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Deep Block Proximal Linearised Minimisation Algorithm for Non-convex Inverse Problems

Chaoyan Huang, Zhongming Wu, Yanqi Cheng, Tieyong Zeng, Carola-Bibiane Schönlieb, Angelica I. Aviles-Rivero

TL;DR

This paper considers a general type of non-convex optimisation model that captures many inverse image problems and presents an inertial block proximal linearised minimisation (iBPLM) algorithm, and extends beyond the Jacobi-type parallel and the Gauss-Seidel-type alternating update rules.

Abstract

Image restoration is typically addressed through non-convex inverse problems, which are often solved using first-order block-wise splitting methods. In this paper, we consider a general type of non-convex optimisation model that captures many inverse image problems and present an inertial block proximal linearised minimisation (iBPLM) algorithm. Our new method unifies the Jacobi-type parallel and the Gauss-Seidel-type alternating update rules, and extends beyond these approaches. The inertial technique is also incorporated into each block-wise subproblem update, which can accelerate numerical convergence. Furthermore, we extend this framework with a plug-and-play variant (PnP-iBPLM) that integrates deep gradient denoisers, offering a flexible and robust solution for complex imaging tasks. We provide comprehensive theoretical analysis, demonstrating both subsequential and global convergence of the proposed algorithms. To validate our methods, we apply them to multi-block dictionary learning problems in image denoising and deblurring. Experimental results show that both iBPLM and PnP-iBPLM significantly enhance numerical performance and robustness in these applications.

Deep Block Proximal Linearised Minimisation Algorithm for Non-convex Inverse Problems

TL;DR

This paper considers a general type of non-convex optimisation model that captures many inverse image problems and presents an inertial block proximal linearised minimisation (iBPLM) algorithm, and extends beyond the Jacobi-type parallel and the Gauss-Seidel-type alternating update rules.

Abstract

Image restoration is typically addressed through non-convex inverse problems, which are often solved using first-order block-wise splitting methods. In this paper, we consider a general type of non-convex optimisation model that captures many inverse image problems and present an inertial block proximal linearised minimisation (iBPLM) algorithm. Our new method unifies the Jacobi-type parallel and the Gauss-Seidel-type alternating update rules, and extends beyond these approaches. The inertial technique is also incorporated into each block-wise subproblem update, which can accelerate numerical convergence. Furthermore, we extend this framework with a plug-and-play variant (PnP-iBPLM) that integrates deep gradient denoisers, offering a flexible and robust solution for complex imaging tasks. We provide comprehensive theoretical analysis, demonstrating both subsequential and global convergence of the proposed algorithms. To validate our methods, we apply them to multi-block dictionary learning problems in image denoising and deblurring. Experimental results show that both iBPLM and PnP-iBPLM significantly enhance numerical performance and robustness in these applications.
Paper Structure (13 sections, 8 theorems, 34 equations, 6 figures, 3 tables, 2 algorithms)

This paper contains 13 sections, 8 theorems, 34 equations, 6 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. BCD-based Optimisation Algorithms
  4. Inertial Acceleration
  5. Plug-and-Play (PnP) Algorithms
  6. Dictionary Learning-based Image Restoration
  7. Proposed Method
  8. Inertial block proximal linearised minimisation algorithm
  9. Plug-and-Play iBPLM Algorithm
  10. Experimental Results
  11. iBPLM for Image Denoising
  12. PnP-iBPLM for Image Restoration
  13. Conclusion

Key Result

Proposition 3.5

\newlabelNSDP0 For the sequence $\{x^k\}$ generated by the proposed iBPLM algorithm, it must satisfy where $\xi_i^k:=\frac{\alpha_i^k\gamma_i^kL_i^k+\alpha_i^k}{\gamma_i^k}$, $\delta_i^k:=\frac{1-\alpha_i^k -\gamma_i^k L_i^k-\alpha_i^k\gamma_i^kL_i^k-\gamma_i^kw'_iL^k}{\gamma_i^k}$, and $w'_i=\sum_{q=i+1}^{p}w_{qi}$.

Figures (6)

  • Figure 1: Image restoration results (PSNR/SSIM) with Gaussian noise level $25$. Visualisation comparing our technique with K-SVD rubinstein2012analysis, with two examples presented in (a)-(d) and (e)-(h), respectively.
  • Figure 2: PSNR, SSIM, and energy curves of the proposed methods with different $\tau_X$ values from iteration 2 to 8.
  • Figure 3: Effect of $\alpha_X$ and $\alpha_D$ in iBPLM algorithm on 'butterfly' with noise level $25$. The first image is the PSNR surface under different $\alpha_X$ and $\alpha_D$. The second one is the energy curves of different $\alpha_X$ with fixed $\alpha_D=0.5$. The third one is the energy curves of different $\alpha_D$ with fixed $\alpha_X=0.5$. The comparisons of the second and third plots are conducted through log-log scale analysis.
  • Figure 4: Image restoration results (PSNR/SSIM) with motion blur kernel MB$(20, 60)$ and Gaussian noise level $25$. We refer to 'Equivariant' as 'Equi.'. Visualisation comparison of our scheme and some state-of-the-art PnP-based methods: (c) DPIR zhang2021plug, (d) DiffPIR zhu2023denoising, (e) Equivariant terris2023equivariant, (f) SNORE renaud2024plug, (g) DYSdiff wu2024extrapolated, and (h) Our PnP-iBPLM.
  • Figure 5: Image restoration results (PSNR/SSIM) with motion blur kernel MB$(20, 60)$ and Gaussian noise level $25$. We refer to 'Equivariant' as 'Equi.'. Visualisation comparison of our scheme and some state-of-the-art PnP-based methods: (c) DPIR zhang2021plug, (d) DiffPIR zhu2023denoising, (e) Equivariant terris2023equivariant, (f) SNORE renaud2024plug, (g) DYSdiff wu2024extrapolated, and (h) Our PnP-iBPLM.
  • ...and 1 more figures

Theorems & Definitions (18)

  • Definition 3.1
  • Definition 3.2
  • Remark 3.3
  • Proposition 3.5
  • Proof 1
  • Lemma 3.6
  • Proof 2
  • Remark 3.7
  • Theorem 3.8
  • Proof 3
  • ...and 8 more