[Paper Review] Image denoising: learning noise distribution via PDE-constrained optimization

Juan-Carlos De Los Reyes, Carola‐Bibiane Schönlieb|arXiv (Cornell University)|Jul 14, 2012

Image and Signal Denoising Methods39 references3 citations

TL;DR

This paper proposes a PDE-constrained optimization framework to learn noise distributions in total variation (TV) image denoising by determining optimal noise model weights. It establishes existence of solutions, derives an optimality system, and numerically computes parameters using quasi-Newton and semismooth Newton methods, enabling robust, data-driven noise modeling in TV-based denoising.

ABSTRACT

We propose a PDE-constrained optimization approach for the determination of noise distribution in total variation (TV) image denoising. An optimization problem for the determination of the weights correspondent to different types of noise distributions is stated and existence of an optimal solution is proved. A tailored regularization approach for the approximation of the optimal parameter values is proposed thereafter and its consistency studied. Additionally, the differentiability of the solution operator is proved and an optimality system characterizing the optimal solutions of each regularized problem is derived. The optimal parameter values are numerically computed by using a quasi-Newton method, together with semismooth Newton type algorithms for the solution of the TV-subproblems.

Motivation & Objective

• To address the challenge of accurately modeling unknown noise distributions in total variation (TV) image denoising.
• To formulate a PDE-constrained optimization problem for determining optimal weights corresponding to different noise types.
• To prove the existence of an optimal solution for the noise distribution parameter estimation problem.
• To develop a tailored regularization approach for approximating optimal parameters and establish its consistency.
• To derive an optimality system characterizing the solution of each regularized problem for numerical computation.

Proposed method

• Formulates a PDE-constrained optimization problem where the state equation is the TV denoising model, and the parameters to be optimized represent noise distribution weights.
• Introduces a regularization strategy to stabilize the approximation of optimal parameter values and proves its consistency in the limit.
• Establishes the differentiability of the solution operator mapping parameters to denoised images, enabling gradient-based optimization.
• Derives a first-order optimality system that characterizes the optimal solution of each regularized problem, providing necessary conditions for optimality.
• Employs a quasi-Newton method for outer-level optimization of the noise model parameters, combined with semismooth Newton-type solvers for the inner TV subproblems.
• Solves the resulting nonlinear system using a nested iterative approach, coupling parameter update and denoising subproblem resolution.

Experimental results

Research questions

• RQ1 Can a PDE-constrained optimization framework be effectively used to learn unknown noise distributions in TV-based image denoising?
• RQ2 Does the proposed optimization problem admit a solution, and can the existence of an optimal parameter set be rigorously proven?
• RQ3 How can regularization be designed to ensure stable and consistent approximation of the optimal noise model parameters?
• RQ4 What is the differentiability property of the solution operator in the context of TV denoising with variable noise parameters?
• RQ5 Can an optimality system be derived to characterize the solution of the regularized problem and enable efficient numerical computation?

Key findings

• The existence of an optimal solution for the noise distribution parameter estimation problem in TV denoising is rigorously proven under the proposed PDE-constrained framework.
• The proposed regularization approach for parameter approximation is shown to be consistent, meaning the approximated solutions converge to the true optimal parameters as regularization is refined.
• The solution operator mapping noise model parameters to denoised images is proven to be differentiable, enabling gradient-based optimization methods.
• An optimality system is derived that characterizes the solution of each regularized problem, providing a necessary condition for optimality.
• Numerical experiments demonstrate that the combination of quasi-Newton and semismooth Newton methods effectively computes optimal noise model parameters.
• The framework enables data-driven selection of noise distributions in TV denoising, improving robustness and accuracy without prior knowledge of noise type.