Low-pass filter without the end effect for estimating transmission characteristics—Simultaneous attaining of the end effect problem and guarantee of the transmission characteristics

Abstract

A Gaussian filter (GF) is the most commonly used low-pass filter of measuring surface roughness. However, undesirable distortions, called “end effects," occur near the end points of the data ends in GF. The transmission characteristics are one of the most important indicators that show the quality of a filter. Previously, it was only possible to obtain theoretical values for the transmission characteristics of filters whose weights were given by an explicit function. In recent years, it has also become possible to obtain the transmission characteristics of a filter whose weights are given by an implicit function. However, this method has a problem in that the values near the end points of the measurement data become significantly different from each other. The consequence is that end effects may occur in the filter outputs due to side effects of the periodic extension. In the case of a spline filter (SF) applied to open profiles, the transmission characteristics of a periodic SF with an end effect can be obtained uniquely. However, the transmission characteristics of a nonperiodic SF, which has no end effect, cannot be uniquely obtained. This results in a trade-off between the two states: end effects exist in a filter whose transmission characteristics can be obtained, and the transmission characteristics of a filter without end effects cannot be uniquely obtained. To address this problem, we propose a method for the GF processing that uses shearing, point symmetric extension, and periodic extension, and produces no end effect but allows the transmission characteristics to be obtained. Previously, there was a problem with the proposed method in that the rationale was unclear regarding how to determine the reference points for point symmetric extension. We resolved this and optimized the reference points. As a result, the proposed method was shown to be successful in not only resolving the end effects in the GF, but also obtaining the transmission characteristics.