An Approach to Colour Morphological Supremum Formation using the
  LogSumExp Approximation
Marvin Kahra Michael Breuß Andreas Kleefeld Martin Welk
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.