We consider the problem of robust deconvolution, and particularly the recovery of an unknown deterministic signal convolved with a known filter and corrupted by additive noise. We present a novel, non-iterative data-driven approach. Specifically, our algorithm works in the frequency-domain, where it tries to mimic the optimal unrealizable non-linear Wiener-like filter as if the unknown deterministic signal were known. This leads to a threshold-type regularized estimator, where the threshold at each frequency is determined in a data-driven manner. We perform a theoretical analysis of our proposed estimator, and derive approximate formulas for its Mean Squared Error (MSE) at both low and high Signal-to-Noise Ratio (SNR) regimes. We show that in the low SNR regime our method provides enhanced noise suppression, and in the high SNR regime it approaches the optimal unrealizable solution. Further, as we demonstrate in simulations, our solution is highly suitable for (approximately) bandlimited or frequency-domain sparse signals, and provides a significant gain of several dBs relative to other methods in the resulting MSE.
翻译:我们考虑了强劲的分解问题,特别是以已知的过滤器和添加剂噪音腐蚀了未知的确定性信号的回收问题。我们提出了一个新颖的、非显示性的数据驱动方法。具体地说,我们的算法在频率域工作,它试图模仿最佳的无法实现的非线性Wiener式过滤器,仿佛不知道的确定性信号是已知的。这导致一个临界型常规估计器,每个频率的阈值都以数据驱动的方式确定。我们从理论上分析了我们提议的测算器,并在低和高信号到噪音比率(SNR)制度下得出其平均偏差(MSE)的近似方公式。我们表明,在低信号到噪音制度中,我们的方法提供了强化的噪音抑制,在高的SNR制度中,它接近了最佳的无法实现的解决方案。此外,我们在模拟中显示,我们的解决办法非常适合(约)带宽或频率偏差的信号,并且为随后的MSE MSE中的其他方法提供了显著的几分贝。