表面计量学中样条滤波器理论的研究
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摘要
在表面计量学中,表面评定基准线——中线的确定是一个关键问题,众多的表面特征参数与之有关。国际标准ISO11562规定用高斯滤波中线作为表面评定的中线。然而,在实际使用中,高斯滤波器存在严重的边缘效应问题,为剔除边缘效应的影响,人们常常将两边的一段表面数据在滤波后舍弃。损失有效分析数据,这是人们所不希望的。为了解决这个问题,国际标准ISO16610建议将平滑样条滤波器引入表面评定,通过选择边界条件,样条滤波器能够有效地抑制边缘效应。
现在两个问题是必须要解决的。一个问题是平滑样条滤波器虽然已经被写入国际标准,但是迄今为止,在表面计量学中仍然没有给出一套完整的样条滤波器设计理论来指导样条滤波器的使用;另一个问题是样条滤波器虽然抑制了边缘效应,弥补了高斯滤波器的不足,但它与高斯滤波器的滤波特性不一致,因此就出现了两条不同的表面评定基准线,如何统一这两种方法所对应的滤波特性,也就成为表面计量学研究所要解决的理论问题。
本文首先对三次样条滤波器理论进行了研究和总结,在此基础上,利用各种样条函数与Whittaker变分法相结合,设计出了一系列样条滤波器和实现算法,并进行了大量实验证明其正确性和可行性,对样条滤波器的设计和使用具有指导意义。围绕着样条滤波器的理论,主要研究工作如下:
(1)对三次样条滤波器的构造原理进行了详细的研究,从而为设计更多应用于表面计量领域的样条滤波器提供理论基础。在此研究过程中,给出了二分迭代法大大加快了三次样条滤波器的计算速度;同时指出了三次样条滤波器设计复杂、计算量大,且三次样条滤波器没有明确的传递函数,这对滤波器研究极为不便。
(2)为解决样条滤波器与高斯滤波器的滤波结果不一致的问题,提出了基于级联逼近样条滤波器的表面轮廓滤波方法,这是有关样条滤波的中心极限定理。研究了平滑样条滤波器的设计原理和实现算法。在平滑样条函数中加入一阶微分项,构造出逼近样条滤波器。它不但继承了平滑样条滤波器的边缘效应抑制能力和计算的高效性,而且其幅度传输特性与高斯滤波器接近。平滑样条滤波器与高斯滤波器的的幅度传输特性偏差最大为10.6%,而二阶级联后的逼近样条滤波器能将偏差控制在1%以内,逼近精度提高了一个数量级。
(3)提出一种能够分别实现高斯滤波器、平滑样条滤波器和理想滤波器滤波特性的样条滤波器——分数阶B样条滤波器,并将其应用于表面形貌评定。B样条滤波器具有极高的计算效率,但因其采用递归实现算法,所以对边缘效应没有抑制能力。基于分数阶B样条函数构建了分数阶B样条滤波器,它的主要特点是具有可调的过渡带,一方面,通过对阶数的调整,它的传输特性可以接近理想滤波器,另一方面,当阶数选为3时,它可以逼近实现高斯滤波器。另外,当逼近系数为零时,它就退化成了平滑样条滤波器。文中给出了基于FFT变换的算法,使得各阶分数阶样条滤波器都可以方便的实现,且效率高。
(4)研究了以B样条函数为基构造的巴特沃思小波及其特性,提出了利用巴特沃思小波进行表面形貌分析,同时还可以利用它建立工程表面轮廓中线。从表面轮廓滤波器的角度考虑,双正交样条小波和插值样条小波存在各自的缺点。双正交样条小波具有良好的传输特性,但是由于卷积算法的限制,实现速度都比较慢。插值样条小波提升算法计算高效,且能有效抑制边缘效应,但它的幅度传输特性不好,在低通带存在过冲,在过渡带存在转折,扭曲了滤波结果。从幅频特性上,二阶巴特沃思小波具有平滑的截止特性和优良的低通特性;从构造上,巴特沃思小波滤波器具有双正交特性,同样具有零相移特性;从算法上,巴特沃思小波滤波器采用提升和递归结合算法,加快了小波变换速度。综合幅频特性、零相移特性和计算效率,可知巴特沃思小波是一个优秀的可用于表面形貌测量的小波滤波器。
理论分析和应用实验表明:文中的样条滤波器的设计理论和实现方法,丰富了样条滤波器的理论研究成果,并具有重要的应用价值。
In surface evaluation tasks, establishment of reference line——mean line is the key approach, based on which many texture parameters are calculated. International standard ISO11562 has proposed the Gaussian filtered mean line as the reference line for profile evaluation. However, in practice, Gaussian filter has serious end effect. In order to remove this effect, people always discard a part of filtered surface data on both sides. This has resulted in the loss of valid data. To solve this problem, the international standard ISO16610 introduces smoothing spline filter into surface evaluation. With the selection of end condition, smoothing spline filter can restrain the end effect effectively.
Now, there are two problems must be addressed. One is that although the smoothing spline filter has been written to the international standard, in surface metrology, there still is not an integrated design theoretics to guide its application; the other is that the inconsistent filtering characteristics of these two filters lead to two different reference lines, which has brought the surface metrology a new theoretical issue, that is, how to unify their filtering characteristics.
In this dissertation, firstly, the theories of cubic spline filter are analyzed and summarized. Based on this step, a series of spline filters are constructed by combination with various kinds of spline functions and Whittaker variational principle, and their feasibility and validity are testified by a lot of experiments. These works provide a guideline for the design and further application of spline filters. Around the theory of spline filter, the main research subjects are explored as follows:
(1) The theories about cubic spline filter’s construction are studied carefully to provide basic principles for more spline filters’construction. In the course of this study, to increase the computation speed, a dimidiate iterative method is introduced into cubic spline filtering. It’s found that cubic spline filter has complex construction theory and intensive computation, even has not an explicit transfer function that is very convenient for the filter’s study.
(2) In order to solve the divarication of the filtering results between spline filter and Gaussian filter, cascaded approximating spline filter is proposed to evaluate the surface as profile filter, which is the implementation of central limit theory about the spline filter. Design principle and algorithm of smoothing spline filter are studied. By adding a first-order differential term in the smoothing spline function, approximating spline filter is constructed, which not only inherits the smoothing spline filter’s qualities including edge restraint and computaion efficiency, but also approximates to Gaussian filter with high precision. The maximal deviation of transmission characteristics of smoothing spline filter and Gaussian filter is 10.6%, but this deviation between two cascaded approximating spline filter and Gaussian filter can be controlled within 1%, the approximation precision has be improved by an order of magnitude.
(3) A new fractional B spline filter, which can realize the filtering characteristics of Gaussian filter, smoothing spline filter and ideal filter arbitrarily, is proposed and implemented in the surface metrology. B spline filter has a very high computational efficiency, but serious end effect owing to the recursive difference algorithm. The fractional spline filter is constructed based on fractional spline B function, and endowed with adjustability of transition characteristics. When order is chosen higher enough, it can approach to ideal low pass filter, and when chosen 3, it also can approximate to Gaussian filter. In addition, it becomes a smoothing spline filter, when approximation coefficient equals zero. An algorithm based on FFT transform is proposed for the general application of fractional spline filter, which is testified convenient and efficient.
(4) Butterworth wavelet constructed by B spline function and its properties are studied, furthermore it is proposed to implement the analysis of surface texture, and then extract the mean line of workpiece profile. The disadvantages of biorthogonal spline wavelet and cubic interpolating spline wavelet are studied from the perspective of profile filter. Biorthogonal wavelet has both linear phase property, and better transmission characteristics, but a slower calculation speed. On the contrary, interpolating spline filter has a fast calculation and a suppression ability to end effect, but unsatisfactory transmission characteristics, because there is an overshoot on its low pass band, and an inflexion on its transition band. In terms of amplitude characteristics, Butterworth wavelet has smooth cut-off and excellent low pass characteristics; because of construction theory, it has biorthogonal and zero phase-shift properties; for algorithm, it integrates lifting and recursive filtering method, and achieves fast computation. Considering of amplitude transmission characteristics, zero phase-shift property and computational efficiency comprehensively, Butterworth wavelet is believed as an excellent wavelet filter suitable for surface measurement.
Theories and experiments all justify that the design method and implement algorithm of spline filters in this paper not only enrich the spline filter’s theory, but also extend more comprehensive applications.
现在两个问题是必须要解决的。一个问题是平滑样条滤波器虽然已经被写入国际标准,但是迄今为止,在表面计量学中仍然没有给出一套完整的样条滤波器设计理论来指导样条滤波器的使用;另一个问题是样条滤波器虽然抑制了边缘效应,弥补了高斯滤波器的不足,但它与高斯滤波器的滤波特性不一致,因此就出现了两条不同的表面评定基准线,如何统一这两种方法所对应的滤波特性,也就成为表面计量学研究所要解决的理论问题。
本文首先对三次样条滤波器理论进行了研究和总结,在此基础上,利用各种样条函数与Whittaker变分法相结合,设计出了一系列样条滤波器和实现算法,并进行了大量实验证明其正确性和可行性,对样条滤波器的设计和使用具有指导意义。围绕着样条滤波器的理论,主要研究工作如下:
(1)对三次样条滤波器的构造原理进行了详细的研究,从而为设计更多应用于表面计量领域的样条滤波器提供理论基础。在此研究过程中,给出了二分迭代法大大加快了三次样条滤波器的计算速度;同时指出了三次样条滤波器设计复杂、计算量大,且三次样条滤波器没有明确的传递函数,这对滤波器研究极为不便。
(2)为解决样条滤波器与高斯滤波器的滤波结果不一致的问题,提出了基于级联逼近样条滤波器的表面轮廓滤波方法,这是有关样条滤波的中心极限定理。研究了平滑样条滤波器的设计原理和实现算法。在平滑样条函数中加入一阶微分项,构造出逼近样条滤波器。它不但继承了平滑样条滤波器的边缘效应抑制能力和计算的高效性,而且其幅度传输特性与高斯滤波器接近。平滑样条滤波器与高斯滤波器的的幅度传输特性偏差最大为10.6%,而二阶级联后的逼近样条滤波器能将偏差控制在1%以内,逼近精度提高了一个数量级。
(3)提出一种能够分别实现高斯滤波器、平滑样条滤波器和理想滤波器滤波特性的样条滤波器——分数阶B样条滤波器,并将其应用于表面形貌评定。B样条滤波器具有极高的计算效率,但因其采用递归实现算法,所以对边缘效应没有抑制能力。基于分数阶B样条函数构建了分数阶B样条滤波器,它的主要特点是具有可调的过渡带,一方面,通过对阶数的调整,它的传输特性可以接近理想滤波器,另一方面,当阶数选为3时,它可以逼近实现高斯滤波器。另外,当逼近系数为零时,它就退化成了平滑样条滤波器。文中给出了基于FFT变换的算法,使得各阶分数阶样条滤波器都可以方便的实现,且效率高。
(4)研究了以B样条函数为基构造的巴特沃思小波及其特性,提出了利用巴特沃思小波进行表面形貌分析,同时还可以利用它建立工程表面轮廓中线。从表面轮廓滤波器的角度考虑,双正交样条小波和插值样条小波存在各自的缺点。双正交样条小波具有良好的传输特性,但是由于卷积算法的限制,实现速度都比较慢。插值样条小波提升算法计算高效,且能有效抑制边缘效应,但它的幅度传输特性不好,在低通带存在过冲,在过渡带存在转折,扭曲了滤波结果。从幅频特性上,二阶巴特沃思小波具有平滑的截止特性和优良的低通特性;从构造上,巴特沃思小波滤波器具有双正交特性,同样具有零相移特性;从算法上,巴特沃思小波滤波器采用提升和递归结合算法,加快了小波变换速度。综合幅频特性、零相移特性和计算效率,可知巴特沃思小波是一个优秀的可用于表面形貌测量的小波滤波器。
理论分析和应用实验表明:文中的样条滤波器的设计理论和实现方法,丰富了样条滤波器的理论研究成果,并具有重要的应用价值。
In surface evaluation tasks, establishment of reference line——mean line is the key approach, based on which many texture parameters are calculated. International standard ISO11562 has proposed the Gaussian filtered mean line as the reference line for profile evaluation. However, in practice, Gaussian filter has serious end effect. In order to remove this effect, people always discard a part of filtered surface data on both sides. This has resulted in the loss of valid data. To solve this problem, the international standard ISO16610 introduces smoothing spline filter into surface evaluation. With the selection of end condition, smoothing spline filter can restrain the end effect effectively.
Now, there are two problems must be addressed. One is that although the smoothing spline filter has been written to the international standard, in surface metrology, there still is not an integrated design theoretics to guide its application; the other is that the inconsistent filtering characteristics of these two filters lead to two different reference lines, which has brought the surface metrology a new theoretical issue, that is, how to unify their filtering characteristics.
In this dissertation, firstly, the theories of cubic spline filter are analyzed and summarized. Based on this step, a series of spline filters are constructed by combination with various kinds of spline functions and Whittaker variational principle, and their feasibility and validity are testified by a lot of experiments. These works provide a guideline for the design and further application of spline filters. Around the theory of spline filter, the main research subjects are explored as follows:
(1) The theories about cubic spline filter’s construction are studied carefully to provide basic principles for more spline filters’construction. In the course of this study, to increase the computation speed, a dimidiate iterative method is introduced into cubic spline filtering. It’s found that cubic spline filter has complex construction theory and intensive computation, even has not an explicit transfer function that is very convenient for the filter’s study.
(2) In order to solve the divarication of the filtering results between spline filter and Gaussian filter, cascaded approximating spline filter is proposed to evaluate the surface as profile filter, which is the implementation of central limit theory about the spline filter. Design principle and algorithm of smoothing spline filter are studied. By adding a first-order differential term in the smoothing spline function, approximating spline filter is constructed, which not only inherits the smoothing spline filter’s qualities including edge restraint and computaion efficiency, but also approximates to Gaussian filter with high precision. The maximal deviation of transmission characteristics of smoothing spline filter and Gaussian filter is 10.6%, but this deviation between two cascaded approximating spline filter and Gaussian filter can be controlled within 1%, the approximation precision has be improved by an order of magnitude.
(3) A new fractional B spline filter, which can realize the filtering characteristics of Gaussian filter, smoothing spline filter and ideal filter arbitrarily, is proposed and implemented in the surface metrology. B spline filter has a very high computational efficiency, but serious end effect owing to the recursive difference algorithm. The fractional spline filter is constructed based on fractional spline B function, and endowed with adjustability of transition characteristics. When order is chosen higher enough, it can approach to ideal low pass filter, and when chosen 3, it also can approximate to Gaussian filter. In addition, it becomes a smoothing spline filter, when approximation coefficient equals zero. An algorithm based on FFT transform is proposed for the general application of fractional spline filter, which is testified convenient and efficient.
(4) Butterworth wavelet constructed by B spline function and its properties are studied, furthermore it is proposed to implement the analysis of surface texture, and then extract the mean line of workpiece profile. The disadvantages of biorthogonal spline wavelet and cubic interpolating spline wavelet are studied from the perspective of profile filter. Biorthogonal wavelet has both linear phase property, and better transmission characteristics, but a slower calculation speed. On the contrary, interpolating spline filter has a fast calculation and a suppression ability to end effect, but unsatisfactory transmission characteristics, because there is an overshoot on its low pass band, and an inflexion on its transition band. In terms of amplitude characteristics, Butterworth wavelet has smooth cut-off and excellent low pass characteristics; because of construction theory, it has biorthogonal and zero phase-shift properties; for algorithm, it integrates lifting and recursive filtering method, and achieves fast computation. Considering of amplitude transmission characteristics, zero phase-shift property and computational efficiency comprehensively, Butterworth wavelet is believed as an excellent wavelet filter suitable for surface measurement.
Theories and experiments all justify that the design method and implement algorithm of spline filters in this paper not only enrich the spline filter’s theory, but also extend more comprehensive applications.
引文
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