Shortest Length Total Orders Do Not Minimize Irregularity in
  Vector-Valued Mathematical Morphology
Samuel Francisco Marcos Eduardo Valle
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.