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Graph Convolutional Networks for Model-Based Learning in Nonlinear Inverse Problems.

William Herzberg1, Daniel B Rowe1, Andreas Hauptmann2

  • 1Department of Mathematical and Statistical Sciences; Marquette University, Milwaukee, WI 53233 USA.

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|July 25, 2022
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Summary
This summary is machine-generated.

This study introduces a Graph Convolutional Newton-type Method (GCNM) for medical image reconstruction on nonuniform meshes. The GCNM framework enables direct learning on complex domains, improving accuracy for nonlinear inverse problems like Electrical Impedance Tomography (EIT).

Keywords:
Finite element methodconductivityelectrical impedance tomographygraph convolutional networksmodel-based deep learning

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Area of Science:

  • Medical Imaging
  • Computational Science
  • Machine Learning

Background:

  • Model-based image reconstruction often relies on uniform domains, necessitating complex interpolation for nonuniform meshes in nonlinear inverse problems.
  • Finite element methods (FEM) are common for solving nonlinear inverse problems but require specialized handling of nonuniform meshes.
  • Existing methods struggle with direct application to nonuniform meshes, limiting the flexibility of learned reconstruction techniques.

Purpose of the Study:

  • To develop a flexible framework for model-based learning directly on nonuniform meshes in medical image reconstruction.
  • To introduce an iterative Graph Convolutional Newton-type Method (GCNM) that integrates the forward model within the network for direct computation on problem-specific meshes.
  • To evaluate the GCNM's performance in Electrical Impedance Tomography (EIT), a nonlinear inverse problem often solved with FEM.

Main Methods:

  • Interpreting nonuniform meshes as graphs and utilizing graph convolutional neural networks (GCNs) for network architecture.
  • Formulating an iterative Graph Convolutional Newton-type Method (GCNM) that incorporates the forward model into the inverse problem solution.
  • Applying the GCNM to Electrical Impedance Tomography (EIT) for absolute imaging, comparing it against standard iterative methods and a graph residual network.

Main Results:

  • The GCNM demonstrates good generalizability across different domain shapes and meshes.
  • The method shows robustness with out-of-distribution data and experimental data, trained solely on simulated data.
  • GCNM achieved comparable or superior results to existing methods in absolute EIT imaging.

Conclusions:

  • The proposed GCNM framework effectively extends model-based learning to nonuniform meshes, overcoming limitations of previous interpolation-based approaches.
  • GCNM offers a flexible and powerful tool for solving complex nonlinear inverse problems in medical imaging, particularly EIT.
  • The method's ability to generalize from simulated to experimental data without transfer training highlights its practical applicability.