Algebraic and PDE Approaches for Multiscale Image Operators with Global Constraints: Reference Semilattice Erosions and Levelings

Abstract

This paper begins with analyzing the theoretical connections between levelings on lattices and scale-space erosions on reference semilattices. They both represent large classes of self-dual morphological operators that exhibit both local computation and global constraints. Such operators are useful in numerous image analysis and vision tasks ranging from simplification, to geometric feature detection, to segmentation. Previous definitions and constructions of levelings were either discrete or continuous using a PDE. We bridge this gap by introducing generalized levelings based on triphase operators that switch among three phases, one of which is a global constraint. The triphase operators include as special cases reference semilattice erosions. Algebraically, levelings are created as limits of iterated or multiscale triphase operators. The sub-class of multiscale geodesic triphase operators obeys a semigroup, which we exploit to find a PDE that generates geodesic levelings. Further, we develop PDEs that can model and generate continuous-scale semilattice erosions, as a special case of the leveling PDE. We discuss theoretical aspects of these PDEs, propose discrete algorithms for their numerical solution which are proved to converge as iterations of triphase operators, and provide insights via image experiments.