Morphological adjunctions represented by matrices in max-plus algebra for signal and image processing

Samy Blusseau (1, 2, 3) , Santiago Velasco-Forero (1, 2, 3) , Jesus Angulo (1, 2, 3) , Isabelle Bloch (4)

1 CMM - Centre de Morphologie Mathématique
2 PSL - Université Paris Sciences et Lettres
3 Mines Paris - PSL (École nationale supérieure des mines de Paris)
4 LFI - Learning, Fuzzy and Intelligent systems

In discrete signal and image processing, many dilations and erosions can be written as the max-plus and min-plus product of a matrix on a vector. Previous studies considered operators on symmetrical, unbounded complete lattices, such as Cartesian powers of the completed real line. This paper focuses on adjunctions on closed hypercubes, which are the complete lattices used in practice to represent digital signals and images. We show that this constrains the representing matrices to be doubly-0-astic and we characterise the adjunctions that can be represented by them. A graph interpretation of the defined operators naturally arises from the adjacency relationship encoded by the matrices, as well as a max-plus spectral interpretation.

Mots clés: Morphological operators; Max-plus algebra; Graph theory

Domaines: Traitement du signal et de l'image [eess.SP]; Théorie des représentations [math.RT]; Algèbres d'opérateurs [math.OA]; Théorie spectrale [math.SP]

Samy Blusseau, Santiago Velasco-Forero, Jesus Angulo, Isabelle Bloch. Morphological adjunctions represented by matrices in max-plus algebra for signal and image processing. DGMM 2022, IAPR Second International Conference on Discrete Geometry and Mathematical Morphology, Oct 2022, Strasbourg, France. ⟨hal-03738665v1⟩