Fast Solvers for Cahn--Hilliard Inpainting

Abstract

The solution of Cahn-Hilliard variational inequalities is of interest in many applications. We discuss the use of them as a tool for binary image inpainting. This has been done before using double-well potentials but not for nonsmooth potentials as considered here. The existing bound constraints are incorporated via the Moreau-Yosida regularization technique. We develop effective preconditioners for the efficient solution of the Newton steps associated with the fast solution of the MoreauYosida regularized problem. Numerical results illustrate the efficiency of our approach. Moreover, precise eigenvalue intervals are given for the preconditioned system using a double-well potential. A comparison between the smooth and nonsmooth Cahn-Hilliard inpainting models shows that the latter achieves better results.

Fast solution of Cahn–Hilliard variational inequalities using implicit time discretization and finite elements

“…In Figure 1 one can clearly see at most two eigenvalue clusters both for varying c and for varying h, which are bounded below by one. With slowly increased ε we get similar eigenvalue distributions, see Figure 2 and [11]. The number of eigenvalue clusters remains constant for varying c and h. Next, we will see that also complex eigenvalues appear, see Figure 3.…”

“…Note the necessity of the time step restriction τ < 4γε 3 for the implicit scheme, which has not to be claimed for the semi-implicit one. Even though, the results obtained for large time steps with the semi-implicit system are highly inaccurate for capturing the evolution of the sharp interface model, see Section 8.2 or [13,5,11].…”

GPU CUDA accelerated Image Inpainting using Fourth Order PDE equation

“…The most recent approach to inpainting is based on fourth order PDE method [8]. There are lots of researches trying to improve the quality and to reduce processing time of inpainting process using high order PDE method [5,8].…”

“…There are lots of researches trying to improve the quality and to reduce processing time of inpainting process using high order PDE method [5,8]. A number of methods were suggested to solve inpainting problem, many of which are based on advanced mathematical techniques [1,2], [4][5][6], 8,9].…”

A Preconditioner for the Ohta--Kawasaki Equation

“…In general it is very difficult to precondition the sum of different matrices. The approach adopted here is the matching strategy of Pearson and Wathen [13,2]: the sum is approximated by the product of matrices, carefully chosen to match as many terms of the sum as possible. We propose the approximation…”