Differential morphology: multiscale image dynamics, max-min difference equations, and slope transforms

We develop 2D max-min difference equations that model the space dynamics of 2D morphological systems and some nonlinear signal transforms, called slope transforms, that can analyze these systems in a transform domain in a way conceptually analogous to the application of Fourier transforms to linear systems. Further, we discuss some nonlinear partial differential equations (PDEs) that model the evolution of multiscale morphological filters of the max-min type. These PDEs are related to the eikonal equation. Solutions of the eikonal PDE are proposed based on 2D min difference equations that can compute distance transforms. We view the analysis of the multiscale morphological PDEs and of the eikonal PDE solved via max-min equations as a unified area in nonlinear image processing which we call differential morphology. Its potential applications include distance-path finding, segmentation, gridless halftoning, and shape from shading