## 4 Robust Treatment  
The reliability of reference data is an important index to characterize surface roughness. When the surface contains outliers, such as scratches, grooves, and missing surface data, filter reference is distorted. The robustness estimation weight function is introduced into the filter to enhance the robustness against the abnormal signals. Also, different weight functions may produce completely different results when processing the same measured signal, thus increasing the difference in surface evaluation between different filters. In 2010, the robust Gaussian regression filter was included in the ISO 16610-31. Also, it has been shown that the combination of the Tukey estimation function and the Gaussian regression filter can improve the robustness of the Gaussian regression filter [18]. However, it increases the time and calculation costs. In the same year, the ISO 16610-32 stipulated a robust spline filter with \(p=1\)(the L1-norm based Robust Spline Filter). Based on the idea of a variational robust spline filter, a compromise function between the sum of the absolute values of the resists and the approximate bending energy of the data has been constructed. However, this function requires many cyclic iterative operations to be solved. It has been proved that the standard robust spline filter cannot achieve satisfactory results in the surface evaluation [23]. Therefore, it is necessary to design a robust filter, which not only has high filtering accuracy and computational efficiency, but also can reduce the evaluation difference between robust spline filter and robust Gaussian regression filter, so as to realize the unification of the two evaluation standards. In this study, the approximating spline is used as a base filter, and several typical weight functions are selected for comparison. The matrix definition of the robust approximating spline is as follows:  
\[(\rho+\tau\mu P+\mu Q)w=\rho z, \tag{11}\]  
where \(\rho\) denotes a robust weight function, and \(\tau\) denotes the approximating coefficient, and \(\mu\) denotes the Lagrangian parameter. \(P\) and \(Q\) are matrices related to boundary conditions. First, the filter results are obtained by using the approximating spline filter, and then the results are substituted into the weight function to obtain the weights of each point. Then, the calculated weights are assigned to the points with small residual errors, and small weights are assigned to the points with large residual errors. Finally, the weights of each point are substituted into the robust approximating spline, and the robust filter values are iteratively calculated. According to M-estimation, different robust weight functions may lead to diametrically different results when one and the same signal observed is processed. An impertinent choice of weight functions can sometimes make the estimating method invalid. In practice, it is rather difficult to grasp the exact distribution of the measured signal. To make approximating spline filter adaptable to widely practical application, several typical weight functions are selected and compared with each other [32]. (1) WLAV  
\[\rho(v)=\frac{1}{v}, \tag{12}\]  
where \(v=\tau/cMAD\), \(MAD=|\operatorname{med}(r)|\). (2) Tukey  
\[\rho(v)=\left\{ \begin{array}{ll}\left(\left(1-v^{2}\right)^{2}\right),&|v| \leq 1\\ 0,&|v|>1\end{array} \right., \tag{13}\]  
where \(v=\tau/cMAD\), and \(\operatorname{c}\) denotes the modulation constant used to balance robustness and efficiency; \(c=4.44\) is adopted as an international standard. (3) ADRF\[\rho(v)=\left\{ \begin{array}{ll}1,&|v|\leq a\\ a/|v|,&|v|\in(a,b]\\ b/v^{2},&|v|\in(b,c]\\ 0,&|v|>c\end{array} \right., \tag{14}\]  
where \(v=r\)/s, \(s\) denotes a scale parameter, and \(a\), \(b\), and \(c\) denote the fine-tuning parameters. (4) [PERSON]  
\[\rho(v)=\left\{ \begin{array}{ll}\frac{1}{\pi v}\sin(\pi v),&|v|\leq 1\\ 0,&|v|>1\end{array} \right., \tag{15}\]  
(5) Hampel  
\[\rho(v)=\left\{