Application of Wavelet Transforms to Experimental Spectra:  Smoothing, Denoising, and Data Set Compression

Abstract

Various methods have been proposed for smoothing and denoising data sets, but a distinction is seldom made between the two procedures. Here, we distinguish between them in the signal domain and its transformed domain. Smoothing removes components (of the transformed signal) occurring in the high end of the transformed domain regardless of amplitude. Denoising removes small-amplitude components occurring in the transformed domain regardless of position. Methods for smoothing and denoising are presented which depend on the recently developed discrete wavelet (DW) transform technique. The DW smoothing and denoising methods are filtering techniques that are applied to the transformed data set, prior to back-transforming it to the signal domain. These DW techniques are compared with other familiar methods of smoothing (Savitzky-Golay) and denoising through filtering (Fourier transform). Preparatory to improving experimental data sets, synthetic data sets, comprised of ideal functions to which a known amount of noise has been added, are examined. The filter cutoffs are systematically explored, and DW techniques are shown to be highly successful for data sets with great dynamic range. In the minority of cases, smoothing and denoising are nearly interchangeable. It is shown that DW smoothing compresses with the most predictable results, whereas DW denoising compresses with minimal distortion of the signal.Experimental spectra are often complicated by noise, which may be due to interfering physical or chemical processes, imperfections in the experimental apparatus, or any of a number of causes which result in random, spurious fluctuations of the signal received at the detector. By definition, noise is instantaneously irreproducible although often it may be characterizable on average if it obeys Gaussian or Poisson distribution statistics, for example. In the present work, an interfering signal of another compound is not regarded as noise, nor are drift, baseline, signal pickup, or "cyclic noise" such as that used in a broader definition by Erickson et al. 1 included.

Signal Processing Methods for Capillary Electrophoresis

“…The triangle is likely to be the earliest peak model used (Dyson 1998) and is perhaps the simplest. It has been used as a peak model in a number of studies (Barclay and Bonner 1997;Stewart et al 2008). One definition for a triangle function is (Couch 1990),…”

“…Wavelet denoising strategies were also used (Ceballos et al 2008) on pattern recognition in CE. Similar wavelet based denoising strategies to those cited above have also been applied to liquid chromatography data (Barclay and Bonner 1997;Shao et al 2004), Raman spectroscopy (Hu et al 2007), mass spectrometry data (Barclay and Bonner 1997;Coombes et al 2005), as well as numerous other areas of research (Jagtiani et al 2008;Komsta 2009). The DWT is sufficient in many scenarios when removing high and low frequency noise from signals.…”

“…However, little effort has been devoted to the comparison of performance from these different methods in spite of the few published work (Barclay and Bonner 1997;Cruz-Marcelo et al 2008;Wee et al 2008;Yang et al 2009). In order to select or develop most efficient method, comparative studies are needed to test the methods on the same data (Leptos et al 2006).…”

Statistical evaluation of electrospray tandem mass spectra for optimized peptide fragmentation

“…For noise filtering of FT-ICR data, a discrete wavelet denoising operation [29] was performed using a 10-point Daubechies wavelet and a hard threshold function that zeroed the lowest 99.9% of points in the wavelet transformed data. This removed much of the noise, leaving signal peaks discernibly above the remaining baseline.…”

“…Closer inspection of the spectra showed that they consist largely of random noise as background to which a few fragment ion signals are superimposed. Consequently, we processed each spectrum to remove the noise using a wavelet denoising function [29] followed by a baseline correction. We then calculated arithmetic mean (Figure 7c) and standard deviation (Figure 7d) for the filtered fragment spectra and plotted them against decreasing ion activation attenuation on the x-axis, which corresponds to increasing kinetic energy of the precursor ions.…”