Graph Convolutional Networks for Model-Based Learning in Nonlinear Inverse Problems

TL;DR

This paper introduces a graph convolutional neural network framework for model-based learning in nonlinear inverse problems on nonuniform meshes, demonstrated on Electrical Impedance Tomography with improved generalizability.

Contribution

It extends model-based learning to nonuniform meshes using graph convolutional networks, enabling direct computation on problem-specific meshes for nonlinear inverse problems.

Findings

GCNM outperforms standard iterative methods in EIT imaging.

The method generalizes well to different domain shapes and unseen data.

It operates effectively without transfer training on experimental data.

Abstract

The majority of model-based learned image reconstruction methods in medical imaging have been limited to uniform domains, such as pixelated images. If the underlying model is solved on nonuniform meshes, arising from a finite element method typical for nonlinear inverse problems, interpolation and embeddings are needed. To overcome this, we present a flexible framework to extend model-based learning directly to nonuniform meshes, by interpreting the mesh as a graph and formulating our network architectures using graph convolutional neural networks. This gives rise to the proposed iterative Graph Convolutional Newton-type Method (GCNM), which includes the forward model in the solution of the inverse problem, while all updates are directly computed by the network on the problem specific mesh. We present results for Electrical Impedance Tomography, a severely ill-posed nonlinear inverse problem that is frequently solved via optimization-based methods, where the forward problem is solved by finite element methods. Results for absolute EIT imaging are compared to standard iterative methods as well as a graph residual network. We show that the GCNM has strong generalizability to different domain shapes and meshes, out of distribution data as well as experimental data, from purely simulated training data and without transfer training.