# Comparison of Machine Learning Methods for Image Reconstruction Using the LSTM Classifier in Industrial Electrical Tomography

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Research Object

#### 2.2. Data Preparation

#### 2.3. The Concept of the Method Oriented Ensemble (MOE)

# | Algorithm 1. The pseudocode algorithm for training MOE |

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. | m = 96 % number of measurements n = 2883 % number of finite elements in reconstruction mesh (pixels) Train n models ${f}_{1}\left({x}_{1\dots m}\right)\to {y}_{1\dots n}$ with method # 1 (e.g., EN) Train n models ${f}_{2}\left({x}_{1\dots m}\right)\to {y}_{1\dots n}$ with method # 2 (e.g., LR-LS) Train n models ${f}_{3}\left({x}_{1\dots m}\right)\to {y}_{1\dots n}$ with method # 3 (e.g., LR-SVM) Train n models ${f}_{4}\left({x}_{1\dots m}\right)\to {y}_{1\dots n}$ with method # 4 (e.g., SVM) Train n models ${f}_{5}\left({x}_{1\dots m}\right)\to {y}_{1\dots n}$ with method # 5 (e.g., ANN) % Assigning the RMSE for each method and pixel for i = 1:5 % for 5 methods: EN, LR-LS, LR-SVM, SVM, ANN for j = 1:n % for n = 2883 pixels calculate RMSE(i, j) % assignment root mean square error for i-th method and j-th pixel end meanRMSE(i) = mean(RMSE(i,:)) % Calculate the mean RMSE for each of the 5 methods. end % Assignment meanRMSE for i-th method and all 2883 pixels. Prepare the training set to train the LSTM classifier. Inputs—96 measurements. Output—5 categories/classes. Select the method with the lowest meanRMSE. Reconstruct all n pixels using the selected method. |

#### 2.4. Elastic Net (EN)

_{1}and L

_{2}norms, or, to put it another way, between Robert Tibshirani’s LASSO (Least Absolute Shrinkage and Selection Operator) and ridge regression is known as Tikhonov regularization. The approach is also successful when there are numerous correlated predictors or when the number of discretized current elements is substantially more significant than the number of measurement points. Task (1) can be used to indicate the problem that determines the elastic net:

_{1}and L

_{2}norms of unknown parameters ${\beta}^{\prime}$, as shown by Equation (2). The trade-off between LASSO and ridge regression is represented by parameter $0\le \alpha \le 1$. It is pure ridge regression if the value is $\alpha =0$, but it is pure LASSO if the value is $\alpha =1$.

#### 2.5. Linear Regression with Least-Squares Learner (LR-LS)

#### 2.6. Linear Regression with Support Vector Machine Learner (LR-SVM)

_{1}LASSO regularization technique, employs a regression model that incorporates the “absolute value of magnitude” into the loss function as a penalty component. In this study, the “learner” was a linear regression model based on the SVM approach, which was utilized [45]. $f\left(x\right)=x\beta +b$ is the loss function for a linear regression model type, where β denotes a vector of pp coefficients, x denotes an observation of p predictor variables, and b denotes a scalar bias. The mean square error (MSE) is determined as a loss function in the implemented algorithm, and it takes the form of the formula $\ell \left[y,f\left(x\right)\right]=\mathrm{max}\left[0,\text{}\left|y-f\left(x\right)\right|-\epsilon \right]$ where $y\in \left(-\infty ,\infty \right)$ is a reconstruction of the response value. The LASSO cost function is represented by the Equation (3)

#### 2.7. Support Vector Machine (SVM)

#### 2.8. Artificial Neural Network (ANN)

#### 2.9. The Long Short-Term Memory (LSTM) Network for Classification

**C**features (channels) of length

**S**through an LSTM layer, as depicted in Figure 5. The output (also known as the hidden state) and the cell state at time step t are represented by the symbols

**h**and

_{t}**c**, respectively, in the diagram [46]. For example, the first LSTM block considers both the network’s starting state and the first time step of the sequence to compute both the first output and the updated cell state. To compute the output and the updated cell state

_{t}**c**at time step t, the block uses the current state of the network (c

_{t}_{t−}

_{1}, h

_{t−}

_{1}) and the next time step in the series (time step t).

**W**, the recurrent weights

**R**, and the biases

**b**.

_{1}of dimension X by Y by Z, then the output of the fully connected layer is an array A

_{2}of size X’ by Y by Z. The proper input to A

_{2}at time step t is ${\mathrm{WA}}_{t}+b$, where A t is the time step t of A and b is the bias. Glorot initializer was used to generate the weights for this layer in this research [47]. The softmax is the penultimate layer. It is a common type of layer in deep categorization neural networks. An ultimately linked layer is always preceding the softmax layer. The formula ${y}_{r}\left(x\right)={e}^{{a}_{r}\left(x\right)}/{\sum}_{j=1}^{k}{e}^{{a}_{j}\left(x\right)}$ denotes the softmax activation function, with $0\le {y}_{r}\le 1$, and ${\sum}_{j=1}^{k}{y}_{j}=1$.

_{c}is the number of correctly rebuilt pixels, and N is the total number of pixels [49]. Equation (9) defines the cross-entropy loss between network predictions and target values

_{i}is the number of patterns, and X

_{i}is the number of network outputs. The training-progress graphic demonstrates the correctness of the training. Indeed, it indicates the accuracy of each minibatch’s classification. This number increases to 100% for optimal training development. At the finish of the training procedure, the classifier’s accuracy oscillates between 99% and 100%. It took approximately 12 min to train. The computation was carried out on a personal computer configured as follows: 2.80 GHz Intel

^{®}CoreTM i5-8400 CPU, 16 GB RAM, NVIDIA GeForce RTX 2070 GPU. Parallel computing with GPU was used [50,51].

## 3. Results and Discussion

#### 3.1. Visualizations of Real Measurements

#### 3.2. Comparison of the Reconstructions Based on Simulation Data

^{−5}).

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**The key elements of the test stand: (

**a**)—electrical impedance tomograph connected to the electrodes; (

**b**)—a physical model of a tank with plastic tubes immersed in water, (

**c**)—cylinder diameter in (cm).

**Figure 3.**A sample training measurement case generated with the Eidors toolbox: (

**a**)—cross-section with a visible inclusion; (

**b**)—values of 96 measurements corresponding to the cross-section containing the inclusion.

**Figure 6.**Gates interaction in the LSTM network [46].

**Figure 10.**Real measurements: (

**a**)—dimensioned test stand for real cases. Three variants of the arrangement of phantoms in the tested tank with 16 electrodes: (

**b**)—2 phantoms, (

**c**)—3 phantoms, (

**d**)—4 phantoms.

**Figure 11.**Image reconstructions based on real measurements: (

**a**)—a case with two inclusions, (

**b**)—a case with three inclusions, (

**c**)—a case with four inclusions.

# | Layer Description | Activations | Learnable Parameters (Weights and Biases) | Total Learnables | States |
---|---|---|---|---|---|

1 | Sequence input with 96 dimensions | 96 | - | 0 | - |

2 | BiLSTM with 128 hidden units | 256 | Input weights: 1024 × 96 Recurrent weights: 1024 × 128 Bias: 1024 × 1 | 230,400 | Hidden state 256 × 1 Cell state 256 × 1 |

3 | Batch normalization | 256 | Offset: 256 × 1 Scale: 256 × 1 | 512 | - |

4 | BiLSTM with 128 hidden units | 256 | Input weights: 1024 × 256 Recurrent weights: 1024 × 128 Bias: 1024 × 1 | 364,240 | Hidden state 256 × 1 Cell state 256 × 1 |

5 | Fully connected layer | 5 | Weights: 5 × 256 Bias: 5 × 1 | 1285 | - |

6 | Softmax | 5 | - | 0 | - |

7 | Classification output (cross entropy) | 5 | - | 0 | - |

**Table 2.**Reconstruction quality indicators for homogenous methods—testing set. The best values of indicators are marked in blue.

Case Number | Indicator | Methods of Reconstruction | Best Homogenous Method (MOE Concept) | ||||
---|---|---|---|---|---|---|---|

EN | LR-LS | LR-SVM | SVM | ANN | |||

1 | RMSE | 0.289 | 0.133 | 0.145 | 0.135 | 0.068 | ANN |

RIE | 0.293 | 0.135 | 0.147 | 0.137 | 0.069 | ANN | |

MAPE | 1610.2 | 1880.7 | 2142.1 | 1988.4 | 126.6 | ANN | |

ICC | 0.875 | 0.536 | 0.392 | 0.530 | 0.914 | ANN | |

2 | RMSE | 0.299 | 0.1519 | 0.123 | 0.117 | 0.132 | SVM |

RIE | 0.306 | 0.155 | 0.126 | 0.120 | 0.135 | SVM | |

MAPE | 2845.92 | 2659.4 | 2191.2 | 1990.1 | 1087.3 | ANN | |

ICC | 0.905 | 0.709 | 0.843 | 0.853 | 0.794 | EN | |

3 | RMSE | 0.304 | 0.157 | 0.077 | 0.084 | 0.140 | LR-SVM |

RIE | 0.318 | 0.164 | 0.081 | 0.087 | 0.147 | LR-SVM | |

MAPE | 4467.8 | 2935.4 | 846.1 | 835.3 | 1239.5 | SVM | |

ICC | 0.986 | 0.826 | 0.960 | 0.953 | 0.870 | EN | |

4 | RMSE | 0.317 | 0.135 | 0.076 | 0.085 | 0.152 | LR-SVM |

RIE | 0.336 | 0.143 | 0.081 | 0.090 | 0.161 | LR-SVM | |

MAPE | 5952.1 | 2536.1 | 916.7 | 988.2 | 1584.6 | LR-SVM | |

ICC | 0.950 | 0.900 | 0.969 | 0.962 | 0.876 | LR-SVM |

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Kłosowski, G.; Rymarczyk, T.; Niderla, K.; Rzemieniak, M.; Dmowski, A.; Maj, M.
Comparison of Machine Learning Methods for Image Reconstruction Using the LSTM Classifier in Industrial Electrical Tomography. *Energies* **2021**, *14*, 7269.
https://doi.org/10.3390/en14217269

**AMA Style**

Kłosowski G, Rymarczyk T, Niderla K, Rzemieniak M, Dmowski A, Maj M.
Comparison of Machine Learning Methods for Image Reconstruction Using the LSTM Classifier in Industrial Electrical Tomography. *Energies*. 2021; 14(21):7269.
https://doi.org/10.3390/en14217269

**Chicago/Turabian Style**

Kłosowski, Grzegorz, Tomasz Rymarczyk, Konrad Niderla, Magdalena Rzemieniak, Artur Dmowski, and Michał Maj.
2021. "Comparison of Machine Learning Methods for Image Reconstruction Using the LSTM Classifier in Industrial Electrical Tomography" *Energies* 14, no. 21: 7269.
https://doi.org/10.3390/en14217269