An adaptive finite element method in $L^2$-TV-based image denoising
Michael Hintermüller Monserrat Rincon-Camacho
Inverse Problems & Imaging 2014, 8(3): 685-711 doi: 10.3934/ipi.2014.8.685
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.
keywords: A posteriori error estimation primal-dual method adaptive finite elements semismooth Newton method. elliptic variational inequality of the second kind total variation
A nonlinear multigrid solver with line Gauss-Seidel-semismooth-Newton smoother for the Fenchel pre-dual in total variation based image restoration
Ke Chen Yiqiu Dong Michael Hintermüller
Inverse Problems & Imaging 2011, 5(2): 323-339 doi: 10.3934/ipi.2011.5.323
Based on the Fenchel pre-dual of the total variation model, a nonlinear multigrid algorithm for image denoising is proposed. Due to the structure of the differential operator involved in the Euler-Lagrange equations of the dual models, line Gauss-Seidel-semismooth-Newton step is utilized as the smoother, which provides rather good smoothing rates. The paper ends with a report on numerical results and a comparison with a very recent nonlinear multigrid solver based on Chambolle's iteration [6].
keywords: multigrid method. Image restoration total variation regularization duality

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