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 is a re-representation of morphology onto the morphological eigenfunctions. As such, the slope transform provides for tangential morphology the same analytical power as the Fourier transform provides for linear signal processing. Under the slope transform 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, Young-Fenchel conjugate, and 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.
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