###### Abstract
At present, there are many methods to solve matrix equation, including the direct method [17] and iterative method in numerical analysis. The iterative method generally requires enough number of iterations to achieve the desired accuracy, and it can also sometimes diverge. The direct rule is an effective method to solve large matrix equations (e.g., large banded equations) [18]. The exact solution of spline matrix can be obtained by a finite step operation. Common direct methods include LU decomposition (Doolittle) method [19], LDLT method [20], and Thomas method [21]. LU decomposition method is an improvement of Gauss elimination method, and its essence is to decompose the coefficient matrix into the product of a unit lower triangular matrix and a unit upper triangular matrix. The LDLT method is to decompose the matrix into the unit lower triangular matrix (i.e., all the elements of the main diagonal are 1), the transpose matrix of the lower triangular matrix, and the diagonal matrix (Only the main diagonal of the matrix has elements, and the rest are zeroes). Thomas method is a formula derivation of band preserving decomposition for banded matrix (especially tridiagonal matrix), which is essentially a kind of LU decomposition. In this work, the direct method is applied to approximating spline filter to compare their performance in the profile and areal filtering. Under the comprehensive action of many factors, including different processing technology and material properties, the surface of parts can produce various shapes and sizes of microscopic geometry, such as scratches, grooves, and other outliers. These outliers do not follow the strict normal distribution strictly, and the interference of outliers can cause the distortion of the filtered mean line. Thus, the authenticity of filtered data is affected. In addition, data distortion caused by surface defect and instrument limitation often appears in the process of surface measurement. These distorted data form outliers on the surface topography, for example, missing data will appear as a blank area on the surface topography. The existence of outliers also affects filtering results. Therefore, robust spline filters have been widely studied. For instance, in 2005, [PERSON] et al. proposed a robust algorithm based on the L2-norm, which has good robustness to outliers [22]. In 2013, [PERSON] et al. proposed a new robust spline profile filter based on Tukey function [23]. Compared with the robust spline filter specified by international standards, this filter has not only better robustness, but also higher computational efficiency. The robust Gaussian regression filter and robust spline filter have also been included in international standards [24; 25]. However, the computational cost of the robust Gaussian regression filter is high. Moreover, the precision of robust spline filter is low and there is no corresponding international standard of three-dimensional. More importantly, the filtering benchmark given by the international standard for robust splines is not satisfactory in all the cases. Therefore, it is necessary to design a robust filter that can not only maintain the precision of the filter but also improve the computational efficiency to replace the standard robust spline filter. At the same time, the difference between the robust Gaussian filter and the robust Gaussian filter should be reduced to obtain a better comparison of measurement results. In this work, the transmission characteristic deviation between the approximating spline and the Gaussian filter is reduced by cascade approximating, and the transmission characteristic deviation value corresponding to a different approximation order is determined, thus providing reference data for approximation order selection. Also, the cascade approximating method also realizes the isotropic areal spline filter. Besides, direct methods for solving an approximate spline matrix are derived and compared in terms of computational efficiency and accuracy, and relevant suggestions are provided.