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© 2004

Parabolic Quasilinear Equations Minimizing Linear Growth Functionals

Book

Part of the Progress in Mathematics book series (PM, volume 223)

Table of contents

  1. Front Matter
    Pages i-xiv
  2. Fuensanta Andreu-Vaillo, José M. Mazón, Vicent Caselles
    Pages 1-30
  3. Fuensanta Andreu-Vaillo, José M. Mazón, Vicent Caselles
    Pages 31-56
  4. Fuensanta Andreu-Vaillo, José M. Mazón, Vicent Caselles
    Pages 57-79
  5. Fuensanta Andreu-Vaillo, José M. Mazón, Vicent Caselles
    Pages 81-123
  6. Fuensanta Andreu-Vaillo, José M. Mazón, Vicent Caselles
    Pages 125-162
  7. Fuensanta Andreu-Vaillo, José M. Mazón, Vicent Caselles
    Pages 163-212
  8. Fuensanta Andreu-Vaillo, José M. Mazón, Vicent Caselles
    Pages 213-269
  9. Back Matter
    Pages 271-342

About this book

Introduction

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.

Keywords

Approximation Calculus of variations Dirichlet problem Image restoration Neumann problem Nonlinear semigroups Parabolic equations partial differential equation

Authors and affiliations

  1. 1.Departamento de Análisis MatemáticoUniversitat de ValenciaBurjassot (Valencia)Spain
  2. 2.Departamento de TecnologíaUniversitat Pompeu FabraBarcelonaSpain

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Reviews

"This book is well written…[and] should be of interest to anyone studying image reconstruction and to anyone in PDEs trying to see what kinds of modern applications abound for the subject."

—Mathematical 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. It is a recommended reading for all researchers interested in this field."

—Zentralblatt Math

"The goal of this mongraph is to present general existence and uniqueness results for quasilinear parabolic equations whose operator is the subdifferential of a convex Lagrangian which has linear growth. Special emphasis is given to the case of the minimizing total variational flow for which the Neumann, Dirichlet, and Cauchy problem are discussed. The developed techniques apply to problems in continuum mechanics, image restoration and faceted crystal growth."

---Monatshefte für Mathematik