Guest session — Image processing invited paper: A Cauchy problem for an inverse problem in image inpainting

Image inpainting consists of reconstructing lost or deteriorated parts of an image. Different techniques can be applied to solve this problem. In particular, partial differential equations (PDEs) are widely used and are proved to be efficient. If we denote by Ω the entire image domain, the image inpainting problem is then to fill in image information in the incomplete/damaged region D ⊂ Ω based on the image information available out side D, i.e., in ΩD. When this information is available in a neighborhood of the whole boundary ∂D, it can be used as a Dirichlet boundary condition for the partial differential equation that propagates the information inside D. The aim of this work is to treat the case when this information is not available near a part of the boundary ∂D. Image inpainting in this case is considered as a linear Cauchy problem for the harmonic inpainting, and a nonlinear Cauchy problem for images containing edges. The novelty of this work has two folds. The first consists in extending the work introduced for a nonlinear elliptic equation. The Cauchy problem is formulated as a two-player Nash game. The first player is given the known Dirichlet data and uses the Neumann condition prescribed over the inaccessible part of the boundary as strategy variable. The second player is given the known Neumann data, and plays with the Dirichlet condition prescribed over the inaccessible boundary. The two players solve in parallel the associated Boundary Value Problems. In the second step, the proposed approach is exploited in image inpaiting. The image information in the incomplete/damaged region is completed by the corresponding Cauchy solution. Some numerical experiments are provided to illustrate the efficiency and stability of our algorithm for several image inpainting examples.