Matrix Morphology with Extremum Principle

  • Martin WelkEmail author
  • Michael Breuß
  • Vivek Sridhar
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11564)


The fundamental operations of mathematical morphology are dilation and erosion. In previous works, these operations have been generalised in a discrete setting to work with fields of symmetric matrices, and also corresponding methods based on partial differential equations have been constructed. However, the existing methods for dilation and erosion in the matrix-valued setting are not overall satisfying. By construction they may violate a discrete extremum principle, which means that results may leave the convex hull of the matrices that participate in the computation. This may not be desirable from the theoretical point of view, as the corresponding property is fundamental for discrete and continuous-scale formulations of dilation and erosion in the scalar setting. Moreover, if such a principle could be established in the matrix-valued framework, this would help to make computed solutions more interpretable.

In our paper we address this issue. We show how to construct a method for matrix-valued morphological dilation and erosion that satisfies a discrete extremum principle. We validate the construction by showing experimental results on synthetic data as well as colour images, as the latter can be cast as fields of symmetric matrices.


Dilation Erosion Matrix valued images Extremum principle 


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institute of Biomedical Image AnalysisPrivate University of Health Sciences, Medical Informatics and TechnologyHall/TyrolAustria
  2. 2.Brandenburg University of Technology Cottbus–SenftenbergCottbusGermany

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