Fuensanta Andreu-Vaillo

Parabolic Quasilinear Equations Minimizing Linear Growth Functionals

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 Lo(?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.

Nonlocal Diffusion Problems

Nonlocal diffusion problems arise in a wide variety of applications, including biology, image processing, particle systems, coagulation models, and mathematical finance. These types of problems are also of great interest for their purely mathematical content. This book presents recent results on nonlocal evolution equations with different boundary conditions, starting with the linear theory and moving to nonlinear cases, including two nonlocal models for the evolution of sandpiles. Both existence and uniqueness of solutions are considered, as well as their asymptotic behaviour. Moreover, the authors present results concerning limits of solutions of the nonlocal equations as a rescaling parameter tends to zero. With these limit procedures the most frequently used diffusion models are recovered: the heat equation, the

Total Variation Based Image Restoration

p

The Neumann Problem for the Total Variation Flow

-Laplacian evolution equation, the porous media equation, the total variation flow, a convection-diffusion equation and the local models for the evolution of sandpiles due to Aronsson-Evans-Wu and Prigozhin. Readers are assumed to be familiar with the basic concepts and techniques of functional analysis and partial differential equations. The text is otherwise self-contained, with the exposition emphasizing an intuitive understanding and results given with full proofs. It is suitable for graduate students or researchers. The authors cover a subject that has received a great deal of attention in recent years. The book is intended as a reference tool for a general audience in analysis and PDEs, including mathematicians, engineers, physicists, biologists, and others interested in nonlocal diffusion problems.

Asymptotic Behaviour and Qualitative Properties of Solutions

For the purpose of image restoration the process of image formation can be modeled in a first approximation by the formula [207]

The Cauchy problem for linear nonlocal diffusion

{u_d} = Q\{ II(k*u) + n\} ,

A nonlocal convection diffusion problem

(1.1) where u represents the photonic flux k is the point spread function of the optical-captor joint apparatus П is a sampling operator, i.e., a Dirac comb supported by the centers of the matrix of digital sensors, n represents a random perturbation due to photonic or electronic noise, and Qis a uniform quantization operator mapping ℝ to a discrete interval of values, typically [0, 255].

Nonlocal models for sandpiles

This chapter is devoted to prove existence and uniqueness of solutions for the minimizing total variation flow with Neumann boundary conditions, namely