This paper aims to establish the theoretical foundation for shift inclusion in mathematical morphology. In this paper, we prove that the morphological opening and closing concerning structuring elements of shift inclusion property would preserve the ordering of images, while this property is important in granulometric analysis and related image processing tasks. Furthermore, we proposed a systematic way, called the decomposition theorem for shift inclusion, to construct sequences of structuring elements with shift inclusion property. Moreover, the influences of the image domain are discussed and the condition named weak shift inclusion is defined, which is proved as an equivalent condition for ensuring the order-preserving property.

翻译：本文旨在为数学形态学中的转换包容奠定理论基础。 在本文中,我们证明,关于转移包容属性结构要素的形态开放和关闭将保留图像的顺序,而这种属性在颗粒度分析和相关的图像处理任务中是重要的。此外,我们提出了一种系统的方法,称为转换包容的分解理论,以构建带有转移包容属性的结构要素序列。此外,讨论了图像域的影响,并界定了称为弱变包容的条件,这被证明是确保顺序保留属性的同等条件。