Aim
The classical method of surface measuring is to move a stylus along a line over a surface to acquire a one-dimensional surface profile. Modern contact-less methods (e.g. white light interferometry) allow to measure the whole surface, not only a line profile. Since the obtained data is now two-dimensional, surface characterization methods developed for the one-dimensional case are not suited to work with the new data. Our aim is to provide methods for the characterization of the two-dimensional height maps.
Methods
We try to describe a surface with statistical parameters. One promising technique is the use of level sets. The surface is cut by a plane at different height levels. Each of these cuts yields a 2D binary image which can by described with method from random set theory. It is known that any characteristic for such random sets which is rotation invariant, additive and convex continuous can be expressed by the Minkowski functionals, that is the area, the contour length and the Euler characteristic.
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| Fig1. - A surface and the corresponding stack of level sets | |
We get three characterizing functions for a surface: The area of the level sets depending on the cutting height, the contour length function and the Euler characteristic function. These three functions can be compared with the expected functions calculated from simulated data or theoretical surface models.
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| Fig 2. - A measured surface (above) and a simulation with similar properties(below) | Fig 3. - The Minkowski functions of the measured data and the simulation | |
The three characterizing functions are also interesting as they provide a natural extension of the so-called Abbott-Firestone function or bearing area curve which is well known and widely used in surface meteorology. This function is nothing but the area function, and together with the contour length and Euler characteristic function, we get a sound and mathematically tractable system for surface characterization.