Overview

Interpolation has been a significant topic in a number of areas including numerical analysis; computational PDEs; approximation theory; real, complex, and harmonic analysis; and signal processing.

In image processing, image interpolation is such a fundamental problem that there have been numerous prior works in existence. In the engineering literature, for instance, one witnesses the following samples from a large pool: image interpolation [13, 181, 182], image replacement [156, 316], error concealment [161, 187], and image editing [111]. In mathematical image and vision analysis, image interpolation has also been studied systematically in the remarkable works of Nitzberg, Mumford, and Shiota for segmentation with depth and edge completion [234], Masnou and Morel for level-line completion [213, 214], and Caselles, Morel, and Sbert for axiomatic interpolation based on second order PDEs [50]. To our best knowledge, the work of Masnou and Morel [214], which won the best-student-paper award in ICIP 98, was the first work performing variational image interpolation as inspired by Nitzberg, Mumford, and Shiota s variational model for edge completion [234].

The word inpainting is an artistic synonym for image interpolation and has been circulated for quite a while among museum restoration artists [312]. It was first transplanted into digital image processing in the remarkable work by Bertalmio et al. [24], which has stimulated the recent wave of interest in numerous problems related to image interpolation, including the works by Chan and Shen and their collaborators.

The current chapter presents several inpainting models based upon the Bayesian,...