Combined Second and Fourth-Order PDEs Model and Associated Variational Problems for Geometric Images Inpainting and Denoising (CSIAM Transactions on Applied Mathematics, 2021)
We consider a Partial Differential Equation model combining second andfourth-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 $Γ$-limitMumford-Shah-Euler like functional. The approximation functionals are build withinan adaptive strategy, based on two ingredients: a fine location of the singular set usingmesh refinement, and second, a local choice of the diffusion coefficients which modifythe reconstruction operator. Unlike the usual methods, mostly based on prior guess onthe continuous solution and leading to complex and nonlinear systems of PDEs, ourmethod consists in solving linear problems and updating the diffusion coefficients.The high order of the operator allows us to perform simultaneously efficient filteringof the data and the interpolation in the damaged regions. The method turns out tobe superior to any second-order model in restoring large gap connections and curvyfeatures. In order to validate this approach, we compare the results of our methodwith those of some existing one in the fields of geometry-oriented inpainting and wepresent several numerical examples.