Eigenfunctions of Ultrametric Morphological Openings and Closings

Abstract

This paper deals with the relationship between spectral analysis in min-max algebra and ultrametric morphological operators. Indeed, morphological semigroups in ultrametric spaces are essentially based on that algebra. Theory of eigenfunctionals in min-max analysis is revisited, including classical applications (preference analysis, percolation and hierarchical segmentation). Ultrametric distance is the fix point functional in min-max analysis and from this result, we prove that the ultrametric distance is the key ingredient to easily define the eigenfunctions of ultrametric morphological openings and closings.