Colour Morphological Distance Ordering based on the Log-Exp-Supremum

TL;DR

The paper tackles the lack of a natural total order in colour morphology by introducing a distance-based log-exp-supremum (DLES) framework that leverages the Loewner semi-order on colour-derived symmetric matrices to compute a reference colour via $S=\lim_{m\to\infty}\frac{1}{m}\log\left(\sum_{i=1}^n e^{m\boldsymbol{X}_i}\right)$. A distance to this reference colour in a modified colour space is then used, together with a lexicographic cascade, to produce a total order for pixel-wise colour comparisons within structuring elements, enabling robust dilation and closing while avoiding false colours. Experiments compare two distance variants, $\Delta E_{\text{mHyAB}}$ and $\Delta E_{\text{polar}}$, against a channel-wise RGB baseline, showing similar visual performance to standard morphologies but without colour artefacts; a third option, $\Delta E_{1H}$, performs worse. The results suggest that DLES provides a practical, colour-consistent alternative for colour morphology and motivate future work on exploring additional distance measures and extending the approach to a broader set of morphological operations.

Abstract

Mathematical morphology, a field within image processing, includes various filters that either highlight, modify, or eliminate certain information in images based on an application's needs. Key operations in these filters are dilation and erosion, which determine the supremum or infimum for each pixel with respect to an order of the tonal values over a subset of the image surrounding the pixel. This subset is formed by a structuring element at the specified pixel, which weighs the tonal values. Unlike grey-scale morphology, where tonal order is clearly defined, colour morphology lacks a definitive total order. As no method fully meets all desired properties for colour, because of this difficulty, some limitations are always present. This paper shows how to combine the theory of the log-exp-supremum of colour matrices that employs the Loewner semi-order with a well-known colour distance approach in the form of a pre-ordering. The log-exp-supremum will therefore serve as the reference colour for determining the colour distance. To the resulting pre-ordering with respect to these distance values, we add a lexicographic cascade to ensure a total order and a unique result. The objective of this approach is to identify the original colour within the structuring element that most closely resembles a supremum, which fulfils a number of desired properties. Consequently, this approach avoids the false-colour problem. The behaviour of the introduced operators is illustrated by application examples of dilation and closing for synthetic and natural images.

Figures (3)

• Figure 1: Comparison of dilation results with channel-wise RGB approaches and the DLES method for different distance functions of a $64 \times 64$ pepper image with a $3 \times 3$ square SE. Top:From left to right: Original image, channel-wise Matlab dilation and distance based dilation with the reference colour white in RGB space. Bottom:From left to right: Dilation with the DLES method using the distance function $\Delta E_{\textup{mHyAB}}$, $\Delta E_{\textup{polar}}$ and $\Delta E_{1H}$.
• Figure 2: Comparison of closing results for a $512 \times 512$ image from the TAMPERE17 noise-free image database imageDB with a $9 \times 9$ SE. From left to right: Original image, closing with the DLES method using the functions $\Delta E_{\textup{mHyAB}}$, $\Delta E_{\textup{polar}}$ and $\Delta E_{1H}$.
• Figure 3: Component-wise comparison of closing results for one hundred randomly generated $32 \times 32$ images with the DLES method using $\Delta E_{\textup{mHyAB}}$ (red), $\Delta E_{\textup{polar}}$ (green) and $\Delta E_{1H}$ (blue) and a $3\times 3$ SE. From left to right: Hue, chroma and modified luminance with the corresponding mean values over all images.