AN ADAPTIVE FINITE ELEMENT METHOD IN L-2-TV-BASED IMAGE DENOISING

作者：Hintermueller Michael; Rincon Camacho Monserrat

来源：Inverse Problems and Imaging, 2014, 8(3): 685-711.

DOI：10.3934/ipi.2014.8.685

• A posteriori error estimation
• elliptic variational inequality of the second kind
• adaptive finite elements
• total variation
• primal-dual method
• semismooth Newton method

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 re finement 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 re finement 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.

• 出版日期2014-8