Discrete Total Variation with Finite Elements and Applications to Imaging
This work provides a novel framework for extending classical TV image reconstruction algorithms to finite element spaces, which is important for imaging problems on irregular domains.
The paper introduces a discrete total variation (DTV) seminorm for finite element functions on simplicial meshes, enabling efficient implementation of TV-based image reconstruction algorithms in low and higher-order finite element spaces with the same efficiency as Cartesian grid methods.
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-$L^2$ and TV-$L^1$, 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.