Coiflet wavelet

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. 

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)). 

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. 

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... 

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. 

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

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). 

Using the method described earlier it was found that the optimal wavelet for this data set was Coiflet 5. The results from the multiscale cluster analysis of this data set are summarised in Fig. 21. 

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.

Figure 7 Solution to the two-scale equation with Coiflet 6 filter coefficients, and its associated mother wavelet...

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