Photographers often need to replace a damaged part of an image by a more ’healthy’ looking patch, such as removing dust, scratches, wrinkles. For this pur- pose there is the Clone Tool in Photoshop and similar tools in other applications. The problem is that as a rule there is no area in the image with exactly the same texture and lighting so that replacement is seamless and natural. Doing this type of work was a difficult job, considered an art by photographers, and sophisticated techniques have been developed [1]. Photoshop 7 introduced the Healing Brush tool [2] as a solution to those problems. But algorithms for seamless ’fill in’ had been independently addressed in the literature in different ways, which we will describe next.

A method called Inpainting [3] is using a combination of second and third order partial differential equation (PDE) for solving it. This approach has two limitations. First, the reconstructed area is too smooth, which follows from the nature of the PDEs used. More recent methods [4] extract a texture component of the image and use it to fill in the texture component of the selected area, while reconstructing the image component based on PDEs. Second, for a low order PDE the reconstructed image in the selected area may not have smooth behavior at the boundary. In the simple case of second order PDE with Dirichlet boundary condition, there is discontinuity in the slope of the reconstructed function.

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