When Convex Analysis Meets Mathematical Morphology on Graphs

  • Laurent NajmanEmail author
  • Jean-Christophe Pesquet
  • Hugues Talbot
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9082)


In recent years, variational methods, i.e., the formulation of problems under optimization forms, have had a great deal of success in image processing. This may be accounted for by their good performance and versatility. Conversely, mathematical morphology (MM) is a widely recognized methodology for solving a wide array of image processing-related tasks. It thus appears useful and timely to build bridges between these two fields. In this article, we propose a variational approach to implement the four basic, structuring element-based operators of MM: dilation, erosion, opening, and closing. We rely on discrete calculus and convex analysis for our formulation. We show that we are able to propose a variety of continuously varying operators in between the dual extremes, i.e., between erosions and dilation; and perhaps more interestingly between openings and closings. This paves the way to the use of morphological operators in a number of new applications.


Optimization Convex analysis Discrete calculus Graphs 


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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Laurent Najman
    • 1
    Email author
  • Jean-Christophe Pesquet
    • 1
  • Hugues Talbot
    • 1
  1. 1.Laboratoire d’Informatique Gaspard Monge - CNRS UMR 8049Université Paris-EstMarne la Vallée Cedex 2, ParisFrance

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