How to Make n-D Plain Maps Defined on Discrete Surfaces Alexandrov-Well-Composed in a Self-Dual Way

Abstract

In 2013, Najman and Geraud proved that by working on a well-composed discrete representation of a gray-level image, we can compute what is called its tree of shapes, a hierarchical representation of the shapes in this image. This way, we can proceed to morphological filtering and to image segmentation. However, the authors did not provide such a representation for the non-cubical case. We propose in this paper a way to compute a well-composed representation of any gray-level image defined on a discrete surface, which is a more general framework than the usual cubical grid. Furthermore, the proposed representation is self-dual in the sense that it treats bright and dark components in the image the same way. This paper can be seen as an extension to gray-level images of the works of Daragon et al. on discrete surfaces.