Technical notePeriodic Gaussian filter according to ISO 16610-21 for closed profiles
Introduction
The standard ISO 16610-21 defines the Gaussian filter for dimensional metrology [1]. Qiang-Qian Jiang as the project leader of ISO 16610-21 and Paul Scott as the convenor of working group 15 of ISO TC 213 (both are international experts for surface metrology and working at the University of Huddersfield) were significantly involved in writing the new standard. A nice guideline dealing with the implementation of the Gaussian filter was published by Michael Krystek [4], an internationally recognized expert for surface metrology as well.
It has to be mentioned that ISO 16610-21 serves as a fundamental basis for industrial and metrological applications. And it is well known that the Gaussian filter not only has nice metrological properties but also has some restrictions. For example, open profiles have a finite length. Hence, the evaluable measuring length is shortened by so-called filter running-in and running-out lengths which are a factor of the width of the Gaussian weighting function. When needed, these sections can be still used by applying ISO 16610, part 28 [3]. However, filtering is then no longer phase-correct and therefore these sections are also called the end effect regions of the filter. ISO 16610-21 suggests a simplistic transfer of filtering of open profiles to filtering of closed profiles (e.g. roundness measurements). But this leads to an unnecessary restriction. Because in case of a closed profile, it has to be taken into consideration that the profile can be extended infinitely and thus any measuring length of an equivalent open profile can be reproduced. The result is a periodic convolution of a closed profile of arbitrary length and a periodic weighting function derived from the Gaussian weighting function with any given width.
Section snippets
Linear profile filtering with the Gaussian filter
The basis of linear profile filtering is the convolution integral. The procedure can be seen as determination of a moving average with a weighting function. The following relation applies to an open, infinitely extended profile:where w(x) is the filter line, z(x) is the unfiltered profile and s(x) is the weighting function of the filter. An important requirement for defining the weighting function s(x) is that the area under the function
Linear periodic Gaussian filter for closed profiles
For closed profiles with a short measuring length (e.g. a cylindrical component with a small diameter), it may not be possible to use the “classical” Gaussian filter when the cut-off wavelength λc is a high value. However, this restriction is not necessary. A closed profile can also be handled as an open profile of infinite length. Then a periodic representation, with the circumference serving as the period length, is chosen. This means that the filter can also be used for geometries with small
Example
If a cosine-shaped profile with a wavelength of λ = 0.8 mm is measured over the measuring length L = 1.6 mm and a cut-off frequency of fc = 1 is assumed, the result is f/fc = 2. ISO 16610-21 does not recommend the evaluation of this profile for precision measurements. With the amplitude transfer from Eq. (1), a value ofcan be determined. When the periodic convolution integral of the cosine profile is calculated, it becomes clear that filtering is easy to accomplish and that the amplitude
Conclusion
Due to the required precision, ISO 16610-21 specifies restrictions for use of the Gaussian filter for roundness profiles with a short measuring length (e.g. cylindrical components with small diameters). These restrictions are assumed by simply applying the filtering process of open profiles to that of closed profiles. But this limitation is not necessary when the periodic Gaussian weighting function is considered. Then the filter can be used for small diameters and small values of the cut-off
References (4)
Geometrical Product Specifications (GPS)—Filtration—Part 21: Linear profile filters: Gaussian filters
(2011)Geometrical Product Specifications (GPS)—Filtration—Part 22: Linear profile filters: Spline filters
(2006)
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