Research On Spline Filtering For Surface Metrology

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.