Constructing the Tree of Shapes of an Image by Fusion of the Trees of Connected Components of Upper and Lower Level Sets

The tree of shapes of an image is an ordered structure which permits an efficient manipulation of the level sets of an image, modeled as a real continuous function defined on a rectangle of $${\mathbb{R}}^N$$, N ≥ 2. In this paper we construct the tree of shapes of an image by fusing both trees of connected components of upper and lower level sets. We analyze the branch structure of both trees and we construct the tree of shapes by joining their branches in a suitable way. This was the algorithmic approach for 2D images introduced by F. Guichard and P. Monasse in their initial paper, though other efficient approaches were later developed in this case. In this paper, we prove the well-foundedness of this approach for the general case of multidimensional images. This approach can be effectively implemented in the case of 3D images and can be applied for segmentation, but this is not the object of this paper.