用对数求和指数近似替代颜色形态学中的最大值运算，提升高维色彩处理的稳定性与效率。
Matrix-Valued LogSumExp Approximation for Colour Morphology
- 以矩阵形式表示颜色，用对数求和指数近似最大值运算
- 实现高维色彩形态学的可结合性，保持运算一致性
- 方法连续依赖输入，适合图像处理与计算机视觉应用
数学形态学是图像处理中通过滑动窗口改变像素的工具，其核心依赖上确界与下确界的定义。然而，对于如色彩这样的高维数据，一般化定义上、下确界仍具挑战。已有方法多有妥协。本文基于Burgeth与Kleefeld的工作，将颜色视为2×2对称矩阵，并利用Loewner序在双锥结构中比较。本文创新地以对数求和指数（LogSumExp）近似最大值，取代传统上确界操作，从而将一维形态学中的膨胀运算可结合性推广至高维。同时，研究了该方法的最小性性质，并引入松弛机制确保对输入数据的连续依赖性。
原文摘要 · Abstract (English)
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., Breuß, 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.
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