# A Link Between the Multiplicative and Additive Functional Asplund's Metrics

## TL;DR

This paper establishes a mathematical link between two types of Asplund's metrics within the Logarithmic Image Processing framework, showing how their distance maps relate and demonstrating robustness to lighting variations.

## Contribution

It demonstrates the relationship between LIP-multiplicative and LIP-additive Asplund's metrics and their distance maps using LIP isomorphism, enhancing understanding of their properties.

## Key findings

- The distance maps of the two metrics are mathematically linked.
- Transforming images allows computation of one metric's distance map from the other's.
- The LIP-additive metric shows robustness to lighting changes.

## Abstract

Functional Asplund's metrics were recently introduced to perform pattern matching robust to lighting changes thanks to double-sided probing in the Logarithmic Image Processing (LIP) framework. Two metrics were defined, namely the LIP-multiplicative Asplund's metric which is robust to variations of object thickness (or opacity) and the LIP-additive Asplund's metric which is robust to variations of camera exposure-time (or light intensity). Maps of distances-i.e. maps of these metric values-were also computed between a reference template and an image. Recently, it was proven that the map of LIP-multiplicative As-plund's distances corresponds to mathematical morphology operations. In this paper, the link between both metrics and between their corresponding distance maps will be demonstrated. It will be shown that the map of LIP-additive Asplund's distances of an image can be computed from the map of the LIP-multiplicative Asplund's distance of a transform of this image and vice-versa. Both maps will be related by the LIP isomorphism which will allow to pass from the image space of the LIP-additive distance map to the positive real function space of the LIP-multiplicative distance map. Experiments will illustrate this relation and the robustness of the LIP-additive Asplund's metric to lighting changes.