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Daubechies family

In the simplest (and most localized) member of the Daubechies family, the four coefficients [Cq, Cj, C2, C3] represent the low-pass filter H that is applied to the odd rows of the transformation matrix. The even rows perform a different convolution by the coefficients [C3, -C2, Cj, -Cq] that represent the high-pass filter G. H acts as a coarse filter (or approximation filter) emphasizing the slowly changing (low-frequency) features, and G is the detail filter that extracts the rapidly changing (high-frequency) part of the data vector. The combination of the two filters H and G is referred to as a filter bank. [Pg.98]

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. [Pg.98]

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. [Pg.63]

Fig. 5 An example of dilating and Iran.dating wavelets from the Daubechies family. Fig. 5 An example of dilating and Iran.dating wavelets from the Daubechies family.
There are few possible strategies of library compression. Each of them has its own advantages and drawbacks. The most efficient method of data set compression, i.e. Principal Component Analysis (PCA), leads to use of global features. As demonstrated in [15] global features such as PCs (or Fourier coefficients) are not best suited for a calibration or classification purposes. Often, quite small, well-localized differences between objects determine the very possibility of their proper classification. For this reason wavelet transforms seem to be promising tools for compression of data sets which are meant to be further processed. However, even if we limit ourselves only to wavelet transforms, still the problem of an approach optimally selected for a particular purpose remains. There is no single method, which fulfills all requirements associated with a spectral library s compression at once. Here we present comparison of different methods in a systematic way. The approaches A1-A4 above were applied to library compression using 21 filters (9 filters from the Daubechies family, 5 Coiflets and 7 Symmlets, denoted, respectively as filters Nos. 2-10, 11-15 and 16-22). [Pg.297]

One might be interested in how the adaptive wavelet performs against predefined filter coefficients. In this section, we perform the 2-band DWT on each data set using filter coefficients from the Daubechies family with Nf = 16. The coefficients from some band (j, x) are supplied to BLDA. We consider four bands - band(3,0), band(3,l), band(4,0) and band(4,l). The results for the training and testing data are displayed in Table 3. The test CCR rates are the same for the seagrass and butanol data, but the AWA clearly produces superior results for the paraxylene data. [Pg.447]

Table 3. Classification results for wavelet and scaling coefficients produced using filter coefficients from the Daubechies family with... Table 3. Classification results for wavelet and scaling coefficients produced using filter coefficients from the Daubechies family with...
This section is similar to Section 1.5.1 in that we perform the 2-band DWT on each data set using filter coefficients from the Daubechies family with Nf= 16. The coefficients xM(0), xf >(l), Xl l(O), Xl l(l) are supplied to MLR. The DWT was performed on the original (uncentred data), but the coefficients and response variables were centred, prior to them entering the MLR model. The R rain response are displayed in... [Pg.455]

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... [Pg.120]

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. [Pg.121]

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. [Pg.76]

The Haar, Daubechies, symmlets and coiflets are wavelet families which exhibit orthogonality and compact support (see Fig. 12). Criteria which the scaling (j)(t) and wavelet vl/(t) must satisfy for orthogonality were discussed in Section 5. Also, in this section the term compact support was briefly mentioned. A wavelet is compactly supported if it is nonzero over a finite interval and zero outside this interval. Such wavelets include the Haar, Daubechies, symmlets and coiflets. [Pg.76]

Fig. 12 An example of a Haar wavelet (a), and wavelets from the Daubechies (b), symmlet (c) and coiflet (d) families. Fig. 12 An example of a Haar wavelet (a), and wavelets from the Daubechies (b), symmlet (c) and coiflet (d) families.
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. [Pg.177]

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... [Pg.225]

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)). [Pg.242]

Although many wavelet applications use orthogonal wavelet basis, others work better with redundant wavelet families. The redundant representation offered by wavelet frames has demonstrated to be good both in signal denoising and compaction (Daubechies et al. 1986, 1992). [Pg.156]

Figure 10 Some families of wavelets used for multiresolution analysis (a) Haar, (b) Daubechies 2, (c) Symmlet 8, and (d) Coiflet. Figure 10 Some families of wavelets used for multiresolution analysis (a) Haar, (b) Daubechies 2, (c) Symmlet 8, and (d) Coiflet.

See other pages where Daubechies family is mentioned: [Pg.406]    [Pg.407]    [Pg.412]    [Pg.226]    [Pg.271]    [Pg.137]   
See also in sourсe #XX -- [ Pg.63 ]




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