Mathematical morphology is a part of image processing that has proven to be fruitful for numerous applications. Two main operations in mathematical morphology are dilation and erosion. These are based on the construction of a supremum or infimum with respect to an order over the tonal range in a certain section of the image. The tonal ordering can easily be realised in grey-scale morphology, and some morphological methods have been proposed for colour morphology. However, all of these have certain limitations. In this paper we present a novel approach to colour morphology extending upon previous work in the field based on the Loewner order. We propose to consider an approximation of the supremum by means of a log-sum exponentiation introduced by Maslov. We apply this to the embedding of an RGB image in a field of symmetric $2\times2$ matrices. In this way we obtain nearly isotropic matrices representing colours and the structural advantage of transitivity. In numerical experiments we highlight some remarkable properties of the proposed approach.

翻译：数学形态学是图像处理中已被证明在许多应用中富有成效的组成部分。其两个主要运算——膨胀与腐蚀——建立在图像某区域色调范围上关于某种序关系的上确界或下确界的构造之上。在灰度形态学中，色调排序易于实现，且已有研究者提出了一些用于彩色形态学的方法，但这些方法均存在一定局限性。本文在基于Loewner序的已有工作基础上，提出了一种新型彩色形态学方法。我们考虑采用Maslov提出的对数求和指数方法对上确界进行近似，并将其应用于RGB图像到对称$2\times2$矩阵场的嵌入中。通过这种方式，我们获得了具有近各向同性特性的颜色表征矩阵，并实现了传递性的结构优势。数值实验凸显了该方法的一些显著特性。