Symmlet wavelet

The restrictions placed on the mother wavelets for multiresolution analysis do not limit the variety of shapes that can be used as mother wavelets different researchers have proposed several different wavelet functions, each with benefits and drawbacks. The wavelet shape tradeoff is between how compactly it can be localized in space and its level of smoothness. For example, the Haar wavelet, which is the simplest wavelet and was identified almost 100 years ago, is well localized in space, but it has an unnatural square-wave oscillation (see Figure 10). Many related wavelets exist, collectively referred to as wavelet families some of these families include the Meyer wavelet, Coiflet wavelet, spline wavelet, orthogonal wavelet, symmlet wavelet, and local cosine basis. Figure 10 depicts several of these wavelets and... 

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. 

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

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). 

The sum of all the elements in each of the normalised u vectors is calculated. This gives an indication of the area Ek below each curve. These curves are monotonously decreasing with only positive values and the one with the smallest area Ek corresponds to the optimal wavelet for a particular data set. For all data sets in this chapter the following six wavelets were tested for Haar, Beylkin, Coiffet, Daubechies, Symmlet and Vai-dyanathan with varying number of vanishing moments. 

Eubact results. For this data set DFA was used in an unsupervised mode. The optimal wavelet was found to be Symmlet 9. Fig. 23 shows the results from the multiscale cluster analysis. 

Various mother wavelets, e.g. Haar, Daubechies 4, 6, 8, 12 and 20, Symmlet 6 and 8, Coiflet 2 and 3, Villasenor 1 to 5 and Antonini, spline wavelets, and different levels of decomposition, were employed in the computations. 

In wavelet analysis, the concept of frequency can be replaced by the idea of scale. Unlike sine waves known from FT analysis (cf. Figure 3.8), a wavelet is a waveform with an average value of zero and limited duration. Wavelets tend to be irregular and asymmetric. In Figure 3.14, the wavelets of Classes Haar, Daubechies, Coiflet, Symmlet, Morlet, and Mexican Hat are given as examples. 

Figure 3.14 Wavelets of classes Haar (a), Daubechies (b), Coiflet (c), Symmlet (d), Morlet (e), and Mexican hat (f).

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

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