Discrete gradient methods for solving variational image regularisation models
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Authors
Grimm, V
McLachlan, RI
McLaren, DI
Quispel, GRW
Schönlieb, CB
Publication Date
2017-06-28Journal Title
Journal of Physics A: Mathematical and Theoretical
ISSN
1751-8113
Publisher
IOP Sciences
Volume
50
Issue
29
Number
295201
Type
Article
This Version
AM
Metadata
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Grimm, V., McLachlan, R., McLaren, D., Quispel, G., & Schönlieb, C. (2017). Discrete gradient methods for solving variational image regularisation models. Journal of Physics A: Mathematical and Theoretical, 50 (29. 295201) https://doi.org/10.1088/1751-8121/aa747c
Abstract
Discrete gradient methods are well-known methods of geometric numerical integration, which preserve the dissipation of gradient systems. In this paper we show that this property of discrete gradient methods can be interesting in the context of variational models for image processing, that is where the processed image is computed as a minimiser of an energy functional. Numerical schemes for computing minimisers of such energies are desired to inherit the dissipative property of the gradient system associated to the energy and consequently guarantee a monotonic decrease of the energy along iterations, avoiding situations in which more computational work might lead to less optimal solutions. Under appropriate smoothness assumptions on the energy functional we prove that discrete gradient methods guarantee a monotonic decrease of the energy towards stationary states, and we promote their use in image processing by exhibiting experiments with convex and non-convex variational models for image deblurring, denoising, and inpainting.
Sponsorship
Engineering and Physical Sciences Research Council (EP/N014588/1)
Engineering and Physical Sciences Research Council (EP/M00483X/1)
Engineering and Physical Sciences Research Council (EP/J009539/1)
Alan Turing Institute (unknown)
European Commission Horizon 2020 (H2020) Marie Sk?odowska-Curie actions (691070)
Leverhulme Trust (RPG-2015-250)
Engineering and Physical Sciences Research Council (EP/H023348/1)
European Commission Horizon 2020 (H2020) Marie Sk?odowska-Curie actions (777826)
Identifiers
External DOI: https://doi.org/10.1088/1751-8121/aa747c
This record's URL: https://www.repository.cam.ac.uk/handle/1810/292706
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