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Mathematical morphology is a theory concerned with non-linear operators forimage processing and analysis. The underlying framework for mathematicalmorphology is a partially ordered set with well-defined supremum and infimumoperations. Because vectors can be ordered in many ways, finding appropriateordering schemes is a major challenge in mathematical morphology forvector-valued images, such as color and hyperspectral images. In this context,the irregularity issue plays a key role in designing effective morphologicaloperators. Briefly, the irregularity follows from a disparity between theordering scheme and a metric in the value set. Determining an ordering schemeusing a metric provide reasonable approaches to vector-valued mathematicalmorphology. Because total orderings correspond to paths on the value space, oneattempt to reduce the irregularity of morphological operators would be defininga total order based on the shortest length path. However, this paper shows thatthe total ordering associated with the shortest length path does notnecessarily imply minimizing the irregularity.

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