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

• A high-order spline filter based on an improved variational principle is proposed.
• Its structural parameters are determined by the aid of the universal Taylor series.
• The greater the order of this spline filter, the better the approximation of the Gaussian filter.
• A cascade algorithm in terms of the low-order filters performs stably and efficiently.

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.