A Proof of the Tree of Shapes in n-D

TL;DR

This paper proves that the self-dual hierarchical structure computed by a quasi-linear time algorithm on n-D images is exactly the tree of shapes, a fundamental structure in mathematical morphology with various applications.

Contribution

It establishes the equivalence between the computed hierarchical structure and the theoretical tree of shapes for n-D images, confirming the algorithm's correctness.

Findings

The algorithm is in quasi-linear time and is optimal.

The hierarchical structure matches the theoretical tree of shapes.

Applications include filtering, shapings, and segmentation.

Abstract

In this paper, we prove that the self-dual morphological hierarchical structure computed on a n-D gray-level wellcomposed image u by the algorithm of G{\'e}raud et al. [1] is exactly the mathematical structure defined to be the tree of shape of u in Najman et al [2]. We recall that this algorithm is in quasi-linear time and thus considered to be optimal. The tree of shapes leads to many applications in mathematical morphology and in image processing like grain filtering, shapings, image segmentation, and so on.