Comparison of Selected Machine Learning Algorithms for Industrial Electrical Tomography
<p>Comparison of the traditional concept with the improved concept: (<b>a</b>) a single prediction system with 96 predictors and 2883 responses; (<b>b</b>) multiple prediction system composed of 2883 separately trained subsystems, each of which has 96 predictors and 1 response.</p> "> Figure 2
<p>Measurement model in electrical impedance tomography: (<b>a</b>) opposite, (<b>b</b>) neighboring method.</p> "> Figure 3
<p>The test stand: (<b>a</b>) the measurement device—a hybrid tomograph made by the Netrix S.A. Research and Development Center, (<b>b</b>) tank with 2 phantoms, (<b>c</b>) tank with 4 phantoms.</p> "> Figure 4
<p>Three variants of the arrangement of phantoms in the tested tank with 16 electrodes: (<b>a</b>) 2 phantoms, (<b>b</b>) 3 phantoms, (<b>c</b>) 4 phantoms.</p> "> Figure 5
<p>Dimensioned model of the EIT tested tank.</p> "> Figure 6
<p>A training case generated with the simulation method with a graph showing the voltages.</p> "> Figure 7
<p>A mathematical neural model for converting electrical signals into images.</p> "> Figure 8
<p>Image reconstruction for 16 measurement electrodes by the Gauss-Newton method with Laplace regularization: (<b>a</b>) 2 objects, (<b>b</b>) 3 objects, (<b>c</b>) 4 objects.</p> "> Figure 9
<p>Image reconstruction for 32 measurement electrodes by the Gauss-Newton method with Laplace regularization: (<b>a</b>) 2 objects, (<b>b</b>) 3 objects, (<b>c</b>) 4 objects.</p> "> Figure 10
<p>Image reconstruction for 16 measurement electrodes by Multiply Neural Networks: (<b>a</b>) 2 objects, (<b>b</b>) 3 objects, (<b>c</b>) 4 objects.</p> "> Figure 11
<p>Image reconstruction for 32 measurement electrodes by Multiply Neural Networks: (<b>a</b>) 2 objects, (<b>b</b>) 3 objects, (<b>c</b>) 4 objects.</p> "> Figure 12
<p>Image reconstruction for 16 measurement electrodes by multiply LARS: (<b>a</b>) 2 objects, (<b>b</b>) 3 objects, (<b>c</b>) 4 objects.</p> "> Figure 13
<p>Image reconstruction for 32 measurement electrodes by multiply LARS: (<b>a</b>) 2 objects, (<b>b</b>) 3 objects, (<b>c</b>) 4 objects.</p> "> Figure 14
<p>Image reconstruction for 16 measurement electrodes by multiply Elastic net: (<b>a</b>) 2 objects, (<b>b</b>) 3 objects, (<b>c</b>) 4 objects.</p> "> Figure 15
<p>Image reconstruction for 32 measurement electrodes by multiply Elastic net: (<b>a</b>) 2 objects, (<b>b</b>) 3 objects, (<b>c</b>) 4 objects.</p> ">
Abstract
:1. Introduction
2. Materials and Methods
2.1. Electrical Tomography
2.2. Measurement Models
Algorithm 1. The pseudo code to generate learning cases | ||
1. | N = 50000; | % The number of cases |
2. | for 1: N | |
3. | random selection of the number of objects; | % set of NumberOfObjects variable |
4. | for 1: NumberOfObjects | |
5. | random selection of the object’s location; | % center and radius |
6. | end | |
7. | adding an output image to the set of training cases; | % saving response data |
8. | determination of voltages and adding Gaussian noise; | % Gaussian noise = randn(1, 96) × 5 × 10−5 |
9. | saving the values of voltages to the training set; | % saving input data |
10. | end |
2.3. Algorithms and Methods
2.3.1. Image Reconstruction
2.3.2. Gauss-Newton Method
- Um—voltages obtained as a result of the measurements
- Us(σ)—voltages received by numerical calculations (FEM) for given conductivity σ
- σ*—conductivity represents known properties
- λ—regularization parameter (positive real number)
- L—regularization matrix.
2.3.3. LARS
- standardize input variables;
- select the most correlated input variable with the output variable. Add input variable to the linear model;
- determine the residual from the obtained model;
- add a variable which is the most correlated with the residual to the model;
- move coefficient β towards its least-squares coefficient;
2.3.4. Elastic Net
2.3.5. Multiply Neural Network
Algorithm 2. The Matlab code for training multiple ANN system |
% - input matrix 96 × 50000 of training cases |
% - output matrix 2883 × 50000 of training cases |
% Choose a Training Function |
trainFcn = 'trainlm'; % In this case Levenberg-Marquardt backpropagation was chosen |
hiddenLayerSize = 10; % Choose a number of hidden layers |
net = fitnet(hiddenLayerSize,trainFcn); % Create a fitting network under variable ‘net’ |
% Choose input and output pre/post-processing functions |
% ‘removeconstantrows’ - remove matrix rows with constant values |
% ‘mapminmax’ - map matrix row minimum and maximum values to [−1 1] |
net.input.processFcns = {'removeconstantrows','mapminmax'}; |
net.output.processFcns = {'removeconstantrows','mapminmax'}; |
% Setup division of data for training, validation, testing |
net.divideFcn = 'dividerand'; % Divide data randomly |
net.divideMode = 'sample'; % Divide up every sample |
net.divideParam.trainRatio = 70/100; % 70% of cases is allocated for training |
net.divideParam.valRatio = 15/100; % 15% of cases is allocated for validation |
net.divideParam.testRatio = 15/100; % 15% of cases is allocated for testing |
net.performFcn = 'mse'; % Mean Squared Error will be used for performance evaluation |
x = X'; |
y = Y'; |
N=2883; % The resolution of output picture grid |
parfor i=1:N % Start ‘for’ loop with parallel computing |
% Assign an i-th row of reference cases to the variable t. Each of the 2883 lines corresponds |
% to one pixel of the output image |
t = y(i,:); |
% Train the network. The variable ‘nets_for_pixels’ is a structure that consists of 2883 |
% separately trained neural networks. |
[nets_for_pixels{i},~] = train(net,x,t); |
end % End ‘parfor’ loop |
3. Results
3.1. Gauss-Newton Method
3.2. Multiply Neural Networks
3.3. Multiply LARS
3.4. Multiply Elastic Net
3.5. Comparison of Image Reconstructions
4. Conclusions
Author Contributions
Acknowledgments
Conflicts of Interest
References
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Quality Indicators for Testing Set | ANN Type | ||
---|---|---|---|
96—10—1 | 96—10—10 | 96—20—10 | |
0.0069 | 0.0087 | 0.0086 | |
0.7548 | 0.6994 | 0.6897 |
Methods | Evaluation Metrics | Tested Cases | |||||
---|---|---|---|---|---|---|---|
E16_O2 | E16_O3 | E16_O4 | E32_O2 | E32_O3 | E32_O4 | ||
ANN | MSE | 0.0074 | 0.0086 | 0.0076 | 0.0060 | 0.0061 | 0.0058 |
RIE | 0.0869 | 0.0936 | 0.0886 | 0.0782 | 0.0785 | 0.0771 | |
ICC | 0.7356 | 0.7371 | 0.8218 | 0.7484 | 0.8163 | 0.7946 | |
Expected time of image reconstruction [s] | 0.1501 | 0.1578 | 0.1574 | 0.2776 | 0.2785 | 0.2787 | |
Elastic net | MSE | 0.0111 | 0.0148 | 0.0197 | 0.0081 | 0.0131 | 0.0174 |
RIE | 0.2466 | 0.3499 | 0.3451 | 0.2120 | 0.2661 | 0.3300 | |
ICC | 0.5024 | 0.4651 | 0.4535 | 0.5090 | 0.4785 | 0.4702 | |
Expected time of image reconstruction [s] | 0.00062 | 0.00066 | 0.00071 | 0.0013 | 0.0014 | 0.0014 | |
LARS | MSE | 0.0115 | 0.0153 | 0.0203 | 0.0074 | 0.0121 | 0.0160 |
RIE | 0.1053 | 0.1216 | 0.1402 | 0.0871 | 0.1113 | 0.1280 | |
ICC | 0.4658 | 0.4586 | 0.4438 | 0.5261 | 0.5072 | 0.5082 | |
Expected time of image reconstruction [s] | 0.00041 | 0.00095 | 0.00092 | 0.0019 | 0.0018 | 0.0018 | |
Gauss-Newton with Laplace regulari-zation | MSE | 0.0199 | 0.0267 | 0.0351 | 0.0110 | 0.0164 | 0.0225 |
RIE | 0.1661 | 0.2524 | 0.3415 | 0.1563 | 0.1755 | 0.2402 | |
ICC | 0.5290 | 0.4643 | 0.4181 | 0.5853 | 0.5984 | 0.5412 | |
Expected time of image reconstruction [s] | 0.01248 | 0.01010 | 0.00940 | 0.01159 | 0.01229 | 0.01197 |
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Rymarczyk, T.; Kłosowski, G.; Kozłowski, E.; Tchórzewski, P. Comparison of Selected Machine Learning Algorithms for Industrial Electrical Tomography. Sensors 2019, 19, 1521. https://doi.org/10.3390/s19071521
Rymarczyk T, Kłosowski G, Kozłowski E, Tchórzewski P. Comparison of Selected Machine Learning Algorithms for Industrial Electrical Tomography. Sensors. 2019; 19(7):1521. https://doi.org/10.3390/s19071521
Chicago/Turabian StyleRymarczyk, Tomasz, Grzegorz Kłosowski, Edward Kozłowski, and Paweł Tchórzewski. 2019. "Comparison of Selected Machine Learning Algorithms for Industrial Electrical Tomography" Sensors 19, no. 7: 1521. https://doi.org/10.3390/s19071521
APA StyleRymarczyk, T., Kłosowski, G., Kozłowski, E., & Tchórzewski, P. (2019). Comparison of Selected Machine Learning Algorithms for Industrial Electrical Tomography. Sensors, 19(7), 1521. https://doi.org/10.3390/s19071521