Neural Correction Operator: A Reliable and Fast Approach for Electrical Impedance Tomography

TL;DR

The paper tackles the severely ill-posed inverse problem of Electrical Impedance Tomography (EIT) by decomposing the inversion into a fast reconstruction step using limited-iteration L-BFGS and a learned correction step. The neural correction operator framework models the inverse as $\mathcal{F}=\mathcal{C}\circ\mathcal{R}_K$, where $\mathcal{R}_K$ provides a rough initial image and $\mathcal{C}$ refines it using either a ResNet or a conditional DDPM to account for nonlinearity and uncertainty. Empirical results on Four Circles and Shepp-Logan phantoms show substantial improvements in reconstruction quality (PSNR, SSIM, and relative $\ell_1$/$\ell_2$ errors) and robustness to noise, along with 6–8× speedups over full L-BFGS solves. The approach offers a generalizable paradigm for applying neural operators to severely ill-posed inverse problems, with practical implications for fast and reliable EIT imaging, particularly in medical contexts.

Abstract

Electrical Impedance Tomography (EIT) is a non-invasive medical imaging method that reconstructs electrical conductivity mediums from boundary voltage-current measurements, but its severe ill-posedness renders direct operator learning with neural networks unreliable. We propose the neural correction operator framework, which learns the inverse map as a composition of two operators: a reconstruction operator using L-BFGS optimization with limited iterations to obtain an initial estimate from measurement data and a correction operator implemented with deep learning models to reconstruct the true media from this initial guess. We explore convolutional neural network architectures and conditional diffusion models as alternative choices for the correction operator. We evaluate the neural correction operator by comparing with L-BFGS methods as well as neural operators and conditional diffusion models that directly learn the inverse map over several benchmark datasets. Our numerical experiments demonstrate that our approach achieves significantly better reconstruction quality compared to both iterative methods and direct neural operator learning methods with the same architecture. The proposed framework also exhibits robustness to measurement noise while achieving substantial computational speedup compared to conventional methods. The neural correction operator provides a general paradigm for approaching neural operator learning in severely ill-posed inverse problems.

Figures (8)

• Figure 1: Convergence of L-BFGS for Shepp-Logan media.
• Figure 2: Left: ResNet architecture used to learn the neural operator $\mathcal{C}_R$. We use 8 residual blocks to learn the overall features of the image, and use a fully-connected layer at the end to upsample back to the input dimensions. Right: Composition of a ResBlock and a DownResBlock. $N$ denotes the number of channels in the input.
• Figure 3: Left: UNet architecture used to learn the neural operator $\mathcal{C}_R$. We use 4 downsampling blocks to learn the overall features of the image, and 4 upsampling with residual connections to return to the input dimensions. Here "$||$" denotes concatenation along the channel dimension and "$+$" denotes addition. Right: Composition of a DownUBlock and an UpUBlock. Here $N$ denotes the number of channels in the input and $D_t$ is the embedding dimension.
• Figure 4: Four Circles Dataset. Four different samples of ground truth (Column 1) and reconstructed media from baseline models (Columns 2-5) and the proposed methods (Columns 6-7). Our proposed methods perform significantly better than baseline models in capturing the shape and sharp boundary of the circles.
• Figure 5: Shepp-Logan Dataset. Four different samples of Ground truth (Column 1) and reconstructed media from baseline models (Columns 2-5) and the proposed methods (Columns 6-7). Our proposed methods perform significantly better than baseline models in finding the interior structures.