Moiré deflection tomography is an important method for detecting high-temperature complex flow fields, providing effective methods and data in fields such as flow field diagnosis^[1–3]. First, the moiré fringes of the flow fields to be measured are obtained by experiments^[4,5]. Then, the moiré fringes are preprocessed^[6], and the phase information is extracted^[7,8]. Finally, the refractive index distributions can be reconstructed^[9–11], based on which of the key parameter distributions of the measured flow fields can be obtained. Therefore, accurate and fast extraction of phase information is a crucial step in the whole process, which can provide important prerequisites for further calculation of deflection angles and the reconstruction of the refractive index. Phase unwrapping consistently serves as a pivotal procedure for attaining accurate phase information. Furthermore, it is extensively employed across diverse domains, including 3D measurements^[12,13], and others. It plays an instrumental role in providing the critical technological underpinning for achieving precision in measurements.

The reported research indicates that phase unwrapping methods include spatial methods^[14,15], temporal methods^[16,17], and deep learning^[18,19]. When the measured object exhibits transient characteristics, temporal phase unwrapping methods may not be well-suited. Therefore, when moiré deflection tomography is used for flow field diagnosis, spatial phase unwrapping methods are more commonly employed. Up to now, spatial phase unwrapping methods mainly include path-dependent and path-independent methods. Among them, the multi-grid method^[20,21] based on the multi-resolution idea is a typical path-independent method. The method decomposes the original phase image into multiple coarse and detailed resolution levels and then performs phase unwrapping calculations at different levels. Therefore, the implementation of this method is relatively complex, requiring interpolation and unwrapping operations on phase images of different resolution levels, which could have some impact on accuracy. In contrast, path-dependent methods have lower computational costs. Up to now, commonly used path-dependent algorithms include the flood-fill method^[22,23] and the region-growing method^[24,25]. The flood-fill method owns the advantages of simplicity and intuitiveness, and it can effectively handle the unwrapping of connected regions with high computational efficiency. However, this method is susceptible to the influence of the filling path, which may lead to instability in the unwrapping results, especially when dealing with complex scenes. The region-growing method’s advantages include simplicity, wide applicability, and strong adaptability. However, it is sensitive to thresholds, which can greatly affect the computational results. Additionally, the basic idea of the unwrapping method based on deep learning is to use convolutional neural network (CNN) models to automatically learn the features of phase images. Then, it obtains the phase unwrapping results based on the learned features. However, the unwrapping method requires a large amount of training data, as a result of which it has the characteristics of high cost and poor interpretability.

Consequently, the current moiré fringe phase unwrapping methods still encounter some challenges when dealing with complex scenes. In this paper, an intelligent phase unwrapping method is proposed based on multi-level grids. This method not only ensures high accuracy when handling complex scenes but also maintains low computational cost and faster processing speed, with a relatively flexible threshold setting range. The intelligent enhancement of moiré fringe phase information extraction holds significant importance.

A multi-level grid phase unwrapping method is proposed to obtain the true phase based on the wrapped phase derived from the Fourier transform. The method depends on the stability and efficiency of the multi-grid method, as well as the low-cost and high-speed characteristics of the row-column method^[26] taken into account. It breaks the limitation of the row-column method, which can only handle the phase of packages from the row and column directions. Compared to the traditional multi-grid method, this method employs different strategies for phase unwrapping and grid iteration. It also offers higher confidence in phase unwrapping, as it does not involve interpolation or fitting during the process.

The multi-level grid method can be roughly divided into two steps: mesh division and phase unwrapping. For the wrapped phase ϕ(x,y) with pixel size 2n×2n, a multi-level grid method is applied to divide it into grid size 2j×2j (where j represents the grid level, j=1,2,…,n) for processing. For the jth level grid, the wrapped phase ϕ2j×2j(x,y) is obtained from that of the previous level. Then, for each coarsened grid at this level, the central four data points, which are from the corresponding four grids of the previous level, are extracted and subjected to multi-directional phase unwrapping, aiming to eliminate newly generated “faults” between the grids. This process can be represented as ϕ2j×2j(m,q)=(a11a12a21a22),(1)where (m,q) represents the coordinates of the jth grid level (m=1,2,…,h, q=1,2,…,h, and h satisfies h=2n−j), while a11, a12, a21, and a22 can be expressed as {a11=ϕ2j×2j(m,q)(2j−1+(m−1)×2j,2j−1+(q−1)×2j)a12=ϕ2j×2j(m,q)(2j−1+(m−1)×2j+1,2j−1+(q−1)×2j)a21=ϕ2j×2j(m,q)(2j−1+(m−1)×2j,2j−1+(q−1)×2j+1)a22=ϕ2j×2j(m,q)(2j−1+(m−1)×2j+1,2j−1+(q−1)×2j+1).(2)

These four data points are subjected to multi-directional phase unwrapping, resulting in ϕ2j×2j(m,q)′=(a11a12+Mr×2πa21+Mc×2πa22+Md×2π),(3)where Mr, Mc, and Md are integers used for phase unwrapping representing the row, column, and diagonal directions, respectively.

Then, we can obtain Δϕ2j×2j(m,q)=ϕ2j×2j(m,q)′−ϕ2j×2j(m,q)=(Δϕ(m,q),1Δϕ(m,q),2Δϕ(m,q),3Δϕ(m,q),4).(4)

In fact, the data within the upper-level grid are unwrapped, and only the “faults” between grids need to be processed. Therefore, to save computational costs and improve speeds, the proposed method transforms phase unwrapping between data points into grids. The process of grid coarsening and iterative phase unwrapping as described above is illustrated in Fig. 1.

As shown in Fig. 1, the four central points represent four data points at the center of the jth level grid, while the four grids represent the regions in the (j−1)th level grid where these data points are located. Our core idea is to extend the multi-directional unwrapping process by these four data points (using the first purple point as a reference, unwrapping toward the three black points) for the four grid regions (using the first yellow region as a reference, unwrapping toward the three blue regions). Further, we assign each difference to all the data belonging to the (j−1)th level grid to achieve phase unwrapping of the data for the jth level grid, which means ϕ2j−1×2j−1(m′,q′)′=ϕ2j−1×2j−1(m′,q′)+Δϕ(m,q),i,(5)where (m′,q′) represents the coordinates of the (j−1)th level grid, while i is related to the position of the (j−1)th level grid in the jth level grid, as shown in Eq. (4).

Finally, the true phase φ(x,y) can be obtained through n iterations such as φ(x,y)=ϕ2n×2n′(x,y).(6)

Based on the above method, the true phase can be rapidly and accurately obtained.

This part will verify the feasibility of the newly proposed method from three perspectives, including theoretical principle, time cost, and accuracy.

1. (1)Theoretical principle: Both the multi-grid method and the row-column method are relatively mature phase unwrapping means. This study sufficiently combines the advantages of them, and corresponding improvements have been made aiming at individual imperfections.
2. (2)Time cost: (a) In each phase unwrapping process of this method, its core object only selects four data points, which means a 2×2 size at the center of the grid, so the entire grid can be unwrapped from three directions: row, column, and diagonal. If using a larger grid, such as a 3×3 size grid, it is necessary to unwrap 8 data points in four directions, including rows, columns, and two diagonals. (b) Based on the above processing process, this method does not need to unwrap all the data within the grid. It only needs to “radiate” the processed four data points into their respective grids to achieve the unwrapping processing of the entire grid. (c) The method in this paper does not require a large number of iterations to improve the accuracy.
3. (3)Accuracy: The proposed method is different from multi-grid method, as it does not involve interpolation and fitting operations, which results in greater accuracy and reliability. To demonstrate this, we conducted a series of numerical simulations for verification. A numerical simulation was conducted to generate a Gaussian surface as shown in Fig. 2(a), and the corresponding wrapped image is shown in Fig. 2(b). Further, the phase unwrapping obtained by the proposed method and the multi-grid method are shown in Figs. 3(a)–3(d), with their respective errors shown in Figs. 3(e)–3(h).

According to Fig. 3, the proposed method has the smallest error, and the error of the multi-grid method decreases as the number of iterations increases. This theoretically demonstrates the feasibility of our method. For better observation and validation of the accuracy and efficiency of the proposed method, a comparative analysis of the results from the data in terms of error and time between the proposed method and the multi-grid method is presented, as shown in Table 1.

Based on the comparative analysis of simulation results and errors shown in Figs. 2 and 3, as well as Table 1, it is evident that the proposed method exhibits smaller errors and lower computational costs. Compared to the multi-grid method, it achieves more precise phase results with fewer iterations. To further validate the advantages of our method, specific experimental comparative analyses will be conducted.

The experimental setup is shown in Fig. 4. A laser with a central wavelength of 532 nm as a light source with a maximum output power of 400 mW. A beam expansion collimation system is formed with a beam expander lens 2 and a collimating lens 3 with a focal length of 30 cm. Lens 4 is the phase object to be measured. Lenses 5 and 6 are two Ronchi gratings with a grating pitch of 0.02 mm, and the distance between them is Δ=17.2mm. Lenses 7 and 9 are imaging lenses with a focal length of 30 cm and a diameter of 75 mm. Obviously, it is a 4f system from the back of grating 6 to the screen 10. A CCD is used to record the moiré fringes on the screen.

To demonstrate the wide applicability of the proposed method, we select experiments with a candle-air flame, heated air around an electric iron, and an alcohol lamp flame. The resulting moiré fringes, with a pixel size of 832×832 and a true size of 50.0mm×50.0mm, are shown in Fig. 5.

In further, an area to be calculated for further discussion is selected with a pixel size of 512×512 and a true size of 30.8mm×30.8mm. Then, grayscale processing and Fourier transform are performed, based on which the wrapped phase results are shown in Fig. 6.

The wrapped phase is executed in nine iterations based on the method proposed in this paper, and the final true phase results are shown in Fig. 7.

The flood-fill method and the multi-grid method are, respectively, chosen as typical representatives of path-dependent and path-independent methods to compare with our proposed method in terms of accuracy and efficiency. Based on Fig. 6, the multi-grid method and the flood-fill method are applied to process three different flow fields separately, and the results are shown in Fig. 8.

Meanwhile, in order to better demonstrate the advantages of the algorithm in terms of precision and speed, the corresponding residual data are shown in Fig. 9, while the corresponding computation time and iteration number are presented in Table 2. The programs are implemented on a computer with the 2019 MacBook Pro running in a 16-inch environment, a CPU of Intel Core i7-9750 H with six cores, and a frequency of 2.6 GHz and 16 GB RAM at 2667 MHz and DDR4.

It is evident that, as the number of iterations increases, the phase results obtained by the multi-grid method gradually approach those obtained by our proposed method. This indicates that even with a limited number of iterations, our method can quickly achieve a level of accuracy similar to that of the multi-grid method, highlighting the advantages of our method in terms of precision, reliability, and efficiency. Additionally, the results obtained by the flood-fill method are nearly identical to those obtained by our proposed method. However, in terms of computation time, our proposed method exhibits a significant advantage. This demonstrates that our proposed method maintains higher efficiency and lower costs when dealing with scenes of varying complexity. This advantage not only saves more time for subsequent experimental data processing but also provides more accurate data support. In essence, for path-dependent methods, the threshold determines which regions are considered reliable and can be unwrapped during the phase unwrapping process, while identifying areas deemed unreliable and requiring further processing. If the threshold is set too low, it may lead to over-unwrapping and inaccurate results. Conversely, if the threshold is set too high, it may prevent certain regions from unwrapping, resulting in the loss of important information. Therefore, selecting an appropriate threshold is crucial.

To demonstrate the flexibility of threshold selection in our method, the results of our method under different threshold settings are compared with two path-dependent methods, as shown in Fig. 10(a). Additionally, the performance of our method on three different flow fields under various thresholds is observed, as shown in Fig. 10(b).

The results show that, compared to the flood-fill method and the region-growing method, our method allows for more flexibility in threshold selection. Meanwhile, within the appropriate threshold range for each respective method, the average computation time of our method under different thresholds is 0.5663 s, significantly less than 14.2069 s for the region-growing method and 36.4006 s for the flood-fill method. Not only that, the method proposed in this paper also provides better processing results for moiré fringes in different flow fields. This further demonstrates that our method not only saves more time but also has a wider threshold range. To further theoretically validate the advantages of the proposed method in terms of accuracy, the deflection angle distributions will be compared.

Figure 7(a) shows the deflection angle distribution obtained from the method proposed in this paper and the results of the multi-grid method. For the convenience of comparison, the central row data are extracted, as shown in Fig. 11.

Distinctly, the deflection angles obtained by the proposed multi-level grid method are closer to zero on both sides of the curve. This theoretical observation verifies the high accuracy of our method and provides more reliable data support for subsequent refractive index reconstruction. In essence, our proposed method, as it does not involve traditional phase unwrapping techniques like interpolation and fitting, can obtain more precise unwrapping results for noise-free data, saving more time in data processing. Additionally, the grid iteration approach effectively allows us to control local noisy data during processing without propagating this effect to other regions.

We propose a multi-level grid method to handle the wrapped phase. It addresses the wrapped phase by dividing it into small grids, unwrapping each grid in multiple directions, and obtaining the true phase distributions by gradually coarsening each grid. First, by simulating the wrapped phase using a Gaussian surface and unwrapping it through the proposed method, the simulation results verify the feasibility and reliability of the proposed method in terms of error and time. Then, three actual measured flow fields (candle-air flame, heated air around an electric iron, and alcohol lamp flame) were selected to obtain moiré fringes and wrapped phase, and phase unwrapping was performed using the proposed method. Furthermore, the calculation results were compared with those obtained from the multi-grid method, the flood-fill method, and the region-growing algorithm. The relevant results indicate that the proposed method has a wider threshold range, faster computational speed, and higher accuracy when dealing with complex scenes. However, the results may be affected to some extent by strong noise. Consequently, we could plan to enhance its ability to handle local noisy regions or incorporate path-independent methods to improve its noise resistance in future work. In a word, the proposed method provides a new solution to the problem of phase unwrapping and offers beneficial exploration and practice for research in related fields.