Abstract
Image denoising using mean curvature leads to the problem of solving a nonlinear fourth-order integro-differential equation. The nonlinear fourth-order term comes from the mean curvature regularization functional. In this paper, we treat this high-order nonlinearity by reducing the nonlinear fourth-order integro-differential equation to a system of first-order equations. Then a cell-centered finite difference scheme is applied to this system. With a lexicographical ordering of the unknowns, the discretization of the mean curvature functional leads to a block pentadiagonal matrix. Our contributions are fourfold: (i) we give a new method for treating the high-order nonlinearity term; (ii) we express the discretization of this term in terms of simple matrices; (iii) we give an analysis for this new method and establish that the error is of first order; and (iv) we verify this theoretical result by illustrating the convergence rates in numerical experiments.
Original language | English |
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Pages (from-to) | 108-127 |
Number of pages | 20 |
Journal | Electronic Transactions on Numerical Analysis |
Volume | 54 |
DOIs | |
Publication status | Published - 8 Jan 2021 |
Keywords
- cell-centered finite difference method
- image denoising
- mean curvature
- numerical analysis
- differential equation