A multiscale fourth‐order model for the image inpainting and low‐dimensional sets recovery

Abstract

We consider a fourth‐order variational model, to solve the image inpainting problem, with the emphasis on the recovery of low‐dimensional sets (edges and corners) and the curvature of the edges. The model permits also to perform simultaneously the restoration (filtering) of the initial image where this one is available. The multiscale character of the model follows from an adaptive selection of the diffusion parameters that allows us to optimize the regularization effects in the neighborhoods of the small features that we aim to preserve. In addition, because the model is based on the high‐order derivatives, it favors naturally the accurate capture of the curvature of the edges, hence to balance the tasks of obtaining long curved edges and the obtention of short edges, tip points, and corners. We analyze the method in the framework of the calculus of variations and the Γ‐convergence to show that it leads to a convergent algorithm. In particular, we obtain a simple discrete numerical method based on a standard mixed‐finite elements with well‐established approximation properties. We compare the method to the Cahn–Hilliard model for the inpainting, and we present several numerical examples to show its performances. Copyright © 2016 John Wiley & Sons, Ltd.

On the image inpainting problem from the viewpoint of a nonlocal Cahn-Hilliard type equation

“…This was the motivation for the development of advanced numerical methods based on finite difference methods [11] , finite element methods with preconditioning [8] , spectral methods [9] , and operator splitting [20] in order to make the practical computation faster and more efficient. For a mathematical analysis of the fourth order models we refer to [35] .…”

“…Note that it is almost a rule that nonlinear PDEs (like Perona-Malik or the CHE mentioned above; see also [35] ) often capture the most interesting phenomena. This increases the computational costs and makes the numerical procedure more complicated.…”

An adaptive Cahn-Hilliard equation for enhanced edges in binary image inpainting

“…In fact, the discontinuities (edges) are contained in regions where the brightness changes abruptly and therefore where this error indicator is large. Thus, quantity is well suited to locally control and select the diffusion coefficient a using an adaptive strategy, see Belhachmi and Fr ed eric 20 ; Theljani et al 21,22 During the adaptation, we use the following formula for each triangle K…”

“…Thus, quantity is well suited to locally control and select the diffusion coefficient α using an adaptive strategy, see Belhachmi and Frédéric 20 ; Theljani et al. 21,22 During the adaptation, we use the following formula for each triangle K where α trh is a threshold parameter and κ is a coefficient chosen to control the rate of decreasing in α . In addition, we use a mesh-adaptation technique allowing a tight location of the singularities.…”

An image inpainting method for object removal based on difference degree constraint

“…The new model can inherit genuine anisotropism of tensor regularization, and better handle subtle details and complex structures. Theljani et al [36] based on a fourth-order variational model, used an adaptive selection of the diffusion parameters to optimize the regularization effects in the neighborhoods of the small features. Mousavi et al [28] considered the effect of spectrum and phase angle of the Fourier transform, generated two regularization parameters and had two degree of freedom, so as to restore an image.…”