Differential Morphology

Morphological image processing has been based traditionally on modeling images as sets or as points in a complete lattice of functions and viewing morphological image transformations as set or lattice operators. Thus, so far, the two classic approaches to analyzing or designing the deterministic systems of mathematical morphology have been (1) geometry, by viewing them as image set transformations in Euclidean spaces, and (2) algebra, to analyze their properties using set or lattice theory. Geometry was used mainly for intuitive understanding, and algebra was restricted to the space domain. Despite their limitations, these approaches have produced a powerful and self-contained broad collection of nonlinear image analysis concepts, operators, and algorithms. In parallel with these directions, there is a recently growing part of morphological image processing that is based on ideas from differential calculus and dynamical systems. It combines some early ideas on using simple morpholological operations to obtain signal gradients with some recent ideas on using differential equations to model nonlinear multiscale processes or distance propagation in images. In this chapter we present a unified view of the various interrelated ideas in this area and develop some systems analysis tools in both the space and a (slope) transform domain. The main tools of morphological image processing are a broad class of nonlinear image operators, of which the two most fundamental are dilation and erosion. The space domain in which images are defined can be either continuous, E = R2 or