Discrete Total Variation with Finite Elements and Applications to Imaging

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 origi...

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Vydáno v:Journal of mathematical imaging and vision Ročník 61; číslo 4; s. 411 - 431
Hlavní autoři: Herrmann, Marc, Herzog, Roland, Schmidt, Stephan, Vidal-Núñez, José, Wachsmuth, Gerd
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York Springer US 01.05.2019
Springer Nature B.V
Témata:
Algorithms
Applications of Mathematics
Approximation
Computer Science
Continuity (mathematics)
Finite element method
Image Processing and Computer Vision
Image reconstruction
Mathematical analysis
Mathematical Methods in Physics
Noise reduction
Partial differential equations
Polynomials
Quadratures
Signal,Image and Speech Processing
Dual problem
68U10
65K05
94A08
Numerical algorithms
49M29
Discrete total variation
Image reconstruction
ISSN:0924-9907, 1573-7683
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Shrnutí: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.
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ISSN:0924-9907
1573-7683
DOI:10.1007/s10851-018-0852-7