Morphological Systems Theory: Slope Transforms, Max—Min Differential Equations, Envelope Filters, and Sampling

Abstract

In this paper we view the basic morphological signal operators as a class of nonlinear systems obeying a supremum/infimum-of-sums superposition and endow them with analytic tools that parallel the functionality of and have many conceptual similarities with ideas and tools used in linear systems. In the time/space domain, these systems are found equivalent to a max-plus or min-plus convolution with their impulse response, and a class of nonlinear difference/differential equations is introduced to describe their dynamics The hyperplanes < α, υ > +b are eigenfunctions of such morphological systems, which leads to developing a slope response for them, as a function of the slope α, and related slope transforms for arbitrary signals. These ideas provide a transform (slope) domain for morphological systems, where dilation and erosion in time/space corresponds to addition of slope transforms, time/space hyperplanes transform into slope impulses, time/space cones become bandpass slope-selective filters, and distance transforms correspond to ideal slope filters. The paper summarizes results for 1D signals/systems and extends them to the 2D case and to sampling.

This is an invited paper. It was written while the anthor was supported by the US National Science Foundation under Grant MIP-9396301.