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

TL;DR

This paper investigates the relationship between shortest length total orders and irregularity minimization in vector-valued mathematical morphology, revealing that shortest paths do not necessarily reduce irregularity.

Contribution

It demonstrates that shortest length total orders do not inherently minimize irregularity in vector-valued morphological operators, challenging previous assumptions.

Findings

Shortest length paths do not guarantee minimal irregularity.

Irregularity depends on the ordering scheme and metric choice.

The study clarifies the limitations of shortest path-based orderings.

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.