Higher-order total directional variation: Analysis
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Peer-reviewed
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Abstract
We analyse a new notion of total anisotropic higher-order variation which, differently from the Total Generalized Variation by Bredies et al., quantifies for possibly non-symmetric tensor fields their variations at arbitrary order weighted by possibly inhomogeneous, smooth elliptic anisotropies. We prove some properties of this total variation and of the associated spaces of tensors with finite variations. We show the existence of solutions to a related regularity-fidelity optimisation problem. We also prove a decomposition formula which appears to be helpful for the design of numerical schemes, as shown in a companion paper, where several applications to image processing are studied.
Description
Keywords
anisotropic total variation, higher-order total variation, variational model
Journal Title
SIAM Journal on Imaging Sciences
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Journal ISSN
1936-4954
1936-4954
1936-4954
Volume Title
13
Publisher
Society for Industrial & Applied Mathematics (SIAM)
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All rights reserved
Sponsorship
EPSRC (1513548)
Engineering and Physical Sciences Research Council (EP/H023348/1)
Engineering and Physical Sciences Research Council (EP/N014588/1)
European Commission Horizon 2020 (H2020) Marie Sk?odowska-Curie actions (691070)
European Commission Horizon 2020 (H2020) Marie Sk?odowska-Curie actions (777826)
Leverhulme Trust (RPG-2018-121)
Leverhulme Trust (PLP-2017-275)
Alan Turing Institute (Unknown)
EPSRC (EP/S026045/1)
Engineering and Physical Sciences Research Council (EP/M00483X/1)
Engineering and Physical Sciences Research Council (EP/L016516/1)
Engineering and Physical Sciences Research Council (EP/H023348/1)
Engineering and Physical Sciences Research Council (EP/N014588/1)
European Commission Horizon 2020 (H2020) Marie Sk?odowska-Curie actions (691070)
European Commission Horizon 2020 (H2020) Marie Sk?odowska-Curie actions (777826)
Leverhulme Trust (RPG-2018-121)
Leverhulme Trust (PLP-2017-275)
Alan Turing Institute (Unknown)
EPSRC (EP/S026045/1)
Engineering and Physical Sciences Research Council (EP/M00483X/1)
Engineering and Physical Sciences Research Council (EP/L016516/1)