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.

翻译：数学形态学是图像处理的一个分支，它通过一个在图像上移动的窗口，依据特定运算改变某些像素。上确界与下确界的概念在此起着关键作用，但事实证明，为高维数据（例如彩色表示）普遍定义这些概念具有挑战性。因此，已有多种方法在做出一定妥协的前提下解决此问题。本文中，我们将分析一种新方法的构建，该方法我们已在论文[Kahra, M., Breu{\ss}, M., Kleefeld, A., Welk, M., DGMM 2024, pp. 325-337]中通过实验展示。该方法基于Burgeth和Kleefeld [Burgeth, B., Kleefeld, A., ISMM 2013, pp. 243-254]提出的一种方法，他们将颜色视为对称的$2\times2$矩阵，并通过Loewner序在双锥中借助不同的上确界进行比较。然而，我们将用LogExp逼近替代最大值运算中的上确界。这使得我们能够将膨胀运算的结合律从一维情形推广到高维情形。此外，我们将研究极小性性质，并规定一种松弛条件以确保我们的方法对输入数据具有连续依赖性。