Enhancing Electrical Impedance Tomography reconstruction using Learned Half-Quadratic Splitting Networks with Anderson Acceleration

TL;DR

This work tackles the nonlinear, ill-posed inverse problem of Electrical Impedance Tomography by marrying physics-based modeling with learning-based priors through a learned half-quadratic splitting network (HQSNet). It further accelerates convergence and suppresses artifacts by applying Anderson acceleration to both the Gauss-Newton data-fit step and the learned proximal gradient step, yielding AA-HQSNet. Experiments on simulated circular-EIT data show AA-HQSNet outperforms state-of-the-art baselines in MSE, SSIM, and robustness to noise, while maintaining sharp anomaly structures across varying numbers of inclusions. The framework is presented as generic and extensible to other physics-embedded deep learning methods for inverse problems.

Abstract

Electrical Impedance Tomography (EIT) is widely applied in medical diagnosis, industrial inspection, and environmental monitoring. Combining the physical principles of the imaging system with the advantages of data-driven deep learning networks, physics-embedded deep unrolling networks have recently emerged as a promising solution in computational imaging. However, the inherent nonlinear and ill-posed properties of EIT image reconstruction still present challenges to existing methods in terms of accuracy and stability. To tackle this challenge, we propose the learned half-quadratic splitting (HQSNet) algorithm for incorporating physics into learning-based EIT imaging. We then apply Anderson acceleration (AA) to the HQSNet algorithm, denoted as AA-HQSNet, which can be interpreted as AA applied to the Gauss-Newton step and the learned proximal gradient descent step of the HQSNet, respectively. AA is a widely-used technique for accelerating the convergence of fixed-point iterative algorithms and has gained significant interest in numerical optimization and machine learning. However, the technique has received little attention in the inverse problems community thus far. Employing AA enhances the convergence rate compared to the standard HQSNet while simultaneously avoiding artifacts in the reconstructions. Lastly, we conduct rigorous numerical and visual experiments to show that the AA module strengthens the HQSNet, leading to robust, accurate, and considerably superior reconstructions compared to state-of-the-art methods. Our Anderson acceleration scheme to enhance HQSNet is generic and can be applied to improve the performance of various physics-embedded deep learning methods.

Figures (8)

• Figure 1: EIT forward and inverse scenarios. After the surface of the object is represented as a finite element mesh, the forward process transforms the mesh data into a measured voltage value; the inverse process then back-performs the reconstruction of ground truth conductivity value from this measured voltage
• Figure 2: Comparison of performance between Newton-AA (NAA) and Gauss-Newton-AA (GNAA) with different initial values, showcasing the variation of the $l_{2}$ norm of $f(x)$ (y-axis) across iterations (x-axis)
• Figure 3: (a)The unrolled AA-HQSNet architecture, where $\mathcal{G}$ represents the Gauss-Newton method summarized in Algorithm \ref{['alg:1']}, and $\Phi_{\theta}$ represents the proximal neural network depicted in Fig.\ref{['fig:network']}. (b) The Gauss-Newton Anderson acceleration algorithm with iteration steps $K^{(1)}$, as shown in Algorithm \ref{['alg:gnaa']}. (c) The proximal gradient descent Anderson acceleration algorithm with iteration steps $K^{(2)}$, introduced in Algorithm \ref{['alg:pgdaa']}
• Figure 4: Unrolled proximal gradient network architecture $\Phi_{\theta}$ used in proximal-gradient decent steps, and its input is the data term and corresponding gradient term
• Figure 5: Visual reconstruction performance comparisons between AA-HQSNet, HQSNet, GN-LM, and D-bar with varying numbers of anomalies are shown in the top row. The associated structure maps (EIEI metric) are reported in the bottom row. The EIEI structure maps are color-coded, with yellow representing artifacts, red representing anomalies, and blue representing the background. The EIEI values for all test cases with varying numbers of anomalies ranging from 1 to 4 are reported, and a higher value of it indicates better performance