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Analytic Existence and Uniqueness Results for PDE-Based Image Reconstruction with the Laplacian

  • Laurent HoeltgenEmail author
  • Isaac Harris
  • Michael Breuß
  • Andreas Kleefeld
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10302)

Abstract

Partial differential equations are well suited for dealing with image reconstruction tasks such as inpainting. One of the most successful mathematical frameworks for image reconstruction relies on variations of the Laplace equation with different boundary conditions. In this work we analyse these formulations and discuss the existence and uniqueness of solutions of corresponding boundary value problems, as well as their regularity from an analytic point of view. Our work not only sheds light on useful aspects of the well posedness of several standard problem formulations in image reconstruction but also aggregates them in a common framework. In addition, the performed analysis guides us to specify two new formulations of the classic image reconstruction problem that may give rise to new developments in image reconstruction.

Keywords

Partial differential equations Laplace equation Mixed boundary conditions Image reconstruction Image inpainting 

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Laurent Hoeltgen
    • 1
    Email author
  • Isaac Harris
    • 2
  • Michael Breuß
    • 1
  • Andreas Kleefeld
    • 3
  1. 1.Institute for MathematicsBrandenburg Technical UniversityCottbusGermany
  2. 2.Department of MathematicsTexas A&M UniversityCollege StationUSA
  3. 3.Institute for Advanced SimulationForschungszentrum Jülich GmbH, Jülich Supercomputing CentreJülichGermany

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