MULTISCALE FOURTH-ORDER MODEL FOR IMAGE INPAINTING AND LOW-DIMENSIONAL SETS RECOVERY
Résumé
We consider a fourth-order variational model, to solve the image inpainting problem, with the emphasis on the recovery of low-dimensional sets (edges, 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 which allows us to optimise the regularization effects in the neighborhoods of the small features that we aim to preserve. In addition, since 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 $\Gamma$-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.
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