Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Data Completion for Electrical Impedance Tomography by Conditional Diffusion Models

Ke Chen, Haizhao Yang, Chugang Yi

TL;DR

This work tackles the data sparsity challenge in Electrical Impedance Tomography by learning a conditional diffusion model that completes partially observed DtN measurements. By modeling $p(\Lambda_\gamma|\Lambda^{\mathrm{o}}_\gamma, \mathbf{M}_s)$ and sampling from it, the method provides high-quality completed DtN data that can be fed into standard inverse solvers, enabling accurate conductivity reconstructions from as little as $1\%$ of the measurements. The authors establish a nonasymptotic end-to-end convergence guarantee for the diffusion completion in the polygonal-conductivity regime and demonstrate through extensive experiments that diffusion-based completion outperforms low-rank matrix completion under highly structured missingness. The approach is plug-and-play with existing solvers, robust to noise, and scalable to flexible measurement configurations, potentially enabling practical EIT deployments with minimal sensing. Overall, the work provides both theoretical guarantees and strong empirical evidence that diffusion priors can substantially alleviate data scarcity in EIT.

Abstract

Data scarcity is a fundamental barrier in Electrical Impedance Tomography (EIT), as undersampled Dirichlet-to-Neumann (DtN) measurements can substantially degrade conductivity reconstructions. We address this bottleneck by completing partially observed DtN measurements using a diffusion based generative model. Specifically, we train a conditional diffusion model to learn the distribution of DtN data and to infer full measurement vectors given partial observations. Our approach supports flexible source receiver configurations and can be used as a plug in preprocessing step with off the shelf EIT solvers. Under mild assumptions on the polygon conductivity class, we derive nonasymptotic end to end bounds on the distributional discrepancy between the completed and ground truth DtN measurements. In numerical experiments, we couple the proposed diffusion completion procedure with a deep learning based inverse solver and compare its performance against the same solver with full measurement data. The results show that diffusion completion enables reconstructions comparable to the full data baseline while using only 1% of the measurements. In contrast, standard baselines such as matrix completion require 30% of the measurements to achieve similar reconstruction quality.

Data Completion for Electrical Impedance Tomography by Conditional Diffusion Models

TL;DR

This work tackles the data sparsity challenge in Electrical Impedance Tomography by learning a conditional diffusion model that completes partially observed DtN measurements. By modeling and sampling from it, the method provides high-quality completed DtN data that can be fed into standard inverse solvers, enabling accurate conductivity reconstructions from as little as of the measurements. The authors establish a nonasymptotic end-to-end convergence guarantee for the diffusion completion in the polygonal-conductivity regime and demonstrate through extensive experiments that diffusion-based completion outperforms low-rank matrix completion under highly structured missingness. The approach is plug-and-play with existing solvers, robust to noise, and scalable to flexible measurement configurations, potentially enabling practical EIT deployments with minimal sensing. Overall, the work provides both theoretical guarantees and strong empirical evidence that diffusion priors can substantially alleviate data scarcity in EIT.

Abstract

Data scarcity is a fundamental barrier in Electrical Impedance Tomography (EIT), as undersampled Dirichlet-to-Neumann (DtN) measurements can substantially degrade conductivity reconstructions. We address this bottleneck by completing partially observed DtN measurements using a diffusion based generative model. Specifically, we train a conditional diffusion model to learn the distribution of DtN data and to infer full measurement vectors given partial observations. Our approach supports flexible source receiver configurations and can be used as a plug in preprocessing step with off the shelf EIT solvers. Under mild assumptions on the polygon conductivity class, we derive nonasymptotic end to end bounds on the distributional discrepancy between the completed and ground truth DtN measurements. In numerical experiments, we couple the proposed diffusion completion procedure with a deep learning based inverse solver and compare its performance against the same solver with full measurement data. The results show that diffusion completion enables reconstructions comparable to the full data baseline while using only 1% of the measurements. In contrast, standard baselines such as matrix completion require 30% of the measurements to achieve similar reconstruction quality.
Paper Structure (44 sections, 18 theorems, 109 equations, 10 figures, 2 tables, 2 algorithms)

This paper contains 44 sections, 18 theorems, 109 equations, 10 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Setup
  3. Finite element discretizations
  4. Partial observations and data completion
  5. Method
  6. Diffusion Model
  7. Conditional Diffusion Models for Data Completion
  8. Solve the inverse problem by data completion
  9. Error Analysis
  10. Notations.
  11. Admissible conductivities and parametrization
  12. Chartwise regularity of the DtN map
  13. Chartwise Lipschitz for the DtN measurements
  14. Metric entropy and bounded support of $F_N(K)$
  15. Convergence guarantee for diffusion completion
  16. ...and 29 more sections

Key Result

Lemma 4.2

Fix $P^\ast\in\mathcal{A}_{1/2}$ and let $v^\ast$ be its counterclockwise ordered vertex list. Then there exists $\rho_{P^\ast}>0$ such that whenever $\|v-v^\ast\|_{\mathrm{poly}}<\rho_{P^\ast}$, the polygonal chain $\partial P(v)$ is simple and $P(v)\in\mathcal{A}_{1/2}$.

Figures (10)

  • Figure 1: Three strategies of solving EIT. Direct reconstruction (first row) from sparse measurements yields degraded reconstruction. We propose to first complete the DtN measurements and then reconstruct (second row), yielding image quality close to those from a direct reconstruction from full measurements (third row).
  • Figure 2: Discretization of conductivity $\gamma$. Left: Triangular mesh over the unit disk where red boundary points mark candidate sensor locations. Middle: A conductivity over the triangular mesh from the Disk distribution. Right: The same conductivity converted to a uniform $128\times128$ Cartesian grid on $[-1,1]\times[-1,1]$ with zero padding.
  • Figure 3: Normalization of DtN measurements. Left column: conductivity fields (top: background; middle: disks; bottom: Shepp--Logan phantom). Middle column: raw DtN matrices computed from the corresponding conductivities. Right column: normalized DtN matrices produced by our preprocessing step.
  • Figure 4: Hierarchical partition with level 3. Diagonal blocks are colored in orange, and off-diagonal blocks are colored in blue.
  • Figure 5: Diffusion completion versus matrix completion on an off-diagonal block under different masking patterns and sampling rate. Note that the matrix-completion baseline is only applied to off-diagonal blocks, where low-rank structure is expected, whereas our diffusion model is applied to the whole matrix and presents the off-block in this plotLeft: ground truth (normalized) upper-right block of DtN matrix. Right, top row: masking matrices with sampling rate $s=1\%$, $s=15\%$ and $s=30\%$. Right, bottom row: corresponding reconstructions under the masks shown above. The diffusion model (first column) achieves high-quality reconstruction under the submatrix masking even at an extremely low sampling rate $s=1\%$ with relative error (RE) $0.9\%$. In contrast, matrix completion fails at $s=1\%$ under both submatrix and random masking (second and third columns). Matrix completion requires substantially higher random sampling rates (fourth and fifth columns) to achieve comparable accuracy, reaching RE $8.2\%$ at $s=15\%$ and RE $1.0\%$ at $s=30\%$. In contrast, our diffusion model performs completion over the entire DtN matrix.
  • ...and 5 more figures

Theorems & Definitions (47)

  • Definition 3.1: Unconditional DDPM sampler
  • Definition 3.2: Conditional DDPM sampler
  • Definition 4.1: Admissible polygon class
  • Remark 4.1
  • Definition 4.2: Relaxed admissible polygon classes
  • Remark 4.2: A priori data
  • Lemma 4.2: Local admissibility radius
  • Lemma 4.3
  • Lemma 4.4: Compactness of $K$
  • Lemma 4.5: Finite patch cover and compact parameter patches
  • ...and 37 more