图像处理中的数学问题
2009-10
世界图书出版公司
（法）奥伯特 著
377
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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，and represent and under． stand the world around US．Just think、for instance、about digital television， medical imagery,and 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 we are entering the domains of computer vision and image processing．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 iustifv the well-founded nature of the methodology．Among the numerous approaches that have been suggested，we focus on partial difierential equations(PDEs)，and variational approaches in this book．Traditionally applied in physics．these methods have been successfully and widely transferred to computer vision over 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 referring to analogue 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 frame work 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．
Introduction、The Image Society、What Is a Digital Image、About Partial Differential Equations（PDEs）、Detailed Plan、Mathematical Preliminaries、How to Read This Chapter、The Direct Method in the Calculus of Vgriations、Topologies on Banach Spaces、Convexity and Lower Semicontinuity、Rclaxation、Aboutr-Convergence、The Space of Functions of Bounded Variation、Basic Definitions on Measures、Definition ofBV（Ω）、Properties ofBV（Ω）、Convex Functions of Measures、Viscosity Solutions in PDEs等等。
作者：(法国)奥伯特(Gjlles Aubert) (法国)Pierre Kornprobst
ForewordPreface to the Second EditionPreface to the First EditionGuide to the Main Mathematical Concepts andTheir ApplicationNotation and Symbols1 Introduction 1.1 The Image Society 1.2 What Is a Digital Image? 1.3 About Partial Differential Equations (PDEs) 1.4 Detailed Plan2 Mathematical Preliminaries How to Read This Chapter 2.1 The Direct Method in the Calculus of Variations... 2.1.1 Topologies on Banach Spaces 2.1.2 Convexity and Lower Semicontinuity . 2.1.3 Relaxation 2.1.4 About F-Convergence 2.2 The Space of Functions of Bounded Variation 2.2.1 Basic Definitions on Measures 2.2.2 Definition of BV(ft) 2.2.3 Properties of BV(f~) 2.2.4 Convex Functions of Measures 2.3 Viscosity Solutions in PDEs 2.3.1 About the Eikonal Equation 2.3.2 Definition of Viscosity Solutions 2.3.3 About the Existence 2.3.4 About the Uniqueness 2.4 Elements of Differential Geometry: Curvature 2.4.1 Parametrized Curves 2.4.2 Curves as Isolevel of a Function u 2.4.3 Images as Surfaces 2.5 Other Classical Results Used in This Book 2.5.1 Inequalities 2.5.2 Calculus Facts 2.5.3 About Convolution and Smoothing 2.5.4 Uniform Convergence 2.5.5 Dominated Convergence Theorem 2.5.6 Well-Posed Problems3 Image Restoration How to Read This Chapter 3.1 Image Degradation 3.2 The Energy Method 3.2.1 An Inverse Problem 3.2.2 Regularization of the Problem 3.2.3 Existence and Uniqueness of a Solution for the Minimization Problem 3.2.4 Toward the Numerical Approximation The Projection Approach The Half-Quadratic Minimization Approach 3.2.5 Some Invariances and the Role of A 3.2.6 Some Remarks on the Nonconvex Case 3.3 PDE-Based Methods 3.3.1 Smoothing PDEs The Heat Equation Nonlinear Diffusion The Alvarez-Guichard-Lions-Morel Scale Space Theory Weickert's Approach Surface Based Approaches 3.3.2 Smoothing-Enhancing PDEs The Perona and Malik Model ……4 The Segmentation Problem5 Other Challenging ApplicationsA Introduction to Finite Difference MethodsB Experiment Yourself!ReferencesIndex
插图：The message we wish to convey iS that the intuition that lcads to certain formulations and the underlying theoretical study are often complementary．Developing a theoretical justification of a problem is not simply“art for art's 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 we show the contribution of mathematics to this domain by studying classical and challenging problems that come from computer vision．It is also the occasion to highlight some difficult and unsolved theoretical questions．The second is the computer vision community：we present a clear． selLC0ntained．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．This book iS divided into seven main parts．Chapter 1 introduces the subject and gives a detailed plan of the book．In Chapter 2，most of the mathematical notions used therein are recalled in an educative fashion and illustrated in detail．In Chapters 3 and 4 we examine how PDES and variational methods can be Successfully applied in the restoration and segmentation of one image．Chapter 5 is more applied，and some challenging computer vision problems are described，such as inpainting，sequence analysis，classification or vector-valued image processing．Since the final goal of any approach is to compute a numerical solution，we propose an introduction to the method of finite difierences in the Appendix．
《图像处理中的数学问题(第2版)(英文版)》由世界图书出版公司出版。
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图像处理中的数学问题
本书是对图像处理的数学理论感兴趣的读者不可错过的一本好书，适合研究生水平的读者阅读。对工程背景的读者可能稍微难了一点。
图像处理方面值得学习
很好的一本书，可以经常翻翻。
老板推荐的
还没仔细看，应该比较经典
经典书籍，值得我们一读
挺好的，顺便练习外语了。
这本书侧重于理论，适合数学系的学生！
书挺好。全英文版，很实用。
没有时间细读，看着挺高深的，又是英文版，所以以后找机会再读吧
数学啊不好啃
就是内容比较难理解。
全英文的，内容本来就难了，更难懂了。
书本有一张破损
不是很完美
书挺好的，一般书店还很难买到
如果英文好，可以瞧瞧
是专业要用的书，只是每次在当当买多本的时候，都有那么一两本有残页。
就几页就不换了。希望改进啊。
书是按图像处理分类的，主要讲了变分法，张弛，收敛和粘性解，有难度。
介绍图像处理中的一些难题，纸张一般，内容很好！
老外的书比较牛逼，而且要看的深入进去
全英文的，但是内容看起来很亲切，装订很好
书是好书，数学功底要求高，要是有中文版就好了。
书还可以，只是有几页纸破了，学习水平集的经典书籍。
第一页就是目录的第三章了，前面的所有都缺页
质量挺好的，书不错，又是我们急需的
内容丰富，知识结构安排也很合理，不过对数学要求较高
很经典的一本书，值得拥有！
还没仔细看，印刷很一般
拿到书了，看了前言，前言作者很不客气地说当年搞图像处理的计算机专家与电子工程师的数学比较差，哈哈。