The goal of this chapter is twofold: first, to provide the basic mathematical background needed to read the rest of this book, and second, to give the reader the basic background and motivation to learn more about the topics covered in this chapter by use of, for example, the referenced books and papers. This background is necessary to better prepare the reader to work in the area of partial differential equations (PDEs) applied to image processing and computer vision. Topics covered include differential geometry, PDEs, variational formulations, and numerical analysis. Extensive treatment on these topics can be found in the following books, which are considered essential for the shelves of everybody involved in this topic:

1. Guggengheimer's book on differential geometry [166]. This is one of the few simple-to-read books that covers affine differential geometry, Cartan moving frames, and basic Lie group theory. A very enjoyable book.

2. Spivak's “encyclopedia” on differential geometry [374]. Reading any of the comprehensive five volumes is a great pleasure. The first volume provides the basic mathematical background, and the second volume contains most of the basic differential geometry needed for the work described in this book. The very intuitive way Spivak writes makes this book a great source for learning the topic.

3. DoCarmo's book on differential geometry [56]. This is a very formal presentation of the topic, and one of the classics in the area.

4. […]