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Numerical analysis of the TV regularization and H-1 fidelity model for decomposing an image into cartoon plus texture

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Elliott, Charles M. and Smitheman, S. A. (2009) Numerical analysis of the TV regularization and H-1 fidelity model for decomposing an image into cartoon plus texture. IMA Journal of Numerical Analysis, Vol.29 (No.3). pp. 651-689. doi:10.1093/imanum/drn025 ISSN 0272-4979.

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Official URL: http://dx.doi.org/10.1093/imanum/drn025

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Abstract

The Osher–Solé–Vese (OSV) model, which is the gradient flow of an energy consisting of the total variation functional plus an H−1 fidelity term, is studied. In this paper, we build on the analysis of the OSV model which we gave in Elliott & Smitheman (2007, Comm. Pure Appl. Anal., in press). We introduce backward Euler finite-element approximations to a regularized version of the OSV initial boundary-value problem (IBVP) and to a weak formulation of the original problem. Well-posedness and unconditional Lyapunov stability of these fully discrete schemes are proved. Convergence results as the spatial mesh parameter, the time step size and the regularization parameter tend to 0 are proved. Rates of convergence as the time step size and the regularization parameter tend to 0 are found. The existence, uniqueness and Lyapunov stability of a solution to a linearly implicit finite-element approximation to the regularized version of the OSV IBVP are also proved.

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science, Engineering and Medicine > Science > Mathematics
Journal or Publication Title: IMA Journal of Numerical Analysis
Publisher: Oxford University Press
ISSN: 0272-4979
Official Date: July 2009
Dates:
DateEvent
July 2009Published
Volume: Vol.29
Number: No.3
Page Range: pp. 651-689
DOI: 10.1093/imanum/drn025
Status: Peer Reviewed
Publication Status: Published
Access rights to Published version: Restricted or Subscription Access

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