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

• URL: http://arxiv.org/abs/2407.17182v3
• Date: Tue, 15 Apr 2025 02:32:00 GMT
• Title: A DeepONet for inverting the Neumann-to-Dirichlet Operator in Electrical Impedance Tomography: An approximation theoretic perspective and numerical results
• Authors: Anuj Abhishek, Thilo Strauss,
• Abstract summary: In this work, we consider the non-invasive medical imaging modality of Electrical Impedance Tomography.<n>The problem is to recover the conductivity in a medium from a set of data that arises out of a current-to-voltage map.<n>We formulate this inverse problem as an operator-learning problem where the goal is to learn the implicitly defined operator-to-function map.
• Score: 2.209921757303168
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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