# Domain independent post-processing with graph U-nets: Applications to Electrical Impedance Tomographic Imaging

arXiv: https://arxiv.org/abs/2305.05020 · HTML: https://mlsanity.com/paper/2305.05020/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 50 references that are in the map.
- First posted on arXiv 8 May 2023.

## Builds on (most cited first, 18 of 18)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Graph Attention Networks](https://mlsanity.com/paper/1710.10903/) (2017)
- [A Comprehensive Survey on Graph Neural Networks](https://mlsanity.com/paper/1901.00596/) (2019)
- [Graph U-Nets](https://mlsanity.com/paper/1905.05178/) (2019)
- [Self-Attention Graph Pooling](https://mlsanity.com/paper/1904.08082/) (2019)
- [Revisiting Graph Neural Networks: All We Have is Low-Pass Filters](https://mlsanity.com/paper/1905.09550/) (2019)
- [ASAP: Adaptive Structure Aware Pooling for Learning Hierarchical Graph Representations](https://mlsanity.com/paper/1911.07979/) (2019)
- [Kernel Transformer Networks for Compact Spherical Convolution](https://mlsanity.com/paper/1812.03115/) (2018)
- [Multi-hop Attention Graph Neural Network](https://mlsanity.com/paper/2009.14332/) (2020)
- [Graph Pooling for Graph Neural Networks: Progress, Challenges, and Opportunities](https://mlsanity.com/paper/2204.07321/) (2022)
- [Deep D-bar: Real time Electrical Impedance Tomography Imaging with Deep Neural Networks](https://mlsanity.com/paper/1711.03180/) (2017)
- [SE(3) symmetry lets graph neural networks learn arterial velocity estimation from small datasets](https://mlsanity.com/paper/2302.08780/) (2023)
- [Beltrami-Net: Domain Independent Deep D-bar Learning for Absolute Imaging with Electrical Impedance Tomography (a-EIT)](https://mlsanity.com/paper/1811.12830/) (2018)
- [Graph Convolutional Networks for Model-Based Learning in Nonlinear Inverse Problems](https://mlsanity.com/paper/2103.15138/) (2021)
- [Approximate k-space models and Deep Learning for fast photoacoustic reconstruction](https://mlsanity.com/paper/1807.03191/) (2018)
- [Model-corrected learned primal-dual models for fast limited-view photoacoustic tomography](https://mlsanity.com/paper/2304.01963/) (2023)

## Built on by (most cited first, 3 of 3)

- [Electrical Impedance Tomography: A Fair Comparative Study on Deep Learning and Analytic-based Approaches](https://mlsanity.com/paper/2310.18636/) (2023)
- [Graph Neural Regularizers for PDE Inverse Problems](https://mlsanity.com/paper/2510.21012/) (2025)
- [Graph convolutional networks enable fast hemorrhagic stroke monitoring with electrical impedance tomography](https://mlsanity.com/paper/2412.07888/) (2024)

## Abstract (arXiv)

Reconstruction of tomographic images from boundary measurements requires flexibility with respect to target domains. For instance, when the system equations are modeled by partial differential equations the reconstruction is usually done on finite element (FE) meshes, allowing for flexible geometries. Thus, any processing of the obtained reconstructions should be ideally done on the FE mesh as well. For this purpose, we extend the hugely successful U-Net architecture that is limited to rectangular pixel or voxel domains to an equivalent that works flexibly on FE meshes. To achieve this, the FE mesh is converted into a graph and we formulate a graph U-Net with a new cluster pooling and unpooling on the graph that mimics the classic neighborhood based max-pooling. We demonstrate effectiveness and flexibility of the graph U-Net for improving reconstructions from electrical impedance tomographic (EIT) measurements, a nonlinear and highly ill-posed inverse problem. The performance is evaluated for simulated data and from three measurement devices with different measurement geometries and instrumentations. We successfully show that such networks can be trained with a simple two-dimensional simulated training set and generalize to very different domains, including measurements from a three-dimensional device and subsequent 3D reconstructions.