Higher-Order Total Directional Variation: Analysis
Abstract
We analyze a new notion of total anisotropic higher-order variation which, differently from total generalized variation in [K. Bredies, K. Kunisch, and T. Pock, SIAM J. Imaging Sci., 3 (2010), pp. 492--526], quantifies for possibly nonsymmetric 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 optimization 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.
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© 2020, Society for Industrial and Applied Mathematics.
History
Submitted: 17 January 2019
Accepted: 6 January 2020
Published online: 24 March 2020
Keywords
MSC codes
Authors
Funding Information
Agence Nationale de la Recherche https://doi.org/10.13039/501100001665 : ANR-14-CE27-0019
Funding Information
Alan Turing Institute https://doi.org/10.13039/100012338 : TU/B/000071
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H2020 Marie Skłodowska-Curie Actions https://doi.org/10.13039/100010665 : 777826
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Horizon 2020 Framework Programme https://doi.org/10.13039/100010661
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Isaac Newton Institute for Mathematical Sciences https://doi.org/10.13039/100012112
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Leverhulme Trust https://doi.org/10.13039/501100000275
Funding Information
Engineering and Physical Sciences Research Council https://doi.org/10.13039/501100000266 : EP/L016516/1
Funding Information
Engineering and Physical Sciences Research Council https://doi.org/10.13039/501100000266 : EP/M00483X/1, EP/K009745/1, EP/N014588/1
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