Chapter Two - Representations for Morphological Image Operators and Analogies with Linear Operators

Abstract

Abstract This chapter deals with representation theoretical issues of nonlinear image operators, mainly based on the methodology of mathematical morphology, and more generally operators on lattices. After a brief overview of developments in morphological image operators both chronologically and thematically, the chapter provides a survey of some main concepts and results in the theory of lattices and morphological operators, especially of the monotone type. It also provides comparisons with linear operator theory. Then, it introduces a nonlinear signal space called complete weighted lattice, which generalizes both mathematical morphology and minimax algebra. Afterwards, it focuses on the representation of translation-invariant and/or increasing operators either on Euclidean spaces (or their discretized versions) or on complete weighted lattices by using a nonlinear basis. The results are operator representations as a supremum or infimum of nonlinear convolutions that are either of the max-plus type or their generalizations in weighted lattices. These representations have several potential applications in computation, imaging and vision, and nonlinear functional analysis.