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When Convex Analysis Meets Mathematical Morphology on Graphs

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

Abstract

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.

Keywords

Optimization Convex analysis Discrete calculus Graphs 

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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Laurent Najman
    • 1
  • 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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