In our approach, we now denote by the image and its boundary, the missing part of the image and its boundary. Let be the image that we want to restore. We assume that is known in , and unknown in .
The idea is to adapt the crack localization method to inpainting: crack detection first allows us to identify the cracks (or edges) of the hidden part of the image, and then we will impose that the Laplacian of the restored image is equal to zero in . For a given crack , as (Dirichlet condition) and (Neumann condition) are known on the boundary of , we can solve two different problems inside .
For a given crack , we denote by the solution of the following Dirichlet problem:
In the same way, if we assume to be enough regular, we can consider the solution of the following Neumann problem:
(2.13) |