# Mathematical Problems in Image Processing

## Partial Differential Equations and the Calculus of Variations

Book

Part of the Applied Mathematical Sciences book series (AMS, volume 147)

1. Front Matter
Pages i-xii
2. Pages 1-21
3. Pages 25-29
4. Pages 31-65
5. Pages 67-135
6. Pages 137-187
7. Pages 189-236
8. Back Matter
Pages 237-294

### Introduction

Partial differential equations and variational methods were introduced into image processing about 15 years ago, and intensive research has been carried out since then. The main goal of this work is to present the variety of image analysis applications and the precise mathematics involved. It is intended for two audiences. The first is the mathematical community, to show the contribution of mathematics to this domain and to highlight some unresolved theoretical questions. The second is the computer vision community, to present a clear, self-contained, and global overview of the mathematics involved in image processing problems.
The book is divided into five main parts. Chapter 1 is a detailed overview. Chapter 2 describes and illustrates most of the mathematical notions found throughout the work. Chapters 3 and 4 examine how PDEs and variational methods can be successfully applied in image restoration and segmentation processes. Chapter 5, which is more applied, describes some challenging computer vision problems, such as sequence analysis or classification. This book will be useful to researchers and graduate students in mathematics and computer vision.

### Keywords

Calculus of Variations Mathematica PDE PDEs in image processing Partial Differential Equations computer computer vision differential equation image analysis image processing image restoration mathematics partial differential equation variational methods in image processing

#### Authors and affiliations

1. 1.Laboratoire J.-A. DieudonnieUniversity of Nice-Sophia Antipolis U.M.R. no. 6621 du C.N.R.S.Nice Cedex 02France
2. 2.INRIASophia AntipolisFrance

### Bibliographic information

• Book Title Mathematical Problems in Image Processing
• Book Subtitle Partial Differential Equations and the Calculus of Variations
• Authors Gilles Aubert
Pierre Kornprobst
• Series Title Applied Mathematical Sciences
• DOI https://doi.org/10.1007/b97428
• Copyright Information Springer-Verlag New York, LLC 2002
• Publisher Name Springer, New York, NY
• eBook Packages
• Hardcover ISBN 978-0-387-95326-7
• Softcover ISBN 978-1-4757-7717-8
• eBook ISBN 978-0-387-21766-6
• Series ISSN 0066-5452
• Edition Number 1
• Number of Pages XXV, 288
• Number of Illustrations 129 b/w illustrations, 0 illustrations in colour
• Topics
• Buy this book on publisher's site
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## Reviews

From the reviews:

MATHEMATICAL REVIEWS

"The authors have substantially contributed to many aspects of the subject…The level of the book is advanced, and the exposition very clear, developed in depth, and to some extent self-contained. The proofs of the main theorems are usually given in detail; when necessary there is also a precise reference to the ample literature on the specific topic. This book will certainly interest both mathematicians (some interesting, still open theoretical questions there are highlighted) and researchers in computer vision…The books substantial and up-to-date body for reference is a detailed guide to the literature on the many aspects of image processing."

G. Aubert and P. Kornprobst

Mathematical Problems in Image Processing

Partial Differential Equations and the Calculus of Variations

"This book is devoted to a detailed presentation of several aspects of mathematical problems in image processing . . . The authors have substantially contributed to many aspects of their subject . . . The level of the book is advanced, and the to some extent self-contained . . . This book will certainly interest both mathematicians (some interesting, still open theoretical questions here are highlighted) and researchers in computer vision."—MATHEMATICAL REVIEWS