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Morphological Signal Processing and the Slope Transform

**Leo Dorst and Rein van den Boomgaard**

*invited paper for Signal Processing, vol. 38, 1994, pp. 79-98.*
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.

Click here for a
postscript version of the entire paper (183k).

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