The purpose of this book is to provide an introduction to the use of partial differential equations (PDEs) for digital image restoration. It is a way of sharing what I have learned while studying these methods for about ten years, defending a Ph.D. thesis on PDE inpainting, teaching courses in Göttingen and Cambridge on the topic and writing a couple of research papers on the use of PDEs in image inpainting along the way. Let me say what this book is and what it is not. It is:

1. An introduction to inpainting methods that use PDEs and local variational approaches to restore lost image contents;

2. An account from an enthusiast on some state-of-the-art inpainting methods from their abilities to their limitations and reference for informed researchers in the field of digital image processing; and

3. A work targeted at readers with basic knowledge in functional analysis, PDEs, measure theory and convex optimisation. Therefore, I recommend this textbook only for students from the graduate level onwards.

This book is not:

1. A book that gives credit to the whole wealth of inpainting methods (in particular, this book will focus only on local inpainting methods and will scratch only non-local inpainting methods such as exemplar-based inpainting); or

2. A book for undergraduate students.

One final comment before we go in medias res: the more I learn and understand about image inpainting, the more the complexity of the matter and the variety of different problems, each requiring specialised methods, become apparent.