Chapter 2 overviews local methods for inpainting, also referred to as geometric methods, starting in 1993. These approaches are typically based on the solution of partial differential equations (PDEs) arising from the minimisation of certain mathematical energies. Geometrical methods have proven to be powerful for the removal of scratches, long tiny lines or small damages such as craquelures in art-related images.

Chapter 4 describes the rise of deep learning inpainting methods in the past ten years. These methods learn an end-to-end mapping from a corrupted input to its estimated restoration. In contrast with traditional methods from the previous chapters, which use model-based or hand-crafted features, learning-based algorithms are able to infer the missing content by training on a large-scale dataset and can capture local or non-local dependencies inside the image and over the full dataset and exploit high-level information inherent in the image itself. In this chapter we present the seminal deep learning inpainting methods up to 2020 together with dedicated datasets designed for the inpainting problem.

Chapter 3 provides a historical view of non-local inpainting methods, also called examplar-based or patch-based methods. These approaches rely on the self-similarity principle, i.e. on the idea that the missing information in the inpainting domain can be copied from somewhere else within the intact part of the image. Over the years. many improvements and algorithms have been proposed, enabling us to offer visually plausible solutions to the inpainting problem, especially for large damages and areas with texture.

Chapter 5 focuses on specific strategies to addess inpainting in real-life cultural heritage restoration cases, such as the colour restoration of old paintings, the inpainting of ancient frescoes, and the virtual restoration of damaged illuminated manuscripts.

Chapter 1 presents a brief overview of the book and the basics on inpainting, visual perception and Gestalt laws, together with a presentation of the Fitzwilliam Museum dataset of illuminated manuscripts, selected to represent different types of damage and consequent restoration challenges, which will be used throughout the book.

Chapter 6 offers a a retrospective view of inpainting methods described in the book and it summarizes key findings with some take home messages.

This chapter provides an introduction to uncertainty relations underlying sparse signal recovery. We start with the seminal work by Donoho and Stark (1989), which defines uncertainty relations as upper bounds on the operator norm of the band-limitation operator followed by the time-limitation operator, generalize this theory to arbitrary pairs of operators, and then develop, out of this generalization, the coherence-based uncertainty relations due to Elad and Bruckstein (2002), plus uncertainty relations in terms of concentration of the 1-norm or 2-norm. The theory is completed with set-theoretic uncertainty relations which lead to best possible recovery thresholds in terms of a general measure of parsimony, the Minkowski dimension. We also elaborate on the remarkable connection between uncertainty relations and the “large sieve,” a family of inequalities developed in analytic number theory. We show how uncertainty relations allow one to establish fundamental limits of practical signal recovery problems such as inpainting, declipping, super-resolution, and denoising of signals corrupted by impulse noise or narrowband interference.