The mean curvature-based image deblurring model is widely used in image restoration to preserve edges and remove staircase effect in the resulting images. However, the Euler–Lagrange equations of the mean curvature model lead to the challenging problem of solving a nonlinear fourth order integro-differential equation. Furthermore the discretization of the Euler–Lagrange equations produces a nonlinear ill-conditioned system which affects the convergence of the numerical algorithms such as Krylov subspace methods (GMRES, etc.) In this paper, we have treated the high order nonlinearity by converting the nonlinear fourth order integro-differential equation into a system of first order equations. To speed up convergence by GMRES method, we have introduced a new circulant preconditioned matrix. Fast convergence is assured by the proved analytical property of our proposed new preconditioner. The first order error estimates are also established for the finite difference discretization. The effectiveness of our algorithm can be observed through fast convergence rates in numerical examples.
|Journal||Computational and Applied Mathematics|
|State||Published - Jun 2022|
Bibliographical noteFunding Information:
The first and the third author would like to acknowledge the DSR (Deanship of Scientific Research) at KFUPM for funding this work through small business project (SB181013).
© 2022, The Author(s) under exclusive licence to Sociedade Brasileira de Matemática Aplicada e Computacional.
- Ill-posed problem
- Image deblurring
- Mean curvature
- Numerical analysis
- Precondition matrix
ASJC Scopus subject areas
- Computational Mathematics
- Applied Mathematics