Matrix-Valued LogSumExp Approximation for Colour Morphology

Journal of Mathematical Imaging and Vision, vol. 67

Abstract

Abstract Mathematical morphology is a part of image processing that employs a moving window to modify pixel values through the application of specific operations. The supremum and infimum are pivotal concepts, yet defining them in a general sense for high-dimensional data such as colour is a challenging endeavour. As a result, a number of different approaches have been taken to try to find a solution, with certain compromises being made along the way. In this paper, we present an analysis of a novel approach that replaces the supremum within a morphological operation with the LogExp approximation of the maximum for matrix-valued colours. This approach has the advantage of extending the associativity of dilation from the one-dimensional to the higher-dimensional case. Furthermore, the minimality property is investigated and a relaxation specified to ensure that the approach is continuously dependent on the input data.

Authors 4

1. Marvin Kahra corresponding

   Brandenburg University of Technology Cottbus-Senftenberg

   Affiliation as printed

   Institute for Mathematics, Brandenburg University of Technology Cottbus-Senftenberg, Platz der Deutschen Einheit 1, 03046, Cottbus, Brandenburg, Germany

2. Michael Breuß

   Brandenburg University of Technology Cottbus-Senftenberg

   Affiliation as printed

   Institute for Mathematics, Brandenburg University of Technology Cottbus-Senftenberg, Platz der Deutschen Einheit 1, 03046, Cottbus, Brandenburg, Germany

3. Andreas Kleefeld Aachen Faculty of Medical Engineering and Technomathematics

   Forschungszentrum Jülich · Jülich Supercomputing Centre · FH Aachen

   Affiliation as printed

   Faculty of Medical Engineering and Technomathematics, University of Applied Sciences Aachen, Heinrich-Mußmann-Str. 1, 52428, Jülich, North Rhine-Westphalia, Germany

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

4. Martin Welk

   UMIT TIROL - Private University for Health Sciences and Health Technology

   Affiliation as printed

   Institute of Biomedical Image Analysis, UMIT TIROL – Private University for Health Sciences and Health, Eduard-Wallnöfer-Zentrum 1, 6060, Hall, Tyrol, Austria