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.
