Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A DeepONet for inverting the Neumann-to-Dirichlet Operator in Electrical Impedance Tomography: An approximation theoretic perspective and numerical results

Anuj Abhishek, Thilo Strauss

TL;DR

The paper tackles the inverse problem in Electrical Impedance Tomography (EIT) of recovering the internal conductivity $\gamma$ from boundary current-to-voltage data by learning an operator-to-function map from the Neumann-to-Dirichlet operator to conductivity using a DeepONet. It establishes an approximation-theoretic guarantee showing the operator map can be approximated to arbitrary accuracy by composing an encoder, a neural approximator, and a reconstructor, while quantifying three error sources: approximator, encoder, and reconstructor. The authors implement the architecture under the Complete Electrode Model with a practical 16-electrode setup, generate a large synthetic dataset via a finite-element PDE solver, and demonstrate that DeepONet yields more accurate and localized reconstructions of anomalies than the Iteratively Regularized Gauss-Newton (IRGN) baseline, even in the presence of noise. This work provides both theoretical guarantees and empirical evidence that neural-operator approaches can effectively solve operator-to-function mappings in EIT, offering a path toward interpretable and robust non-invasive conductivity imaging.

Abstract

In this work, we consider the non-invasive medical imaging modality of Electrical Impedance Tomography, where the problem is to recover the conductivity in a medium from a set of data that arises out of a current-to-voltage map (Neumann-to-Dirichlet operator) defined on the boundary of the medium. We formulate this inverse problem as an operator-learning problem where the goal is to learn the implicitly defined operator-to-function map between the space of Neumann-to-Dirichlet operators to the space of admissible conductivities. Subsequently, we use an operator-learning architecture, popularly called DeepONets, to learn this operator-to-function map. Thus far, most of the operator learning architectures have been implemented to learn operators between function spaces. In this work, we generalize the earlier works and use a DeepONet to actually {learn an operator-to-function} map. We provide a Universal Approximation Theorem type result which guarantees that this implicitly defined operator-to-function map between the space of Neumann-to-Dirichlet operator to the space of conductivity function can be approximated to an arbitrary degree using such a DeepONet. Furthermore, we provide a computational implementation of our proposed approach and compare it against a standard baseline. We show that the proposed approach achieves good reconstructions and outperforms the baseline method in our experiments.

A DeepONet for inverting the Neumann-to-Dirichlet Operator in Electrical Impedance Tomography: An approximation theoretic perspective and numerical results

TL;DR

The paper tackles the inverse problem in Electrical Impedance Tomography (EIT) of recovering the internal conductivity from boundary current-to-voltage data by learning an operator-to-function map from the Neumann-to-Dirichlet operator to conductivity using a DeepONet. It establishes an approximation-theoretic guarantee showing the operator map can be approximated to arbitrary accuracy by composing an encoder, a neural approximator, and a reconstructor, while quantifying three error sources: approximator, encoder, and reconstructor. The authors implement the architecture under the Complete Electrode Model with a practical 16-electrode setup, generate a large synthetic dataset via a finite-element PDE solver, and demonstrate that DeepONet yields more accurate and localized reconstructions of anomalies than the Iteratively Regularized Gauss-Newton (IRGN) baseline, even in the presence of noise. This work provides both theoretical guarantees and empirical evidence that neural-operator approaches can effectively solve operator-to-function mappings in EIT, offering a path toward interpretable and robust non-invasive conductivity imaging.

Abstract

In this work, we consider the non-invasive medical imaging modality of Electrical Impedance Tomography, where the problem is to recover the conductivity in a medium from a set of data that arises out of a current-to-voltage map (Neumann-to-Dirichlet operator) defined on the boundary of the medium. We formulate this inverse problem as an operator-learning problem where the goal is to learn the implicitly defined operator-to-function map between the space of Neumann-to-Dirichlet operators to the space of admissible conductivities. Subsequently, we use an operator-learning architecture, popularly called DeepONets, to learn this operator-to-function map. Thus far, most of the operator learning architectures have been implemented to learn operators between function spaces. In this work, we generalize the earlier works and use a DeepONet to actually {learn an operator-to-function} map. We provide a Universal Approximation Theorem type result which guarantees that this implicitly defined operator-to-function map between the space of Neumann-to-Dirichlet operator to the space of conductivity function can be approximated to an arbitrary degree using such a DeepONet. Furthermore, we provide a computational implementation of our proposed approach and compare it against a standard baseline. We show that the proposed approach achieves good reconstructions and outperforms the baseline method in our experiments.
Paper Structure (20 sections, 8 theorems, 24 equations, 3 figures, 3 tables)

This paper contains 20 sections, 8 theorems, 24 equations, 3 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Mathematical preliminaries and background of the EIT problem
  3. Related Research
  4. The Learning Problem for EIT
  5. Main Approximation Theorem
  6. Approximator error
  7. Encoding error
  8. Reconstruction error
  9. Network Architecture
  10. Experiments
  11. Training the Neural Network
  12. Training Data
  13. Training and Testing
  14. Performance
  15. Baseline: Iteratively Regularized Gauss-Newton Method
  16. ...and 5 more sections

Key Result

Theorem 3.1

\newlabellem:20 Consider the sets $\mathbb{H}_{r}, D_{\Lambda}$, $\tilde{ \Gamma}$, and $\mathcal{G}^{\dagger}_{\text{ext}}$ as above. Note $D_{\Lambda}$ is a compact set of $\mathbb{H}_{r}$. We will show that $\forall \epsilon >0$, there exist numbers $M \text{ and }P$ and continuous maps $\mathc

Figures (3)

  • Figure 1: The true map $\mathcal{G}^{\dagger}_{\text{ext}}$ is approximated by a composition of three maps, encoder $\mathcal{E}$, approximator $\mathcal{A}$ and reconstructor $\mathcal{R}$. The resultant error in the approximation thus comprises of encoder, approximator, and reconstructor errors.
  • Figure 1: Comparison of reconstructions of our DeepONet with the classical IRGN method. We observe that the DeepONet obtains much closer $\sigma$ values to the ground Truth in the anomalous regions. The white line in the ground truth is the cut at which the plot on the right hand side is displayed.
  • Figure 2: DeepONet architecture used in this manuscript. It feeds the EIT measurements into a Branch Net and 2d point on $\Omega$ into a Trunk Net. The outputs of both networks are combined via an inner product to predict the $\sigma_{x, y}$ at points $(x, y)$.

Theorems & Definitions (11)

  • Definition 1.1
  • Theorem 3.1
  • Proof 1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 7.1
  • Proof 2
  • Theorem 7.2
  • Lemma 7.3
  • Lemma 7.4
  • ...and 1 more