Shortest Length Total Orders Do Not Minimize Irregularity in Vector-Valued Mathematical Morphology

Abstract: Mathematical morphology is a theory concerned with non-linear operators for image processing and analysis. The underlying framework for mathematical morphology is a partially ordered set with well-defined supremum and infimum operations. Because vectors can be ordered in many ways, finding appropriate ordering schemes is a major challenge in mathematical morphology for vector-valued images, such as color and hyperspectral images. In this context, the irregularity issue plays a key role in designing effective morphological operators. Briefly, the irregularity follows from a disparity between the ordering scheme and a metric in the value set. Determining an ordering scheme using a metric provide reasonable approaches to vector-valued mathematical morphology. Because total orderings correspond to paths on the value space, one attempt to reduce the irregularity of morphological operators would be defining a total order based on the shortest length path. However, this paper shows that the total ordering associated with the shortest length path does not necessarily imply minimizing the irregularity.