Parapolic Quasilinear Equations Minimizing Linear Growth Functionals (Progress in Mathematics Vol.223) (2004. 368 p.)

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Parapolic Quasilinear Equations Minimizing Linear Growth Functionals (Progress in Mathematics Vol.223) (2004. 368 p.)

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  • 製本 Hardcover:ハードカバー版/ページ数 368 p.
  • 商品コード 9783764366193

Full Description


This book details the mathematical developments in total variation based image restauration. From the reviews:"This book is devoted to PDE's of elliptic and parabolic type associated to functionals having a linear growth in the gradient, with a special emphasis on the applications related to image restoration and nonlinear filters....The book is written with great care, paying also a lot of attention to the bibliographical and historical notes."-- ZENTRALBLATT MATH

Table of Contents

Preface                                            xi
1 Total Variation Based Image Restoration 1 (30)
1.1 Introduction 1 (7)
1.2 Equivalence between Constrained and 8 (5)
Unconstrained Restoration
1.3 The Partial Differential Equation 13 (5)
Satisfied by the Minimum of (1.17)
1.4 Algorithm and Numerical Experiments 18 (7)
1.5 Review of Numerical Methods 25 (6)
2 The Neumann Problem for the Total Variation 31 (26)
Flow
2.1 Introduction 31 (2)
2.2 Strong Solutions in Lイ(Ω) 33 (2)
2.3 The Semigroup Solution in Lケ(Ω) 35 (7)
2.4 Existence and Uniqueness of Weak Solutions 42 (5)
2.5 An LN-L Regularizing Effect 47 (3)
2.6 Asymptotic Behaviour of Solutions 50 (5)
2.7 Regularity of the Level Lines 55 (2)
3 The Total Variation Flow in RN 57 (24)
3.1 Initial Conditions in Lイ(RN) 57 (5)
3.2 The Notion of Entropy Solution 62 (1)
3.3 Uniqueness in Lケloc(RN) 63 (6)
3.4 Existence in Lケloc(RN) 69 (2)
3.5 Initial Conditions in Lケ(IEB") 71 (1)
3.6 Time Regularity 72 (5)
3.7 An LN-L Regularizing Effect 77 (1)
3.8 Measure Initial Conditions 77 (4)
4 Asymptotic Behaviour and Qualitative 81 (44)
Properties of Solutions
4.1 Radially Symmetric Explicit Solutions 81 (6)
4.2 Some Qualitative Properties 87 (5)
4.3 Asymptotic Behaviour 92 (11)
4.4 Evolution of Sets in Rイ: The Connected 103(11)
Case
4.5 Evolution of Sets in Rイ: The Nonconnected 114(4)
Case
4.6 Some Examples 118(2)
4.7 Explicit Solutions for the Denoising 120(5)
Problem
5 The Dirichlet Problem for the Total Variation 125(38)
Flow
5.1 Introduction 125(1)
5.2 Definitions and Preliminary Facts 126(3)
5.3 The Main Result 129(1)
5.4 The Semigroup Solution 129(14)
5.5 Strong Solutions for Data in Lイ(Ω) 143(5)
5.6 Existence and Uniqueness for Data in 148(11)
Lケ(Ω)
5.7 Regularity for Positive Initial Data 159(4)
6 Parabolic Equations Minimizing Linear Growth 163(50)
Functionals: Lイ-Theory
6.1 Introduction 163(4)
6.2 Preliminaries 167(4)
6.3 The Existence and Uniqueness Result 171(2)
6.4 Strong Solution for Data in Lイ(Ω) 173(26)
6.5 Asymptotic Behaviour 199(1)
6.6 Proof of the Approximation Lemma 200(13)
7 Parabolic Equations Minimizing Linear Growth 213(58)
Functionals: Lケ-Theory
7.1 Introduction 213(3)
7.2 The Main Result 216(1)
7.3 The Semigroup Solution 217(15)
7.4 Existence and Uniqueness for Data in 232(32)
Lケ(Ω)
7.5 A Remark for Strictly Convex Lagrangians 264(4)
7.6 The Cauchy Problem 268(3)
Appendix
A Nonlinear Semigroups 271(26)
A.1 Introduction 271(1)
A.2 Abstract Cauchy Problems 272(3)
A.3 Mild Solutions 275(3)
A.4 Accretive Operators 278(7)
A.5 Existence and Uniqueness Theorem 285(5)
A.6 Regularity of Mild Solutions 290(1)
A.7 Completely Accretive Operators 291(6)
B Functions of Bounded Variation 297(14)
B.1 Definitions 297(1)
B.2 Approximation by Smooth Functions 298(2)
B.3 Traces and Extensions 300(1)
B.4 Sets of Finite Perimeter and the Coarea 301(1)
Formula
B.5 Some Isoperimetric Inequalities 302(1)
B.6 The Reduced Boundary 303(2)
B.7 Connected Components of Sets of Finite 305(6)
Perimeter
C Pairings Between Measures and Bounded 311(12)
Functions
C.1 Trace of the Normal Component of 311(3)
Certain Vector Fields
C.2 The Measure (z, Du) 314(3)
C.3 Representation of the Radon-Nikodym 317(6)
Derivative theta(z, Du,  
Bibliography 323(16)
Index 339