Building the Topological Tree of Shapes from the Tree of Shapes

The topological tree of shapes was recently introduced as a new hierarchical structure within the family of morphological trees. Morphological trees are efficient models for image processing and analysis. For such applications, it is of paramount importance that these structures be built and manipulated with optimal complexity. In this article, we focus on the construction of the topological tree of shapes. We propose an algorithm for building the topological tree of shapes from the tree of shapes. In particular, a cornerstone of this algorithm is the construction of the complete tree of shapes, another recently introduced tree unifying both the tree of shapes and the topological tree of shapes. We also discuss the cost of the computation of these structures.