# Morphological adjunctions represented by matrices in max-plus algebra for signal and image processing

arXiv: https://arxiv.org/abs/2207.13926 · HTML: https://mlsanity.com/paper/2207.13926/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Jul 2022.

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

- [Training morphological neural networks with gradient descent: some theoretical insights](https://mlsanity.com/paper/2403.12975/) (2024)

## Abstract (arXiv)

In discrete signal and image processing, many dilations and erosions can be written as the max-plus and min-plus product of a matrix on a vector. Previous studies considered operators on symmetrical, unbounded complete lattices, such as Cartesian powers of the completed real line. This paper focuses on adjunctions on closed hypercubes, which are the complete lattices used in practice to represent digital signals and images. We show that this constrains the representing matrices to be doubly-0-astic and we characterise the adjunctions that can be represented by them. A graph interpretation of the defined operators naturally arises from the adjacency relationship encoded by the matrices, as well as a max-plus spectral interpretation.