Andreu-Vaillo, Fuensanta, Caselles, Vicent, Mazon, José M.
2004, XIV, 342 p.
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Award-winning monograph of the Ferran Sunyer i Balaguer Prize 2003.
This book contains a detailed mathematical analysis of the variational approach to image restoration based on the minimization of the total variation submitted to the constraints given by the image acquisition model. This model, initially introduced by Rudin, Osher, and Fatemi, had a strong influence in the development of variational methods for image denoising and restoration, and pioneered the use of the BV model in image processing. After a full analysis of the model, the minimizing total variation flow is studied under different boundary conditions, and its main qualitative properties are exhibited. In particular, several explicit solutions of the denoising problem are computed.
Content Level »Research
Keywords »Approximation - Calculus of variations - Dirichlet problem - Image restoration - Neumann problem - Nonlinear semigroups - Parabolic equations - partial differential equation
1 Total Variation Based Image Restoration.- 1.1 Introduction.- 1.2 Equivalence between Constrained and Unconstrained Restoration.- 1.3 The Partial Differential Equation Satisfied by the Minimum of (1.17).- 1.4 Algorithm and Numerical Experiments.- 1.5 Review of Numerical Methods.- 2 The Neumann Problem for the Total Variation Flow.- 2.1 Introduction.- 2.2 Strong Solutions in L2(?).- 2.3 The Semigroup Solution in L1(?).- 2.4 Existence and Uniqueness of Weak Solutions.- 2.5 An LN-L? Regularizing Effect.- 2.6 Asymptotic Behaviour of Solutions.- 2.7 Regularity of the Level Lines.- 3 The Total Variation Flow in ?N.- 3.1 Initial Conditions in L2(?N).- 3.2 The Notion of Entropy Solution.- 3.3 Uniqueness in Ló(?N).- 3.4 Existence in Lloc1.- 3.5 Initial Conditions in L2(?N).- 3.6 Time Regularity.- 3.7 An LN-L? Regularizing Effect.- 3.8 Measure Initial Conditions.- 4 Asymptotic Behaviour and Qualitative Properties of Solutions.- 4.1 Radially Symmetric Explicit Solutions.- 4.2 Some Qualitative Properties.- 4.3 Asymptotic Behaviour.- 4.4 Evolution of Sets in ?2: The Connected Case.- 4.5 Evolution of Sets in ?2: The Nonconnected Case.- 4.6 Some Examples.- 4.7 Explicit Solutions for the Denoising Problem.- 5 The Dirichlet Problem for the Total Variation Flow.- 5.1 Introduction.- 5.2 Definitions and Preliminary Facts.- 5.3 The Main Result.- 5.4 The Semigroup Solution.- 5.5 Strong Solutions for Data in L2(?).- 5.6 Existence and Uniqueness for Data in L1(?).- 5.7 Regularity for Positive Initial Data.- 6 Parabolic Equations Minimizing Linear Growth Functionals: L2-Theory.- 6.1 Introduction.- 6.2 Preliminaries.- 6.3 The Existence and Uniqueness Result.- 6.4 Strong Solution for Data in L2(?)).- 6.5 Asymptotic Behaviour.- 6.6 Proof of the Approximation Lemma.- 7 Parabolic Equations Minimizing Linear Growth Functionals: L1-Theory.- 7.1 Introduction.- 7.2 The Main Result.- 7.3 The Semigroup Solution.- 7.4 Existence and Uniqueness for Data in L1(?).- 7.5 A Remark for Strictly Convex Lagrangians.- 7.6 The Cauchy Problem.- A Nonlinear Semigroups.- A.1 Introduction.- A.2 Abstract Cauchy Problem.- A.3 Mild Solutions.- A.4 Accretive Operators.- A.5 Existence and Uniqueness Theorem.- A.6 Regularity of Mild Solutions.- A.7 Completely Accretive Operators.- B Functions of Bounded Variation.- B.2 Approximation by Smooth Functions.- B.3 Traces and Extensions.- B.4 Sets of Finite Perimeter and the Coarea Formula.- B.5 Some Isoperimetric Inequalities.- B.6 The Reduced Boundary.- B.7 Connected Components of Sets of Finite Perimeter.- C Pairings Between Measures and Bounded Functions.- C.1 Trace of the Normal Component of Certain Vector Fields.- Dankwoord/ Acknowledgements.