IDEAS home Printed from https://ideas.repec.org/h/spr/spochp/978-1-4419-9569-8_8.html
   My bibliography  Save this book chapter

Non-Local Functionals for Imaging

In: Fixed-Point Algorithms for Inverse Problems in Science and Engineering

Author

Listed:
  • Jérôme Boulanger

    (Austrian Academy of Sciences)

  • Peter Elbau
  • Carsten Pontow
  • Otmar Scherzer

Abstract

Non-local functionals have been successfully applied in a variety of applications, such as spectroscopy or in general filtering of time-dependent data. We mention the patch-based denoising of image sequences [Boulanger et al. IEEE Transactions on Medical Imaging (2010)]. Another family of non-local functionals considered in these notes approximates total variation denoising. Thereby we rely on fundamental characteristics of Sobolev spaces and the space of functions of finite total variation (see [Bourgain et al. Journal d’Analyse Mathématique 87, 77–101 (2002)] and several follow up papers). Standard results of the calculus of variations, like for instance the relation between lower semi-continuity of the functional and convexity of the integrand, do not apply, in general, for the non-local functionals. In this paper we address the questions of the calculus of variations for non-local functionals and derive relations between lower semi-continuity of the functionals and separate convexity of the integrand. Moreover, we use the new characteristics of Sobolev spaces to derive novel approximations of the total variation energy regularisation. All the functionals are well-posed and reveal a unique minimising point. Even more, existing numerical schemes can be recovered in this general framework.

Suggested Citation

  • Jérôme Boulanger & Peter Elbau & Carsten Pontow & Otmar Scherzer, 2011. "Non-Local Functionals for Imaging," Springer Optimization and Its Applications, in: Heinz H. Bauschke & Regina S. Burachik & Patrick L. Combettes & Veit Elser & D. Russell Luke & Henry (ed.), Fixed-Point Algorithms for Inverse Problems in Science and Engineering, chapter 0, pages 131-154, Springer.
    • Handle: RePEc:spr:spochp:978-1-4419-9569-8_8
    DOI: 10.1007/978-1-4419-9569-8_8
    as

    Download full text from publisher

    To our knowledge, this item is not available for download. To find whether it is available, there are three options:
    1. Check below whether another version of this item is available online.
    2. Check on the provider's web page whether it is in fact available.
    3. Perform a search for a similarly titled item that would be available.

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Fuensanta Andrés & Damián Castaño & Julio Muñoz, 2023. "Minimization of the Compliance under a Nonlocal p -Laplacian Constraint," Mathematics, MDPI, vol. 11(7), pages 1-22, March.
    2. Fuensanta Andrés & Julio Muñoz, 2017. "On the Convergence of a Class of Nonlocal Elliptic Equations and Related Optimal Design Problems," Journal of Optimization Theory and Applications, Springer, vol. 172(1), pages 33-55, January.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:spochp:978-1-4419-9569-8_8. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no bibliographic references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.