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

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.     

一种新型稳健样条轮廓滤波器

在表面计量学中,通常利用轮廓滤波器提取轮廓中线作为表面评定的基准线。由于传统轮廓滤波器极易受到划痕和深谷等奇异信号影响而产生失真甚至错误的中线,为了增强其稳健性并保证评定精度,国际标准中给出了两种替代滤波器,即稳健型样条滤波器和高斯回归滤波器。然而,这两种滤波器的构造原理和实现算法迥异,其滤波特性差异较大,从而导致对同一表面具有不同的评定结果。为解决这一问题,提出将M估计权函数与样条滤波器相结合构造了一种新型稳健样条滤波器。该滤波器不但具有更强的抗差能力,其计算效率也远高于国际标准推荐的稳健滤波器。更重要的是,大量实验表明,其滤波特性与高斯回归滤波器接近,该滤波器的应用有助于表面评定标准的传递和测量结果比对。     

High-order spline filter and ideal low-pass filter at the limit of its order

We propose a method of obtaining a more vertical step-edge-like cut-off characteristic of a low-pass filter, in which a mean line for the roughness profile is extracted and the computation time is significantly reduced. In order to make the cut-off characteristic more step-edge-like, attention is focused on the relationship between the spline filter order and mean line smoothness, and the order is made larger. This filter, i.e., the high-order spline filter, has extremely low end effects because it is an extension of the spline filter. Raising the order of the high-order spline filter to ∞ can also realize an ideal low-pass filter that has a step-edge-shaped cut-off characteristic. In addition, in order to reduce computation time, the high-order spline filter is calculated in the frequency domain. When several thousand data points are used, the spline filter requires a maximum of 1 h, whereas using the proposed high-order spline filter allows computation to be completed in less than 1 s.     

Filter with variable transmission characteristics for determination of three-dimensional roughness

In this study, we propose a low-pass filter whose transmission characteristics are determined to be more than 99.99% identical to the designed transmission characteristics at wavelengths greater than the cut-off wavelength. This low-pass filter is realized using a frequency-domain method which uses down-sampling, a frequency filter, and a cubic B-spline. When the frequency characteristics of a Gaussian filter (GF) were used as the designed transmission characteristics, this low-pass filter perfectly replaced the GF. In addition, no end effects and directional characteristics developed, and high-speed computation was possible. Furthermore, we can easily realize a low-pass filter having transmission characteristics that rise sharply at a right angle like a step edge.     

A universal spline filter for surface metrology

In this study, a novel fractional spline filter is proposed for surface metrology. The filter combines arbitrary order fractional spline function with variational principle, and is endowed with an adjustability of transition characteristic. If the order is high enough, the fractional spline filter can obtain more step-edge-like cut-off characteristic of the low-pass filter, and it can also realize the transmission characteristic of the Gaussian filter with high accuracy if the order equals to two. In addition, if the approximation parameter is set to zero, this filter will be simplified to a form of ordinary spline filter. Therefore, this fractional spline filter is called the universal spline filter. In order to reduce the computation time, the universal spline filter is calculated by a fast Fourier transform (FFT) algorithm. Furthermore, for practical use, an improved method prior to the FFT algorithm is proposed to suppress the end effect. Finally, the efficiency of the whole algorithm in comparison with the ordinary spline filter is demonstrated by the experiments, and it is also shown that the universal spline filter is able to alleviate the end effect significantly.     

High-order spline filter: Design and application to surface metrology

The differences between the transmission characteristics of the standard cubic spline filter and the Gaussian filter lead to different evaluation results even for the same profile. There is indeed an adverse impact on the comparison of measurement results and the applications of the related international standards. A novel high-order spline filter is proposed to resolve this practical problem of approximating the Gaussian filtering characteristic. The design of the new filter is based on an improved variational approach by adding the high order derivative terms to the bending energy part whose structural parameters are determined by the aid of the universal Taylor series, so as to realize the convergence to the function of the Gaussian filter. In addition, a cascade algorithm in terms of the low-order filters is also developed in order to ensure stable performance of the high-order filter. Its effectiveness and application were verified by the experiments.     

Periodic Gaussian filter according to ISO 16610-21 for closed profiles

It is well known that the Gaussian filter according to ISO 16610, part 21 has nice metrological properties but also has some restrictions. The evaluable measuring length of open profiles is shortened by so-called filter running-in and running-out lengths which are a factor of the width of the Gaussian weighting function. ISO 16610-21 suggests a simplistic transfer of filtering of open profiles to filtering of closed profiles (e.g. roundness measurements). And this leads, due to the running-in and running-out lengths, 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. Hence, as this paper shows, the measuring length for closed profiles can be arbitrary for any given width of the Gaussian weighting function.     

The effects of cutter eccentricity on the cutting force in the ball-end finish milling

Ball-end mill are usually used for finish machining with multi-axis CNC milling centers. Cutter eccentricity (offset magnitude e _c, angular location of the offset vector λ _c) is very important when predicting cutting forces accurately. In the process of cutting force modeling, not only the e _c, but also the λ _c, affects the cutting forces by associating with the rotation angle, the chip thickness, and the entry and exit angles of the cutting forces. The e _c affects the values of the cutting forces, while λ _c causes the delay phenomenon of the cutting forces mainly. Experiment results show that the cutting forces, considering the cutter eccentricity, more especially considering the influence of the λ _c, agree better with the measured cutting forces than the cutting forces without considering the cutter eccentricity.     

巴特沃思小波在表面形貌信号分离中的应用

为了快速准确地分析工程表面,提出了一种基于巴特沃思小波的滤波方法。首先,讨论了小波滤波器的低通幅度传输特性,并以此作为选择适合表面形貌分析小波滤波器的依据。然后,介绍了巴特沃思小波滤波器的构造原理,并给出了它的快速实现算法;综合其优良的传输特性和高效的实现算法,以巴特沃思小波作为表面分析的滤波器,对实际表面轮廓进行了多层分解。最后,利用巴特沃思小波建立了表面轮廓评定的基准线,并给出了确定小波分解层数的方法。实验结果表明,巴特沃思小波能够快速准确地实现表面轮廓的多尺度分析,稳定可靠地提取表面中线;在普通计算机上提取11200点数据表面中线仅耗时60ms,利用该中线计算所得R值相对误差仅比利用高斯中线所得R值相对误差大0.12%。     

An in-line digital filtering algorithm for surface roughness profiles

The authors develop a fast algorithm for low- and high-pass Gaussian digital filters, executed while one discrete datum of a traced profile is sampled. In this report, the 8-step, modified running–box-function filter is presented. The transmission characteristics of this new filter are very close to those of the Gaussian digital filter given in ISO 468-1982. In addition, the conditions for using this filter are given for practical use.     

对金刚石刀具刀尖圆弧波纹度评价的滤波分析

为准确评价超精密金刚石刀具刀尖圆弧波纹度,对金刚石刀具刀尖圆弧波纹度的评价进行了滤波分析。介绍了刀尖圆弧波纹度的评价方法,针对关键的滤波环节,分析了高斯滤波、高斯回归滤波、样条滤波和小波滤波的原理,根据实际采样数据设计模拟实验对比了上述4种滤波器的滤波特性。结果表明,对于不同轮廓中线但高频分量完全相同的轮廓,样条滤波器的滤波效果优于高斯滤波器,并通过实际应用对比了高斯滤波器和样条滤波器。实验结果表明:对具有不同轮廓中线但高频分量完全相同的模拟轮廓,分别采用高斯滤波器和样条滤波器,当截止波长分别取0.25mm和0.8mm时,高斯滤波器滤波结果的标准差分别为0.4366nm和9.6544nm,样条滤波器滤波结果的标准差分别为0.0271nm和1.1844nm,说明样条滤波器更适合于超精密金刚石刀具刀尖圆弧波纹度的评价。     

广义B样条滤波器在表面形貌测量中的应用

摘 要：为了提取工件表面轮廓中线,给出一种基于三阶广义B样条函数的数字光滑滤波器。首先由线性方程微分方程推导出广义B样条函数。接着,利用广义B样条函数和变分公式结合推导出数字平滑滤波器,该数字平滑滤波器由一IIR滤波器和间接广义B样条变换组成。然后,给出了数字平滑滤波器的具体的递归算法,并且证明了该滤波器的通用特性。最后提取了一实际工件表面中线。实验结果表明：该滤波器具有比高斯滤波器更好的截止特性,该算法利用普通计算机提取11200点数据表面中线仅耗时0.002秒。该滤波器具有可调平滑性,零相移特性,克服了传统三阶样条插值的曲率低和振荡的缺陷,其递归滤波器的具体算法,算法简洁,效率高。     

表面粗糙度测量中的稳健样条滤波算法

研究了表面异常信号对国际标准样条滤波器的影响.引入稳健统计学思想对样条滤波器进行了稳健处理,增强样条滤波的稳健性.利用该算法对实际表面进行测量,其结果表明:能够可靠地获得评定中线,实现表面信息的无失真提取和分析,与原有国际标准样条滤波器相比,稳健性更强.     

随动磨削曲轴表面异常轮廓误差去除方法

在高精度随动曲轴磨床中,为控制曲轴加工轮廓误差常采用预补偿的方法,如果测量时工件上有磨屑、毛刺等干扰将在测量结果中引入明显的异常轮廓,采用传统的高斯滤波器对数据进行处理将极大地影响轮廓误差的补偿精度,甚至导致废品出现。针对这一问题,给出了适用于闭轮廓的高斯滤波器、Rk滤波器和鲁棒高斯回归滤波器的理论模型。分别应用3种滤波器,对比分析结果可知,鲁棒高斯回归滤波器去除异常轮廓误差效果最理想,并通过人为改变异常轮廓的尺度,进一步验证鲁棒高斯回归滤波器的适应性和可靠性。该滤波方法集成到随动曲轴磨床软件中,实现异常轮廓的自动去除,提高了补偿效率,有效保证了曲轴磨削轮廓的误差精度。     

Energy resolution of an Omega-type monochromator and imaging properties of the MANDOLINE filter

We report on the energy resolution properties of an Omega-type monochromator. In a TEM/STEM setup with a MANDOLINE filter and extreme stability of the high voltage and the filter current, an energy resolution of 43 meV for 0.1 s exposure time and 87 meV for 100 s exposure time was measured at 200 kV with 40 meV monochromator slit width. The monochromized zero-loss peaks are additionally characterized by their edge steepness. Moreover a drop in the monochromized zero-loss peak by 10³ after 260 meV can be obtained even without deconvolution. For small fields of view, the energy resolution mostly does not depend on the MANDOLINE filter. With the Corrected OMEGA filter an energy resolution of 41 meV was measured for 0.03 s exposure time at 200 kV with 30 meV monochromator slit width and 77 meV for 50 s exposure time at 80 kV with 40 meV monochromator slit width. Furthermore, the MANDOLINE filter’s setup and imaging properties are presented such as isochromaticity (<5 meV) and transmissivity (T_(1 eV)=17,400 nm²), which set a new standard for imaging energy filters and allow EFTEM spectrum imaging with energy windows ≤200 meV and reasonable fields of view.     

A two-dimensional isotropic spline filter

The work deals with the development of a two-dimensional isotropic spline filter with application to the separation of components of the surface topography, i.e. roughness, waviness and form. An appropriate variational problem was formulated to define the mapping of primary surface data into filtered surface data with a transfer function of a circularly symmetric low-pass Butterworth filter of a given order. The variational problem was solved by approximating the filtered surface data by means of two-dimensional B-spline functions. The Fourier transform of the filter impulse response was determined to estimate the quality of approximation of the Butterworth filter and to select the appropriate filter cutoff. The paper presents one application of the designed filter to determine 3D roughness of an inner ring race of a ball bearing.     

Design of UWB Band-pass Filters with GPS Band Rejection

This paper presents a compact Ultra-Wideband(UWB)band-pass filter using a high-pass filter and a low-pass one,and the resonator with lumped elements.The structure of our proposed band-pass filter is very simple and the Defected Ground Structure(DGS)structure is used to get the low-pass filter characteristics.This proposed band-pass filter can be much smaller than a cascaded type filter.As a result of simulation,the insertion loss is less than 0.3 dB throughout the pass-band of 2.2 GHz～10.6 GHz,while the return loss is more than 18 dB.And it has rejection level of 36 dB at GPS band.     

Research on error averaging effect of hydrostatic guideways

Hydrostatic guideways can obtain high motion accuracy due to their error averaging ability. In order to optimize the structural and accuracy design of hydrostatic guideways, the mechanism and affecting factors of the error averaging effect are investigated in detail in this paper. A new method is proposed to quantitatively analyze the motion errors of a typical closed hydrostatic guideway with four pads, and the restrictor is also taken into consideration. A hydrostatic guideway in an ultra-precision grinding machine LGF750 is analyzed with the method. The results show that the wavelength of each profile error component is a main affecting factor on the error averaging effect. Fluid films usually show obvious error averaging effect on each profile error component when the wavelength λ is shorter than twice of the pad length 2l_u. However, for each component with the wavelength λ over 2l_u, the error averaging effect becomes obvious only when the ratio of the pad length to the evaluation length of motion straightness l_u/l_e is greater than 0.5. Furthermore, the instability of the oil supply system is taken into consideration, and possible influences on the error averaging effect due to the supply pressure fluctuation are investigated preliminarily.     

A new concept of harmonic current detection for shunt active power filters control

The paper deals with measurement and control aspects of active power filtering in ship systems. The main focus of the paper is concentrated on improvement of harmonic current measurement algorithms based on i_p–i_q method with application of low-pass filter (LPF) module and, alternatively, on i_p–i_q method with application of mean value module.     

Edge burr development when using a cemented carbide micro-drill: Formation and control mechanisms

To explore the mechanism of formation and methods of control of edge burr in the grinding and forming of cemented carbide micro-drill, the morphology, material composition, and the edge structure of the generated area of the edge burrs are studied. Through indentation and scratch experiments, the critical grinding depth (h_c) of grinding machining is calculated to be 0.793–1.052 μm. The grinding experiments have verified the effects of the actual grinding depth h₀ of different single abrasive particles on the edge burr and the effectiveness of the method of controlling the burr by increasing the cutting angle of the abrasive particles. The experimental results show that edge burrs are mainly concentrated on the cutting edge close to the outer cylinder of the micro-drill. When the actual grinding depth h₀ of a single abrasive particle is less than the critical grinding depth h_c, workpiece material is mainly subjected to plastic deformation removal, the length L of the edge burr along the edge direction is 10–20 μm, the width W of the burr perpendicular to the edge direction is 1–3 μm. The formation of edge burrs is mainly related to the actual grinding depth and the abrasive grain cutting angle γ of abrasive grains. With the increase of h₀, the length L of the edge burr decreases, and the width W thereto first increases, then decreases. Increasing the cutting angle of abrasive particles can control the edge burr. By changing the grinding direction of the grinding wheel, the cutting angle of the abrasive particles can be changed from an acute angle to an obtuse angle, thereby eliminating the edge burr without affecting the performance of micro-drilling.