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Oracle-Net for nonlinear compressed sensing in Electrical Impedance Tomography reconstruction problems

Damiana Lazzaro, Serena Morigi, Luca Ratti

TL;DR

This work tackles nonlinear, ill-posed inverse problems by integrating a sparsity-promoting variational framework with a data-driven support Oracle for nonlinear measurements. The Oracle-Net, a Graph-U-Net, predicts the support mask from electrode-based EIT measurements and feeds this prior into a proximal-gradient solver that minimizes a constrained, sparsity-regularized objective. Under mild nonlinearity and RIP-like Jacobian conditions, the approach yields convergence with a sqrt(δ) rate when the regularization parameter scales with the noise level, and numerical results on EIT demonstrate improved reconstructions with undersampled data. The method offers a practical pathway to reduce measurement requirements in EIT and potentially other nonlinear inverse problems by combining learned priors with rigorous variational optimization.

Abstract

Sparse recovery principles play an important role in solving many nonlinear ill-posed inverse problems. We investigate a variational framework with support Oracle for compressed sensing sparse reconstructions, where the available measurements are nonlinear and possibly corrupted by noise. A graph neural network, named Oracle-Net, is proposed to predict the support from the nonlinear measurements and is integrated into a regularized recovery model to enforce sparsity. The derived nonsmooth optimization problem is then efficiently solved through a constrained proximal gradient method. Error bounds on the approximate solution of the proposed Oracle-based optimization are provided in the context of the ill-posed Electrical Impedance Tomography problem. Numerical solutions of the EIT nonlinear inverse reconstruction problem confirm the potential of the proposed method which improves the reconstruction quality from undersampled measurements, under sparsity assumptions.

Oracle-Net for nonlinear compressed sensing in Electrical Impedance Tomography reconstruction problems

TL;DR

This work tackles nonlinear, ill-posed inverse problems by integrating a sparsity-promoting variational framework with a data-driven support Oracle for nonlinear measurements. The Oracle-Net, a Graph-U-Net, predicts the support mask from electrode-based EIT measurements and feeds this prior into a proximal-gradient solver that minimizes a constrained, sparsity-regularized objective. Under mild nonlinearity and RIP-like Jacobian conditions, the approach yields convergence with a sqrt(δ) rate when the regularization parameter scales with the noise level, and numerical results on EIT demonstrate improved reconstructions with undersampled data. The method offers a practical pathway to reduce measurement requirements in EIT and potentially other nonlinear inverse problems by combining learned priors with rigorous variational optimization.

Abstract

Sparse recovery principles play an important role in solving many nonlinear ill-posed inverse problems. We investigate a variational framework with support Oracle for compressed sensing sparse reconstructions, where the available measurements are nonlinear and possibly corrupted by noise. A graph neural network, named Oracle-Net, is proposed to predict the support from the nonlinear measurements and is integrated into a regularized recovery model to enforce sparsity. The derived nonsmooth optimization problem is then efficiently solved through a constrained proximal gradient method. Error bounds on the approximate solution of the proposed Oracle-based optimization are provided in the context of the ill-posed Electrical Impedance Tomography problem. Numerical solutions of the EIT nonlinear inverse reconstruction problem confirm the potential of the proposed method which improves the reconstruction quality from undersampled measurements, under sparsity assumptions.
Paper Structure (13 sections, 6 theorems, 58 equations, 6 figures)

This paper contains 13 sections, 6 theorems, 58 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Related works
  3. EIT Forward and Inverse models
  4. A regularized constrained model for the EIT inverse problem
  5. Proximal-Gradient Method to solve \ref{['eq:regularization']}
  6. Some explicit regularization models
  7. Support Estimation via Oracle-Net
  8. Numerical Results
  9. Oracle-Net validation
  10. EIT reconstruction results
  11. Experimental compressed sensing issues
  12. Conclusions
  13. Appendix

Key Result

Theorem 1

(Properties of $F_{p,h}$ and $F'_{p,h}$, from lechleiter2008newton)

Figures (6)

  • Figure 1: Oracle-Net architecture
  • Figure 2: Oracle-Net, weight vector setup: (a) adjacent injection-adjacent measurements; (b) opposite injection-adjacent measurements
  • Figure 3: Ideal Oracle (solid) overlapped to the learned Oracle mask $M_{\mathcal{O}}$ (in transparency). Top to bottom: three different thresholds $\sigma_{th}$ applied to six different samples.
  • Figure 4: Noise-free datasets, reconstructions with the different algorithms applying opposite-adjacent protocol and $\sigma_{th}=0.8$ for the threshold of the Oracle-Net result.
  • Figure 5: Noisy dataset, reconstructions with the different algorithms applying opposite-adjacent protocol and $\sigma_{th}=0.8$ for the threshold of the Oracle-Net result.
  • ...and 1 more figures

Theorems & Definitions (11)

  • Theorem 1
  • Proposition 1
  • proof
  • Remark 1
  • Theorem 2
  • Corollary 1
  • proof
  • Corollary 2
  • proof
  • Proposition 2
  • ...and 1 more