# Double-sided probing by map of Asplund's distances using Logarithmic Image Processing in the framework of Mathematical Morphology

arXiv: https://arxiv.org/abs/1701.08092 · HTML: https://mlsanity.com/paper/1701.08092/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 27 Jan 2017.

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

- [Logarithmic Morphological Neural Nets robust to lighting variations](https://mlsanity.com/paper/2204.09319/) (2022)
- [Logarithmic Mathematical Morphology: Theory and applications](https://mlsanity.com/paper/2309.02007/) (2023)

## Abstract (arXiv)

We establish the link between Mathematical Morphology and the map of Asplund's distances between a probe and a grey scale function, using the Logarithmic Image Processing scalar multiplication. We demonstrate that the map is the logarithm of the ratio between a dilation and an erosion of the function by a structuring function: the probe. The dilations and erosions are mappings from the lattice of the images into the lattice of the positive functions. Using a flat structuring element, the expression of the map of Asplund's distances can be simplified with a dilation and an erosion of the image; these mappings stays in the lattice of the images. We illustrate our approach by an example of pattern matching with a non-flat structuring function.