Optimal Wavelet Selection for Signal Denoising

Wavelet denoising plays a key role in removing noise from signals and is widely used in many applications. In denoising, selection of the mother wavelet is desirable for maximizing the separation of noise and signal coefficients in the wavelet domain for effective noise thresholding. At present, wavelet selection is carried out in a heuristic manner or using a trial-and-error that is time consuming and prone to error, including human bias. This paper introduces a universal method to select optimal wavelets based on the sparsity of Detail components in the wavelet domain, an empirical approach. A mean of sparsity change (
$\mu _{sc}$
) parameter is defined that captures the mean variation of noisy Detail components. The efficacy of the presented method is tested on simulated and experimental signals from Electron Spin Resonance spectroscopy at various SNRs. The results reveal that the
$\mu _{sc}$
values of signal vary abruptly between wavelets, whereas for noise it displays similar values for all wavelets. For low Signal-to-Noise Ratio (SNR) data, the change in
$\mu _{sc}$
between highest and second highest value is
$\approx 8-10\%$
and for high SNR data it is around 5%. The mean of sparsity change increases with the SNR of the signal, which implies that multiple wavelets can be used for denoising a signal, whereas, the signal with low SNR can only be efficiently denoised with a few wavelets. Either a single wavelet or a collection of optimal wavelets (i.e., top five wavelets) should be selected from the highest
$\mu _{sc}$
values. The code is available on GitHub and the signalsciencelab.com website.