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A Preconditioned Version of a Nested Primal-Dual Algorithm for Image Deblurring

Stefano Aleotti, Marco Donatelli, Rolf Krause, Giuseppe Scarlato

TL;DR

The paper addresses variational image deblurring by solving $\min_u f(u)+h(Wu)$ with $f(u)=\tfrac{1}{2}\|Au-b^{\delta}\|^{2}$, introducing a left preconditioned, inexact proximal-gradient method (PNPD). It shows that variable-metric schemes are equivalent to right preconditioning in this setting and develops PNPD with both stationary and non-stationary preconditioners, along with a polynomial preconditioner that preserves a commutative structure for efficient inner iterations. Convergence is established for the PNPD framework, and extensive numerical results on blurred and noisy images demonstrate that PNPD can achieve comparable reconstruction quality to NPDIT while significantly reducing CPU time. The findings highlight the practical advantage of left preconditioning and non-stationary preconditioners for fast, stable image deblurring, with potential extensions to other imaging inverse problems like CT.

Abstract

Variational models for image deblurring problems typically consist of a smooth term and a potentially non-smooth convex term. A common approach to solving these problems is using proximal gradient methods. To accelerate the convergence of these first-order iterative algorithms, strategies such as variable metric methods have been introduced in the literature. In this paper, we prove that, for image deblurring problems, the variable metric strategy can be reinterpreted as a right preconditioning method. Consequently, we explore an inexact left-preconditioned version of the same proximal gradient method. We prove the convergence of the new iteration to the minimum of a variational model where the norm of the data fidelity term depends on the preconditioner. The numerical results show that left and right preconditioning are comparable in terms of the number of iterations required to reach a prescribed tolerance, but left preconditioning needs much less CPU time, as it involves fewer evaluations of the preconditioner matrix compared to right preconditioning. The quality of the computed solutions with left and right preconditioning are comparable. Finally, we propose some non-stationary sequences of preconditioners that allow for fast and stable convergence to the solution of the variational problem with the classical $\ell^2$--norm on the fidelity term.

A Preconditioned Version of a Nested Primal-Dual Algorithm for Image Deblurring

TL;DR

The paper addresses variational image deblurring by solving with , introducing a left preconditioned, inexact proximal-gradient method (PNPD). It shows that variable-metric schemes are equivalent to right preconditioning in this setting and develops PNPD with both stationary and non-stationary preconditioners, along with a polynomial preconditioner that preserves a commutative structure for efficient inner iterations. Convergence is established for the PNPD framework, and extensive numerical results on blurred and noisy images demonstrate that PNPD can achieve comparable reconstruction quality to NPDIT while significantly reducing CPU time. The findings highlight the practical advantage of left preconditioning and non-stationary preconditioners for fast, stable image deblurring, with potential extensions to other imaging inverse problems like CT.

Abstract

Variational models for image deblurring problems typically consist of a smooth term and a potentially non-smooth convex term. A common approach to solving these problems is using proximal gradient methods. To accelerate the convergence of these first-order iterative algorithms, strategies such as variable metric methods have been introduced in the literature. In this paper, we prove that, for image deblurring problems, the variable metric strategy can be reinterpreted as a right preconditioning method. Consequently, we explore an inexact left-preconditioned version of the same proximal gradient method. We prove the convergence of the new iteration to the minimum of a variational model where the norm of the data fidelity term depends on the preconditioner. The numerical results show that left and right preconditioning are comparable in terms of the number of iterations required to reach a prescribed tolerance, but left preconditioning needs much less CPU time, as it involves fewer evaluations of the preconditioner matrix compared to right preconditioning. The quality of the computed solutions with left and right preconditioning are comparable. Finally, we propose some non-stationary sequences of preconditioners that allow for fast and stable convergence to the solution of the variational problem with the classical --norm on the fidelity term.
Paper Structure (17 sections, 12 theorems, 71 equations, 14 figures, 1 table, 3 algorithms)

This paper contains 17 sections, 12 theorems, 71 equations, 14 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Inexact proximal gradient methods
  4. Nested Primal-Dual Method (NPD)
  5. A variable metric nested primal--dual framework
  6. Variable metric approach as right preconditioning
  7. Preconditioned Nested Primal-Dual (PNPD)
  8. A polynomial choice for P
  9. PNPD as an inexact version of FISTA and ITTA
  10. PNPD with a non-stationary preconditioner
  11. Numerical results
  12. Example 1
  13. Stability
  14. Non--stationary Preconditioning
  15. Example 2: cameraman with 2% Gaussian noise
  16. ...and 2 more sections

Key Result

lemma thmcounterlemma

Let $\varphi_1,\varphi_2 : \mathbb{R}^{d'}\rightarrow\mathbb{R}\cup\{\infty\}$ be proper, convex and lower semicontinuous functions and $W\in\mathbb{R}^{d'\times d}$.

Figures (14)

  • Figure 1: Example 1: (\ref{['fig:example_1 ground_truth']}) Ground truth image of a cameramen. (\ref{['fig:example_1 psf']}) PSF used to blur the ground truth (center crop of size $20\times20$). (\ref{['fig:example_1 observed']}) Observed image $b^\delta$ obtained by adding white Gaussian noise on top of the discrete circular convolution of the ground truth and the PSF. The Gaussian noise $\eta_{\delta}$ is such that $\|\eta_{\delta}\| = 0.01\|b^\delta\|$.
  • Figure 2: Example 1: Comparison of the reconstructions obtained with NPD, PNPD, and NPDIT after 10 iterations. The preconditioner parameter is $\nu=10^{-1}$. The number of nested loop iterations is $k_{\text{max}}=3$. The regularization parameter is $\lambda=2\cdot10^{-4}$ for NPD and NPDIT, while $\lambda=2\cdot10^{-3}$ for PNPD.
  • Figure 3: Example 1: Comparison of the RREs and SSIMs between PNPD, NPD, and NPDIT. The preconditioner parameter is $\nu=10^{-1}$. The number of nested loop iterations is $k_{\text{max}}=1$ for NPD and $k_{\text{max}}=3$ for NPDIT and PNPD. The regularization parameter is $\lambda=2\cdot10^{-4}$ for NPD and NPDIT, while $\lambda=2\cdot10^{-3}$ for PNPD.
  • Figure 4: Example 1: Comparison of the SSIMs of PNPD with $k_{\text{max}}=1$ for different values of $\nu$ and $\lambda$.
  • Figure 5: Example 1: Comparison of the SSIMs of PNPD for different values of $\nu$, $\lambda$, and $k_{\text{max}}$ set high enough to fix instability.
  • ...and 9 more figures

Theorems & Definitions (23)

  • definition thmcounterdefinition
  • definition thmcounterdefinition
  • lemma thmcounterlemma
  • definition thmcounterdefinition
  • lemma thmcounterlemma
  • definition thmcounterdefinition
  • lemma thmcounterlemma
  • lemma thmcounterlemma
  • lemma thmcounterlemma
  • proof
  • ...and 13 more