# A Comparative Study of Variational Autoencoders, Normalizing Flows, and Score-based Diffusion Models for Electrical Impedance Tomography

arXiv: https://arxiv.org/abs/2310.15831 · HTML: https://mlsanity.com/paper/2310.15831/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Placed in Inverse problems and image restoration, one of 14 directions in Diffusion models.
- Builds on 19 of its 36 references that are in the map.
- First posted on arXiv 24 Oct 2023.

## Placed in

- Diffusion models / [Inverse problems and image restoration](https://mlsanity.com/diffusion/inverse-problems-and-image-restoration/)

## Builds on (most cited first, 19 of 19)

- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Generative Adversarial Networks](https://mlsanity.com/paper/2203.00667/) (2022)
- [Density estimation using Real NVP](https://mlsanity.com/paper/1605.08803/) (2016)
- [Image Super-Resolution via Iterative Refinement](https://mlsanity.com/paper/2104.07636/) (2021)
- [Diffusion Posterior Sampling for General Noisy Inverse Problems](https://mlsanity.com/paper/2209.14687/) (2022)
- [NICE: Non-linear Independent Components Estimation](https://mlsanity.com/paper/1410.8516/) (2014)
- [Maximum Likelihood Training of Score-Based Diffusion Models](https://mlsanity.com/paper/2101.09258/) (2021)
- [Solving Inverse Problems in Medical Imaging with Score-Based Generative Models](https://mlsanity.com/paper/2111.08005/) (2021)
- [Come-Closer-Diffuse-Faster: Accelerating Conditional Diffusion Models for Inverse Problems through Stochastic Contraction](https://mlsanity.com/paper/2112.05146/) (2021)
- [Learning Likelihoods with Conditional Normalizing Flows](https://mlsanity.com/paper/1912.00042/) (2019)
- [Guided Image Generation with Conditional Invertible Neural Networks](https://mlsanity.com/paper/1907.02392/) (2019)
- [JPEG Artifact Correction using Denoising Diffusion Restoration Models](https://mlsanity.com/paper/2209.11888/) (2022)
- [Conditional Invertible Neural Networks for Medical Imaging](https://mlsanity.com/paper/2110.14520/) (2021)
- [On Conditioning the Input Noise for Controlled Image Generation with Diffusion Models](https://mlsanity.com/paper/2205.03859/) (2022)
- [Physics Embedded Machine Learning for Electromagnetic Data Imaging](https://mlsanity.com/paper/2207.12607/) (2022)
- [Graph Convolutional Networks for Model-Based Learning in Nonlinear Inverse Problems](https://mlsanity.com/paper/2103.15138/) (2021)
- [Bayesian Inversion for Nonlinear Imaging Models using Deep Generative Priors](https://mlsanity.com/paper/2203.10078/) (2022)
- [Missing Cone Artifacts Removal in ODT using Unsupervised Deep Learning in Projection Domain](https://mlsanity.com/paper/2103.09022/) (2021)
- [Differentiable Gaussianization Layers for Inverse Problems Regularized by Deep Generative Models](https://mlsanity.com/paper/2112.03860/) (2021)

## Built on by (most cited first, 4 of 4)

- [Diff-INR: Generative Regularization for Electrical Impedance Tomography](https://mlsanity.com/paper/2409.04494/) (2024)
- [Data-driven approaches for electrical impedance tomography image segmentation from partial boundary data](https://mlsanity.com/paper/2407.01559/) (2024)
- [A Conditional Diffusion Model for Electrical Impedance Tomography Image Reconstruction](https://mlsanity.com/paper/2412.16979/) (2024)
- [Conditional Diffusion Model for Electrical Impedance Tomography](https://mlsanity.com/paper/2501.05769/) (2025)

## Abstract (arXiv)

Electrical Impedance Tomography (EIT) is a widely employed imaging technique in industrial inspection, geophysical prospecting, and medical imaging. However, the inherent nonlinearity and ill-posedness of EIT image reconstruction present challenges for classical regularization techniques, such as the critical selection of regularization terms and the lack of prior knowledge. Deep generative models (DGMs) have been shown to play a crucial role in learning implicit regularizers and prior knowledge. This study aims to investigate the potential of three DGMs-variational autoencoder networks, normalizing flow, and score-based diffusion model-to learn implicit regularizers in learning-based EIT imaging. We first introduce background information on EIT imaging and its inverse problem formulation. Next, we propose three algorithms for performing EIT inverse problems based on corresponding DGMs. Finally, we present numerical and visual experiments, which reveal that (1) no single method consistently outperforms the others across all settings, and (2) when reconstructing an object with 2 anomalies using a well-trained model based on a training dataset containing 4 anomalies, the conditional normalizing flow model (CNF) exhibits the best generalization in low-level noise, while the conditional score-based diffusion model (CSD*) demonstrates the best generalization in high-level noise settings. We hope our preliminary efforts will encourage other researchers to assess their DGMs in EIT and other nonlinear inverse problems.