Mathematical morphology, a field within image processing, includes various filters that either highlight, modify, or eliminate certain information in images based on an application's needs. Key operations in these filters are dilation and erosion, which determine the supremum or infimum for each pixel with respect to an order of the tonal values over a subset of the image surrounding the pixel. This subset is formed by a structuring element at the specified pixel, which weighs the tonal values. Unlike grey-scale morphology, where tonal order is clearly defined, colour morphology lacks a definitive total order. As no method fully meets all desired properties for colour, because of this difficulty, some limitations are always present. This paper shows how to combine the theory of the log-exp-supremum of colour matrices that employs the Loewner semi-order with a well-known colour distance approach in the form of a pre-ordering. The log-exp-supremum will therefore serve as the reference colour for determining the colour distance. To the resulting pre-ordering with respect to these distance values, we add a lexicographic cascade to ensure a total order and a unique result. The objective of this approach is to identify the original colour within the structuring element that most closely resembles a supremum, which fulfils a number of desired properties. Consequently, this approach avoids the false-colour problem. The behaviour of the introduced operators is illustrated by application examples of dilation and closing for synthetic and natural images.

翻译：数学形态学作为图像处理领域的一个分支，包含多种滤波器，这些滤波器根据应用需求对图像中的特定信息进行增强、修改或消除。这些滤波器中的关键操作是膨胀和腐蚀，它们根据像素周围图像子集上色调值的排序，为每个像素确定上确界或下确界。该子集由指定像素处的结构元素构成，该元素对色调值进行加权。与灰度形态学中色调顺序明确定义不同，彩色形态学缺乏明确的全序关系。由于没有方法能完全满足彩色的所有理想性质，因此始终存在一些局限性。本文展示了如何将采用Loewner半序的彩色矩阵对数-指数-上确界理论与一种已知的彩色距离方法（以预序形式）相结合。因此，对数-指数-上确界将作为确定彩色距离的参考色。针对由此产生的基于距离值的预序，我们添加了字典级联以确保全序关系和唯一结果。该方法的目标是识别结构元素中最接近上确界的原始彩色，这满足了一系列理想性质。因此，该方法避免了伪彩色问题。通过合成图像和自然图像的膨胀与闭运算应用实例，阐明了所引入算子的行为特性。