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

Samuel Francisco, Marcos Eduardo Valle

TL;DR

The paper investigates whether ordering schemes for vector-valued images in mathematical morphology can reduce irregularity by aligning the total order with the value-space metric. It evaluates the shortest-length Hamiltonian path approach (the TSP order) against the standard lexicographical RGB order, showing that shorter Hamiltonian paths do not guarantee lower irregularity, as measured by an irregularity index. The findings demonstrate that metric-informed total orders alone may not suffice to minimize irregularity and highlight the need for optimization strategies, such as evolutionary methods, to search for orderings that better reconcile topology and metric in vector-valued 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.

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

TL;DR

The paper investigates whether ordering schemes for vector-valued images in mathematical morphology can reduce irregularity by aligning the total order with the value-space metric. It evaluates the shortest-length Hamiltonian path approach (the TSP order) against the standard lexicographical RGB order, showing that shorter Hamiltonian paths do not guarantee lower irregularity, as measured by an irregularity index. The findings demonstrate that metric-informed total orders alone may not suffice to minimize irregularity and highlight the need for optimization strategies, such as evolutionary methods, to search for orderings that better reconcile topology and metric in vector-valued 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.
Paper Structure (5 sections, 9 equations, 2 figures)

This paper contains 5 sections, 9 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Mathematical Morphology and the Irregularity Issue
  3. Shortest-Length Hamiltonian Path
  4. Computational Experiments
  5. Concluding Remarks

Figures (2)

  • Figure 1: Dilation, erosion, opening, and closing of the tiny images 1, 2, 3, and 4, with TSP and RGB approaches, follow of the global irregularity measure $\Phi_1^g$ and the length of the Hamiltonian path.
  • Figure 2: Distribution of the input data obtained by the $V(\mathbf{I})$, and the respective Hamiltonian paths based on the TSP and lexicographical orderings.