@Article{CSIAM-AM-2-4,
author = {Theljani, Anis and Zakaria, Belhachmi,  and Moez, Kallel,  and Moakher, Maher},
title = {Combined Second and Fourth-Order PDEs Model and Associated Variational Problems for Geometric Images Inpainting and Denoising},
journal = {CSIAM Transactions on Applied Mathematics},
year = {2021},
volume = {2},
number = {4},
pages = {652--679},
abstract = {<p style="text-align: justify;">We consider a Partial Differential Equation model combining second and
fourth-order operators for solving the geometry inpainting and denoising problems.
The model allows the accurate recovery of curvatures and the singular set of the reconstructed image (edges, corners). The approach proposed permits a dynamical modelling by constructing a family of simple discrete energies that admit as a $Γ$-limit
Mumford-Shah-Euler like functional. The approximation functionals are build within
an adaptive strategy, based on two ingredients: a fine location of the singular set using
mesh refinement, and second, a local choice of the diffusion coefficients which modify
the reconstruction operator. Unlike the usual methods, mostly based on prior guess on
the continuous solution and leading to complex and nonlinear systems of PDEs, our
method consists in solving linear problems and updating the diffusion coefficients.
The high order of the operator allows us to perform simultaneously efficient filtering
of the data and the interpolation in the damaged regions. The method turns out to
be superior to any second-order model in restoring large gap connections and curvy
features. In order to validate this approach, we compare the results of our method
with those of some existing one in the fields of geometry-oriented inpainting and we
present several numerical examples.</p>},
issn = {2708-0579},
doi = {https://doi.org/10.4208/csiam-am.SO-2020-0007},
url = {https://global-sci.com/article/82359/combined-second-and-fourth-order-pdes-model-and-associated-variational-problems-for-geometric-images-inpainting-and-denoising}
}