###### Abstract  
Isotropy is an important feature of an area filter in the three-dimensional surface roughness evaluation. First, the transmission characteristic deviation between the approximating spline filter and the Gaussian filter is reduced by cascading approximating. Second, the approximating spline filter is superimposed on the orthogonal direction to obtain an isotropic areal filter. Then, four direct methods for the solving approximating spline matrix are applied. Based on the profile filtering and areal filtering, the computational efficiency and accuracy are compared. The experimental results show that the improved square root method (LDLT decomposition) combines both computational efficiency and filtering precision, and is a good choice for solving the approximating spline matrix. Finally, six kinds of robust approximating spline filters are constructed. Taking the output value of robust Gaussian regression filter (RGRF) as reference, and the honing profile and step surface with deep valley characteristics were used as test surfaces to compare their robustness and iteration time. The experimental results show that the approximating spline filter based on the ADRF function has the shortest iteration times, while its roughness is close to the robust Gaussian regression filter. surface roughness; approximating spline; cascade approximating; isotropy; matrix algorithm; robustness Article  
[PERSON]; [PERSON]; [PERSON]; [PERSON] Study on the Performances of an Approximating Spline Filter Based on the ADRF Function in Surface Roughness Evaluation  
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the advantage of suppressing the end effect, there is a great difference in the transmission characteristics between the spline filter and the Gaussian filter. In other words, when these filters are applied to the same surface, their surface metrological evaluation results cannot be compared. Moreover, the areal spline filter formed by superposition has anisotropic amplitude frequency characteristics in the orthogonal direction [10], which limits its wide application in the three-dimensional surface topography. Therefore, many studies on the isotropic areal spline filters have been conducted. For instance, in 2008, [PERSON] et al. proposed a new frequency-domain areal filtering method, which can obtain transmission characteristics of an arbitrary amplitude [11]. In 2015, [PERSON] et al. designed a high-order spline profile filter and constructed a spline areal filter with isotropic amplitude characteristics [12]. In 1989, [PERSON] proposed the approximating spline theory model in order to approximate the transmission characteristics of the Gaussian filters with high accuracy [13]. The difference of transmission characteristics of spline filter and Gaussian filter can be further reduced by cascading approximating. Approximating spline is a further improvement of cubic spline by adding the first derivative to the bending energy of cubic spline. The cascade approximating takes the single step of approximating spline filter as the basic process prototype, and then uses approximating spline filter in series, which is essentially to achieve further approximation of Gaussian transmission characteristics through iteration. In 2009, [PERSON] et al. further studied the approximating spline theory and realized cascade approximating spline filter [14]. [PERSON] first applied the numerical solution method to the spline function matrix, which greatly improved the computational efficiency of spline filters [15]. [PERSON] et al. applied Gaussian elimination to the cubic spline matrix [16]. Similar to the spline filters, the approximating splines also need to be converted to the matrix form. Therefore, matrix decomposition efficiency and numerical stability are important factors to be considered in the surface roughness evaluation.