# Stability of the Tree of Shapes to Additive Noise

## Abstract

The tree of shapes (ToS) is a famous self-dual hierarchical structure in mathematical morphology, which represents the inclusion relationship of the shapes (i.e. the interior of the level lines with holes filled) in a grayscale image. The ToS has already found numerous applications in image processing tasks, such as grain filtering, contour extraction, image simplificationand so on. Its structure consistency is bound to the cleanliness of the level lines, which are themselves deeply affected by the presence of noise within the image. However, according to our knowledge, no one has measured before how resistant to (additive) noise this hierarchical structure is. In this paper, we propose and compare several measures to evaluate the stability of the ToS structure to noise.

## Bibtex (lrde.bib)

@InProceedings{	  boutry.21.dgmm.3,
author	= {Nicolas Boutry and Guillaume Tochon},
title		= {Stability of the Tree of Shapes to Additive Noise},
booktitle	= {Proceedings of the IAPR International Conference on
Discrete Geometry and Mathematical Morphology (DGMM)},
year		= 2021,
month		= may,
publisher	= {Springer},
series	= {Lecture Notes in Computer Science},
volume	= {12708},
pages		= {365--377},
abstract	= {The tree of shapes (ToS) is a famous self-dual
hierarchical structure in mathematical morphology, which
represents the inclusion relationship of the shapes
(\textit{i.e.} the interior of the level lines with holes
filled) in a grayscale image. The ToS has already found
numerous applications in image processing tasks, such as
grain filtering, contour extraction, image simplification,
and so on. Its structure consistency is bound to the
cleanliness of the level lines, which are themselves deeply
affected by the presence of noise within the image.
However, according to our knowledge, no one has measured
before how resistant to (additive) noise this hierarchical
structure is. In this paper, we propose and compare several
measures to evaluate the stability of the ToS structure to
noise.},
doi		= {10.1007/978-3-030-76657-3_26}
}