Mathematical Morphology on the Spherical CIELab Quantale with an Application in Color Image Boundary Detection

Abstract

Mathematical morphology is a theory with applications in image processing and analysis. This paper presents a quantale-based approach to color morphology based on the CIELab color space in spherical coordinates. The novel morphological operations take into account the perceptual difference between color elements by using a distance-based ordering scheme. Furthermore, the novel approach allows for the use of non-flat structuring elements. An illustrative example reveals that non-flat dilations and erosions may preserve more features of a color image than their corresponding flat operations. Furthermore, the novel non-flat morphological operators yielded promising results on experiments concerning the detection of the boundaries of objects on color images.

Three-Dimensional Generalized Discrete Fuzzy Number and Applications in Color Mathematical Morphology

“…A conditionally invariant mathematical morphological framework for color images was presented in [33], and a vector ordering method based on linear transformations from RGB to other color spaces and principal component analysis (PCA) were developed. Valle and Valente [34] proposed a quantal-based approach to color mathematical morphology based on the CIEL*a*b* color space in spherical coordinates. The definition of morphological operators used a distance-based ordering scheme and non-flat structuring elements.…”

Characterization and Statistics of Distance-Based Elementary Morphological Operators

“…In the last decades, different approaches to distance-based color MM have been proposed by prominent researchers including [1,2,8,14,20,26]. Despite successful applications, these approaches often face the difficult task of choosing a suitable reference for sorting vector values.…”

“…In this paper, we focus on a distance-based approach which have been investigated and generalized by many researchers including [1,2,8,14,20,[26][27][28]. Precisely, we focus on the distance-based approach to multivalued MM obtained by first comparing the distance between the value of a pixel and a reference value r followed by a lexicographical cascade to resolve ambiguities [2].…”

“…given by (4) with the reference fixed on black and white, respectively [2]. At this point, we would like to recall that some computational experiments using distance-based operators revealed that an achromatic color, such as black or white, has been recommended for an edge detection task if no information on the reference is available [26].…”

Measuring the Irregularity of Vector-Valued Morphological Operators Using Wasserstein Metric

“…In the case of vector-valued images, vector spaces endowed with a total order is one of the most comfortable frameworks for the extension of morphological processing [2,1,22]. Approaches that have been recently formulated using total orderings include [9,12,[18][19][20][21]. Despite their successful applications for color and hyperspectral image processing, Chevallier and Angulo showed that the information contained in a total order is too weak to reproduce the natural topology of the value space [5].…”