Volume 2, Issue 4
Combined Second and Fourth-Order PDEs Model and Associated Variational Problems for Geometric Images Inpainting and Denoising

Anis Theljani, Zakaria Belhachmi, Moez Kallel & Maher Moakher

CSIAM Trans. Appl. Math., 2 (2021), pp. 652-679.

Published online: 2021-11

Export citation
  • Abstract

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.

  • AMS Subject Headings

35G15, 34K28, 65M32, 65M50, 94A08

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

thaljanianis@gmail.com (Anis Theljani)

  • BibTex
  • RIS
  • TXT
@Article{CSIAM-AM-2-652, author = {Theljani , AnisZakaria Belhachmi , 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 = {

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.

}, issn = {2708-0579}, doi = {https://doi.org/10.4208/csiam-am.SO-2020-0007}, url = {http://global-sci.org/intro/article_detail/csiam-am/19987.html} }
TY - JOUR T1 - Combined Second and Fourth-Order PDEs Model and Associated Variational Problems for Geometric Images Inpainting and Denoising AU - Theljani , Anis AU - Zakaria Belhachmi , AU - Moez Kallel , AU - Moakher , Maher JO - CSIAM Transactions on Applied Mathematics VL - 4 SP - 652 EP - 679 PY - 2021 DA - 2021/11 SN - 2 DO - http://doi.org/10.4208/csiam-am.SO-2020-0007 UR - https://global-sci.org/intro/article_detail/csiam-am/19987.html KW - Moving mesh method, conservative interpolation, iterative method, $l^ 2$ projection. AB -

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.

Anis Theljani, Zakaria Belhachmi, Moez Kallel & Maher Moakher. (2021). Combined Second and Fourth-Order PDEs Model and Associated Variational Problems for Geometric Images Inpainting and Denoising. CSIAM Transactions on Applied Mathematics. 2 (4). 652-679. doi:10.4208/csiam-am.SO-2020-0007
Copy to clipboard
The citation has been copied to your clipboard