A learning-based method for solving ill-posed nonlinear inverse problems: a simulation study of Lung EIT

TL;DR

This paper introduces a novel learning-based approach using variational autoencoders to transform ill-posed nonlinear inverse problems, exemplified by lung EIT, into well-posed problems through low-dimensional manifold learning, validated by simulations.

Contribution

It presents a new method that leverages training data and deep learning to improve the solution of ill-posed nonlinear inverse problems, moving beyond traditional regularization techniques.

Findings

The method effectively reconstructs lung EIT images from simulated data.

The low-dimensional latent space captures essential image features.

Numerical simulations demonstrate improved stability and accuracy.

Abstract

This paper proposes a new approach for solving ill-posed nonlinear inverse problems. For ease of explanation of the proposed approach, we use the example of lung electrical impedance tomography (EIT), which is known to be a nonlinear and ill-posed inverse problem. Conventionally, penalty-based regularization methods have been used to deal with the ill-posed problem. However, experiences over the last three decades have shown methodological limitations in utilizing prior knowledge about tracking expected imaging features for medial diagnosis. The proposed method's paradigm is completely different from conventional approaches; the proposed reconstruction uses a variety of training data sets to generate a low dimensional manifold of approximate solutions, which allows to convert the ill-posed problem to a well-posed one. Variational autoencoder was used to produce a compact and dense representation for lung EIT images with a low dimensional latent space. Then, we learn a robust connection between the EIT data and the low-dimensional latent data. Numerical simulations validate the effectiveness and feasibility of the proposed approach.