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Solving Electrical Impedance Tomography with Deep Learning

Yuwei Fan, Lexing Ying

arXiv:1906.03944v2physics.comp-phcs.LGcs.NAmath.NA

TL;DR

EIT seeks to recover conductivity from boundary measurements, but its inverse map is ill-conditioned and numerical solutions can be costly. This paper uses perturbative analysis and a low-rank observation to design neural networks for the forward and inverse maps; numerical results show reasonable accuracy.

  • Problem

    EIT requires recovering conductivity from boundary observations, while the inverse map is numerically unstable from ill-conditioning and can require regularization and many iterations.

    Source: §1 · p. 2

  • Method

    Perturbative analysis motivates compact neural-network architectures for approximating the forward and inverse maps between conductivity and the DtN map.

    Source: §1 · p. 2, §5 · p. 18

  • Results

    Numerical results show that the proposed networks approximate the forward or inverse maps with reasonable accuracy.

    Source: §5 · p. 19

  • Takeaways & Limitations

    The proposed architectures provide a neural-network approach to approximating EIT’s forward and inverse maps in the studied settings.

    Source: §5 · pp. 18–19

    Adversarial robustness remains a subject for future study, despite the authors’ expectation that the compact networks may be less vulnerable.

    Source: §5 · p. 20

Abstract

from arXiv · show

This paper introduces a new approach for solving electrical impedance tomography (EIT) problems using deep neural networks. The mathematical problem of EIT is to invert the electrical conductivity from the Dirichlet-to-Neumann (DtN) map. Both the forward map from the electrical conductivity to the DtN map and the inverse map are high-dimensional and nonlinear. Motivated by the linear perturbative analysis of the forward map and based on a numerically low-rank property, we propose compact neural network architectures for the forward and inverse maps for both 2D and 3D problems. Numerical results demonstrate the efficiency of the proposed neural networks.

1 Introduction

EIT uses boundary voltage and current measurements to image an unknown conductivity distribution, but its forward and inverse maps are high-dimensional, and inversion is ill-conditioned. The paper motivates compact neural networks with a perturbative, numerically low-rank view of the maps and applies them to 2D and 3D problems.

1 Introduction

  • EIT determines an unknown medium’s conductivity from boundary voltage and current measurements, enabling repeated, non-invasive monitoring in medical and industrial applications.The paper cites ventilation distribution, lung overdistension, pneumothorax [18], and industrial applications.Source: §1 · p. 1
  • The inverse problem recovers η from the DtN kernel λη, but remains severely ill-conditioned even when a solution exists and is unique under certain conditions.The ill-conditioning is attributed to the elliptic nature of EIT.Source: §1 · p. 2
  • The forward map takes a d-dimensional function to a 2(d−1)-dimensional function, making full 3D DtN-map computation and representation expensive; inverse solvers often need regularization and many iterations.The inverse map is numerically unstable due to ill-conditioning.Source: §1 · p. 2
  • Perturbative analysis indicates that, after reparameterizing the DtN map, the forward and inverse maps are locally numerically low-rank, reducing the mapping to two (quasi) (d−1)-dimensional functions.This observation motivates the proposed neural-network architecture.Source: §1 · p. 2
  • BCR-Net approximates both maps with compact multiscale networks, using about 10^4–10^5 parameters in 2D and 10^5–10^6 in 3D, which supports training on limited datasets.The paper discusses the 2D design and tests in Section 3 and extends the results to 3D in Section 4.Source: §1 · p. 3

2 Mathematical analysis of the DtN map

The section relates the DtN kernel to the Green function and analyzes how it changes with the potential. For sufficiently small perturbations, a first-order approximation of the difference map is linear and motivates the neural-network designs.

2 Mathematical analysis of the DtN map

  • The DtN kernel is expressed in terms of the Green function by applying the divergence theorem and taking a boundary normal derivative.The derivation uses G as the Green function of L = −Δ + η and u as the solution of the governing equation.Source: §2 · p. 3
  • For η sufficiently small, the Green operator is expanded as a Neumann series around the background operator G0.The series begins G = G0 + G0EG0 + G0EG0EG0 + … .Source: §2 · p. 3
  • Difference imaging subtracts the background kernel λ0 from λη, and for sufficiently small η the difference operator is approximated by its first, E-linear term G0EG0.With E = −ηI, this yields an approximation for the difference DtN map μ.Source: §2 · pp. 3–4
  • The first-order difference-map approximation provides the motivation for the neural-network architectures.The analysis is framed as a perturbation around a fixed background potential η0, taken as zero for simplicity.Source: §2 · pp. 3–4

3 Neural network for the 2D case

This section develops forward and inverse networks for 2D EIT using dimension reduction with one-sided and two-sided detection, and examines reconstructions of smooth inclusions and shapes under noise.

3 Neural network for the 2D case

  • The 2D setup uses periodic left and right boundaries, with electrodes on the top alone or on both top and bottom; one-sided detection assumes zero Dirichlet data at the bottom.The analysis first considers one-sided detection and then extends the architecture to two-sided detection.Source: §3 · p. 4

3.1 Forward map for the one-sided detection

  • For the one-sided forward map, kernel smoothness in h and z enables a low-rank approximation that reduces computation to 1D convolutions, implemented by encoding, BCR-Net processing, and decoding.The linearized approximation applies when η is sufficiently small; BCR-Net provides the intermediate multiscale convolution module.Source: §3.1 · pp. 5–7

3.2 Inverse map for the one-sided detection

  • For the inverse map, combining the components suggests a suitable architecture: the neural-network architecture of the forward map followed by a 2d convolutional neural network.Its intermediate layers use ReLU, while its final layer uses a linear activation because η can take any real number.Source: §3.2 · p. 8

3.3 Inverse map for the two-sided detection

  • Two-sided detection produces four DtN components, and each forward component can be factorized by the one-sided dimension-reduction procedure when η is sufficiently small.The components correspond to top-to-top, top-to-bottom, bottom-to-top, and bottom-to-bottom measurements.Source: §3.3 · pp. 8–9

3.3.1 Architecture for the inverse map

  • For the two-sided inverse map, the four measurement branches are combined into a single Nx × 4c array before joint decoding; symmetry also permits weight sharing between mirrored branches.The authors report that combining before decoding performs better than summing decoded results, and use shared layers with flipped outputs for mirrored boundary pairs.Source: §3.3.1 · p. 9

3.4 Numerical results

  • The 2D experiments use a 160 × 80 grid with 10K training and 10K test pairs, evaluate relative ℓ2 error, and repeat training five times with different random seeds.The test error is averaged over the test samples.Source: §3.4 · p. 10

3.4.1 Smooth potential case.

  • With shallow inclusions, the one-sided forward network is relatively accurate at c=6 using 28K parameters, while c=8 balances accuracy and efficiency.The comparison is reported for the forward problem; the cited figure presents test errors and parameter counts across channel choices.Source: Figure 5 · p. 11, §3.4.1 · p. 11
  • For shallow inclusions, c = 10 uses 78K parameters and produces accurate Gaussian locations, shapes, and magnitudes.These results demonstrate that the proposed architectures can represent the forward and inverse maps for shallow inclusions.Source: §3.4.1 · p. 12
  • For deep inclusions, one-sided detection predicts the forward map comparably to shallow cases but fails on the inverse map, with large errors near the bottom boundary.The passage attributes the failure to inverse-problem instability and notes that adding bottom-boundary electrodes can provide more information.Source: §3.4.1 · p. 12, Figure 8 · p. 13
  • Two-sided detection significantly lowers deep-inclusion test error, slightly below the shallow one-sided case; at c = 10, the network uses 177K parameters.The test error is reported as relatively small even at this channel count.Source: §3.4.1 · p. 12, Figure 9 · p. 13

3.4.2 Shape reconstruction

  • With noisy measurements, shallow-inclusion shape boundaries become blurrier as noise grows, but the network still predicts the shapes’ positions and number correctly.The tests use noise levels δ = 0, 0.5%, and 1%.Source: §3.4.2 · p. 13, Figure 11 · p. 14
  • For deep shapes, two-sided detection improves predictions of shapes near z = 0 compared with one-sided shallow detection by using additional mixed-boundary data.The additional data are µ+− and µ−+ alongside µ++ and µ−−; the same noise-related conclusions apply.Source: §3.4.2 · p. 14, Figure 12 · p. 15
  • Across the tested setups, the networks in Algorithms 1–3 learn the forward and inverse EIT problems.Source: §3.4.2 · p. 14

4 Neural network for 3D the case

The paper extends its EIT forward and inverse-map approach to 3D, where the measurement data are four-dimensional and the potential is three-dimensional. It develops architectures for one-sided and two-sided detection and tests them on shallow and deep inclusions.

4.1 Analysis and NN architecture

  • The 3D forward-map approximation retains the three-step procedure, with interpolation over h becoming two-dimensional and convolution operating in 2D.The 3D data μ are four-dimensional, unlike the three-dimensional potential η.Source: §4.1 · pp. 15–16
  • The one-sided 3D inverse architecture encodes the measurements, processes them with BCR-Net2d, decodes to a 3D estimate, then applies CNN3d post-processing.The encoding compresses locally across the first two dimensions, while decoding extends the representation to three dimensions.Source: §4.1 · pp. 16–17
  • For two-sided detection, the network processes four one-sided measurement cases separately, concatenates their representations, then applies convolution before decoding and 3D post-processing.The four representations are concatenated along the channel direction.Source: §4.1 · p. 17

4.2 Numerical results

  • 3D inverse-network test accuracy was comparable to the corresponding 2D cases for shallow one-sided and deep two-sided inclusions; one-sided error decayed before stagnating around c = 10.The one-sided test used 10K training and 5K test pairs; the two-sided test used 8K training and 2K test pairs.Source: Figure 14 · p. 18, §4.2 · p. 18

5 Conclusions

The paper proposes neural networks for EIT’s forward and inverse maps, with weak-inclusion perturbative analysis motivating dimension reduction. It reports reasonable approximation accuracy and highlights efficiency, compactness, and learned inverse regularization.

5 Conclusions

  • The proposed neural networks approximate EIT’s forward and inverse maps, with weak-inclusion perturbative analysis suggesting dimension reduction for the architecture design.Source: §5 · pp. 18–19
  • Test samples show relative errors of 5.2e −2 for one-sided shallow-inclusion detection and 8.1e −2 for two-sided deep-inclusion detection, both with c = 8 in 3D.Source: Figure 15 · p. 19, Figure 17 · p. 19
  • Well-trained networks predict efficiently and accurately, learn inverse-map regularization from training data, and are compact and easy to train for limited-data applications.Source: §5 · p. 20
  • The discussion focuses on rectangle and cuboid domains; extension to arbitrary convex bounded Lipschitz domains may require careful reparameterization and technical effort.Source: §5 · p. 20
  • The authors expect compact networks may be less vulnerable to adversarial attacks, but leave detailed study of that possibility to future work.Source: §5 · p. 20

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