Abstract. To compute the morphological self-dual representation of images, namely the tree of shapes, the state-of-the-art algorithms do not have a satisfactory time complexity. Furthermore the proposed algorithms are only effective for 2D images and they are far from being simple to implement. That is really penalizing since a self-dual representation of images is a structure that gives rise to many powerful operators and applications, and that could be very useful for 3D images. In this paper we propose a simple-to-write algorithm to compute the tree of shapes; it works for nD images and has a quasi-linear complexity when data quantization is low, typically 12 bits or less. To get that result, this paper introduces a novel representation of images that has some amazing properties of continuity, while remaining discrete.

The tree of shapes is a self-dual tree-based image representation belonging to the field of mathematical morphology. This representation is highly interesting since it is invariant to contrast changes and inversion, and allows for numerous and powerful applications. A new algorithm to compute the tree of shapes has been recently presented: it has a quasilinear complexity; it is the only known algorithm that is also effective for nD images with n > 2; yet it is sequential. With the increasing size of data to process, the need of a parallel algorithm to compute that tree is of prime importance; in this paper, we present such an algorithm. We also give some benchmarks that show that the parallel version is computationally effective. As a consequence, that makes possible to process 3D images with some powerful self-dual morphological tools.

International audienceIn 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