Approaches to Multivalued Mathematical Morphology Based on Uncertain Reduced Orderings

Mateus Sangalli (1) , Marcos Eduardo Valle (1)

1 IMECC - Instituto de Matemática, Estatística e Computação Científica [Brésil]

Mathematical morphology (MM) is a powerful non-linear theory that can be used for signal and image processing and analysis. Although MM can be very well defined on complete lattices, which are partially ordered sets with well defined extrema operations, there is no natural ordering for multivalued images such as hyper-spectral and color images. Thus, a great deal of effort has been devoted to ordering schemes for multivalued MM. In a reduced ordering, in particular , elements are ranked according to the so-called ordering mapping. Despite successful applications, morphological operators based on reduced orderings are usually too reliant on the ordering mapping. In many practical situations, however , the ordering mapping may be subject to uncertainties such as measurement errors or the arbitrariness in the choice of the mapping. In view of this remark, in this paper we present two approaches to multivalued MM based on an uncertain reduced ordering. The new operators are formulated as the solution of an optimization problem which, apart from the uncertainty, can circumvent the false value problem and deal with irregularity issues.

Mots clés: Mathematical morphology; multivalued image; Optimization problems; uncertainty

Domaines: Statistiques [math.ST]; Traitement du signal et de l'image [eess.SP]; Vision par ordinateur et reconnaissance de formes [cs.CV]; Optimisation et contrôle [math.OC]

Mateus Sangalli, Marcos Eduardo Valle. Approaches to Multivalued Mathematical Morphology Based on Uncertain Reduced Orderings. Mathematical Morphology and Its Applications to Signal and Image Processing, Jul 2019, Saarbrucken, Germany. ⟨10.1007/978-3-030-20867-7_18⟩. ⟨hal-02144471⟩