Self-duality and Digital Topology: Links Between the Morphological Tree of Shapes and Well-Composed Gray-Level Images

• Main Authors: Géraud, Thierry, Carlinet, Edwin, Crozet, Sébastien
• Other Authors: Laboratoire de Recherche et de Développement de l'EPITA (LRDE), Ecole Pour l'Informatique et les Techniques Avancées (EPITA), 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)
• Format: Conference Object
• Language: English
• Published: HAL CCSD 2017
• Subjects: self-dual operators; tree of shapes; vertex-valued graph; well-composed gray-level images; digital topology; [INFO.INFO-TI]Computer Science [cs]/Image Processing [eess.IV]; Iceland
• Online Access: https://inria.hal.science/hal-01476218; https://inria.hal.science/hal-01476218/document; https://inria.hal.science/hal-01476218/file/article.pdf; https://doi.org/10.1007/978-3-319-18720-4_48

International audience In digital topology, the use of a pair of connectivities is required to avoid topological paradoxes. In mathematical morphology, self-dual operators and methods also rely on such a pair of connectivities. There are several major issues: self-duality is impure, the image graph structure depends on the image values, it impacts the way small objects and texture are processed, and so on. A sub-class of images defined on the cubical grid, well-composed images, has been proposed, where all connectivities are equivalent, thus avoiding many topological problems. In this paper we unveil the link existing between the notion of well-composed images and the morphological tree of shapes. We prove that a well-composed image has a well-defined tree of shapes. We also prove that the only self-dual well-composed interpolation of a 2D image is obtained by the median operator. What follows from our results is that we can have a purely self-dual representation of images, and consequently, purely self-dual operators.