Total variation regularization, for reconstruct images inpainting. Regularización por variación total, para reconstruir imágenes con dominios perdidos

Abstract

Digital image processing uses numerical techniques aimed at improving image quality, deleting objects in the image or extracting information; among the most used techniques is the reconstruction of lost parts that consists of modifying an image to rebuild deteriorated or deleted areas of it. Currently, image reconstruction algorithms occupy a wide field of research and development; this article analyzes, describes, poses and solves the problem of image reconstruction with missing regions, from a theoretical and practical point of view; you get a mathematical model that represents the problem, which is shown to be a poorly posed inverse problem in the sense of Hadamard. To solve the problem, the total variation method that is deduced from the variational representation of the Tikhonov regularization method is used; it is shown that the solution of the problem also solves the Neumann problem for the Euler-Lagrange equation, which is solved numerically. Finally, the solution proposal is validated in some images with lost domains. The novelty of the work is that the method used allows the implementation numerically and show the influence of the regularization parameter.