1. Introduction

Schlieren and shadowgraph techniques have been extensively employed for imaging and measuring flow-field structures. The history of schlieren and shadow photography dates back to the 17th century [1]; however, for a long time, they were primarily qualitative visualization methods that lacked the quantitative capabilities of interferometric measurements [2].

Background-oriented schlieren (BOS) has been one of the most significant developments in this field since the beginning of the 21st century. The optical setup of BOS was introduced by L.M. Weinstein [3], and was then further developed and applied to very large fields of view by Gary Settles [4]. Dalziel et al. [5] analytically described the principle of BOS in 2000. In the same year, Meier [6] suggested that BOS could be applied to both the flow visualization and CT reconstruction of flow fields. BOS simplifies the optical setup required to obtain information on light deflection, requiring only the flow under study to be placed between the camera and the textured background on which the camera is focused. BOS is technically easy to implement, has relatively low equipment costs and a wide field of view, and can perform reliable measurements under extreme conditions, making it highly valuable for research purposes. Multigroup BOS synchronized recordings enable the three-dimensional reconstruction of nonaxisymmetric unsteady flows [7].

In 2000, Raffel et al. further refined the BOS and demonstrated its applicability to flow field measurements by visualizing the density field of tip vortices in a hovering helicopter flight [8]. In 2004, Venkatakrishnan et al. [9] used a BOS to obtain the density field of an axisymmetric supersonic flow over a cone-cylinder model and found an excellent correlation between the densities obtained from the BOS and those obtained from the cone surface data. In 2007, Atcheson et al. [10] assessed the performance of optical flow algorithms in BOS and pointed out that combining optical flow with a multiscale background could significantly improve BOS performance.

One limitation of this method is that it is still a two-dimensional flow measurement technique that only detects path-integrated information projected onto an image plane. For axisymmetric flows, a single camera can be used for measurements. Time-averaged two-dimensional displacements are converted into line-averaged densities using the Poisson equation, and then, two-dimensional slices are reconstructed from the three-dimensional density field using Abel inversion. Several subsequent tests have applied this single-camera method to axisymmetric targets [11,12]. For example, Sourgen et al. [13] compared numerical simulations with the BOS results using an inverse Abel transformation.

To measure the three-dimensional flow characteristics of complex, non-axisymmetric turbulence, light deflection information from multiple angles can be combined with tomographic imaging algorithms to reconstruct the three-dimensional refractive index field. This is referred to as background-oriented schlieren tomography (BOST). However, purchasing multiple cameras with sufficient acquisition rates to obtain optical deflection information from multiple angles may sometimes be prohibitively expensive. Therefore, Mateo Gomez et al. [14] coupled an ultrahigh-speed camera to a viewsplitter and illuminated the background with an adequate light source, accomplishing the megahertz time resolution and quantitative reconstruction without symmetry assumptions. Bathel et al. [15] also placed frame splitters in front of each of the high-speed camera’s lenses so that two independent views of the flow could be acquired with each camera, achieving a high-speed, turbulent jet visualization. Classical BOST methods typically consist of two steps: the first being CT reconstruction from displacement images. One fundamental approach to CT reconstruction is the use of back-projection methods, such as filtered back-projection [16–18]; however, these inevitably produce artifacts, especially when the projection angles and the amount of projection data are limited. Therefore, starting in 2010, Ota et al. [19–21] replaced filtered back-projection with an algebraic reconstruction technique (ART) on the basis of using rotating cameras. ART yields better reconstruction results than filtered back projection when the amount of data is limited.

The second step in BOST integrates the results after CT reconstruction, because the direct outcome of CT reconstruction is the distribution of refractive index gradients. The simplest method is to perform line integration directly along each direction [8,22]; however, this approach leads to the accumulation of line noise. Alternatively, the Poisson equation can be employed; Atkinson and Hancock [23] proposed the first time-resolved BOST model, deriving the three-dimensional unsteady flow reconstruction process for the refractive index field from gradients through Poisson integration.

In recent years, more one-step reconstruction methods have been proposed than traditional two-step reconstruction methods. Nicolas et al. [24] introduced a model for estimating the density field directly from image displacement fields, avoiding the intermediate integration step of density gradient reconstruction, and employed regularization techniques to address ill-posed problems. Cai et al. [25] developed a 3D radial basis function-based BOST reconstruction method without integration or additional finite differences. In addition, Masahito Akamine et al. [26] proposed a new extension, using a wall as a mirror to provide sufficient light paths to address limitations in measuring near-wall regions, where most of the light paths are blocked.

Meanwhile, CT reconstruction can be performed using deep learning. Jin et al. proposed flame chemiluminescence tomography (FCT) based on convolutional neural networks (CNNs) [27], which demonstrated rapid combustion monitoring capabilities and computational efficiency in three-dimensional FCT measurements. Lei et al. applied an extreme learning machine (ELM) to speed up and improve the quality of reconstruction for electrical capacitance tomography [28]. Yu et al. further applied the ELM to tomographic absorption spectroscopy [29], dramatically reducing the computational time compared with classical iterative methods. However, this CNN-based FCT approach lacks a clear physical interpretation and treats deep learning as a black box. In contrast, BOS-CT reconstructs the measured field using multidirectional displacements, where each projection observation from different angles refers to the same target, and relations exist between adjacent projections. This physical correlation forms the foundation for CT reconstruction.

Therefore, deep learning models should also have the ability to learn and capture this type of correlation in the data; for example, Huang et al. [30] captured the correlation of adjacent frames to achieve time-resolved prediction of 3D flame evolution based on long short-term memory (LSTM). In recent years, with significant achievements in the field of natural language processing, Recurrent Neural networks (RNNs) [31] have attracted increasing research attention and applications. Compared to traditional neural networks, RNNs are better suited for tasks involving time-series inputs because they can retain the influence of previous inputs in the model and participate in calculating subsequent outputs. Because of their unique structure, RNNs have played an important role in language modeling [32], speech recognition [33,34], machine translation [35], audio and video data analysis [36], and image caption modeling [37]. Theoretically, RNNs can utilize time-series information of any length. However, in practice, gradient vanishing or exploding phenomena may occur quickly when the steps between two inputs become too large [38]. To address gradient vanishing and explosion issues in RNNs, Chung et al. [39] proposed a gated recurrent unit (GRU). As variants of RNNs, GRUs can learn long-term dependencies and exhibit a simpler structure with fewer parameters than another RNN variant, LSTM [40], making them popular in current research. Cahuantzi et al. [41] demonstrated that GRUs outperform LSTMs on low-complexity sequences.

In the field of 3D reconstruction, RNNs have been increasingly utilized owing to the accumulated contextual information in multiview inputs. Choy et al. [42] proposed an early image reconstruction network based on 3D RNNs: 3D-R2N2. However, its 3D model resolution and accuracy were limited. Le et al. [43] introduced a multi-viewpoint recursive neural network (MV-RNN) for 3D mesh segmentation. In 2021, Sun et al. [44] presented a real-time 3D reconstruction network called NeuralRecon, which uses a GRU to guide the network in fusing features from previous segments. This allows the network to capture both local information and global shape priors when sequentially reconstructing surfaces, ultimately achieving real-time incremental reconstruction by merging the fragment feature volumes over time. Zuo et al. [45] adopted a multi-view stereo (MVS) network to avoid the inability to extract the most relevant features from an entire video before feature volume fusion and processing. It can be observed that the GRU method can directly learn spatial correlation knowledge in physics. For optical CT reconstruction, classical iterative methods generate a weight-projection matrix by performing 3D ray tracing using the projection relationship between cameras. Using the GRU model, these computational steps are not required to implement the physical projection process, without the artifacts caused by ray-tracing errors and inefficient iterative calculations caused by the large-scale projection matrix.

To achieve real-time, high-precision 3D flow field reconstruction utilizing the inherent contextual information of BOST multiangle projection sequences, this study proposes a fast BOST reconstruction method based on the GRU model. Section 2 provides a detailed introduction to the principles of the BOST. In Section 3, the design philosophy and overall structure of our model are introduced in detail, along with the architecture of the ResNet and GRU modules. In Section 4, numerical simulations are performed by creating a simulated three-dimensional flow field structure for methane combustion. The performance of the GRU model was validated by calculating the root-mean-square error (RMSE) and structure similarity index measure (SSIM) values of the 3D reconstruction results based on the established model. Finally, in Section 5, the validation results for hot airflow above a candle flame measured by a BOST measurement system consisting of 12 AVT cameras are presented. Through experimental verification, the GRU model was able to reconstruct the 3D refractive index distribution above a candle flame from real-time displacement data. Compared to the ART algorithm, this model can directly obtain reconstruction results with higher accuracy and computational efficiency without the need for weight projection matrices and ray tracing.

2. BOS theory

A schematic diagram of the BOS is shown in Fig. 1, which mainly consists of three parts: the background pattern, test flow field, and camera. When light rays from the background pattern pass through the non-uniform test flow field, they are refracted at a certain deflection angle owing to the variation in the refractive index. Consequently, the position at which the light rays pass through the lens and reach the imaging plane is offset from that of the reference light rays. The deflection angle of the light rays is the integral of the refractive index gradient along the path of the light rays and can be expressed as

Here, ${\varepsilon ^{(\alpha )}}$ represents the deflection angle along the direction of the light ray, ${n_0}$ is the refractive index of the surrounding medium of the test flow field, and $ray$ represents the path of the light ray. In general, because of the small magnitude of the deflection angle $\varepsilon $ and the fulfillment of the paraxial approximation, the integral path of the light ray can be approximated as the undisturbed path. Under this assumption, the relationship between the deflection angle $\varepsilon $ and the background pattern displacement $\varDelta \alpha $ can be described by [46]

 

Fig. 1. Schematic diagram of the BOS theory.

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Here, ${l_A}$, ${l_B}$, and ${l_C}$ are the distances between the background pattern and test flow field, test flow field and lens, and lens and imaging plane, respectively. Based on background image-processing algorithms, $\Delta u$ and $\Delta v$ are the displacements in the imaging plane. Using Eqs. (1) and (2), we can obtain the refractive index distribution function of the test flow field from the displacements. In certain situations, it may be necessary to obtain additional parameters of the flow field, such as density and temperature. According to the Gladstone-Dale equation, the relationship between the refractive index and density can be expressed as [47]

Here, n represents the refractive index and $\rho $ represents the density. Additionally, ${K_{GD}}$ is the Gladstone–Dale constant, which is a function of the wavelength of light. Based on the ideal gas law, temperature T can be expressed as a function of the density:

3. 3D BOS based on GRU

3.1 Overall architecture

The architecture of the implemented model is shown in Fig. 2. The entire network was divided into three parts: feature extraction, reprojection, and a GRU module. First, in the encoder, input data with multiple angular offsets were fed into a feature extraction network to extract multilevel features ranging from sparse to dense. The feature extraction network adopted a residual network (ResNet), as shown in Fig. 4. ResNet not only addresses the issues of vanishing gradients, exploding gradients, and model degradation that occur with increasing network depth, but also naturally allows for multidimensional outputs, facilitating multidimensional reconstruction in the subsequent stages of the model.

 

Fig. 2. Architecture of the overall model.

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After encoding, a pyramid-like reconstruction was observed. Starting from the sparsely extracted features, the two-dimensional feature maps were back-projected and projected onto the corresponding three-dimensional grids. They were then input into the GRU module through a 3D sparse convolution. In the GRU module, the features generated by previous inputs are uploaded to a global hidden state, thereby integrating global prior knowledge from other perspectives. Simultaneously, the obtained sparse flow field distribution serves as the input for the next GRU fusion module, eventually leading to dense global prediction results.

3.2 Gated recurrent unit (GRU)

The GRU is a variant of the RNN. Compared with traditional neural networks, RNNs are better equipped to handle sequential data with temporal dependencies. This is because the internal recurrent units of the RNNs can utilize contextual information from previous inputs and retain the impact of these inputs in the model’s decision-making process with the current input.

Theoretically, RNNs can leverage information from sequences of arbitrary length. However, in practice, when the time steps between the two inputs are too large, vanishing or exploding gradients may arise. This phenomenon causes the RNN to lose its ability to learn long-term dependencies as the time interval increases. GRU addresses this issue by improving the hidden layer nodes of the RNN. Its special gate structure effectively solves the problem of processing longer sequential data and offers a simpler alternative to another variant of the RNN called LSTM. The GRU requires fewer parameters and performs better on small-scale datasets [39].

The internal structure of the GRU is illustrated in Fig. 3. Its core components are two gate units: the update gate and the reset gate. The update gate determines the extent to which the previous hidden state affects the current hidden state. This is responsible for capturing long-term dependencies in the sequence. By contrast, the reset gate determines the degree to which the current input is combined with the previous hidden state. This is responsible for capturing the short-term dependencies in a sequence. The hidden layer output ${h_t}$ of the GRU can be obtained as follows:

Here, ${x_t}$, ${h_{t - 1}}$, ${Z_t}$, and ${r_t}$ represent the input to the GRU hidden layer node, previous hidden state output, updated gate output, and reset gate output, respectively. ${x_t}$ and ${h_{t - 1}}$ jointly determine the candidate activation value ${\tilde{h}_t}$ during the process. w and b represent the weight and bias parameters used during the training, respectively. ${\odot} $ represents the element-wise multiplication of the matrices at the corresponding positions. $\sigma $ and $\tanh $ represent the sigmoid activation function and hyperbolic tangent function, respectively.

 

Fig. 3. Architecture of the GRU cell.

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3.3 Image encoder (feature extraction)

In the image encoder, ResNet50 [48] was used as the feature extraction network. Deeper network structures are often used in deep learning to extract richer features. However, simply increasing the number of layers can lead to problems, such as vanishing gradients, exploding gradients, and model degradation. This problem can be partially addressed using techniques such as normalized initialization and intermediate normalization [49] to ensure the convergence of networks with dozens of layers. However, with even deeper networks, the accuracy saturates, and the performance deteriorates.

To address this issue, the ResNet introduces residual blocks. By adding a “shortcut connection” between the input and output in the feedforward network, the input is passed across layers, ensuring that the model’s performance does not deteriorate with increasing depth. In addition, it allows for multidimensional outputs, facilitating multidimensional reconstruction in the subsequent stages of the model.

The overall structure of ResNet50 is shown in Fig. 4.

 

Fig. 4. Architecture of the ResNet50.

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4. Numerical simulation

First, the feasibility of the method was validated using simulated data. Twelve simulated cameras were uniformly distributed within a 165° range, as shown in Fig. 5. A synthetic dataset was generated. The reference image was randomly cropped from the captured images to a size of 185 × 126 pixels. Computational fluid dynamics (CFD) employs a large eddy simulation (LES) to simulate the combustion flow field. With the simulated flow field and the determined positions and orientations of the cameras, the displacements can be calculated using Eqs. (1) and (2). The flow images can then be obtained by remapping the reference image using the calculated displacement.

 

Fig. 5. Locations of camera arrays in the numerical simulation. The red, green and blue arrows indicate the x, y, and z axes respectively (z axis is perpendicular to paper surface outward).

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To ensure the diversity of the training data for effective GRU training, the turbulent flow fields of the flame plumes with different time steps were used to calculate the displacements. LES models have been utilized to simulate displacements based on nonpremixed methane combustion in natural environments. The simulated geometry is shown in Fig. 6(a). The entire solution domain was cylindrical and enclosed by an inlet, outlet, and walls. Prior to the calculations, the solution domain was filled with air at 300 K. The diameter of the solution domain was 500 mm, and the height was 1000 mm. The methane entered the inlet at a temperature of 1000 K. The inlet had a diameter of 2 mm. The simulation timestep was set to 0.005 s. The 3D view of the simulation results is shown in Fig. 6(e). The figure shows that complex turbulent structures were captured in the upper part of the flame plume during the simulation.

 

Fig. 6. (a) Geometric structure for the simulation. (b-d) Simulated displacements in the first, sixth, and eleventh cameras. (e) Three-dimensional simulation results.

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The flow field in the initial stage of the simulation was relatively stable and exhibited a simple structure with similar flow fields between adjacent frames. To avoid using similar or simple-structured flow fields when generating the dataset, the flow fields were saved every five steps starting from the 200th step.

A total of 4332 flow fields were saved and used to generate the synthetic dataset. The positions and orientations of the cameras were simulated based on the results of actual camera calibration. The distances between the cameras and flow fields ranged from 1500 to 5000 mm. The cameras were distributed along the Z-axis between 400 and 600 mm. The Z-axis of the camera coordinate system faced the inside of a sphere with a diameter of 100 mm at the center of the flow field. The X- and Y-axes of the camera coordinate system were perpendicular to each other and selected within a plane perpendicular to the Z-axis. The magnitudes of the BOS displacements are typically on the order of one pixel [50]; therefore, in our dataset, the maximum displacement was set to three pixels to avoid system errors. For each saved flow field, simulated BOS image pairs and their corresponding displacements were generated using 12 different positions and orientations of the cameras, as described earlier and shown in Fig. 6(b-d). The displacements are visualized using color maps, as shown in Fig. 6(b). Thus, a synthetic dataset consisting of 51984 pairs of BOS images and corresponding displacements was generated. The dataset was divided into training, validation, and testing sets at a ratio of 7:1.5:1.5.

The model used in the simulated data is shown in the diagram, except for the ART reconstruction, which is indicated by dashed lines. During training, a smooth L1 loss was selected as the loss function, which was introduced in the fast R-CNN paper [51]. Its characteristics include addressing the problem of zero-point non-smoothness compared to the L1 loss and being less sensitive to outliers when x is large compared to the L2 loss. This is a slow-changing loss. Assuming that x represents the numerical difference between the predicted box and the ground truth box, it is defined in the following equation:

The proposed network was implemented using the PyTorch library and trained on a PC with an NVIDIA RTX 4090 GPU. The Adam optimizer was used with ${\beta _1} = 0.9$ and ${\beta _2} = 0.999$ to optimize the proposed network. The initial learning rate was set to 1 × 10⁻³, and the learning rate was reduced by half every 12 epochs. To prevent overfitting, training and validation were stopped when the change in the loss function value was less than 1 × 10⁻⁵ or when the maximum predefined number of iterations was reached. A total of 100 epochs were trained, requiring approximately 20 h to converge and validate the test set, as shown in Fig. 7.

 

Fig. 7. Loss function variation curve.

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To quantitatively evaluate the accuracy of the CNN structure, the mean relative error (MRE) was used in the numerical simulations, as defined by the equation

Here, ${\mu _x}$ and ${\mu _y}$ represent the mean of $x$ and y, respectively, and ${\sigma _x}$ and ${\sigma _y}$ indicate the variance. Besides, ${\sigma _{xy}}$ stands for the covariance of x and y. Additionally, ${c_1}$ and ${c_2}$ are regularization parameters, avoiding the instability of results when the mean and variance approach 0, as defined by the equation:

Here, the parameters are default constants. ${k_1}$ and ${k_2}$ are equal to 0.01 and 0.03, respectively. L is the dynamic range of pixel values with a default value of 255 [52].

The horizontal and vertical slice distributions of one instantaneous reconstruction result are shown in Fig. 8. The reconstruction results are presented in Fig. 9. Figure 9 shows five representative 3D reconstruction results retrieved from the uniform sampling of the 650 test dataset samples. In addition to the qualitative comparisons, a quantitative analysis of the reconstruction results for these five samples was conducted based on Eqs. (7) and (8). Table 1 indicates that the proposed GRU model can reconstruct the three-dimensional distribution of the flow field with high accuracy.

 

Fig. 8. Slices of one instantaneous reconstruction result: slices along the vertical direction from −3 mm to 4 mm, and slices along the horizontal direction from −70 mm to 10 mm.

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Fig. 9. Three-dimensional refractive index distribution of different frames.

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Table 1. The verification results of the GRU model on the simulation dataset

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5. 3D BOS based on GRU

Finally, to further evaluate the generalization ability of the proposed network, the GRU model was tested using BOS images captured in actual experiments. The thermal plume above the candle flame distorts the background image. The experimental setup is shown in Fig. 10. Fifteen AVT Guppy F-125B cameras were distributed in a 160° area surrounding the thermal plume above the candle flame. Each camera was connected to a Sony ICX445 lens with a focal length of 12 mm and resolution of 1292 × 964 pixels. The shutter time of the cameras was set to 500 µs. The images were captured at a frame rate of 30 Hz.

 

Fig. 10. Experimental setup. Fifteen cameras were arranged around the hot air flow above the candle flame. The cameras numbered with blue are for reconstruction, and the cameras numbered with red are for validation. The background plates were illuminated using light-emitting diode light sources.

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Three background plates with a sinusoidal pattern were fixed approximately 750 mm away from the candle, at a distance of approximately 650 mm from the candle to the cameras. The background board was illuminated by LED light sources.

After calibration, the GRU model was used for the BOS reconstruction in the actual experiment. A total of 150 projection frames were captured by the cameras, with each dataset consisting of distorted images influenced by the flow field from 15 views. By comparing distorted and undistorted images, the displacement in the imaging plane can be calculated using a cross-correlation algorithm [53]. The cross correlation algorithm aims to identify a flow image window that matches the reference image’s interrogation window. Specifically, the interrogation window slides in the search area of the flow image pixel by pixel horizontally and vertically, and the cross correlation coefficients of the corresponding areas are calculated. The cross correlation coefficient is defined by the following equation [54]:

As shown in Fig. 11, 126 × 185 sample points were selected from each image. The probe region had dimensions of 120 × 120 × 192 mm and was divided into a grid of 100 × 100 × 160 cells, as shown in Fig. 11.

 

Fig. 11. Displacements of camera (left) and the probe region of flow (right).

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The GRU model employs the architecture shown in the Fig. 2, with the ART-reconstructed results used as the ground truth for the input data during training. This resulted in a dataset consisting of 2250 pairs of displacement maps and the corresponding ART reconstruction results. The dataset was split into training, validation, and testing sets at a ratio of 8:1:1. A smooth L1 loss function was selected during training. The proposed network was implemented using the PyTorch library and was trained on a PC with an NVIDIA RTX 4090 GPU. The Adam optimizer was used with ${\beta _1} = 0.9$ and ${\beta _2} = 0.999$ to optimize the proposed network. The initial learning rate was set to 1 × 10⁻³, and the learning rate was reduced by half every 12 epochs. To prevent overfitting, training and validation were stopped when the change in the loss function value was less than 1 × 10⁻⁵ or when the maximum predefined number of iterations was reached. A total of 150 epochs were trained, which required approximately 32 h to converge and validate the test set, as shown in Fig. 12.

 

Fig. 12. Loss function variation curve.

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To verify the accuracy of the method, 12 cameras (shown in blue in Fig. 10) were selected to participate in the GRU reconstruction, and the other three cameras (shown in red in Fig. 10) were used for validation. After obtaining the reconstruction results, the re-projected displacements of the other three cameras were calculated using ray tracing based on Eqs. (1) and (2). A comparison between the measured displacements calculated by the cross-correlation algorithm and the re-projected displacements calculated by ray tracing is shown in Fig. 13 (a). The correlation coefficient (CC) was used to indicate the similarity between the measured displacements and re-projected displacements. The CC between images A and B is expressed by the following equation:

 

Fig. 13. (a). Comparisons between measurement displacements and reprojection displacements: the first row is the reprojection displacements of ART method. The second row is the reprojection displacements of our method (the rendering scalar is consistent with Fig. 11). (b). The Euclidean distance error between measurement displacements and reprojection displacements.

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The distributions of the horizontal and vertical slices of the instantaneous reconstruction results are shown in Fig. 14(a). The figure demonstrates that the proposed method can be used to reconstruct the complex structure of a flow field. However, artifacts still existed in the slice images and were primarily concentrated at the top of the thermal plume above the flame. This is because the training data for the GRU model still use the ART reconstruction results, and ART has difficulty in completely reducing noise [55]; therefore, noise is almost unavoidable. ART reconstruction results can have petal-shaped artifacts in the surrounding areas, as shown in red box in Fig. 14(b), and the number of petal-shaped artifacts is related to the number of projection angles in linear tomography [56]. Artifacts often have wide-ranging and evident effects. Compared to the ART reconstruction results, the range of artifacts in our reconstruction results was smaller and closer to the background, greatly reducing the impact of artifacts and improving the reconstruction effect.

 

Fig. 14. (a). Slices of one instantaneous GRU reconstruction result: slices along the vertical direction from −16.8 mm to 0 mm, and slices along the horizontal direction from −72 mm to 24 mm. (b). Slices of ART reconstruction result along the vertical and the horizontal direction (artifacts are highlighted in red box)

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Fig. 15. Distribution of errors between measurement displacements and reprojection displacements: (a) scatter plot of errors, (b) frequency of different errors along the u-axis direction, and (c) frequency of different errors along the v-axis direction.

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Finally, we selected the reconstruction results for the five different frames, as shown in Fig. 16. As mentioned above, the quantitative reprojection errors were calculated. Detailed information is provided in Table 2. The reprojection errors for different frames are approximately the same, thereby verifying the stability of the method. Additionally, it is worth noting that compared to the ART algorithm (with 200 iterations), the proposed GRU model demonstrates outstanding advantages in terms of computational efficiency. Table 3 shows the time taken for reconstruction using different methods, with an average of only 1.043 s per frame, compared with the ART method (200 iterations) with an average of 120 min per frame.

 

Fig. 16. Three-dimensional refractive index distribution of different frames predicted by GRU model. The distributions of consecutive frames are shown in Visualization 1. The display of distribution under different viewing directions is shown in Visualization 2.

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Table 2. The reprojection errors of the GRU model on the real dataset

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Table 3. Time consumption of the GRU model and ART

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6. Summary

In summary, we proposed a GRU-based BOS-CT model that enables fast three-dimensional reconstruction of real flow fields. First, the accuracy of the method was validated using methane combustion simulations by calculating the MRE of the reconstructed results. Additionally, the GRU model was used to reconstruct the refractive index of hot air flow above a candle flame using a system composed of 12 cameras. The reprojection displacements of the three additional cameras were obtained via ray tracing. The reliability of the reprojection displacements compared to the measured displacements was verified using cross-correlation algorithms. Our method established a correlation between the projection data and the three-dimensional distribution based on the physical nature of multidirectional projections, eliminating the need for differential steps and ray tracing required in traditional reconstruction methods. The reconstruction time per frame was reduced from two hours to 1 s, greatly reducing radial artifacts. However, in experimental scenarios where actual flow field distribution data are lacking, the training of the model still requires ART reconstruction results as ground truth values. Therefore, the noise introduced by the ART is unavoidable and becomes the main limitation to the prediction accuracy of the model. In future research, we will focus on overcoming the limitations of the ART and improving the generalization capability of the model.