Lattice calculus of the morphological slope transform

This paper presents a study of the morphological slope transform in the complete lattice framework. It discusses in detail the interrelationships between the slope transform at one hand and the (Young-Fenchel) conjugate and Legendre transform, two well-known concepts from convex analysis, at the other. The operators and transforms of importance here (hull operations, slope transform, support function, polar, gauge, etc.) are complete lattice operators with interesting properties also known from theoretical morphology. For example, the slope transform and its ‘inverse’ form an adjunction. It is shown that the slope transform for sets (binary signals) coincides with the notion of support Function, known from the theory of convex sets. Two applications are considered: the first application concerns an alternative approach to the distance transform. The second application deals with evolution equations for multiscale morphology using the theory of Hamilton-Jacobi equations.