Exemplar-Based Inpainting from a Variational Point of View

Among all methods for reconstructing missing regions in a digital image, the so-called exemplar-based algorithms are very efficient and often produce striking results. They are based on the simple idea—initially used for texture synthesis—that the unknown part of an image can be reconstructed by simply pasting samples extracted from the known part. Beyond heuristic considerations, there have been very few contributions in the literature to explain from a mathematical point of view the performances of these purely algorithmic and discrete methods. More precisely, a recent paper by Levina and Bickel [Ann. Statist., 34 (2006), pp. 1751–1773] provides a theoretical explanation of their ability to recover very well the texture, but nothing equivalent has been done so far for the recovery of geometry. Our purpose in this paper is twofold: (1) to propose well-posed variational models in the continuous domain that can be naturally associated to exemplar-based algorithms; (2) to investigate their ability to recons...