Morphological systems: Slope transforms and max-min difference and differential equations

Abstract

Linear time-invariant systems are well understood in the time domain either as convolutions with their impulse response or by describing their dynamics via linear differential equations. Their analysis in the frequency domain using their exponential eigenfunctions and related frequency response is also greatly facilitated via Fourier transforms. Attempting to extend such ideas to nonlinear systems, we present in this paper a theory for a broad class of nonlinear systems and a collection of related analytic tools, which parallel the functionality of and have many conceptual similarities with ideas and tools used in linear systems. These nonlinear systems are time-invariant dilations or erosions, in continuous and discrete time, and obey a supremum- or infimum-of-sums superposition. In the time domain, their equivalence with morphological dilation or erosion by their impulse response is established, and their causality and stability are examined. A class of nonlinear difference did differential equations based on max-min operations is also introduced to describe their dynamics. After finding that the affine signals αt + b are eigenfunctions of morphological systems, their slope response is introduced as a function of the slope α, and related slope transforms for arbitrary signals are developed. These ideas provide a transform (slope) domain for morphological systems, where dilation and erosion in time corresponds to addition of slope transforms. Recursive morphological systems, described by max-min difference equations, are also investigated and shown to be equivalent to dilation or erosion by infinite-support structuring elements. Their analysis is significantly aided by using slope transforms. These recursive morphological systems are applied to the design of ideal-cutoff slope-selective filters which are useful for signal envelope estimation.