## Digital inpainting based on the Mumford-Shah-Euler image model.(English)Zbl 1017.94505

Summary: Image inpainting is an image restoration problem in which image models play a critical role, as demonstrated by T. F. Chan and J. Shen’s [SIAM J. Appl. Math. 62, 1019–1043 (2002; Zbl 1050.68157)] recent inpainting schemes based on the bounded variation and the elastica image models [T. F. Chan, S.-H. Kang and J. Shen, SIAM J. Appl. Math. 63, 564–592 (2002; Zbl 1028.68185)]. In this paper, we propose two novel inpainting models based on the Mumford-Shah image model [D. Mumford and J. Shah, Commun. Pure Appl. Math. 42, 577–685 (1989; Zbl 0691.49036)] and its high-order correction – the Mumford-Shah-Euler image model. We also present their efficient numerical realization based on the $$\Gamma$$-convergence approximations of L. Ambrosio and V. M. Tortorelli [ibid. 43, 999–1036 (1990; Zbl 0722.49020) and Boll. Unione Mat. Ital. (7) B 6, 105–123 (1992; Zbl 0776.49029)] and E. De Giorgi [Frontiere orientate di misura minima, Sem. Mat., Scuola Norm. Super. Pisa (1960-61)].

### MSC:

 94A08 Image processing (compression, reconstruction, etc.) in information and communication theory 49Q20 Variational problems in a geometric measure-theoretic setting 49J45 Methods involving semicontinuity and convergence; relaxation 68U10 Computing methodologies for image processing

### Keywords:

variational image restoration
Full Text: