Discrete Total Variation with Finite Elements and Applications to Imaging

• Published in: Journal of mathematical imaging and vision Vol. 61; no. 4; pp. 411 - 431
• Main Authors: Herrmann, Marc, Herzog, Roland, Schmidt, Stephan, Vidal-Núñez, José, Wachsmuth, Gerd
• Format: Journal Article
• Language: English
• Published: New York Springer US 15.05.2019; Springer Nature B.V
• Subjects: Algorithms; Applications of Mathematics; Computer Science; Continuity (mathematics); Finite element method; Image Processing and Computer Vision; Image reconstruction; Mathematical analysis; Mathematical Methods in Physics; Noise reduction; Polynomials; Signal,Image and Speech Processing; Dual problem; 68U10; 65K05; 94A08; Numerical algorithms; 49M29; Discrete total variation; Image reconstruction
• ISSN: 0924-9907; 1573-7683
• DOI: 10.1007/s10851-018-0852-7

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.