## 2 Target Filters  
Currently, no robust SF-based filter is established in ISO because ISO/TS 16610-32 was rejected in the Technical Specifications stage. Furthermore, the new ISO 16610-32 was being redeveloped from the first stage, and its filter specifications are not yet provided. Due to these circumstances, the filter specifications of the new ISO 16610-32 are unknown. Therefore, these filters are not verified in this paper. The RSFs that are verified in this paper are the L2-norm type regression RSF proposed by [PERSON] et al. [16] (hereinafter referred to as RL2-RSF: Regression-type L2-norm Robust Spline Filter) and a re-quantization type RSF using the fast M-estimation method proposed by [PERSON] et al. [17] (hereinafter referred to as FMSF: Fast M-estimation Spline Filter). The RL2-RSF filter has periodic and aperiodic types as a normal SF and calculates the solution of an inverse matrix similar to that of a normal SF. The RL2-RSF realizes robustness by regression calculation which changes weights based on the deviation between the primary profile and the mean line of each repetition, the same as ISO 16610-31. In the RL2-RSF, weights are defined by:  
\[\omega_{k}^{(m)}=\left\{ \begin{bmatrix}1-\left(\frac{z_{k}-s_{k}^{(m)}}{\beta c }\right)^{2}\\ 0\end{bmatrix} ^{2}&\left|z_{k}-s_{k}^{(m)}\right|<\beta c\\ \left|z_{k}-s_{k}^{(m)}\right|<\beta c\end{bmatrix}\right. \tag{1}\]  
where \(z_{k}\) is the input value, \(s_{k}^{(m)}\) is the output value in the \(m\)-th iteration step, \(\sigma\) is the standard deviation of residuals \(z_{k}-s_{k}^{(m)}\), and \(c\) is the parameter that should be set at the appropriate constant value depending on target data characteristics. The RL2-RSF continues the regression calculation until the median deviation between the primary profile and the mean line of each repetition is below the set repetition threshold. Therefore, the results of RL2-RSF vary depending on the value of the repetition threshold. The FMSF is the re-quantization type RSF using the fast M-estimation method [18; 19]. The re-quantization type filter executes convolution with the primary profile and the weight function. For example, the FMSF uses the following weight function. \[s(x)=\frac{\pi}{\lambda c}\sin\Bigl{(}\frac{\pi}{\lambda c}|x|+\frac{\pi}{4} \Bigr{)}\exp\Bigl{(}-\sqrt{2}\frac{\pi}{\lambda c}|x|\Bigr{)} \tag{2}\]  
where \(\lambda c\) is the cutoff wavelength. The FMSF uses the characteristics of the fast M-estimation method to realize robustness only to outliers. When outliers are not included in the primary profile, the output of the FMSF is equal to the output of the normal SF with high accuracy. This characteristic is rare in robust filters. The FMSF uses a convolution method, unlike normal SF, because of using the fast M-estimation method. The convolution method causes the end-effect problem at both ends of the data [20]. However, the FMSF solves the end effect problem by using point symmetric expansion with the calculated optimal reference point [21]. Furthermore, the filter width is equivalent to infinity in calculating the solution of the inverse matrix, such as normal SF; however, because the filter width is finite in the convolutional method such as FMSF, deviation due to the filter width occurs. However, FMSF realizes a characteristic that matches the output of normal SF with high precision when no outlier is included by changing the filter width of the FMSF to be wider than the \(\lambda\)c used in the normal convolution method. The calculation method of the FMSF is unique compared to other robust filters. Therefore, the calculation method of the FMSF is shown below. First, create the 2-dimensional array (measurement \(x\)-direction and height \(z\)-direction). Second, re-quantize in the height \(z\)-direction and vote the weight to the cell corresponding to the height value. Then, the convolution calculation is executed along the measurement \(x\)-direction using the voted weights and the weight function of SF.