When Convex Analysis Meets Mathematical Morphology on Graphs

• Main Authors: Najman, Laurent, Pesquet, Jean-Christophe, Talbot, Hugues
• Other Authors: Laboratoire d'Informatique Gaspard-Monge (LIGM), Université Paris-Est Marne-la-Vallée (UPEM)-École des Ponts ParisTech (ENPC)-ESIEE Paris-Fédération de Recherche Bézout (BEZOUT), Centre National de la Recherche Scientifique (CNRS)-Centre National de la Recherche Scientifique (CNRS)-Centre National de la Recherche Scientifique (CNRS), Benediktsson, J.A., Chanussot, J., Najman, L., Talbot, H., ANR-14-CE27-0001,GRAPHSIP,Traitement de signaux sur graphes(2014)
• Format: Conference Object
• Language: English
• Published: HAL CCSD 2015
• Subjects: Optimization; convex analysis; discrete calculus; graphs; [INFO.INFO-TI]Computer Science [cs]/Image Processing [eess.IV]; Iceland
• Online Access: https://hal.science/hal-01168801; https://hal.science/hal-01168801/document; https://hal.science/hal-01168801/file/morphvar.pdf; https://doi.org/10.1007/978-3-319-18720-4_40

International audience 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.