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An inexact primal-dual method with correction step for a saddle point problem in image debluring

Changjie Fang, Liliang Hu, Shenglan Chen

TL;DR

The paper addresses saddle-point problems arising in image restoration via a first-order primal-dual framework. It introduces an inexact primal-dual method with a correction step based on extended proximal operators with a symmetric positive definite matrix $D$, and it relaxes the usual step-size constraints by enforcing that $R - \tau_k \lambda_k S^{-1}$ is positive definite. It proves convergence of the method and an ergodic rate of $O(1/N)$, with refined rates when the error tolerances decay as $O(1/k^{2\alpha+1})$. It demonstrates practical efficacy by applying the method to TV-L$_1$ image deblurring and showing favorable performance against established primal-dual schemes.

Abstract

In this paper,we present an inexact primal-dual method with correction step for a saddle point problem by introducing the notations of inexact extended proximal operators with symmetric positive definite matrix $D$. Relaxing requirement on primal-dual step sizes, we prove the convergence of the proposed method. We also establish the $O(1/N)$ convergence rate of our method in the ergodic sense. Moreover, we apply our method to solve TV-L$_1$ image deblurring problems. Numerical simulation results illustrate the efficiency of our method.

An inexact primal-dual method with correction step for a saddle point problem in image debluring

TL;DR

The paper addresses saddle-point problems arising in image restoration via a first-order primal-dual framework. It introduces an inexact primal-dual method with a correction step based on extended proximal operators with a symmetric positive definite matrix , and it relaxes the usual step-size constraints by enforcing that is positive definite. It proves convergence of the method and an ergodic rate of , with refined rates when the error tolerances decay as . It demonstrates practical efficacy by applying the method to TV-L image deblurring and showing favorable performance against established primal-dual schemes.

Abstract

In this paper,we present an inexact primal-dual method with correction step for a saddle point problem by introducing the notations of inexact extended proximal operators with symmetric positive definite matrix . Relaxing requirement on primal-dual step sizes, we prove the convergence of the proposed method. We also establish the convergence rate of our method in the ergodic sense. Moreover, we apply our method to solve TV-L image deblurring problems. Numerical simulation results illustrate the efficiency of our method.
Paper Structure (7 sections, 12 theorems, 105 equations, 9 figures, 2 tables, 3 algorithms)

This paper contains 7 sections, 12 theorems, 105 equations, 9 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Main results
  4. Numerical experiments
  5. Sensitivity of parameters
  6. Image denoising
  7. Conclusions

Key Result

Lemma 2.1

Suppose $z\approx_1^{\varepsilon}\mathop{\arg\min}_{x \in X}\{h(x)+\frac{1}{2\tau}\|x-y\|_D^2\}$, then $z\in$ dom $h$ and $z\approx_0^{\varepsilon} \arg\min_X\{h(x)+\frac{1}{2\tau}\|x-y\|_D^2\}$.

Figures (9)

  • Figure 1: Original cammeraman.png(256$\times$256)
  • Figure 2: Original man.png(1024$\times$1024)
  • Figure 3: Cammeraman.png with noise
  • Figure 4: Man.png with noise
  • Figure 5: Sensitivity of $\alpha$
  • ...and 4 more figures

Theorems & Definitions (28)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 18 more