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About the second edition (2006)
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.
Reviews of the
earlier edition
“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.”
“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.”
 |
Gilles Aubert
Professor in Mathematics Université de Nice Sophia Antipolis http://math1.unice.fr/~gaubert/ |
 |
Pierre Kornprobst
Research scientist INRIA, Odyssée lab http://www-sop.inria.fr/odyssee/team/Pierre.Kornprobst |
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 decade. 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.
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.
Methodology
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)
- 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)
C++: Do it yourself!
(Under
construction)
This part will contain source code and informations to run experiment
yourself easily some classical PDE-based approaches.
The chosen programming language is the object-oriented language C++,
which is freeware and a very efficient language. Bjarne Stroustrup
is the designer and original implementor of C++. The interested reader
will
also find a wide variety of books and online tutorials on this
language.
Many image processing libraries are proposed online. We chose the CImg library, which stands for
"Cool Image" developed by David Tschumperle
in
2000. The CImg library is simple to use and efficient. The CImg package
includes full documentation and many examples to help the developer in
his first steps.
We propose a plugin containing some well known PDE-based approaches. To
have an idea, you may download this preliminary
version which contains the CImg library and the plugin, called
Pde.h. If you would like to know recent updates, please send me a message
with "plugin tracker" as subject.
You are welcome to
participate!
External contributors are encouraged to submit their own C++ source codes. Please contact me if you are interested. We hope that such an initiative will enable readers to experiment and compare different approaches without too much effort. We thank in advance new contributors, who will help us to develop this free source code database for PDE-based approaches.
Links
(under
construction: this part will contain links related to the area.
Personal homepages, online support material etc.)
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}
We indicate the official Springer web site for the book where complementary information is also available.
