Discrete gradient methods for solving variational image regularisation models. (28th June 2017)

• Record Type: Journal Article
• Authors: Grimm, V; McLachlan, Robert I; McLaren, David I; Quispel, G R W; Schönlieb, C-B
• Is Part Of: Journal of physics. Volume 50:Number 29(2017)
• Publication Date: 2017-06-28
• Subjects: gradient flow -- discrete gradient method -- variational image processing models -- image deblurring -- image denoising -- total variation -- geometric numerical integration; Mathematical physics -- Periodicals; Statistical physics -- Periodicals; Quantum theory -- Periodicals; Matter -- Properties -- Periodicals; 530.105
• DOI: 10.1088/1751-8121/aa747c
• Languages: English
• ISSNs: 1751-8113

Abstract

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.