## 2 Transmission Characteristics  
In order to reduce the difference of transmission characteristics between spline filter and Gaussian filter, the international standard ISO 16610-22 also recommends the approximating spline filter with a tension parameter \(\beta\) = 0.62524. As shown in Figure 1, the approximating spline filter in the standard is closer to the transmission characteristics of Gaussian filter compared with the cubic spline filter. The maximum deviation is 4.35%, however, the deviation of 4.35% cannot meet all the engineering requirements. According to [26], the spline function consists of two parts. One of them is the residual norm, which ensures that the result is close to the evaluation of profile. The other part is bending energy, which controls the smoothness of the spline curve. Lagrange parameters control the tradeoff between the curve fitting and bending. According to the principle of the variational method [27], adding the first derivative term to the bending energy can improve the transmission characteristics of a spatial frequency function. This method can be used to approximate the transmission characteristics to those of the Gaussian filter. The approximating spline function is based on the cubic spline function and adds the first derivative term to the bending energy. The approximating coefficient is introducedto provide better control of the approximating degree of the transmission characteristics. The filtering characteristic of the approximating spline can be expressed as follows. \[ \begin{array}{c}G(\omega)=\frac{1}{1+2\tau\mu(1-\cos\omega)+4\mu(1-\cos\omega)^ {2}}\\ \\ \mu={(\sqrt{5}-1)}^{2}/64\sin^{4}(\pi\Delta x/\lambda_{c})\end{array}  \tag{1}\]  
where \(\tau\) denotes the approximating coefficient, and \(\mu\) denotes the Lagrangian parameter, and \(\Delta x\) denotes the sampling interval, and \(\lambda_{c}\) denotes the cut-off wavelength. As shown in Figure 2a, the transmission characteristics of the approximating spline and Gaussian functions is enumerated when \(\tau=1/\sqrt{\mu}\), \(\tau=2/\sqrt{\mu}\), or \(\tau=3/\sqrt{\mu}\). From Figure 2b, it can be seen that the transmission characteristics of the approximating spline filter are the closest to those of the Gaussian filter when \(\tau=1/\sqrt{\mu}\)[13]. The maximum transmission characteristic deviation is 4.263%. This value is in line with the deviation range (\(-5.0\)%, \(5.0\)%) of the simplified Gaussian filtering algorithm specified in the ISO 11562 [28]. Compared with the approximating spline recommended by international standard, the improved approximating spline is closer to the transmission characteristics of Gaussian filter as shown in Figure 3.  
Figure 1: (**a**) Transmission characteristics and (**b**) transmission characteristic deviation of the Gaussian filter, and cubic spline filter, and std-approximating spline filter. Figure 2: (**a**) Transmission characteristics and (**b**) transmission characteristic deviation of the approximating spline filter (\(\tau=1/\sqrt{\mu}\), \(2/\sqrt{\mu}\), and \(3/\sqrt{\mu}\)), and Gaussian filter. In order to obtain higher approximation accuracy, many approximating spline filter are connected in series. The transmission characteristic of the cascade approximating spline filter can be expressed as  
\[ \begin{split} G(\omega)=\frac{1}{\left(1+2\pi\mu\left(1-\cos\omega \right)+4\mu\left(1-\cos\omega\right)^{2}\right)^{n}}\\ \mu=\left(\sqrt{4\sqrt[n]{2}-3}-1\right)^{2}/64\sin^{4}(\pi \Delta x/\lambda_{c})\end{split}  \tag{2}\]  
where \(n\) denotes the cascade order. The transmission characteristic deviation for different cascade orders is presented in Figure 4, where it can be seen that with the increase in the cascade order, the approximation degree also increases. However, the improvement in the approximation accuracy results in higher calculation cost. Therefore, considering the effectiveness, the cascade order should be controlled in a reasonable range.