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Adaptive Image Processing: First Order PDE Constraint Regularizers and a Bilevel Training Scheme.

Elisa Davoli1, Irene Fonseca2, Pan Liu3

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Summary
This summary is machine-generated.

A new bilevel training method unifies Total Generalized Variation (TGV^2) and Non-smooth TGV (NsTGV^2) regularizers. This approach proves solution existence for imaging data under specific conditions, offering a novel framework for regularization techniques.

Keywords:
First order differential operatorsImage processingOptimal training scheme

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Area of Science:

  • Image processing and computational mathematics.
  • Development of novel regularization techniques for inverse problems.

Background:

  • Standard regularizers like TGV^2 and NsTGV^2 are crucial in image processing.
  • A unified framework for these regularizers is lacking, hindering broader application.

Purpose of the Study:

  • To introduce a novel class of regularizers using a bilevel training scheme.
  • To provide a unified approach encompassing TGV^2 and NsTGV^2.
  • To establish theoretical guarantees for the existence of solutions in imaging problems.

Main Methods:

  • A bilevel training scheme is employed to derive and optimize regularizers.
  • The method unifies existing regularizers, including TGV^2 and NsTGV^2.
  • Mathematical proofs using Gamma-convergence establish solution existence under specific operator conditions.

Main Results:

  • A novel class of regularizers is introduced, unifying TGV^2 and NsTGV^2.
  • Optimal parameters and regularizers are identified through the bilevel training.
  • Existence of a solution is proven for any training imaging data set, given specific mathematical conditions.

Conclusions:

  • The proposed bilevel training scheme offers a unified and theoretically sound approach to regularization in imaging.
  • The identified regularizers and proven solution existence advance the field of inverse problems and image reconstruction.
  • Numerical examples demonstrate the practical applicability of the novel regularization framework.