An adaptive finite element method in $L^2$-TV-based image denoising

Michael Hintermüller,Monserrat Rincon-Camacho

doi:10.3934/ipi.2014.8.685

Abstract

The first order optimality system of a total variation regularization based variational model with $L^2$-data-fitting in image denoising ($L^2$-TV problem) can be expressed as an elliptic variational inequality of the second kind. For a finite element discretization of the variational inequality problem, an a posteriori error residual based error estimator is derived and its reliability and (partial) efficiency are established. The results are applied to solve the $L^2$-TV problem by means of the adaptive finite element method. The adaptive mesh refinement relies on the newly derived a posteriori error estimator and on an additional heuristic providing a local variance estimator to cope with noisy data. The numerical solution of the discrete problem on each level of refinement is obtained by a superlinearly convergent algorithm based on Fenchel-duality and inexact semismooth Newton techniques and which is stable with respect to noise in the data. Numerical results justifying the advantage of adaptive finite elements solutions are presented.

Highlights

Image denoising is one of the fundamental tasks in mathematical image processing and aims at reconstructing an image from given noisy data
In view of (1.3), our aim is to extend these estimates to situations where j models the total variation (TV)-regularization term in image denoising and where the variational inequality is yet formulated in a di↵erent primal-dual fashion which is in particular suitable for numerical solution schemes
We observe that the algorithm using the adaptive mesh strategy outperforms the one using a uniform mesh

Summary

Introduction

Image denoising is one of the fundamental tasks in mathematical image processing and aims at reconstructing an image from given noisy data. Often this is achieved by employing so-called variational methods which require to minimize an objective functional typically composed of a data fidelity and a regularization term; see, e.g., [44] for further details. We utilize an objective functional which was previously studied in [34] and several papers mentioned therein and is given by (P ) minimize μ 2 krvk2L2 (⌦) + 1 2 kv z k2L2(⌦). We always assume that 0 < μ ⌧ ↵ such that this model is a very close approximation to the renown total variation (TV) regularization model proposed by Rudin, Osher and Fatemi in their seminal work [35]: (1.1)

Objectives

Results

Conclusion