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In this paper we study a class of fast geometric image inpainting methodsbased on the idea of filling the inpainting domain in successive shells fromits boundary inwards. Image pixels are filled by assigning them a color equalto a weighted average of their already filled neighbors. However, there isflexibility in terms of the order in which pixels are filled, the weights usedfor averaging, and the neighborhood that is averaged over. Varying thesedegrees of freedom leads to different algorithms, and indeed the literaturecontains several methods falling into this general class. All of them are veryfast, but at the same time all of them leave undesirable artifacts such as"kinking" (bending) or blurring of extrapolated isophotes. Our objective inthis paper is to build a theoretical model, based on a continuum limit and aconnection to stopped random walks, in order to understand why these artifactsoccur and what, if anything, can be done about them. At the same time, weconsider a semi-implicit extension in which pixels in a given shell are solvedfor simultaneously by solving a linear system. We prove (within the continuumlimit) that this extension is able to completely eliminate kinking artifacts,which we also prove must always be present in the direct method. Although ouranalysis makes the strong assumption of a square inpainting domain, it makesweak smoothness assumptions and is thus applicable to the low regularityinherent in images.

Numerical analysis of shell-based geometric image inpainting algorithms and their semi-implicit extension