Matrix-Valued LogSumExp Approximation for Colour Morphology

TL;DR

This paper introduces a novel matrix-valued LogSumExp approximation method for colour morphology, enabling higher-dimensional supremum operations with improved continuity and associative properties.

Contribution

The paper develops a new LogSumExp-based approach for matrix-valued supremum approximation in colour morphology, extending the applicability of morphological operations to higher-dimensional data.

Findings

Enables associative dilation in higher dimensions

Provides a continuous dependency on input data

Improves supremum approximation accuracy

Abstract

Mathematical morphology is a part of image processing that uses a window that moves across the image to change certain pixels according to certain operations. The concepts of supremum and infimum play a crucial role here, but it proves challenging to define them generally for higher-dimensional data, such as colour representations. Numerous approaches have therefore been taken to solve this problem with certain compromises. In this paper we will analyse the construction of a new approach, which we have already presented experimentally in paper [Kahra, M., Breu{\ss}, M., Kleefeld, A., Welk, M., DGMM 2024, pp. 325-337]. This is based on a method by Burgeth and Kleefeld [Burgeth, B., Kleefeld, A., ISMM 2013, pp. 243-254], who regard the colours as symmetric $2\times2$ matrices and compare them by means of the Loewner order in a bi-cone through different suprema. However, we will replace the supremum with the LogExp approximation for the maximum instead. This allows us to transfer the associativity of the dilation from the one-dimensional case to the higher-dimensional case. In addition, we will investigate the minimality property and specify a relaxation to ensure that our approach is continuously dependent on the input data.