When Convex Analysis Meets Mathematical Morphology on Graphs

Laurent Najman (1) , Jean-Christophe Pesquet (1) , Hugues Talbot (1)

1 LIGM - Laboratoire d'Informatique Gaspard-Monge

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.

Mots clés: Optimization; convex analysis; discrete calculus; graphs

Domaines: Traitement des images [eess.IV]

Laurent Najman, Jean-Christophe Pesquet, Hugues Talbot. When Convex Analysis Meets Mathematical Morphology on Graphs. Mathematical Morphology and Its Applications to Signal and Image Processing, Benediktsson, J.A.; Chanussot, J.; Najman, L.; Talbot, H., May 2015, Reykjavik, Iceland. pp.473-484, ⟨10.1007/978-3-319-18720-4_40⟩. ⟨hal-01168801⟩