Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An Improved Optimal Proximal Gradient Algorithm for Non-Blind Image Deblurring

Qingsong Wang, Shengze Xu, Xiaojiao Tong, Tieyong Zeng

TL;DR

This work tackles non-blind image deblurring by formulating the problem as min_x \frac{1}{2}\|Ax-b\|^{2}+h(x) with a known blur operator $A$ and convex regularizers $h(x)$ such as $\lambda\|x\|_1$ or $\lambda\|x\|_{TV}$. It introduces the improved OptISTA (IOptISTA), which embeds a weighting matrix $W_n$ into the proximal-gradient framework to accelerate convergence, with updates $y_{k+1}=\mathrm{Prox}_{\gamma_k\eta_k h}(y_k-\gamma_k\eta_k W_n\nabla f(x_k))$ and step parameters $\eta_k=1/L$, $\alpha_k$, and $\gamma_k$ inherited from prior optimal-gradient analyses. Numerical experiments on six images under disk and Gaussian blur with noise show that IOptISTA outperforms ISTA, FISTA, EFISTA, IISTA, and OptISTA in PSNR and SSIM for both $\ell_1$ and TV regularizations, validating its practical advantage. The results indicate that incorporating a carefully chosen weighting matrix yields faster convergence and better restoration in non-blind deblurring, with potential for broader use in convex imaging problems.

Abstract

Image deblurring remains a central research area within image processing, critical for its role in enhancing image quality and facilitating clearer visual representations across diverse applications. This paper tackles the optimization problem of image deblurring, assuming a known blurring kernel. We introduce an improved optimal proximal gradient algorithm (IOptISTA), which builds upon the optimal gradient method and a weighting matrix, to efficiently address the non-blind image deblurring problem. Based on two regularization cases, namely the $l_1$ norm and total variation norm, we perform numerical experiments to assess the performance of our proposed algorithm. The results indicate that our algorithm yields enhanced PSNR and SSIM values, as well as a reduced tolerance, compared to existing methods.

An Improved Optimal Proximal Gradient Algorithm for Non-Blind Image Deblurring

TL;DR

This work tackles non-blind image deblurring by formulating the problem as min_x \frac{1}{2}\|Ax-b\|^{2}+h(x) with a known blur operator and convex regularizers such as or . It introduces the improved OptISTA (IOptISTA), which embeds a weighting matrix into the proximal-gradient framework to accelerate convergence, with updates and step parameters , , and inherited from prior optimal-gradient analyses. Numerical experiments on six images under disk and Gaussian blur with noise show that IOptISTA outperforms ISTA, FISTA, EFISTA, IISTA, and OptISTA in PSNR and SSIM for both and TV regularizations, validating its practical advantage. The results indicate that incorporating a carefully chosen weighting matrix yields faster convergence and better restoration in non-blind deblurring, with potential for broader use in convex imaging problems.

Abstract

Image deblurring remains a central research area within image processing, critical for its role in enhancing image quality and facilitating clearer visual representations across diverse applications. This paper tackles the optimization problem of image deblurring, assuming a known blurring kernel. We introduce an improved optimal proximal gradient algorithm (IOptISTA), which builds upon the optimal gradient method and a weighting matrix, to efficiently address the non-blind image deblurring problem. Based on two regularization cases, namely the norm and total variation norm, we perform numerical experiments to assess the performance of our proposed algorithm. The results indicate that our algorithm yields enhanced PSNR and SSIM values, as well as a reduced tolerance, compared to existing methods.

Paper Structure

This paper contains 8 sections, 3 theorems, 30 equations, 12 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Algorithm
  3. Numerical experiments
  4. Setting of n in Wn
  5. Case one: l1 norm regularization
  6. Case two: Total variation (TV) regularization
  7. Conclusion and future works
  8. Proof of Lemma \ref{['lemma1']}

Key Result

Lemma 1

In Algorithm IOptISTA, we have $x_K = y_K$.

Figures (12)

  • Figure 1: The illustration of the IOptISTA and OptISTA algorithms. Where "$x_{k+1} (W_{1})$" is the output of OptISTA, "$x_{k+1}(W_{n})$" is the output of IOptISTA.
  • Figure 2: The numerical results for Figure \ref{['details_images']}(a) with "$k = \text{fspecial}(\text{'disk'}, 13)$" and $\varepsilon\sim\mathcal{N}(0, 10^{-3})$.
  • Figure 3: Details of the test images.
  • Figure 4: The Tol, PSNR and SSIM results for different images, noises and $n$ in $W_{n}$. First row: Figure \ref{['details_images']}(d) with "$k = \text{fspecial}(\text{'disk'}, 7)$" and $\varepsilon\sim\mathcal{N}(0,1e-4)$. Second row: Figure \ref{['details_images']}(a) with "$k = \text{fspecial}(\text{'disk'}, 13.5)$" and $\varepsilon\sim\mathcal{N}(0,5e-4)$. Third row: Figure \ref{['details_images']}(d) with "$k = \text{fspecial}(\text{'disk'}, 11)$" and $\varepsilon\sim\mathcal{N}(0,5e-4)$.
  • Figure 5: The Tol, PSNR and SSIM results for Figure \ref{['details_images']}(b) with "$k = \text{fspecial}(\text{'disk'}, 12)$" and $\varepsilon\sim\mathcal{N}(0, 1e-4)$.
  • ...and 7 more figures

Theorems & Definitions (8)

  • Remark 1
  • Remark 2
  • Lemma 1
  • proof
  • Lemma 2
  • Lemma 3
  • proof
  • proof