Morphological signal processing and the slope transform

This paper presents the operation of tangential dilation, which describes the touching of differentiable surfaces. It generalizes the classical dilation, but is invertible. It is shown that line segments are eigenfunctions of this dilation, and are parallel transported, and that curvature is additive. We then present the slope transform which provides tangential morphology with the analytical power which the Fourier tansform lends to linear signal processing, in particular: dilation becomes addition (just as under a Fourier transform, convolution becomes multiplication). We give a discrete slope transform suited for implementation, and discuss the relationships to the Legendre transform, the Young-Fenchel conjugate, and the ¢A-transform. We exhibit a logarithmic correspondence of this tangential morphology to linear systems theory, and touch on the consequences for morphological data analysis of a scanning tunnelling microscope