Currently, the approximation methods of the Gaussian filter by some other spline filters have been developed. However, these methods are only suitable for the study of one-dimensional filtering, when these methods are used for three-dimensional filtering, it is found that a rounding error and quantization error would be passed to the next in every part. In this paper, a new and high-precision implementation approach for Gaussian filter is described, which is suitable for three-dimensional reference filtering. Based on the theory of generalized B-spline function and the variational principle, the transmission characteristics of a digital filter can be changed through the sensitivity of the parameters(t1, t2), and which can also reduce the rounding error and quantization error by the filter in a parallel form instead of the cascade form. Finally, the approximation filter of Gaussian filter is obtained. In order to verify the feasibility of the new algorithm, the reference extraction of the conventional methods are also used and compared. The experiments are conducted on the measured optical surface, and the results show that the total calculation by the new algorithm only requires 0.07 s for 480′480 data points; the amplitude deviation between the reference of the parallel form filter and the Gaussian filter is smaller; the new method is closer to the characteristic of the Gaussian filter through the analysis of three-dimensional roughness parameters, comparing with the cascade generalized B-spline approximating Gaussian. So the new algorithm is also efficient and accurate for the implementation of Gaussian filter in the application of surface roughness measurement.
光学元件表面面形误差属于非平稳空间信号,为了在分离光学元件表面各频段面形误差的同时尽可能保留原始信号各频段的细节特征,结合超精密抛光的球面光学元件表面特点,提出一种基于双树复小波变换(dual tree complex wavelet transform,简称DT-CWT)的自适应分离法。利用DT-CWT的多分辨分析、方向性好和良好的时频局部化分析能力等特点,对实测的抛光光学元件表面进行DT-CWT的多尺度分解,并在重构时加入自适应影响因子,成功分离了各频段的面形误差。通过对经DT-CWT直接分离法与DT-CWT自适应分离法得到的高、中、低频面形误差进行参数表征,实验证明基于DT-CWT自适应分离法更为有效地分离光学元件各个频段的面形误差及误差特征,便于后续的识别与评定工作。
