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An Efficient Cumulative Edge-Detection Method for Image Reconstruction

Toluwani Okunola, Mirjeta Pasha, Misha E. Kilmer

Abstract

When reconstructing images from noisy measurements, such as in medical scans or scientific imaging, we face an inverse problem: recovering an unknown image from indirect, corrupted observations. These problems are typically ill-posed, meaning small amounts of noise can lead to inaccurate reconstructions. Regularization techniques address this by incorporating prior assumptions about the solution, such as smoothness or sparsity. However, standard methods often blur sharp edges--the boundaries between tissues or structures--losing critical detail. A powerful strategy for edge preservation is iterative reweighting, which solves a sequence of weighted subproblems with adaptively updated weights. Non-cumulative schemes derive weights from the current iterate alone and can be solved efficiently using the Recycled Majorization-Minimization Generalized Krylov Subspace method (RMM-GKS). The cumulative approach of Gazzola et al. progressively accumulates edge information across iterations, achieving superior edge preservation but at high computational cost. This work introduces CR-$\ell_q$-RMM-GKS, which combines cumulative edge detection with computational efficiency. We integrate Gazzola's cumulative weighting with RMM-GKS, which handles general $\ell_q$ penalties ($0 < q \le 2$), automatically selects regularization parameters, and recycles Krylov subspaces between iterations, reducing the nested structure to two levels. Numerical experiments in signal deblurring and tomography demonstrate that CR-$\ell_q$-RMM-GKS produces significantly sharper edge reconstructions than standard non-cumulative methods. In particular, CR-$\ell_1$-RMM-GKS outperforms both standard $\ell_1$ methods and CR-$\ell_2$-RMM-GKS, indicating that cumulative weighting and $\ell_1$ penalties are highly complementary.

An Efficient Cumulative Edge-Detection Method for Image Reconstruction

Abstract

When reconstructing images from noisy measurements, such as in medical scans or scientific imaging, we face an inverse problem: recovering an unknown image from indirect, corrupted observations. These problems are typically ill-posed, meaning small amounts of noise can lead to inaccurate reconstructions. Regularization techniques address this by incorporating prior assumptions about the solution, such as smoothness or sparsity. However, standard methods often blur sharp edges--the boundaries between tissues or structures--losing critical detail. A powerful strategy for edge preservation is iterative reweighting, which solves a sequence of weighted subproblems with adaptively updated weights. Non-cumulative schemes derive weights from the current iterate alone and can be solved efficiently using the Recycled Majorization-Minimization Generalized Krylov Subspace method (RMM-GKS). The cumulative approach of Gazzola et al. progressively accumulates edge information across iterations, achieving superior edge preservation but at high computational cost. This work introduces CR--RMM-GKS, which combines cumulative edge detection with computational efficiency. We integrate Gazzola's cumulative weighting with RMM-GKS, which handles general penalties (), automatically selects regularization parameters, and recycles Krylov subspaces between iterations, reducing the nested structure to two levels. Numerical experiments in signal deblurring and tomography demonstrate that CR--RMM-GKS produces significantly sharper edge reconstructions than standard non-cumulative methods. In particular, CR--RMM-GKS outperforms both standard methods and CR--RMM-GKS, indicating that cumulative weighting and penalties are highly complementary.
Paper Structure (32 sections, 11 equations, 7 figures, 2 tables, 2 algorithms)

This paper contains 32 sections, 11 equations, 7 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Edge-Preserving Regularization via Iterative Reweighting
  3. Non-cumulative weighting
  4. Cumulative weighting
  5. Contribution
  6. Background and Related Work
  7. Discrete Gradient Operator
  8. lq-Norm Regularization and Majorization-Minimization
  9. Recycled MM-GKS Solver
  10. Subspace Recycling
  11. Cumulative Reweighted l2 Regularization
  12. Joint Bidiagonalization for General-Form Tikhonov Problems
  13. Three-Level Nested Iteration
  14. Cumulative Reweighted lq Regularization with Subspace Recycling
  15. Proposed Method
  16. ...and 17 more sections

Figures (7)

  • Figure 1: Visualizing the effect of the discrete gradient operator $\mathbf{L}$. From left to right: a clean image, its horizontal and vertical gradient magnitudes, a noisy version, and its gradient magnitudes. Both actual edges and noise contribute to large gradient values, complicating edge preservation in standard $\ell_2$ regularization.
  • Figure 1: Experiment 1: 1D deblurring. CR-$\ell_1$-RMM-GKS achieves the lowest RRE and stabilizes faster than competing methods through cumulative edge memory.
  • Figure 2: Experiment 1: Final reconstructions after 600 iterations. Left to right: noisy blurred data $\mathbf{b}$, ground truth $\mathbf{x}_{\text{true}}$, and reconstructions from $\ell_2$-RMM-GKS, $\ell_1$-RMM-GKS, CR-$\ell_2$-RMM-GKS, and CR-$\ell_1$-RMM-GKS. CR-$\ell_1$-RMM-GKS best preserves sharp jumps with minimal smoothing artifacts.
  • Figure 3: Experiment 2: Convergence. CR-$\ell_1$-RMM-GKS converges smoothly to the lowest error. CR-$\ell_2$-RMM-GKS shows a pronounced staircase, requiring more outer iterations; standard $\ell_2$ flattens earliest due to oversmoothing.
  • Figure 4: Experiment 2: Final reconstructions after 600 iterations. Left to right: sinogram $\mathbf{b}$, ground truth $\mathbf{x}_{\text{true}}$, and reconstructions from $\ell_2$-RMM-GKS, $\ell_1$-RMM-GKS, CR-$\ell_2$-RMM-GKS, and CR-$\ell_1$-RMM-GKS. CR-$\ell_1$-RMM-GKS produces the sharpest tissue boundaries; $\ell_2$-RMM-GKS exhibits significant blurring of anatomical interfaces.
  • ...and 2 more figures