Fast Calculation Method of A Robust Spline Filter by Using Frequency-Domain Method

Abstract

The spline filter (SF) used the inverse matrix can measure the roughness without wasting the data at both ends of the input data. But when input data include outliers, the output of the SF greatly fluctuates. Therefore, “Fast M-estimation Spline filter (FMSF)” was proposed. In order to apply the SF to the fast M-estimation method, convolution-type SF is used instead of inverse matrix-type SF. FMSF performs robustly to input data including outliers, and gives the same output as that of convolution-type SF to input data without outliers. However, the convolution-type SF is made from an approximate formula. It needs a larger filter to improve accuracy of SF. This require a huge computational cost for convolution. There is a trade-off between approximation error and computational cost. To solve these problems, we propose “frequency-domain-type FMSF.” This allows the use of the SF transmission characteristics function instead of the convolution-type SF. The SF transmission characteristics function has no approximation error and exactly matches the SF output. Furthermore, the FMSF requires O(N²) computational cost. However, by using Chirp-Z transform to discrete Fourier transform, the proposed method can reduce the computational cost to O(N logN).