Gilles Aubert, Professor in Mathematics, Université de Nice Sophia Antipolis
Pierre Kornprobst, INRIA Researcher

It is surprising when we realize just how much we are surrounded by images. Images allow us not only to perform complex tasks on a daily basis, but also to communicate, transmit information, represent and understand the world around us. Just think, for instance about digital television, medical imagery, video-surveillance,... The tremendous development in information technology accounts for most of this. We are now able to handle more and more data. Many day to day tasks are now fully or partially accomplished with the help of computers. Whenever images are involved this phenomena can be called Computer Vision. The requirements for this are reliability and speed. Efficient algorithms have to be proposed to process these digital data. It is also important to rely on a well-established theory to justify the well-founded nature of the methodology.

Amongst the numerous approaches which have been suggested, we focus on Partial Differential Equations (PDE's), and Variational Approaches in this book. Traditionally applied in physics, these methods have been successfully and widely transferred in Computer Vision other the last decades. One of the main interests in using PDEs is that the theory behind the concept is well-established. Of course, PDEs are written in a continuous setting refering to analog images, and once the existence and the uniqueness have been proven, we need to discretize them in order to find a numerical solution. It is our conviction that reasoning within a continuous framework makes the understanding of physical realities easier and stimulates the intuition necessary to propose new models. We hope that this book will illustrate this idea effectively.

The message we wish to put over is that the intuition which leads to certain formulations and the underlying theoretical study are often complementary. Developing a theoretical justification of a problem is not simply ``art for art sake''. In particular, a deep understanding of the theoretical difficulties may lead to the development of suitable numerical schemes or different models.

During the four years since the publication of the first edition, there has been substantial progress in the range of image-processing applications covered by the PDE framework. The main goals of the second edition are to update the first edition by giving a coherent account of some of the recent challenging applications, and to update the existing material. In addition, this book provides the reader with the opportunity to make his own simulations with a minimal effort. To this end, programming tools are made available, which will allow the reader to implement and test easily some classical approaches.

“Mathematical Problems in Image Processing is a major, elegant, and unique contribution to the applied mathematics literature, oriented toward applications in image processing and computer vision. ... Researchers and practitioners working in the field will benefit by adding this book to their personal collection. Students and instructors will benefit by using this book as a graduate course textbook.”

—SIAM Review

“The Mathematician—and he doesn't need to be a ‘die-hard’ applied mathematician—will love it because there are all these spectacular applications of nontrivial mathematical techniques and he can even find some open theoretical questions. The numerical analyst will discover many challenging problems and implementations. The image processor will be an eager reader because the book provides all the mathematical elements, including most of the proofs. . . . Both content and typography are a delight. I can recommend the book warmly for theoretical and applied researchers.”

—Bulletin of the Belgian Mathematics

This book is concerned with the mathematical study of certain image processing problems. Thus we target two audiences:

• The first is the mathematical community and is achieved by showing the contribution of mathematics to this domain by studying classical and challenging problems which come from Computer Vision. It is also the occasion to highlight some difficult and unsolved theoretical questions.
• The second is the Computer Vision community: this is done by presenting a clear, self-contained and global overview of the mathematics involved for the problems of image restoration, image segmentation, sequence analysis and image classification.

We hope that this work will serve as a useful source of reference and inspiration for fellow researchers in Applied Mathematics and Computer Vision, as well as being a basis for advanced courses within these fields.

For the different topics studied, the methodology is as follows:

• Review existing approaches.
• Modelize the given problem by rewriting it in terms of PDEs or variational approaches.
• Perform the mathematical study of the model: existence and uniqueness. Details of proof which are the most representative are given in detail.
• Consideration of algorithms and numerical implementation. In the case of optimization problems on BV for example, we will explain how to consider energies on more regular spaces that can be minimized numerically. As for discretization, an introduction to finite differences is proposed in the Appendix where main notions are explained and the discretization of certain equations from the book are given.
• Give some experimental results, essentially to show the reader the behavior of the models.

An effort has been made to make this book as educative and self-contained as possible. Most of the mathematical results used are recalled and discussed.

• Some numbers to start: First edition (2002) was approx. 315 pp. Second  edition (2006) is approx 410 pp.
• Foreword by Olivier Faugeras (INRIA, France)
• Preface to the second edition: you can see what's new! :-)
• Preface to the first edition
• Table of Contents
• Pages with color images
• Videos (Thanks to Francois Lauze)
• Sequence "frankenstein" (original, with_blotches, inpainted)
  
• Sequence "Mobile and calendar" (original, with blotches, TV inpainting, MC inpainting)
• Sequennce "Yosemite" (original, degraded, TV inpainting, MC inpainting)
  
• Main typos (please do not hesitate to email us if you find some additional mistakes)

As indicated in the second edition, some C++ code is available (see Appendix B, page 343)

• README file giving the current status of the library
• ZIP file containing the latest version of the code

If you have any suggestions about it or wish to contribute, please contact me.

Book{ aubert-kornprobst:06,
author = {Aubert, G. and Kornprobst, P.},
title = {Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations (second edition)},
year = {2006},
volume = {147},
publisher = {Springer-Verlag},
series = {Applied Mathematical Sciences}