Progress in Mathematics

Parabolic Quasilinear Equations Minimizing Linear Growth Functionals

Authors: Andreu-Vaillo, Fuensanta, Caselles, Vicent, Mazon, José M.

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About this book

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.

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


Table of contents (7 chapters)

  • Total Variation Based Image Restoration

    Andreu-Vaillo, Fuensanta (et al.)

    Pages 1-30

  • The Neumann Problem for the Total Variation Flow

    Andreu-Vaillo, Fuensanta (et al.)

    Pages 31-56

  • The Total Variation Flow in ℝN

    Andreu-Vaillo, Fuensanta (et al.)

    Pages 57-79

  • Asymptotic Behaviour and Qualitative Properties of Solutions

    Andreu-Vaillo, Fuensanta (et al.)

    Pages 81-123

  • The Dirichlet Problem for the Total Variation Flow

    Andreu-Vaillo, Fuensanta (et al.)

    Pages 125-162

Buy this book

eBook $109.00
price for USA (gross)
  • ISBN 978-3-0348-7928-6
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Hardcover $139.00
price for USA
  • ISBN 978-3-7643-6619-3
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
Softcover $139.00
price for USA
  • ISBN 978-3-0348-9624-5
  • Free shipping for individuals worldwide
  • Usually dispatched within 3 to 5 business days.
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Bibliographic Information

Bibliographic Information
Book Title
Parabolic Quasilinear Equations Minimizing Linear Growth Functionals
Authors
Series Title
Progress in Mathematics
Series Volume
223
Copyright
2004
Publisher
Birkhäuser Basel
Copyright Holder
Springer Basel AG
eBook ISBN
978-3-0348-7928-6
DOI
10.1007/978-3-0348-7928-6
Hardcover ISBN
978-3-7643-6619-3
Softcover ISBN
978-3-0348-9624-5
Series ISSN
0743-1643
Edition Number
1
Number of Pages
XIV, 342
Topics