Algebraic Foundations of Morphology
Christian Ronse, Jean Serra · 2013
This chapter presents the former approach, which was extended through the 1970s to numerical functions under the impulse of macro- and microscopic medical imaging. During the 1980s, when morphological operations were acting on increasingly varied objects, the need for unification led to the concept of complete lattice as a common framework. This algebraic structure lies at the basis of lattice theory, a branch of algebra introduced more than a century ago by Dedekind who studied modular lattices. Algebraists have long studied the structure of lattices, but have however given scant attention to the concepts of complete lattice and operators. The theory of mathematical morphology is based on the structure of complete lattice. The chapter is devoted to a few lattices whose elements are subsets of a given arbitrary set E. Controlled Vocabulary Terms algebra; biomedical imaging; lattice theory; mathematical morphology