An Approach to Colour Morphological Supremum Formation using the LogSumExp Approximation

arXiv (Cornell University)

Abstract

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.

Authors 4

1. Marvin Kahra

   Brandenburg University of Technology Cottbus-Senftenberg

   Affiliation as printed

   Institute for Mathematics , Brandenburg University of Technology Cottbus -Senftenberg , 03046 Cottbus , Germany

2. Michael Breuß
3. Andreas Kleefeld Aachen Faculty of Medical Engineering and Technomathematics

   Forschungszentrum Jülich · Jülich Supercomputing Centre · FH Aachen

   Affiliation as printed

   Forschungszentrum Jülich GmbH , Jülich Supercomputing Centre , Wilhelm-Johnen-Straße , 52425 Jülich , Germany

   University of Applied Sciences Aachen , Faculty of Medical Engineering and Technomathematics , Heinrich-Mußmann-Straße 1 , 52428 Jülich , Germany

4. Martin Welk

   UMIT TIROL - Private University for Health Sciences and Health Technology

   Affiliation as printed

   UMIT TIROL -Private University for Health Sciences and Health Technology , Eduard-Wallnöfer-Zentrum 1 , 6060 Hall/ Tyrol , Austria