No abstract

Image inpainting is an image restoration problem, in which image models play a critical role, as demonstrated by Chan, Kang and Shen's recent inpainting schemes based on the bounded variation [10] and the elastica [9] image models. In the present paper, we propose two novel inpainting models based on the Mumford-Shah image model [37], and its high order correction-the Mumford-Shah-Euler image model. We also present their efficient numerical realization based on the ¡-convergence approximations of Ambrosio and Tortorelli [1, 2], and De Giorgi [18].

Motivated by the classical TV (total variation) restoration model, we propose a new nonlinear filter-the digital TV filter for denoising and enhancing digital images, or more generally, data living on graphs. The digital TV filter is a data dependent lowpass filter, capable of denoising data without blurring jumps or edges. In iterations, it solves a global total variational (or L(1)) optimization problem, which differs from most statistical filters. Applications are given in the denoising of one dimensional (1-D) signals, two-dimensional (2-D) data with irregular structures, gray scale and color images, and nonflat image features such as chromaticity.

Inspired by the recent work of Bertalmio, Sapiro, Caselles, and Ballester Technical report, ECE-University of Minnesota (1999)] on digital inpaintings, we develop general mathematical models for local non-texture inpaintings. Inside smooth regions, inpaintings are connected to the harmonic and bi-harmonic extensions, and inpainting orders are de ned and analyzed. For inpaintings involving the recovery of edges, we propose variational models that are closely connected to the total variation (TV) restoration model of Rudin, Osher, and Fatemi Physica D, 60 (1992), pp. 259-268] and the Mumford-Shah segmentation model Comm. Pure Appl. Math., XLII (1989), pp 577-685]. Emphasis is put on the TV inpainting model due to its simplicity in theory and its e ciency in computation and applications. We demonstrate the applications of the inpainting models in restoring scratched old photos, disocclusions in vision analysis, text removal from images, and digital zoomings.