig. 7.

Accuracy comparison. (a–c) Three sets of captured fringes, (d–f) three reconstructed results without fringe order correction, (g–i) three reconstructed results after fringe order correction.

Table 2.

Number of error points.

Method	Error point
3 + 1 method	16,889
Proposed	6924
PC	6377

Further, to evaluate the phase accuracy, the error fringe orders are suppressed with the correction algorithm in all three models. The results after correction, alongside the average height of each step, are shown in Fig. 7g–i. By comparing with a simulated standard stair, the MAE and RMSE of the three models are also calculated and shown in Table 3. Notably, the results of the three methods demonstrate comparable accuracy.

Table 3.

The phase accuracy comparison.

Method	Fringe number	MAE (mm)	RMSE (mm)
3 + 1 method	4	0.0251	0.0386
Proposed	4	0.0249	0.0394
PC	6	0.0247	0.0362

In summary, the proposed method exhibits similar accuracy to the 3 + 1 method but has greater robustness in generating the fringe order map. It also demonstrates comparable accuracy to PC while requiring two fewer fringes. This reduction in fringes contributes to a potential 33% increase in speed when considering high-speed reconstruction.

Moreover, to verify the conciseness of the proposed method, the execution time of each method is shown in Table 4. As evident from the data, our proposed method requires the shortest execution time among the three methods. Notably, this efficiency is achieved while maintaining similar accuracy, as evidenced by the conclusion from Table 3. Therefore, our method also has the advantages of low computational cost and conciseness.

Table 4.

The computational cost comparison.

Method	Execution time (s)
3 + 1 method	0.67
Proposed	0.093
PC	0.22

Complex object reconstruction

To evaluate the robustness of the proposed method, a workpiece with rich abrupt changes is measured. Figure 8a and b show the captured deformed fringes and the extracted background light pattern respectively. Figure 8c shows the wrapped phase. According to the correction process, Fig. 8d displays the purified discontinuous location map obtained in Step1, utilizing a threshold of Th = 1.26. Guided by the locations, the interval map generated in Step 2 is presented in Fig. 8e. Finally, after identifying the most frequently occurring fringe order in each period, the corrected fringe order map produced in Step 3 is shown in Fig. 8f. Figure 8g shows a comparative visualization of the cross sections from both uncorrected and corrected fringe order maps. It can be seen the proposed algorithm effectively suppresses the slight jump errors and misalignment. The final model as shown in Fig. 7h is successfully reconstructed, which well retains the abrupt changes of the workpiece.

Fig. 8.

Reconstruction results. (a) One deformed fringe of the workpiece, (b) the average intensity of (a), (c) the wrapped phase, (d) the discontinuous point map, (e) new period map, (f) corrected fringe order map, (g) cross section comparison, (h) the reconstructed workpiece, (i) one deformed fringe of the tools, (j) the average intensity of (i), (k) the wrapped phase, (l) the discontinuous point map, (m) new period map, (n) corrected fringe order map, (o) cross section comparison, (p) the reconstructed tools.

In addition, variously colored tools are also measured by the same fringes. Figure 8i shows one of the deformed fringes. Figure 8j displays the extracted background light pattern, showcasing the high dynamic range of reflectivity resulting from the diverse colors. Figure 8k shows the extracted wrapped phase. Still by the proposed correction algorithm, the discontinuous location map with Th = 1.39, the interval map, and the corrected fringe order map are respectively shown in Fig. 8l–n. From Fig. 8o, notably, although the uncorrected fringe orders suffer from severe jump errors and misalignment caused by the low-reflectivity pixels, the correction algorithm can still effectively solve the problems, showing strong universality. The reconstructed results shown in Fig. 8p clearly reflect the morphology of the tools with various colors, thereby strongly demonstrating the high robustness of the proposed method.

Furthermore, a white flat board with a surface roughness of 0.2 mm is measured for objectively testing the ability to retain details. Figure 9a shows one of the captured fringes. Figure 9b presents the extracted wrapped phase. The results of the correction algorithm are displayed in Fig. 9c–e in sequence. The final reconstructed model is shown in Fig. 9f. The arithmetic mean deviation of the model, a key indicator of the surface roughness, is calculated to be 0.2033 mm, which closely approximates the actual parameter.

Fig. 9.

The flat board reconstruction. (a) One captured fringe, (b) wrapped phase, (c) discontinuous location map, (d) interval map, (e) corrected fringe order map, (f) reconstructed model.

Conclusion

This paper proposes a concise absolute phase extraction algorithm based on just four fringes for measuring multi-isolated complex objects. In this method, we utilize two continuous sinusoidal fringes to collect reliable height information, and use another two fringes that contain unique phase shifts in each 2π period to provide a large number of fringe orders. In fringe decoding process, the proposed average-intensity one-time extraction algorithm promises the rapid acquisition of the absolute phase. Moreover, the uneven-reflectivity distribution can also be eliminated effectively when extracting the wrapped phase. Besides, the universal fringe order correction algorithm is also developed in order to solve the misalignment and error jump at the same time. The accuracy experiment verifies the proposed method performs almost equivalent high accuracy to PC, but the fringe number is reduced from six to four, which enables the method to offer lower computational cost and have great potential in dynamic scene measurements. Besides, the experimental results of the isolated objects, the complex objects, and the color objects systematically prove the high-robustness of the proposed method.

Author contributions

Hechen Zhang: Conceptualization, Methodology, Software, Data Curation, Writing-original draft preparation. Jin Zhou: Methodology, Investigation, Data Curation, Supervision. Dan Jia and Jinlong Huang: Investigation, Formal Analysis. Jin Yuan: Validation, Visualization.

Data availability

The datasets used and/or analysed during the current study available from the corresponding author on reasonable request.

Competing interests

The authors declare no competing interests.