# A Proof of the Tree of Shapes in n-D

arXiv: https://arxiv.org/abs/2206.05109 · HTML: https://mlsanity.com/paper/2206.05109/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 10 Jun 2022.

## Built on by (most cited first, 1 of 1)

- [MorphoSHAP: Rethinking the Unit of Attribution in Explanation for Deep Visual Models](https://mlsanity.com/paper/2609.25815/) (2026)

## Abstract (arXiv)

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'eraud 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.