Mathematical models for local nontexture inpaintings

Inspired by the recent work of Bertalmio et al. on digital inpaintings [SIGGRAPH 2000], we develop general mathematical models for local inpaintings of nontexture images. On smooth regions, inpaintings are connected to the harmonic and biharmonic extensions, and inpainting orders are analyzed. For inpaintings involving the recovery of edges, we study a variational model that is closely connected to the classical total variation (TV) denoising model of Rudin, Osher, and Fatemi [Phys. D, 60 (1992), pp. 259 268]. Other models are also discussed based on the Mumford-Shah regularity [Comm. Pure Appl. Math., XLII (1989), pp. 577 685] and curvature driven diffusions (CDD) of Chan and Shen [J. Visual Comm. Image Rep., 12 (2001)]. The broad applications of the inpainting models are demonstrated through restoring scratched old photos, disocclusion in vision analysis, text removal, digital zooming, and edge-based image coding.