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The total variation (TV)-seminorm is considered for piecewise polynomial,globally discontinuous (DG) and continuous (CG) finite element functions onsimplicial meshes. A novel, discrete variant (DTV) based on a nodal quadratureformula is defined. DTV has favorable properties, compared to the originalTV-seminorm for finite element functions. These include a convenient dualrepresentation in terms of the supremum over the space of Raviart--Thomasfinite element functions, subject to a set of simple constraints. It cantherefore be shown that a variety of algorithms for classical imagereconstruction problems, including TV-$L^2$ and TV-$L^1$, can be implemented inlow and higher-order finite element spaces with the same efficiency as theircounterparts originally developed for images on Cartesian grids.

Discrete Total Variation with Finite Elements and Applications to Imaging