Electrical Impedance Tomography for Anisotropic Media: a Machine Learning Approach to Classify Inclusions

TL;DR

The paper addresses identifying inclusions and their anisotropy in Electrical Impedance Tomography by leveraging a discretized Dirichlet-to-Neumann map input to machine learning models. It combines a 16-electrode continuum-model D-N matrix with Artificial Neural Networks and Support Vector Machines to detect inclusion presence, count inclusions, estimate radii, and classify anisotropy types in a 2D disk Ω. Key findings show robust radii detection (100% test accuracy on real data) and high anisotropy-detection accuracy (up to 100% for some configurations and 94%+ for others), while inclusion counting remains challenging with the tested methods. The work demonstrates the potential of integrating ML with classical EIT analyses, offering practical implications for rapid anisotropy assessment and size estimation, and outlines clear directions for extending to 3D, more complex conductivity models, and realistic electrode models.

Abstract

We consider the problem in Electrical Impedance Tomography (EIT) of identifying one or multiple inclusions in a background-conducting body $Ω\subset\mathbb{R}^2$, from the knowledge of a finite number of electrostatic measurements taken on its boundary $\partialΩ$ and modelled by the Dirichlet-to-Neumann (D-N) matrix. Once the presence of one inclusion in $Ω$ is established, our model, combined with the machine learning techniques of Artificial Neural Networks (ANN) and Support Vector Machines (SVM), may be used to determine the size of the inclusion, the presence of multiple inclusions, and also that of anisotropy within the inclusion(s). Utilising both real and simulated datasets within a 16-electrode setup, we achieve a high rate of inclusion detection and show that two measurements are sufficient to achieve a good level of accuracy when predicting the size of an inclusion. This underscores the substantial potential of integrating machine learning approaches with the more classical analysis of EIT and the inverse inclusion problem to extract critical insights, such as the presence of anisotropy.

Figures (17)

• Figure 1: Illustration of the function of a single neuron $z$ (highlighted in a blue circle) in a neural network. The green circles represent neurons from the previous (input) layer, and the lines indicate the weights and biases connecting each input neuron to the current neuron $z$. Neuron $z$ receives the input $\mathbf{x}^{(1)} = (x_i^{(1)})_{i=1}^n$ (as in \ref{['eq:input_nn']}) from the previous layer, where each $\textbf{W}_i^{(1)}$ denotes column $i$ of matrix $\textbf{W}^{(1)}\in\mathbb{R}^{n\times m}$ ($\textbf{W}_i^{(1)}$ corresponds to the weight associated with input $x_i^{(1)}$). $\textbf{B}^{(1)}$represents the bias associated with the current neurons in $\mathbf{x}^{(1)}$, contributing to the neuron output, $z$. The activation function $\phi$ is then applied to $z$ as in \ref{['eq:input_nn']}, producing the output $\phi(z)$. If there are subsequent layers, $\phi(z)$ acts as the input $\mathbf{x}^{(2)}$ to the next layer, with an associated weight.
• Figure 2: Simplified illustration of the construction of the optimal hyperplane (line) in SVM. In $2D$/$3D$ the support lines/planes passing through the support vectors and parallel to the optimal hyperplane provide a simple illustration of the regions of classification and the margin in the SVM classification process.
• Figure 3: Confusion matrix for Quadratic Kernel, SVM, for a single inclusion detection showing 72.9% accuracy in prediction. Correct classifications are shown in blue and light blue along the main diagonal of the confusion matrix, while incorrect ones are displayed in red in the off-diagonal entries of the matrix. The darker the colour (blue if the classification is correct), the higher the number of classifications in that category. In an ideal confusion matrix, we would see a dark diagonal (indicating high accuracy) and light red off-diagonal elements (indicating fewer misclassifications).
• Figure 4: Confusion matrix for the test data for detecting the number of $19.4$mm inclusions in a tank showing $32.2\%$ accuracy in prediction.
• Figure 5: CF for detection the number of inclusions of radius $=38.8$mm showing $31.3\%$ accuracy in prediction.

Theorems & Definitions (12)

• Definition 3.1
• Definition 3.2
• Remark 3.1
• Remark 3.2
• Remark 3.3
• Definition 3.3
• Remark 5.1
• Remark 5.2
• Remark 5.3
• Remark 5.4