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Multiresolution Adaptive Block-Coordinate Forward-Backward for Image Reconstruction

Edgar Desainte-Maréville, Marion Foare, Paulo Gonçalves, Nelly Pustelnik, Elisa Riccietti

TL;DR

This work proposes an adaptive multiresolution block coordinate Forward-Backward algorithm for image restoration that automatically adapts to varying blur and noise levels without relying on a predefined hierarchical update scheme.

Abstract

Classical first-order optimization methods for imaging inverse problems scale poorly with image resolution. Wavelet based multilevel strategies can accelerate convergence under strong blur, but their fixed coarse-to-fine schedules lose effectiveness in moderate-blur or noise-dominated regimes. In this work, we propose an adaptive multiresolution block coordinate Forward-Backward algorithm for image restoration. Multiresolution block selection is driven by the local magnitude of the proximal update via a stochastic non-smooth Gauss-Southwell rule applied to the wavelet decomposition of the image. This adaptive selection strategy dynamically balances updates across scales, emphasizing coarse or fine blocks according to the degradation regime. As a result, the proposed method automatically adapts to varying blur and noise levels without relying on a predefined hierarchical update scheme.

Multiresolution Adaptive Block-Coordinate Forward-Backward for Image Reconstruction

TL;DR

This work proposes an adaptive multiresolution block coordinate Forward-Backward algorithm for image restoration that automatically adapts to varying blur and noise levels without relying on a predefined hierarchical update scheme.

Abstract

Classical first-order optimization methods for imaging inverse problems scale poorly with image resolution. Wavelet based multilevel strategies can accelerate convergence under strong blur, but their fixed coarse-to-fine schedules lose effectiveness in moderate-blur or noise-dominated regimes. In this work, we propose an adaptive multiresolution block coordinate Forward-Backward algorithm for image restoration. Multiresolution block selection is driven by the local magnitude of the proximal update via a stochastic non-smooth Gauss-Southwell rule applied to the wavelet decomposition of the image. This adaptive selection strategy dynamically balances updates across scales, emphasizing coarse or fine blocks according to the degradation regime. As a result, the proposed method automatically adapts to varying blur and noise levels without relying on a predefined hierarchical update scheme.
Paper Structure (16 sections, 9 equations, 2 figures, 1 algorithm)

This paper contains 16 sections, 9 equations, 2 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Wavelet-based block-coordinate Forward--Backward
  3. Wavelet-based imaging inverse problem
  4. Wavelet representation and separable regularization
  5. General algorithm and state-of-the-art update rules
  6. MAGIC-FB
  7. Proposed multiresolution Gauss--Southwell update
  8. Computational efficiency of MAGIC-FB
  9. Experiments
  10. Experimental setup
  11. Test dataset
  12. Degradation model
  13. Multiresolution representation and regularization
  14. Algorithms and parameters
  15. Numerical results
  16. ...and 1 more sections

Figures (2)

  • Figure 1: Convergence behavior and block-selection patterns for the considered methods on an image deblurring problem. Left: $\sigma_\mathrm{blur} = 7$ and $\sigma_\mathrm{noise} = 0.01$. Right: $\sigma_\mathrm{blur} = 1$ and $\sigma_\mathrm{noise} = 0.1$. Top left and right panels: objective function value as a function of time. Centered top panels: block activation patterns across wavelet scales. Bottom panels: image reconstructions at $t = 1\,\mathrm{s}$ for each method.
  • Figure 2: Performance profiles of the considered methods across the 100 test instances. The $x$-axis represents the factor $\beta$ by which a method's performance is compared to the best method for each problem, while the $y$-axis shows the proportion of problems for which the method is within this factor of the best performance (the closer to the upper left corner, the best).