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Discretization of Learned NETT Regularization for Solving Inverse Problems.

Stephan Antholzer1, Markus Haltmeier1

  • 1Department of Mathematics, University of Innsbruck, Technikerstrasse 13, 6020 Innsbruck, Austria.

Journal of Imaging
|November 25, 2021
PubMed
Summary
This summary is machine-generated.

This study enhances Network Tikhonov Regularization (NETT) for inverse problems by accounting for discretization errors. The findings demonstrate asymptotic convergence and derive convergence rates for practical deep learning-based reconstruction.

Keywords:
convergence analysisdeep learningdiscretization of NETTinverse problemslearned regularizerlimited dataphotoacoustic tomographyregularization

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Area of Science:

  • Applied Mathematics
  • Computer Science
  • Medical Imaging

Background:

  • Deep learning reconstruction methods excel at solving inverse problems.
  • Network Tikhonov Regularization (NETT) integrates neural networks into Tikhonov regularization.
  • Existing NETT analysis is limited to fixed operators and noise-level convergence.

Purpose of the Study:

  • Extend NETT analysis to include practical aspects like discretization.
  • Investigate convergence properties of discretized NETT methods.
  • Analyze convergence rates for NETT in real-world applications.

Main Methods:

  • Developed a generalized framework for analyzing discretized NETT.
  • Incorporated data space, solution space, forward operator, and neural network discretization.
  • Applied asymptotic convergence analysis and derived convergence rates.

Main Results:

  • Demonstrated asymptotic convergence of discretized NETT as noise and discretization errors decrease.
  • Derived theoretical convergence rates for the NETT approach.
  • Presented numerical results for photoacoustic tomography, a limited data problem.

Conclusions:

  • The enhanced NETT framework provides a more robust theoretical foundation for deep learning-based inverse problem solving.
  • Discretization errors can be systematically analyzed and managed in NETT.
  • NETT shows promise for practical applications like photoacoustic tomography reconstruction.