@misc{WachsmuthHerrmannHerzogetal., author = {Wachsmuth, Gerd and Herrmann, Marc and Herzog, Roland and Schmidt, Stephan and Vidal-N{\´u}{\~n}ez, Jos{\´e}}, title = {Discrete Total Variation with Finite Elements and Applications to Imaging}, series = {Journal of Mathematical Imaging and Vision : (JMIV)}, volume = {61}, journal = {Journal of Mathematical Imaging and Vision : (JMIV)}, number = {4}, issn = {0924-9907}, doi = {10.1007/s10851-018-0852-7}, pages = {411 -- 431}, abstract = {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-L2 and TV-L1, 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.}, language = {en} }