This book is concerned with digital image processing techniques that use partial differential equations (PDEs) for the task of image inpainting. Image inpainting is an artistic term for virtual image restoration or image interpolation whereby missing or occluded parts of images are filled in based on information provided by the intact parts of the image. Computer graphic designers, artists and photographers have long used manual inpainting to digitally restore damaged paintings or manipulate photographs. Today, mathematicians apply powerful methods based on PDEs to automate this task. They operate in much the same way that trained restorers do: they propagate information from the structure around a hole into the hole to fill it in.

Virtual image restoration is an important challenge in our modern computerised society. From the reconstruction of crucial information in satellite images of the Earth to the renovation of digital photographs and ancient artwork, virtual image restoration is ubiquitous. The example in Figure 1.1 is entitled Mathematical Analysis Can Make You Fly, and it should give you a first impression of the idea of image inpainting with PDEs. The PDE model used for this example is called TV-H−1inpainting and will be discussed in great detail in Section 5.3.

Digital Image Restoration in Modern Society

Digital images are one of the main sources of information today. The vast number of images and videos that exist in digital form nowadays makes their unaided processing and interpretation by humans impossible. Automatic storage management, processing and analysis algorithms are needed to be able to retrieve only the essence of what the visual world has up its sleeve.