# Logarithmic Mathematical Morphology: Theory and applications

arXiv: https://arxiv.org/abs/2309.02007 · HTML: https://mlsanity.com/paper/2309.02007/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 9 of its 62 references that are in the map.
- First posted on arXiv 5 Sep 2023.

## Builds on (most cited first, 9 of 9)

- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [RV-GAN: Segmenting Retinal Vascular Structure in Fundus Photographs using a Novel Multi-scale Generative Adversarial Network](https://mlsanity.com/paper/2101.00535/) (2021)
- [Study Group Learning: Improving Retinal Vessel Segmentation Trained with Noisy Labels](https://mlsanity.com/paper/2103.03451/) (2021)
- [Double-sided probing by map of Asplund's distances using Logarithmic Image Processing in the framework of Mathematical Morphology](https://mlsanity.com/paper/1701.08092/) (2017)
- [Logarithmic mathematical morphology: a new framework adaptive to illumination changes](https://mlsanity.com/paper/1806.02998/) (2018)
- [Region homogeneity in the Logarithmic Image Processing framework: application to region growing algorithms](https://mlsanity.com/paper/1904.12597/) (2019)
- [A Link Between the Multiplicative and Additive Functional Asplund's Metrics](https://mlsanity.com/paper/1907.07509/) (2019)
- [Retinal vessel segmentation by probing adaptive to lighting variations](https://mlsanity.com/paper/2004.13992/) (2020)
- [Logarithmic Morphological Neural Nets robust to lighting variations](https://mlsanity.com/paper/2204.09319/) (2022)

## Abstract (arXiv)

In Mathematical Morphology for grey-level functions, an image is analysed by another image named the structuring function. This structuring function is translated over the image domain and summed to the image. However, in an image presenting lighting variations, when the structuring function is added to this image, the amplitude of this translated structuring function should vary according to the image intensity. Such a property is not verified in Mathematical Morphology for grey level functions, when the structuring function is summed to the image with the usual additive law. This issue has been addressed by defining a framework that uses an additive law, for which the amplitude of the structuring function added to the image varies according to the amplitude of the image itself. This additive law is chosen within the Logarithmic Image Processing framework and models the lighting variations with a physical cause such as a change of light intensity. The framework is named Logarithmic Mathematical Morphology (LMM) and allows the definition of operators which are robust to such lighting variations.