B-spline wavelets

In voltammetry, the spline wavelet was chosen as the major wavelet function for data de-noising. The function has been applied successfully to analyse voltammetric data since 1994 by Lu and Mo [9]. Mo and his co-workers have published more than fifteen papers on this topic in various journals. The spline wavelet is different from the Daubechies wavelet functions. Mathematically, the mth order basis spline (B-spline) wavelet, Nm, is defined recursively by convolution of the Harr wavelet function as follows [10] ... 

Ip when the SNR value is reduced. Finally, they tested the effect of the number of sampling points from 2 to 2 °. The deviations in Ep and Ip are reduced with a higher value of this parameter. The authors concluded that the third-order B-spline wavelet basis, and truncation frequency L = 3, are the optimum parameters for processing voltammetric signals. 

B-Spline wavelet analysis is a very good technique for processing volta-mmetric signals, but it also suffers a drawback. For a signal with a low SNR value, peak-potential shifting is always observed in the de-noised signal [25]. Therefore, Mo and his co-workers developed a new procedure by combining... 

Fig. 6 Results of the (a) differential pulse stripping voltammetric signal of the Zn(II)-KCl system and (bj the background current signal of KCl. solution that was de-noised with the combined B-spline wavelet and Fourier transform analysis. Reproduced from reference [25] with kind permission of Science in China Press.

S. Sakakibara, A Practice of Data Smoothing by B-Spline Wavelets, In Wavelets Theory. Algorithms, and Applications (C.K. Chui, L. Montefusco and L. Puccio, Eds) Academic Press, San Diego. CA. (1994). pp. 179-196. 

X.P. Zheng and J.Y Mo, The Coupled Application of B-Spline Wavelet and RLT Filtration in Staircase Voltammetry, In New Trends in Chemometrics, First International Conference on Chemometrics in China, Zhangjiajie, China, October 17-22, 1997 (Y.Z. Liang, R. Nortvedt. O.M. Kvalheim. H.L. Shen. Eds), Hunan University Press, Changsha, (1997), pp. 199-200. 

As stated in the previous section, most workers confine their wavelet functions in the Daubechies wavelet series only. For example, we have adopted the Daubechies wavelet function to denoise spectral data from a UV-VIS spectrophotometer [43]. In order to make use of the other available wavelet functions for chemical data analysis, Lu and Mo [44] suggested employing spline wavelets in their work for denoising UV-VIS spectra. The spline wavelet is another commonly used wavelet function in chemical studies. This function has been applied successfully in processing electrochemical signals [9,10] which will be discussed in detail in another chapter of this book. The mth order basis spline (B-spline) wavelet, Nm. is defined as follows [44] ... 

Remarks (i) For conventional FS-TARMA models, the functional subspaces include linearly independent basis functions selected from an ordered set, such as Qiebyshev, trigonometric, b-splines, wavelets, and other functions. For simplicity a functional subspace is often selected to include consecutive basis functions up to a maximum index. Yet, for purposes of model parsimony (economy) and effective estimation, some functions may not be necessary and may be dropped, (ii) An FS-TARMA ( a> c) p ... 

Fig. 11. Wavelet decomposition (a) dyadic sampling using Daubechies-6 wavelet (b) uniform sampling using cubic spline wavelet.

In spline wavelet computation, optimization needs to be carried out on two parameters, namely the order of B-spline, m, and the truncation frequency or frequency scale, L, which represents the cut-off (or truncation) frequency value between the useful signal and noise. Details of the B-spline theory can be found in the references [12,13]. 

Generally, WT is superior to FT in many respects. In Fourier analysis, only sine and cosine functions are available as filters [13], However, many wavelet filter families have been proposed. They include the Meyer wavelet, Coiflet wavelet, spline wavelet, the orthogonal wavelet, and Daubechies wavelet [14,15]. Both Daubechies and spline wavelets are widely employed in chemical studies. Furthermore, there is a well-known drawback in Fourier analysis (Fig. 1). Since the filters chosen for the Fourier analysis are localized in the frequency domain, the time-information is hidden after transformation. It is impossible to tell where a particular signal, for example as that shown in Fig. 1(b), takes place [13]. A small frequency change in FT produces changes everywhere in the Fourier domain. On the other hand, wavelet functions are localized both in frequency (or scale) and in time, via dilations and translations of the mother wavelet, respectively. Both time and frequency information are maintained after transformation (Figs. 1(c) and (d)). 

In spline wavelet computation, two parameters, namely the order of B-spline, m, and truncation frequency, L, which represents the cut-off (or truncation) frequency value between the true signal and noise, need to be optimized. In Lu and Mo s study [44], they found that the best result for denoising UV-VIS spectra with high noise level was obtained with m = 3 and L = 4. Zhao and Wang [45] proposed a technique called the wavelet transform K-factor three-wavelength method to determine simultaneously the concentrations of vanadium, molybdenum and titanium with UV-VIS spectroscopy. In their study, WT was adopted to denoise the spectra acquired. The concentrations of individual ions were determined from the UV-VIS spectra at three selected wavelengths. 

Fig. 8. Typical wavelets and scaling functions (a) Haar, (b) Daubechies-6, (c) cubic spline.

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