Iterative methods for solving the dual formulation arising from image restoration

作者：Chan Tony F^*; Chen Ke; Carter Jamylle L

来源：Electronic Transactions on Numerical Analysis, 2007, 26: 299-311.

• image restoration
• nonlinear partial differential equations
• singularity
• nonlinear iterations
• Fourier analysis
• multigrid method

Many variational models for image denoising restoration are formulated in primal variables that are directly linked to the solution to be restored. If the total variation related semi-norm is used in the models, one consequence is that extra regularization is needed to remedy the highly non-smooth and oscillatory coefficients for effective numerical solution. The dual formulation was often used to study theoretical properties of a primal formulation. However as a model, this formulation also offers some advantages over the primal formulation in dealing with the above mentioned oscillation and non-smoothness. This paper presents some preliminary work on speeding up the Chambolle method [J. Math. Imaging Vision, 20 (2004), pp. 89-97] for solving the dual formulation. Following a convergence rate analysis of this method, we first show why the nonlinear multigrid method encounters some difficulties in achieving convergence. Then we propose a modified smoother for the multigrid method to enable it to achieve convergence in solving a regularized Chambolle formulation. Finally, we propose a linearized primal-dual iterative method as an alternative stand-alone approach to solve the dual formulation without regularization. Numerical results are presented to show that the proposed methods are much faster than the Chambolle method.

• 出版日期2007