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Flexible Krylov Methods for Edge Enhancement in Imaging.

Silvia Gazzola1, Sebastian James Scott1, Alastair Spence1

  • 1Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK.

Journal of Imaging
|October 22, 2021
PubMed
Summary

New flexible Krylov methods accelerate edge-enhancing image reconstruction for inverse problems. These solvers efficiently handle iteratively reweighted least squares (IRLS) problems, reducing computational costs.

Keywords:
computed tomographyedge enhancementflexible Golub–Kahan decompositionimage deblurringimage inpaintingiteratively reweighted least squares

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Area of Science:

  • Computational Imaging
  • Applied Mathematics
  • Numerical Analysis

Background:

  • Variational regularization methods, particularly those using non-smooth terms like total variation, are crucial for solving inverse problems in imaging (e.g., deblurring, inpainting, CT).
  • These methods are often formulated as iteratively reweighted least squares (IRLS) problems, typically solved using inner-outer iterative schemes with Krylov projection methods.
  • Existing inner-outer schemes can be computationally intensive due to repeated Krylov solver applications.

Purpose of the Study:

  • To extend flexible Krylov algorithms for enhanced edge recovery in image reconstruction.
  • To incorporate computationally convenient adaptive regularization parameter selection.
  • To address both square and rectangular linear systems arising in inverse problems.

Main Methods:

  • Development and application of flexible Krylov solvers that integrate iteration-dependent weights within a single subspace, avoiding inner-outer iterations.
  • Utilizing flexible Golub-Kahan decomposition to handle diverse linear system dimensions (square and rectangular).
  • Theoretical analysis, including convergence proofs, and numerical comparisons with existing edge-enhancing solvers.

Main Results:

  • The proposed flexible Krylov methods demonstrate significant computational speedup compared to traditional inner-outer approaches.
  • New methods achieve solutions of comparable or superior quality in edge-enhancing image reconstruction tasks.
  • The flexible Golub-Kahan based methods effectively handle various edge-enhancing regularization terms and system types.

Conclusions:

  • Flexible Krylov solvers offer a computationally efficient alternative for solving IRLS problems in variational regularization for imaging.
  • The extended algorithms provide a robust and fast approach for edge-enhancing image reconstruction with adaptive parameter choice.
  • These advancements contribute to faster and more accurate solutions for a range of inverse problems in computational imaging.