Slope transforms: theory and application to nonlinear signal processing

Fourier transforms are among the most useful linear signal transformations for quantifying the frequency content of signals and for analyzing their processing by linear time-invariant systems. Some nonlinear signal transforms are developed that can provide information about the slope content of signals and are useful analytic tools for large classes of nonlinear systems. Many of their theoretical properties are examined, showing a striking conceptual resemblance to Fourier transforms and their application to linear systems. These novel transforms, called slope transforms, are originally derived from the eigenvalues of morphological dilation and erosion systems, where the corresponding eigenfunctions are lines αt+b parameterized by their slope α. They obey a nonlinear superposition principle of the supremum- or infimum-of-sums type. Applied to the impulse response of dilation or erosion systems, the slope transforms provide a slope response function for these systems, which allows their analysis and design in a transform domain, the slope domain. Applied to arbitrary signals, the slope transforms provide information about upper or lower tangents to the signal´s graph at varying slopes. The upper or lower envelopes of the signal can be obtained from the inverse transforms. Overall, the slope transforms provide a new transform domain for signals and morphological systems where time lines become slope impulses, time cones become slope bandpass filters, and time dilation/erosion transform into addition of slope transforms. Their application to the design of slope-selective filters is also presented