Daubechies 6 wavelet

FIGURE 10.20 A mother Daubechies wavelet and two of its shifted and scaled children fitting a noisy signal. Note that the interval of the fit depends on the scale of the 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. 

According to a proposal of Ingrid Daubechies, the notation DK will be used for a Daubechies Wavelet transform with K coefficients. Actually, D2 is identical to the simplest Wavelet of all, the so-called Haar Wavelet, and, thus, is not originally a member of the Daubechies family. 

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. 

Having the three-dimensional coordinates of atoms in the molecules, we can convert these into Cartesian RDF descriptors of 128 components (B = 100 A ). To simplify the descriptor we can exclude hydrogen atoms, which do not essentially contribute to the skeleton structure. Finally, a wavelet transform can be applied using a Daubechies wavelet with 20 filter coefficients (D20) to compress the descriptor. A low-pass filter on resolution level 1 results in vectors containing 64 components. These descriptors can be encoded in binary format to allow fast comparison during descriptor search. 

The descriptors are transformed by Daubechies wavelet decomposition with 20 filter coefficients (D20). 

This chapter has briefly eluded to two wavelet families, the Haar and Daubechies wavelets. In fact when Nf = 2 the Daubechies wavelet is identical to the Haar wavelet. In this section we would like to discuss in greater detail more about these wavelet families and other wavelet families. We will also provide a brief comparison between the different properties possessed by these wavelets and other wavelet families. This is important because depending on your application, you may need to choose a wavelet that satisfies special properties. We first review the terms orthogonal and compact support. Following this, we will introduce some more properties, namely smoothness and symmetry of wavelets and also discuss the term vanishing moments. 

Table 1. Properties summary for the Haar, Daubechies, wavelets. 

Fig. 12 WPT results (.baling coefficients at the first decomposition level shown only) for an NMR spectrum with periodic boundary extension (Daubechies wavelet, Ny = 6).

Fig. 19 WPT results for a linear ramp signal (N = 90, Daubechies wavelet, N f = 6) with boundary handling using polynomial extrapolation extension. First five levels. shown only...

There exists many different kinds or families of wavelets. These wavelet families are defined by their respective filter coefficients which are readily available for the situation when m = 2, and include for example the Daubechies wavelets, Coiflets, Symlets and the Meyer and Haar wavelets. One basic issue to overcome is deciding which set (or family) of filter coefficients will produce the best results for a particular application. It is possible to trial different sets of filter coefficients and proceed with the family of filter coefficients which produces the most desirable results. It can be advantageous however, to design your own task specific filter coefficients rather than using a predefined set. 

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)). 

Fig. I (a) Experimental (h) Fourier transformed and (c) wandet-transformed IR. spectrum of benzoic acid. Spectra (c) and Id) were derived from (a with a Daubechies > (, wavelet filter at re.solutioti levels J - I and J - 2. respectively.

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] ... 

A higher Rfgg, is obtained for the brix response using Daubechies wavelets. 

ABSTRACT In the discrete wavelet transform approach, a choice of wavelet has a direct impact on the decomposed image, which indicates that the selection of the wavelet is closely related to the detection performance. Since a choice of standard wavelets, e.g. Daubechies wavelets, Coiflets, biorthogonal wavelets etc., may not guarantee efficient discrimination of fabric defects, some researchers suggested methods based on a construction of wavelets adapted to the detection or classification of these defects. We propose a novel method to design adaptive wavelet filters. These filters are constructed to minimize /2-norm of the undecimated discrete wavelet transform of the defect free textile with the aim to enhance the wavelet response in the defect region. Examples show efficiency in the fault detection. 

In the wavelet transform approach, the choice of wavelet has a direct impact on the decomposed image, which indicates that the selection of the wavelet is closely related to the detection and classification performance. And standard wavelets, e. g. Haar wavelet, Daubechies wavelets, Coiflets etc., may not guarantee efficient discrimination of fabric defects. 

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