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Abstract
This paper demonstrates a new method for solving nonlinear tomographic problems, combining calibration-free wavelength modulation spectroscopy (CF-WMS) with a dual-branch deep learning network (Y-Net). The principle of CF-WMS, as well as the architecture, training and performance of Y-Net have been investigated. 20000 samples are randomly generated, with each temperature or H2O concentration phantom featuring three randomly positioned Gaussian distributions. Non-uniformity coefficient (NUC) method provides quantitative characterizations of the non-uniformity (i.e., the complexity) of the reconstructed fields. Four projections, each with 24 parallel beams are assumed. The average reconstruction errors of temperature and H2O concentration for the testing dataset with 2000 samples are 1.55% and 2.47%, with standard deviations of 0.46% and 0.75%, respectively. The reconstruction errors for both temperature and species concentration distributions increase almost linearly with increasing NUC from 0.02 to 0.20. The proposed Y-Net shows great advantages over the state-of-the-art simulated annealing algorithm, such as better noise immunity and higher computational efficiency. This is the first time, to the best of our knowledge, that a dual-branch deep learning network (Y-Net) has been applied to WMS-based nonlinear tomography and it opens up opportunities for real-time, in situ monitoring of practical combustion environments.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Tunable diode laser absorption spectroscopy (TDLAS) has served as a powerful species diagnostic tool in energy and power systems over the past decades for non-intrusive, highly sensitive and quantitative measurements of various gas properties [1–3]. Direct absorption spectroscopy (DAS) and wavelength modulation spectroscopy (WMS) are the two most common schemes for TDLAS sensing. In particular, WMS is more advantageous for situations with low absorbance, high pressure, or blended absorption features [4,5].
However, as a line-of-sight (LOS) technique, TDLAS has been limited to path-integrated sensing along the laser beam. Various approaches have been adopted to retrieve spatially-resolved distributions, including single-beam absorption thermometry [6,7], tomographic absorption spectroscopy (TAS) with multiple laser beams [8,9], and laser absorption imaging (LAI) with a camera [10–12]. Among these approaches, tomographic absorption spectroscopy (TAS) has evolved into a well-established imaging technique from laboratory specialties to real-field combustion environments [8].
Tomography is a canonical inverse problem, such that a forward model (a set of projection measurements) needs to be inverted to estimate the spatial distribution of the quantity of interest (e.g., gas-phase tomography) [13]. TAS can be divided into linear [8,14–16] and nonlinear [8,17–21] modalities according to the basic mathematics defining the problem. Linear TAS, an implementation of classical tomography, records typically one or two absorption transitions with narrow-bandwidth diode lasers. Usually a large number of projections (i.e., a large number of laser beams) are required to alleviate the ill-posedness of the linear equation system and achieve reasonable reconstruction. This largely restricts the application of linear TAS in practical combustion environments where optical access is highly limited. For nonlinear tomographic absorption spectroscopy (NTAS), hyperspectral laser sources [17] or multiplexed techniques [18] are typically adopted, constituting a set of nonlinear equations from which gas property distribution can be recovered. The additional spectral or modulation harmonics information in NTAS complements the reduction of laser beam paths and therefore effectively mitigate the ill-posedness of the inverse problem in linear TAS. The current work focuses on the demonstrations with NTAS.
Extensive research has been dedicated to NTAS using DAS or calibration-free WMS (CF-WMS) technique for the reconstruction of temperature and species concentration [8,17–21]. The DA scheme is commonly used in NTAS due to relatively simple tomographic inversion from linearly aggregated absorbance along the LOS. Ma et al. exploited hyperspectral laser source for tomographic imaging using DAS technique, with a significant reduction in the number of required projections and enhanced stability of the reconstruction [17]. Recently, Cai et al. proposed a multiplexed absorption tomography technique with CF-WMS for simultaneous imaging of temperature and species concentration in harsh environments [18]. Different from linear TAS, NTAS using DAS and CF-WMS typically leads to a sizable system of nonlinear equations. Conventional tomographic algorithms developed for the linear TAS problems cannot solve the inverse problem of nonlinear tomography. It has been proven that the nonlinear tomographic problems can be converted to an optimization problem, and then solved by heuristic algorithms such as the simulated annealing (SA) algorithm [8], or evolution strategy based on covariance matrix adaption algorithm (CMA-ES) [21]. However, the performance of the aforementioned algorithms relies on problem-specific knowledge in heuristics and can also be difficult to implement due to the high computational cost.
Over the past few years, deep learning has gained a tremendous amount of attention for image restoration and reconstruction tasks [22,23], which coincidentally offers a promising alternative for inverse ill-posed problems. A dual-branch convolutional network, namely, CSTNet, was developed by Jiang et al. for high-fidelity, rapid, and simultaneous imaging of temperature and species concentration in linear tomographic problems [16]. For solving nonlinear tomographic problems, Huang et al. proposed a new inversion method based on convolutional neural network (CNN) and the results show that temperature distribution can be rapidly reconstructed using CNN [19]. Deng et al. further investigated the performance of other sophisticated deep neural networks such as deep belief networks (DBN) and recurrent neural network (RNN) in solving nonlinear NTAS problems [20]. However, only preliminary study of DAS-based deep learning methods for nonlinear tomographic problems has been demonstrated, and the reported deep learning networks can only be used to reconstruct either temperature or species concentration alone. Under the inspiration of the pioneering study of DAS-based deep learning method in linear TAS problems by Jiang et al. [16], it is appealing to investigate the nonlinear tomographic problems with more advanced deep learning network. In view of the diagnostics needs and progress, we here propose a novel nonlinear absorption tomography technique, combining a dual-branch deep learning network (Y-Net) with CF-WMS to simultaneously reconstruct temperature and species concentration in harsh environments. This is the first time, to the best of the authors’ knowledge, that a deep learning network is applied to WMS-based nonlinear tomography for simultaneous reconstruction of temperature and species concentration.
2. WMS-based nonlinear tomography
The principle of wavelength modulation spectroscopy has been well documented [1–5], and only relevant spectroscopic fundamentals are briefly reviewed here. In scanned-WMS scheme, a high-frequency sinusoidal modulation at frequency fm is used to modulate the laser wavelength around the targeted absorption transition. The instantaneous optical frequency and intensity can be expressed as:
The transmitted laser intensity It(t) can be obtained by the Beer-Lambert relation.
By normalizing S2f by S1f, G and $\overline {{I_0}}$ can be eliminated, enabling calibration-free WMS-2f/1f scheme. As illustrated in Fig. 1, the tomographic field (i.e., the region of interest, ROI) is discretized into $D \times D$ square pixels. A number of laser diodes with center output frequency ${v_l}(l = 1,2, \ldots L)$ are combined using a wavelength multiplexer and each is modulated at a different frequency fm,l. The lasers are then split into multiple beams to traverse the ROI and then received by the detectors. As illustrated in [8], three types of beam arrangements including parallel-beam, fan-beam and irregular-beam arrangements, are typically implemented for tomographic schemes. Fan-beam arrangement usually suffers lower energy density than line illumination in a parallel-beam arrangement, while irregular-beam arrangement is typically preferred in practical scenarios where optical access is extremely limited. Here we choose to implement parallel-arrangement, and without loss of generality, each set of parallel beams is specified by an angle $\alpha ,$ a distance d, and multiplexed frequencies $({v_1},{v_2}, \ldots {v_L}).$
Fig. 1. Schematic diagram of multiplexed nonlinear absorption tomography based on calibration-free wavelength modulation spectroscopy.
The detected signals are then post-processed using lock-in detection to recover L nonlinear equations for S2f/1f along each measurement projection. The sinogram ${[{({S_{2f/1f}})_v}]_{L \times M}}$ can be expressed as in Eq. (6), with M the number of parallel beams.
In this work, six H2O absorption lines located between 7294.0 cm-1 and 7444.4 cm-1 are chosen for H2O sensing according to Ref. [24]. The spectroscopic parameters for the targeted H2O lines provided by HITRAN 2020 database [25] including wavenumber v0 (cm-1), linestrength S (cm-2·atm-1), and lower-state energy E'‘ (cm-1) are listed in Table 1.
Table 1. Spectroscopic data for the H2O transitions of interest taken from HITRAN 2020
Figure 2 shows the calculated S2f/1f at 7444.37 cm-1 under different temperature and H2O concentration conditions ${a_m},{i_0},{\varphi _1},{i_2},{\varphi _2}$ are set to $0.06{\kern 1pt} {\kern 1pt} \textrm{c}{\textrm{m}^{ - 1}},0.50,1.2\pi ,0.015$ and $1.4\pi$ respectively, according to typical near-infrared diode laser characteristics [4]. The modulation depth am may be further optimized to maximize 2f/1f signal intensity, but is not the main focus of this work. Unlike the DAS absorbance that varies linearly with concentration under optically thin conditions, the calculated WMS-2f/1f signal changes nonlinearly with both temperature and H2O concentration. This will further increase the complexity in the nonlinear tomographic inversion when using conventional heuristic algorithms.
With 24 parallel beams used, the ROI is meshed into $24 \times 24$ pixels at four projection angles $(\alpha \textrm{ = }{\textrm{0}^\circ },{45^\circ },9{\textrm{0}^\circ },{135^\circ })$ resulting in a total number of 96 LOS measurement projections. The “true” field to be reconstructed is represented by assumed phantoms, each featuring three randomly positioned Gaussian distributions to characterize the multi-modal nature of the temperature or H2O concentration distributions in practical combustion environments [16]. The distributions of temperature and H2O concentration can be mathematically expressed as:
The performance of tomographic reconstruction is closely related to the complexity of temperature and concentration distributions. However, quantitative characterization of such spatial non-uniformity is challenging and case-dependent. Recently, Xiao et al. proposed a novel approach, namely the non-uniformity coefficient (NUC) method based on uniform design theory (i.e., star discrepancy measure), to quantify the non-uniformity of temperature or concentration distributions [26]. It has been demonstrated that the NUC method outperforms existing methods such as the standard deviation (SD) method [27], the coefficient of variation (CoV) method [28], and the image entropy (IE) method [29], by taking into account the time space and position information (TSPI) of targeted distributions. Here in our work, NUC method offers a benchmark to quantify the non-uniformity (i.e., complexity) of the reconstructed temperature and species concentration fields in tomographic problems. Take temperature distribution as example, the temperature local discrepancy function (TLDF), ${\boldsymbol d}_{PQ}^ \ast ({\boldsymbol \theta }),$ of the local rectangular region ${\boldsymbol \theta }\textrm{ = [}0,{\theta _1}\textrm{]} \times [0,{\theta _2}]$ in the ROI shown in Fig. 1, can be expressed as:
A large NUCT indicates temperature deviations from uniformity such as local heterogeneity or lop-sidedness, while small discrepancy indicates that the temperature is more uniform in the whole ROI. Non-uniformity coefficient of species concentration distribution is similarly denoted by NUCX.
As depicted in Fig. 3, two examples of representative phantoms with different multi-modal distributions of temperature and H2O concentration are demonstrated. The three Gaussian distributions in phantoms (a) and (b) overlap with each other, and the 96 LOS measurement projections of WMS-2f/1f at the six absorption transition peaks are shown in Fig. 3(c). NUC values of phantoms (a) and (b) are 0.096 and 0.105, and the corresponding standard deviations are 442.36 K and 0.022, respectively. Another example with three distributions separated further apart with different locations and sizes is depicted in phantoms (d) and (e) and the accumulated WMS-2f/1f is shown in Fig. 3 (f). Phantoms (d) and (e) present NUC values of 0.055 and 0.062, with standard deviations of 204.97 K and 0.010, respectively. The horizontal axes of Fig. 3 (c) and (f) correspond to the 24 LOS measurement projections at each projection angle.
Fig. 3. Two examples of representative phantoms with different multi-modal distributions of temperature and H2O concentration, and the corresponding WMS-2f/1f. Example 1: (a) ∼ (c), Example 2: (d) ∼ (f). The red crosses are at the center of the phantoms. Horizontal axes of Fig. (c) and (f) correspond to 24 LOS measurement projections at each projection angles, i.e., $(\alpha \textrm{ = }{\textrm{0}^\circ },{45^\circ },9{\textrm{0}^\circ },{135^\circ }),$ respectively.
The resultant sinograms, i.e., the 96 LOS measurement projections in Fig. 3 (c) and (f), are closely related to the characteristics of temperature and H2O concentration distributions. As can be seen in Fig. 3 (c), the accumulated WMS-2f/1f appears more symmetric about the center of the ROI at all four projection angles when the three phantoms closely overlap with each other, whereas the example of Fig. 3 (f) exhibits more asymmetric and complexed features with the phantoms further separated apart. A key benefit of the deep learning methods is the automated extraction of complex data representations, making it a valuable tool especially when dealing with more complicated raw sinogram datasets. High-fidelity tomographic reconstructions can therefore be achieved with deep learning methods by recasting the reconstruction task into a data-driven problem and learning the intricate features from the large amount of tomography data.
3. Y-Net: a dual-branch deep learning network
Deep learning network offers a promising alternative for solving nonlinear tomographic inversion problems. However, as discussed in Section 1, early work by Jiang et al. have only explored such potential for linear tomographic problems [16]. For nonlinear tomographic problems, previous attempts to apply deep learning method to tomographic combustion diagnostics have not considered the inter-dependence of temperature and species concentration as reflected in the molecular absorption spectroscopy data, and only temperature or species concentration reconstruction alone with DAS-based nonlinear tomographic problems has been demonstrated. For the first time, we propose a dual-branch deep learning network (Y-Net) for nonlinear absorption tomography with CF-WMS considering the cross-correlation between temperature and species concentration distributions. Benefiting from the architecture of two up-sampling paths in Y-Net, the associated temperature and species concentration can be simultaneously reconstructed with high accuracy and low computational cost.
Y-Net is conceptually simple and generalizes the U-Net, and the latter has been proven to be an effective segmentation and classification network in biomedical images [30–32]. Simplified U-Net architecture for 2-D signal implementation is illustrated in Fig. 4. The U-Net architecture consists of a contracting path and an expansive path, serving as the encoder and decoder paths, respectively. Here, the input ${I_{\textrm{in}}} \in {{\mathbb R}^{n \times n}}$ is first filtered by several repeated convolutional filters $\bar{\phi },$ followed by max pooling operation $\varphi \in {{\mathbb R}^{\frac{n}{2} \times n}}$ to generate down-sampled feature maps. Mathematically, the down-sampling process can be described as:
where ${I_{\textrm{in}}} \otimes \bar{\phi }$ denotes the multi-channel convolution in U-Net. The max pooling operation $\varphi$ is given by:The Y-Net, an improved version of U-Net, adds a second parallel branch and exploits a dual-branch feature extraction module. Previous works based on a Y-shape-structured neural network have been demonstrated for poly detection [33], photoacoustic imaging [34], digital holographic reconstruction [35,36], etc. The proposed dual-branch Y-Net in this work, enables simultaneous reconstructions of both temperature and species concentration.
The global architecture of Y-Net is Sillustrated in Fig. 5, consisting of an encoder-decoder network with sum-skip-concatenation connections. The sinograms ${[{({S_{2f/1f}})_v}]_{6 \times 96}}$ obtained in Fig. 1 is reshaped into a matrix with size of $24 \times 24,$ serving as the input image of the proposed Y-Net. It is worth noting that variably shaped and sized images (sinograms) can be given as the input to the deep learning network with an updated Y-Net architecture.
The encoding network in Y-Net, i.e., the down-sampling path on the left side, contains three repeated stages to down-sample the feature maps. The number noted on the top (or bottom) of a block denotes the number of filter channels, i.e., the feature depth, and the number on the left indicates the size of the feature map. The arrows as explained in the legend indicate the operations between two layers. Each stage of the down-sampling path has double $3 \times 3$ convolution kernel with a stride of 1, followed by a batch normalization (BN) and a rectified linear unit (ReLU). A $2 \times 2$ max pooling operation layer with a stride of 2 is applied between the stages in the down-sampling path. As illustrated in Fig. 5, the number of filter channels is doubled after each stage of down-sampling, with the dimension of the input image reduced by half each time. The bridge path is similar to the down-sampling path. The decoding network in Y-Net on the right side consists of two up-sampling paths for either temperature or H2O concentration inference. The skip connection concatenates the feature maps from down-sampling path to up-sampling path, providing an alternative way to ensure feature reusability and recover spatial information loss during down-sampling. Each stage in the up-sampling paths includes repeated and transposed convolution layers to up-sample the feature maps, followed by concatenations from the down-sampling path and double $3 \times 3$ convolutions. Finally, with a fully connected layer, the $576 \times 1$ feature vector is converted into $24 \times 24$ temperature or H2O concentration distribution. The proposed Y-Net has a total of 87 layers and 24,052,512 (24 M) learnable parameters including weights and bias at each layer.
The total mean square error (MSE) loss function to evaluate the reconstruction error of both temperature and H2O concentration is defined as:
where LT and LX denote the MSE of temperature and species concentration reconstruction, respectively, and $\lambda$ is a weighting hyper-parameter. LT and LX can be expressed as: where $T_n^G(X_n^G)$ and $T_n^R(X_n^R)$ are the normalized ground truth (G) and reconstruction (R) of temperature (H2O concentration) distributions, respectively. n is the number of the training dataset, and N is the mini-batch size.4. Results and discussions
4.1 Training of Y-Net
A total number of 20000 samples are randomly generated, of which 80% are used for training, 10% for validation, and 10% for testing. Each sample consists of a temperature phantom, a species concentration phantom and a sinogram with the dimension of $6 \times 96.$ As discussed in Section 2, the complexity of the reconstructed temperature and species concentration distributions in this work is quantitively characterized by the NUC method. The histograms in Fig. 6 (a) and (b) show the NUC value distributions of 20000 temperature and species concentration phantoms, respectively. The average NUC value of 20000 temperature phantoms is 0.0887 and the standard deviation is 0.0266. 20000 species concentration phantoms present an average NUC value of 0.0971 with a standard deviation of 0.0294.
Fig. 6. NUC value distributions of 20000 temperature (a) and species concentration (b) phantoms, respectively.
MATLAB R2021a environment (Mathwork, Natick) is used to implement the proposed Y-Net. The training of the network is performed on a PC with Core i9-10900 K CPU (3.7 GHz) and 32 GB of RAM, using NVIDIA GeForce GTX 1050 Ti GPU. Adaptive moment estimation (ADAM) algorithm is chosen to perform the optimization [37]. The initial learning rate is set to 0.0002 determined by range test [19,20] and halved every 10 epochs. The network is trained with a total epoch of 20 and a mini-batch size of 32. To prevent over-fitting of the neural network, the training stops when the network performance on the validation dataset begins to deteriorate. $\lambda$ is set to 1, indicating that the reconstruction of H2O concentration is of equal importance to that of temperature. Figure 7 shows the evolution of the loss function for training dataset and validation dataset in the Y-Net training process, and it is clear that overfitting has not appeared. As illustrated in the enlarged portion of Fig. 7, LT and LX contribute almost equally to LTX and gradually converge, indicating that satisfactory reconstructions of temperature and H2O concentration can be simultaneously achieved.
Fig. 7. Evolution of the loss function for training dataset (total loss function of temperature and H2O concentration reconstruction (black solid line), loss function of temperature reconstruction (blue dashed dot line), and loss function of H2O concentration reconstruction (magenta dashed dot dot line), respectively), validation dataset (red dashed line) in the Y-Net training process.
4.2 Evaluation of Y-Net
The complete training process takes typically 50 ∼ 60 mins for the training dataset with 16000 samples. To further evaluate the performance of this deep learning network, relative errors between the ground truth and reconstructed temperature (H2O concentration) are defined:
Shown in Fig. 8 are three demonstrative cases with the phantoms and corresponding reconstruction errors in the absence of noise. A typical reconstruction time for the test dataset with 2000 samples is on the order of milliseconds using the proposed Y-Net. Reconstructions of temperature and concentration for the three cases with different multi-modal distributions exhibit satisfactory similarity (i.e., locations, sizes, magnitudes) with the ground truth. Much larger discrepancies are observed in the distributions of $\Delta T$ and $\Delta {X_{{H_2}O}}$ due to some low values on the edges of the sinograms, as illustrated in Fig. 3. The average reconstruction errors of temperature and H2O concentration for the testing dataset with 2000 samples are 1.55% and 2.47%, with standard deviations of 0.46% and 0.75%, respectively. Fig. 9 shows the NUC value distributions of the testing dataset with 2000 samples, together with the performance of tomographic reconstructions in the absence of noise with the proposed Y-Net under different NUC conditions. The error bars shown in Fig. 9 are calculated from the standard deviations of the reconstruction errors. It is apparent that the reconstruction errors for both temperature and species concentration distributions increase almost linearly with increasing NUC. The linear relationship may stem from the fact that the phantoms are generated as a combination of a set of Gaussian distributions. The results indicate that the well-designed Y-Net can effectively extract the feature maps from the sinograms and simultaneously achieve satisfactory reconstructions of temperature and H2O concentration within the NUC range of 0.02∼0.20. Assuming that the linear relationship holds between NUTX and eX, the proposed Y-Net still allows satisfactory reconstruction with eX equal to 5.0% of H2O concentration distribution at NUCX = 0.64.
Fig. 8. Three cases with representative phantoms and the corresponding reconstruction errors in the absence of noise. Case 1: (a) ∼ (d), Case 2: (e) ∼ (h), Case 3: (i) ∼ (l).
Fig. 9. NUC value distributions of the testing dataset with 2000 samples, together with the performance of tomographic temperature (a) and species concentration (b) reconstructions with the proposed Y-Net under different NUC conditions.
However, as discussed in [38], for nonlinear tomographic problem, the projections are nonlinear and sensitive to temperature, but linear and relatively insensitive to concentration. This further increase the complexity and difficulty in the reconstruction of concentration. In this work, Y-Net with a dual-branch symmetrical architecture is utilized for proof-of-concept demonstrations and more sophisticated neural network or fine-tuning of the hyper-parameter $\lambda (0 < \lambda < 1)$ could be implemented to further improve the reconstruction of H2O concentration.
Since the U-Net is tailored for applications with small datasets, the performance of the proposed Y-Net with different sizes of dataset needs to be further investigated. In addition, four projections, each with 24 parallel beams are assumed in this work, resulting in a total number of 96 LOS measurements projections. However, in practical combustion environments, the achievable number of projections can be limited. Therefore, the same reconstruction process is repeated and tested with smaller datasets and limited projections. The latter is implemented by increasing the distance between each parallel laser beam from d to 2d or using only two orthogonal projections $(\alpha \textrm{ = }{\textrm{0}^\circ },9{\textrm{0}^\circ }).$ More specifically, three sets of different optical configurations (two orthogonal projections each with 12 or 24 parallel beams, and four projections each with 12 parallel beams) are implemented. Accordingly, this will result in three different sizes of input matrices of $12 \times 12,12 \times 24,24 \times 12$ for the proposed Y-Net.
Fig. 10 (a) and (b) show the resulted reconstruction errors of temperature and H2O concentration for different numbers of dataset (2000, 12000, 20000) and optical configurations (sinograms with size of $12 \times 12,24 \times 12,12 \times 24,24 \times 24$). Each dataset is again split into 80%, 10% and 10% for training, validation and testing, respectively. Improved reconstructions for both temperature and H2O concentration are observed in Fig. 10 (a) and (b) with the number of datasets increasing. It is worth noting that even training the Y-Net with a dataset of as small as 2000 samples, the reconstruction errors for all four optical configurations are within 4.1 ∼ 5.4% and 6.0 ∼ 7.4% for temperature and H2O concentration, respectively. The performances of tomographic reconstruction based on the proposed Y-Net are on similar levels for the two configurations with sizes of $12 \times 24$ and $24 \times 24,$ indicating that satisfactory reconstructions can be achieved with only two orthogonal projections utilized. The proposed Y-Net works well for nonlinear tomographic absorption spectroscopy under small dataset and limited-projections conditions. However, such performance of the proposed Y-Net is closely related to the complexity, i.e., NUC value distributions, of the reconstructed fields, and further research is underway to reveal more deep-level relationship between the complexity and performance of tomographic problems other than the simple tendency already revealed here.
Fig. 10. Reconstruction errors of temperature (a) and H2O concentration (b) for various numbers of dataset and optical configurations.
4.3 Comparison between Y-Net and SA
It has been verified previously that SA algorithm is capable of solving a nonlinear minimization problem in tomographic reconstruction problems [18]. To further investigate the advantages of the proposed Y-Net, comparison with previous state-of-the-art SA algorithm is demonstrated. Three representative cases shown in Fig. 8, with different multi-modal distributions (i.e., locations, sizes, magnitudes), are selected for demonstration. Random Gaussian noise with zero mean and standard deviations of 5%, 10% and 15%, are added to the sinograms of the three cases. The reconstruction errors of temperature and H2O concentration are estimated from 10 realizations at a given noise level for each reconstruction. SA takes about 15∼20 hours for each reconstruction, while a typical reconstruction time is on the order of milliseconds for the Y-Net, demonstrating its apparent superiority in computational efficiency. Take eT as an example, as seen from the upper panels of Fig. 11, the reconstruction errors for both Y-Net and SA are almost equal in the absence of noise. With the increase of noise level, the reconstruction errors of SA increase significantly, while Y-Net exhibits better noise immunity. The same stands true for H2O concentration reconstruction, as illustrated in the lower panels of Fig. 11. The uncertainties of the reconstruction errors for both Y-Net and SA, are estimated from 10 repeated realizations at each Gaussian noise level. The temperature reconstruction error of Y-Net stays below 3.7% even at 15% relative noise level. This is consistent with the results reported elsewhere [39,40] that deep-learning based methods have better noise suppression and removal capability.
Fig. 11. Comparison of reconstruction errors of temperature and H2O concentration between the proposed Y-Net and SA algorithm for three representative cases shown in Fig. 7. Case 1 (Left panels), Case 2 (Middle panels) and Case 3 (Right panels).
Moreover, as discussed in [38], when solving a nonlinear minimization problem with SA, it is critical to determine the regularization parameters in order to incorporate the constraints for smoothness in tomographic reconstruction. Nevertheless, such a task is not always trivial to accomplish. In this work, the optimal regularization parameters are determined based on the two-step methods developed in [38]. However, it has to be noted that for the reconstruction tasks of different temperature and H2O concentration distributions, the optimal parameters may not always be the same and thus need to be determined separately. On the other hand, the proposed Y-Net has greater adaptability and robustness in tomographic reconstruction of various cases within the NUC range of 0.02∼0.20 compared to the SA method. More detailed comparisons between Y-Net and SA can be implemented for various temperature and concentration distributions, but it is out of the technical scope of the current study.
In summary, in addition to the inherited advantages of deep-learning based methods, such as reconstruction accuracy, noise immunity, and computational efficiency, the Y-Net enables simultaneous reconstruction of both temperature and species concentration. It suggests that tomographic absorption sensor with Y-Net architecture is a promising technique for real-time, in situ monitoring of practical combustion environments. Further work could be conducted on more advanced neural network design such as the incorporation of inception modules to simplify the network architecture [41], investigations between the complexity and performance of tomographic problems, and experimental demonstrations in practical flames. In addition, variants of Y-Net developed for multi-output tomographic problems have been reported [36]. For more complicated practical combustion environments with unknown temperature, species concentration and pressure distributions, more advanced neural network architecture can be conceivably devised such as incorporating another branch for pressure distribution reconstruction.
5. Conclusions
This work reports the first dual-branch Y-Net, incorporating calibration-free WMS with deep learning network, to solve nonlinear tomographic problems. Previous demonstration of multi-field reconstruction has only explored linear tomographic problems, and only preliminary study of nonlinear DAS-based tomographic deep learning methods has been demonstrated to reconstruct either temperature or species concentration alone, neglecting the inter-dependence of temperature and species concentration. Some key features of the proposed method are worth highlighting: (1) In addition to inheriting the advantages of deep learning methods such as better noise immunity, higher reconstruction accuracy and computational efficiency, the proposed Y-Net enables simultaneous reconstruction of both temperature and species concentration; (2) NUC method can serve as a benchmark to quantify the complexity (i.e., the non-uniformity) of the reconstructed temperature and species concentration fields in tomographic problems, and only minor performance deterioration is seen with increased field complexity. The results provide guidelines to further reveal more deep-level relationship between the complexity and performance of tomographic problems.
Specifically, a total number of 20000 samples are randomly generated, of which 80% are used for training, 10% for validation, and 10% for testing. Each temperature or H2O concentration phantom contains three randomly positioned Gaussian distributions. The ROI is discretized into $24 \times 24,$ pixels and 24 parallel beams are used at each projection angle $(\alpha \textrm{ = }{\textrm{0}^\circ },{45^\circ },9{\textrm{0}^\circ },{135^\circ }),$ resulting in a total number of 96 LOS measurements projections. A dual-branch Y-Net consisting of an encoder-decoder network with sum-skip-concatenation connections, is proposed with a total of 87 well-designed layers. Temperature and species concentration distributions can be simultaneously reconstructed with high accuracy, and the average reconstruction errors of temperature and H2O concentration are 1.55% and 2.47% for the testing dataset with 2000 samples. With NUC increasing from 0.02 to 0.20, the reconstruction errors for both temperature and species concentration distributions increase moderately and almost linearly. A relatively satisfactory reconstruction of H2O concentration distribution with eX equal to 5.0%, using the proposed Y-Net, can be achieved at NUCX = 0.64.
The performance of the proposed Y-Net under small dataset and limited-projection-number conditions is further investigated and the reconstruction errors for all four optical configurations with a dataset of as small as 2000 samples, are within 4.1 ∼ 5.4% and 6.0 ∼ 7.4% for temperature and H2O concentration, respectively. Moreover, Y-Net shows its superiority in noise immunity and computational efficiency in nonlinear tomographic problems over the state-of-the-art simulated annealing algorithm. In summary, a dual-branch deep learning network (Y-Net) has been demonstrated for WMS-based nonlinear tomography to simultaneously reconstruct temperature and species concentration fields. It is envisioned that such method can be an effective solution for achieving spatially-resolved, real-time, in situ monitoring in complicated practical combustion environments.
Funding
National Natural Science Foundation of China (51976105, 61627804, 91841302).
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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