Discrete Total Variation with Finite Elements and Applications to Imaging - Herrmann, Marc; Herzog, Roland; Schmidt, Stephan; Vidal-Núñez, José; Wachsmuth, Gerd (Journal of mathematical imaging and vision, 2019)
The total variation (TV)-seminorm is considered for piecewise polynomial, globally discontinuous (DG) and continuous (CG) finite element functions on simplicial meshes. A novel, discrete variant (DTV) based on a nodal quadrature formula is defined. DTV has favorable properties, compared to the original TV-seminorm for finite element functions. These include a convenient dual representation in terms of the supremum over the space of Raviart–Thomas finite element functions, subject to a set of simple constraints. It can therefore be shown that a variety of algorithms for classical image reconstruction problems, including TV- L 2 denoising and inpainting, can be implemented in low- and higher-order finite element spaces with the same efficiency as their counterparts originally developed for images on Cartesian grids.