Function Daubechies 4 wavelet

Density functional theory with Daubechies wavelets 30... 

Another effort in the physics community should be mentioned here. The BigDFT software [35] is based on Daubechies wavelets instead of Gaussian basis functions and offers support within the CUDA programming framework. It was shown to achieve a high parallel efficiency of 90% on parallel computers in which the cross-sectional bandwidth scales well with the number of processors. It uses a parallelized hybrid CPU/GPU programming model and compared to the full CPU implementation, a constant speedup of up to six was achieved with the GPU-enabled version [35]. 

The best known wavelets are the Daubechies wavelets (dbe) and the Coif-man wavelets (coife). In both cases, e is the number of vanishing moments of the functions. Daubechies also suggested the symlets as the nearly symmetric wavelet family as a modification of the db family. The family Haar is the well-known Haar basis [95]. Figure 6.4 shows a number of wavelet functions. As can be seen, the Haar functions are discontinuous and may not provide good approximation for smooth functions. 

FWTs can be implemented quite efficiently the calculation time of algorithms performing wavelet transformations increases only linearly with the length of the transformed vector. A special kind of wavelet was developed by Ingrid Daubechies [63]. Daubechies wavelets are base functions of finite length and represent sharp edges by a small number of coefficients. They have a compact support that is, they are zero outside a specific interval. There are many Daubechies wavelets, which are characterized by the length of the analysis and synthesis filter coefficients. 

FIGURE 4.5 Shape of Daubechies wavelet and scaling functions with different numbers of coefficients. Both functions become smoother with increasing number of coefficients. With more coefficients, the middle of the wavelet functions and the left side of the scaling function deviate more and more from zero. The number of coefficients defines the filter length and the number of required calculations. 

Daubechies Wavelets are basic functions for the wavelet transform, which are selfsimilar and have a fractal structure, used to represent polynomial behavior. 

In WT computation, many wavelet functions have been proposed by different workers. The simplest one, the Harr wavelet - which is also the first member of the family of Daubechies wavelets [7] - has been known for more... 

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] ... 

Oscillographic chronopotentiometry is a new type of electroanalytical technique, developed in the P. R. China [35]. This technique is based on the change of oscillographic signal on the cathode ray oscilloscope. Harr and Daubechies wavelet functions were employed by another group in the P. R. China to de-noise the oscillographic signals of Pb(II) ions in NaOH solution and multi-components systems such as Cu(II) and Al(III) ions in LiCl solution and Cd(II) and In(III) in NaOH solution [34,35]. They found that this... 

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)). 

Chau and his co-workers have proposed some wavelet-based methods to compress UV-VIS spectra [24,37]. In their work, a UV-VIS spectrum was processed with the Daubechies wavelet function, Djfi. Then, all the Cj elements and selected Dj coefficients at different) resolution levels were stored as the compressed spectral data. A hard-thresholding method was adopted for the selection of coefficients from Dj. A compression ratio up to 83% was achieved. As mentioned in the previous section, the choice of mother wavelets is vast in WT, so one can select the best wavelet function for different applications. Flowever, most workers restrict their choices to the orthogonal wavelet bases such as Daubechies wavelet. Chau et al. chose the biorthogonal wavelet for UV VIS spectral data compression in another study [37]. Unlike the orthogonal case, which needs only one mother wavelet (p(t), the biorthogonal one requires two mother wavelets. (p(t) and (p(t), which satisfy the following biorthogonal property [38] ... 

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] ... 

Figure 7 Each wavelet basis is actually a pair of functions the wavelet and its scaling function. These are the two functions of the Daubechies 6 wavelet.

There exist different pairs of wavelets and scaling functions. One such pair is shown in Fig. 4. This is the Mexican hat pair (Daubechies, 1992), which draws its name by the fact that the scaling function looks like the... 

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

FIGURE 10.19 Some example mother wavelet functions. From left to right a coiflet (coif), a symlet (sym), and two Daubechies (db) wavelets. The numbers relate to the number of vanishing moments of the wavelet. 

There are several families of wavelets, proposed by different authors. Those developed by Daubechies [46] are extensively used in engineering applications. Wavelets from these families are orthogonal and compactly supported, they possess different degrees of smoothness and have the maximum number of vanishing moments for a given smoothness. In particular, a function f t) has e vanishing moments if... 

Figure 6.4. The scaling (left) and wavelet (right) functions for four wavelets, Haar, Daubechies 4, Symlet 3 and Coiflet 3.

Fig. 5 shows some wavelet functions which are translated and dilated by different amounts. Notice that they all possess the same shape and differ by the amount by which they are translated and dilated. There exist many kinds or families of wavelets. The wavelets shown in Fig. 5 are wavelets from the Daubechies family, named after Ingrid Daubechies. 

Figure 3.15 Shifting the Daubechies-7 wavelet function )y(x) (a) and by the value b (b).

It has been proved (Daubechies, 1990) that the accuracy of wavelet approximation of function (signal) f t) in the vicinity of Iq has the following upper bound. 

For the dyadic case, p = 2, we can take ao = f = 1, and use the Mexican hat wavelet and dual functions. Also, u = /2 S(0). For the Mexican hat wavelet and dual function, i/ = 3.410 (as computed through Daubechies (1991) frame based analysis ). The explicitly non-frame based analysis of Handy and Murenzi (1999) gives p = 3.427. Both are acceptable estimates. 

Zubarev M, Dumler A, Shutov V, Popov N. Assessment of left ventricular systolic function and diastolic time intervals by the bioimpedance polyrheocardi( ra]Aic system in Electrical Bioimpedance Methods Applications To Medicine And Biotechnology (Riu, PJ and Rosell, J and Bragos, R and Casas, O., ed.) 873 of Annals Of The New York Academy Of Sciences 19l-l96Nev/ York Acad Sciences 1999. Daubechies I.. Ten lectures on wavelets Applied Mathematics. 1992 61. 

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