Inpainting in Discrete Sobolev Spaces: Structural Information for Uncertainty Reduction

Abstract

In this article, we introduce a new mathematical functional whose minimization determines the quality of the solution for the exemplar-based inpainting-by-patch problem. The new functional expression includes finite difference terms in a similar fashion to what happens in the theoretical Sobolev spaces: its use reduces the uncertainty in the choice of the most suitable values for each point to inpaint. Moreover, we introduce a probabilistic model by which we prove that the usual principal directions, generally employed for continuous problems, are not enough to achieve consistent reconstructions in the discrete inpainting asset. Finally, we formalize a new priority index and new rules for its dynamic update. The quality of the reconstructions, achieved using a reduced neighborhood size of more than <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>95</mn><mo>%</mo></mrow></semantics></math></inline-formula> with respect to the current state-of-the-art algorithms based on the same inpainting approach, further provides the experimental validation of the method.