Meyer wavelet

AA Petrosian and FG Meyer. Wavelets in Signal and Image Analysis From Theory to Practice. Kluwer Academic, Boston, MA, 2001. 

It is worthwhile to mention a few other wavelets which have gone unmentioned in the discussion thus far. These are the morlet. mexican hat and Meyer wavelets (see Fig. 13). The Meyer wavelet is one which does not have compact support, stated another way, the support width of the wavelet is infinite. The Meyer wavelet is orthogonal and symmetrical as well. 

Y. Meyer, Wavelets Algorithms and Applications. SIAM, Philadelphia (1993). 

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)). 

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... 

Y. Meyer, ed. Wavelets and Their Applications, Springer, Berlin, 1992. 

This would avoid any of the unpleasant side-effects such as singularities arising from many extension methods. Various approaches have been suggested including, Meyer s boundary wavelets [3] and dyadic boundary wavelets [4]. Though these wavelet construction methods are more elegant compared with the simpler boundary extension techniques, their implementation is considerably more involved. They also have some of their own... 

R.R. Coifman, Y. Meyer and M.V. Wickerhauser. Wavelet Analysis and Signal Processing, Wavelets and their Applications, (M.B. Ruskai et al. Eds), Jones and Bartlett, Boston, (1992). 

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. 

R.R. Coifman, Y. Meyer, S. Quake, and M.V. Wickerhauser, Signal Processing and Compression with Wavelet Packets, In Wavelets and Their Application J.S. Byrnes, J.L. Byrnes, K.A. Hargreaves and K. Berry (Eds) Kluwer Academic Publishers, The Netherlands, (1994), pp. 363-379. 

P. Guillemain, R. Kronland-Martinet and B. Martens. Estimation of Spectral Line with the Help of the Wavelet Transform - Application in NMR Spectroscopy, ir Wavelets and Applications, Proceedings of the Second International Conference or Wavelets and Their Applications, Marseilles, France. May 1989. (Y. Meyer Ed.) Springer-Verlag, Paris, 1992, pp. 38-60. 

The WNN is based on the similarity found between the inverse WT Stromberg s equation (Eq. 9.19) and a hidden layer in the Multi-Layer Perceptron (MLP) network structure (Meyer 1993). In fact, the wavelet decomposition can be seen like a neuronal network model, where the wavelets are indexed by i = instead of the double... 

MaUat, S. A wavelet tour of signal processing, 2nd edn. Academic Press, San Diego (1999) Meyer, Y. Wavelets Algorithms and Applications. Society for Industrial and Applied Mathematics. SIAM, PhUadelphia (1993)... 

Y. Meyer and S. Roques (eds.), Proc. Int. Conf. Wavelets and Applications , Toulouse, France, 8-13 June, 1992, Editions Frontiferes, Gif-sur-Yvette, France, 1993. 

Coifman RR, Meyer Y, Wickerhauser MV. In Meyer Y, Roques S, editors. Progress in Wavelet analysis and applications. France Editions Frontieres 1993. 

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