Different discrete wavelet transforms applied to denoising analytical data

Discrete wavelet transform (DWT) denoising contains three steps: forward transformation of the signal to the wavelet domain, reduction of the wavelet coefficients; and inverse transformation to the native domain. Three aspects that should be considered for DWT denoising include selecting the wavelet type, selecting the threshold, and applying the threshold to the wavelet coefficients. Although there exists an infinite variety of wavelet transformations, 22 orthonormal wavelet transforms that are typically used, which include Haar, 9 daublets, 5 coiflets, and 7 symmlets, were evaluated. Four threshold selection methods have been studied: universal, minimax, Stein's unbiased estimate of risk (SURE), and minimum description length (MDL) criteria. The application of the threshold to the wavelet coefficients includes global (hard, soft, garrote, and firm), level-dependent, data-dependent, translation invariant (TI), and wavelet package transform (WPT) thresholding methods. The different DWT-based denoising methods were evaluated by using synthetic data containing white Gaussian noise. The results of comparison have shown that most DWTs are very powerful methods for denoising and that the MDL and the TI methods are practical. The MDL criterion is the only method that can select a threshold-for wavelet coefficients and select an optimal transform type. The TI method is insensitive to the wavelet filter so that for a variety of wavelet filters equivalent results were obtained. Savitzky-Golay and Fourier transform denoising results were used as reference methods. IR and HPLC data were used to compare denoising methods.