Ismail A R, Asfour S S
Department of Industrial Engineering, University of Miami, Coral Gables, FL 33124-0623, USA.
J Biomech. 1999 Mar;32(3):317-21. doi: 10.1016/s0021-9290(98)00171-7.
Motion analysis systems typically introduce noise to the displacement data recorded. Butterworth digital filters have been used to smooth the displacement data in order to obtain smoothed velocities and accelerations. However, this technique does not yield satisfactory results, especially when dealing with complex kinematic motions that occupy the low- and high-frequency bands. The use of the discrete wavelet transform, as an alternative to digital filters, is presented in this paper. The transform passes the original signal through two complementary low- and high-pass FIR filters and decomposes the signal into an approximation function and a detail function. Further decomposition of the signal results in transforming the signal into a hierarchy set of orthogonal approximation and detail functions. A reverse process is employed to perfectly reconstruct the signal (inverse transform) back from its approximation and detail functions. The discrete wavelet transform was applied to the displacement data recorded by Pezzack et al., 1977. The smoothed displacement data were twice differentiated and compared to Pezzack et al.'s acceleration data in order to choose the most appropriate filter coefficients and decomposition level on the basis of maximizing the percentage of retained energy (PRE) and minimizing the root mean square error (RMSE). Daubechies wavelet of the fourth order (Db4) at the second decomposition level showed better results than both the biorthogonal and Coiflet wavelets (PRE = 97.5%, RMSE = 4.7 rad s-2). The Db4 wavelet was then used to compress complex displacement data obtained from a noisy mathematically generated function. Results clearly indicate superiority of this new smoothing approach over traditional filters.
运动分析系统通常会给记录的位移数据引入噪声。巴特沃斯数字滤波器已被用于平滑位移数据，以获得平滑后的速度和加速度。然而，这种技术并不能产生令人满意的结果，尤其是在处理占据低频和高频频段的复杂运动时。本文提出使用离散小波变换作为数字滤波器的替代方法。该变换将原始信号通过两个互补的低通和高通FIR滤波器，并将信号分解为一个近似函数和一个细节函数。对信号的进一步分解会将信号转换为一组正交近似和细节函数的层次结构。采用反向过程从其近似函数和细节函数完美地重建信号（逆变换）。离散小波变换被应用于佩扎克等人在1977年记录的位移数据。对平滑后的位移数据进行二次微分，并与佩扎克等人的加速度数据进行比较，以便在最大化保留能量百分比（PRE）和最小化均方根误差（RMSE）的基础上选择最合适的滤波器系数和分解级别。在第二分解级别上的四阶Daubechies小波（Db4）比双正交小波和Coiflet小波都显示出更好的结果（PRE = 97.5%，RMSE = 4.7弧度秒-2）。然后，Db4小波被用于压缩从有噪声的数学生成函数中获得的复杂位移数据。结果清楚地表明这种新的平滑方法优于传统滤波器。
