# An Approach to Colour Morphological Supremum Formation using the LogSumExp Approximation

arXiv: https://arxiv.org/abs/2312.13792 · HTML: https://mlsanity.com/paper/2312.13792/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 0 of its 19 references that are in the map.
- First posted on arXiv 21 Dec 2023.

## Built on by (most cited first, 2 of 2)

- [Matrix-Valued LogSumExp Approximation for Colour Morphology](https://mlsanity.com/paper/2411.10141/) (2024)
- [Colour Morphological Distance Ordering based on the Log-Exp-Supremum](https://mlsanity.com/paper/2503.11329/) (2025)

## Abstract (arXiv)

Mathematical morphology is a part of image processing that has proven to be fruitful for numerous applications. Two main operations in mathematical morphology are dilation and erosion. These are based on the construction of a supremum or infimum with respect to an order over the tonal range in a certain section of the image. The tonal ordering can easily be realised in grey-scale morphology, and some morphological methods have been proposed for colour morphology. However, all of these have certain limitations. In this paper we present a novel approach to colour morphology extending upon previous work in the field based on the Loewner order. We propose to consider an approximation of the supremum by means of a log-sum exponentiation introduced by Maslov. We apply this to the embedding of an RGB image in a field of symmetric 2×2 matrices. In this way we obtain nearly isotropic matrices representing colours and the structural advantage of transitivity. In numerical experiments we highlight some remarkable properties of the proposed approach.