A Savitzky-Golay (SG) filter, widely used in signal processing applications, is a finite-impulse-response low-pass filter obtained by a local polynomial regression on noisy observations in the least-squares sense. The problem addressed in this paper is one of optimal order (or filter length) selection of SG filter in the presence of non-Gaussian noise, such that the mean-squared-error (or risk) between the underlying clean signal and the SG filter estimate is minimized. Since mean-squared-error (MSE) depends on the unknown clean signal, direct minimization is impractical. We circumvent the problem within a risk-estimation framework, wherein, instead of minimizing the original MSE, an unbiased estimate of the MSE (which depends only on the noisy observations and noise statistics) is minimized in order to obtain the optimal order. The proposed method gives an unbiased estimate of the MSE considering SG filtering in the presence of additive noise following any distribution with finite first- and second-order statistics and independent of the signal. The SG filter's order and length are optimized by minimizing the unbiased estimate of MSE. The denoising performance of the optimal SG filter is demonstrated on real-world electrocardiogram (ECG) signals as well as signals from the WaveLab Toolbox under Gaussian, Laplacian, and Uniform noise conditions. The proposed denoising algorithm is superior to four benchmark algorithms in low-to-medium input signal-to-noise ratio (SNR) regions ( $-5$ dB to 12.5 dB) in terms of the SNR gain.

中文翻译：

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非高斯噪声中的自适应 Savitzky-Golay 滤波

Savitzky-Golay (SG) 滤波器广泛用于信号处理应用，是一种有限脉冲响应低通滤波器，通过对最小二乘意义上的噪声观测进行局部多项式回归获得。本文解决的问题是在存在非高斯噪声的情况下，SG 滤波器的最优阶数（或滤波器长度）选择之一，使得底层干净信号和 SG 滤波器之间的均方误差（或风险）估计被最小化。由于均方误差 (MSE) 取决于未知的干净信号，因此直接最小化是不切实际的。我们在风险估计框架内规避了这个问题，其中，不是最小化原始 MSE，而是最小化 MSE 的无偏估计（仅取决于噪声观察和噪声统计）以获得最佳顺序。所提出的方法给出了在存在附加噪声的情况下考虑 SG 滤波的 MSE 的无偏估计，这些噪声遵循具有有限一阶和二阶统计数据且独立于信号的任何分布。通过最小化 MSE 的无偏估计来优化 SG 滤波器的阶数和长度。最佳 SG 滤波器的去噪性能在真实世界的心电图 (ECG) 信号以及来自高斯、拉普拉斯和均匀噪声条件下的 WaveLab 工具箱。所提出的去噪算法在中低输入信噪比（SNR）区域优于四种基准算法（ $-5$ dB 至 12.5 dB）在 SNR 增益方面。