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

## TL;DR

This paper links Mathematical Morphology with Asplund's distances using Logarithmic Image Processing, showing how the distance map relates to dilations and erosions, and applies it to pattern matching.

## Contribution

It introduces a novel connection between Asplund's distances and morphological operations within the Logarithmic Image Processing framework, simplifying computations with flat structuring elements.

## Key findings

- The distance map is the logarithm of the ratio of dilation and erosion.
- The approach simplifies with flat structuring elements.
- Application demonstrated in pattern matching.

## Abstract

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.