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

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.

Matrix-Valued LogSumExp Approximation for Colour Morphology

Mathematical morphology is a part of image processing that uses a window that moves across the image to change certain pixels according to certain operations. The concepts of supremum and infimum play a crucial role here, but it proves challenging to define them generally for higher-dimensional data, such as colour representations. Numerous approaches have therefore been taken to solve this problem with certain compromises. In this paper we will analyse the construction of a new approach, which we have already presented experimentally in paper [Kahra, M., Breu{\ss}, M., Kleefeld, A., Welk, M., DGMM 2024, pp. 325-337]. This is based on a method by Burgeth and Kleefeld [Burgeth, B., Kleefeld, A., ISMM 2013, pp. 243-254], who regard the colours as symmetric $2\times2$ matrices and compare them by means of the Loewner order in a bi-cone through different suprema. However, we will replace the supremum with the LogExp approximation for the maximum instead. This allows us to transfer the associativity of the dilation from the one-dimensional case to the higher-dimensional case. In addition, we will investigate the minimality property and specify a relaxation to ensure that our approach is continuously dependent on the input data.

An Approach to Colour Morphological Supremum Formation using the LogSumExp Approximation

Mathematical morphology is a part of image processing that has proven to be fruitful for numerous applications. Two main operations in mathematical morphology are dilation and erosion. These are based on the construction of a supremum or infimum with respect to an order over the tonal range in a certain section of the image. The tonal ordering can easily be realised in grey-scale morphology, and some morphological methods have been proposed for colour morphology. However, all of these have certain limitations. In this paper we present a novel approach to colour morphology extending upon previous work in the field based on the Loewner order. We propose to consider an approximation of the supremum by means of a log-sum exponentiation introduced by Maslov. We apply this to the embedding of an RGB image in a field of symmetric $2\times2$ matrices. In this way we obtain nearly isotropic matrices representing colours and the structural advantage of transitivity. In numerical experiments we highlight some remarkable properties of the proposed approach.