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

TL;DR

This paper explores how max-plus algebra matrices can represent morphological adjunctions in digital signal and image processing, focusing on practical bounded lattices and their spectral and graph interpretations.

Contribution

It characterizes the matrices representing adjunctions on hypercubes and links these to graph and spectral interpretations, extending previous unbounded lattice results.

Findings

Matrices are constrained to be doubly-0-astic for adjunctions on hypercubes

Provides a characterization of representable adjunctions in practical signal/image lattices

Establishes a graph and spectral interpretation of the operators

Abstract

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.