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This book addresses the mathematical aspects of modern image processing methods, with a special emphasis on the underlying ideas and concepts. It discusses a range of modern mathematical methods used to accomplish basic imaging tasks such as denoising, deblurring, enhancing, edge detection and inpainting. In addition to elementary methods like point operations, linear and morphological methods, and methods based on multiscale representations, the book also covers more recent methods based on partial differential equations and variational methods.
Review of the German Edition: The overwhelming impression of the book is that of a very professional presentation of an appropriately developed and motivated textbook for a course like an introduction to fundamentals and modern theory of mathematical image processing. Additionally, it belongs to the bookcase of any office where someone is doing research/application in image processing. It has the virtues of a good and handy reference manual. (zbMATH, reviewer: Carl H. Rohwer, Stellenbosch)

### Chapter 1. Introduction

We omit the philosophical aspect of the question “What are images?” and aim to answer the question “What kind of images are there?” instead. Images can be produced in many different ways:
Kristian Bredies, Dirk Lorenz

### Chapter 2. Mathematical Preliminaries

Mathematical image processing, as a branch of applied mathematics, is not a self-contained theory of its own, but rather builds on a variety of different fields, such as Fourier analysis, the theory of partial differential equations, and inverse problems. In this chapter, we deal with some of those fundamentals that commonly are beyond the scope of introductory lectures on analysis and linear algebra. In particular, we introduce several notions of functional analysis and briefly touch upon measure theory in order to study classes of Lebesgue spaces. Furthermore, we give an introduction to the theory of weak derivatives as well as Sobolev spaces. The following presentation is of reference character, focusing on the development of key concepts and results, omitting proofs where possible. We also give references for further studies of the respective issues.
Kristian Bredies, Dirk Lorenz

### Chapter 3. Basic Tools

In this book we regard, admittedly slightly arbitrarily, as basic tools histograms and linear and morphological filters. These tools belong to the oldest methods in mathematical image processing and are discussed in early books on digital image processing as well (cf. [67, 114, 119]).
Kristian Bredies, Dirk Lorenz

### Chapter 4. Frequency and Multiscale Methods

Like the methods covered in Chap. 3, the methods based on frequency or scale-space decompositions belong to the older methods in image processing. In this case, the basic idea is to transform an image into a different representation in order to determine its properties or carry out manipulations. In this context, the Fourier transformation plays an important role.
Kristian Bredies, Dirk Lorenz

### Chapter 5. Partial Differential Equations in Image Processing

Our first encounter with a partial differential equation is this book was Application 3.​23 on edge detection according to Canny: we obtained a smoothed image by solving the heat equation. The underlying idea was that images contain information on different spatial scales and one should not fix one scale a priori.
Kristian Bredies, Dirk Lorenz

### Chapter 6. Variational Methods

To motivate the general approach and possibilities of variational methods in mathematical imaging, we begin with several examples.
Kristian Bredies, Dirk Lorenz