Orthogonal wavelet

As a general rule, all orthogonal wavelets lack symmetry. This becomes an issue in applications such as image processing where symmetric wavelets are preferable. The symmetric wavelets also facilitate the handling of image boundaries. 

Esteban-Diez, I., Gonzalez-Saiz, J.M., G6mez-Camara, D. and Pizarro Mdlan, C. (2006a) Multivariate calibration of near infrared spectra by orthogonal WAVElet correction using a genetic algorithm. Anal. Chim. Acta, 555, 84—95. 

Much of the literature on wavelets tends to be biased towards discussions on orthogonal wavelets because they are convenient and simple to implement. However, we feel that is necessary to make the reader aware that wavelets need not be orthogonal and that wavelets with other properties can be quite useful too. In this section we discuss biorthogonal wavelets as one alternative to orthogonal wavelets. We direct the reader to [1,7,10,14,15] for more information on other kinds of wavelets. 

As for orthogonal wavelets, a fast wavelet transform exists for computing the scaling and wavelet coefficients. Provided the scaling and wavelet coefficients are known for some scale, the scaling and wavelet coefficients at the next lower scale j - 1, are computed as follows ... 

Unfortunately, the presence of additional coefficients can introduce dependencies in the coefficients close to the boundaries as well as prevent perfect reconstruction upon back-transformation. Furthermore, the symmetric extension property should be applied at each level of the decomposition. Perfect reconstruction can only be achieved through the use of symmetric wavelets such as bi-orthogonal wavelets [2]. 

This may be helpful when back-transforming to identify the important features in the original data sequence. Bi-orthogonal wavelets ensure the perfect reconstruction of the original data sequence. 

The restrictions which are imposed on the filter coefficients so that an MRA and orthogonal wavelet basis exist are summarized as follows [6]... 

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)). 

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] ... 

Orthogonal wavelets are related with theory of multiresolution analysis and usually cannot be expressed in an informal context they must fulfill stringent orthogonal conditions, on the other hand, wavelet frames are constructed by simple operations of translation and dilation and are the easiest to use (Akay 1997, Heil 1989, Gutes et al. 

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). 

This decomposition algorithm forms the first half of the discrete orthogonal wavelet transform. The reconstruction of coefficients 35+U from coefficients andy,- is performed with... 

The discrete orthogonal wavelet transform can be achieved in 0(n) operations, where n is the number of processed data. 

The scaling and wavelet equation are key components to derive the discrete orthogonal wavelet transform. If/eZ,2(IK), and if we denote scaling coefficients of function/ by yj, and wavelet coefficients of function / by then... 

Thus the decomposition algorithms is the same as in the case of discrete orthogonal wavelet transform... 

Jasper et al. 1996) suggested a new method to capture the texture information using adaptive orthogonal wavelets. Texture constraint were used to adapt the wavelets to better characterize specific textures. These wavelets have higher sensitivity to the changes in the texture caused by defects. They minimized a quadratic cost function J defined by... 

Sometimes real, symmetric wavelets or wavelets with more vanishing moments and at the same time with smaller support than corresponding orthogonal wavelets or simply more regular", in closed form defined wavelets are required. One way to obtain them is to construct two sets of biorthogonal wavelets r/r and its dual x). One of these two biorthogonal wavelets is consequently used to decompose the signal and the second one to reconstruct it. The numbers M,N of... 

And it also changes in a clear way the form of the discrete wavelet transform. In contrast to the construction o f orthogonal wavelets, we have two filters to solve and we have the freedom to choose one filter and solve for the second one. It means that we have at disposal more degrees of freedom in contrast to orthogonal wavelets. 

In this paper we use biorthogonal wavelets on the interval constructed by (Cema Finek 2008 Cerna Finek 2009) which outperforms similar construction in the sense of better conditioning of base functions as well as in better conditioning of wavelet transform. The condition number seems to be nearly optimal most especially in the case of cubic spline wavelets. From the viewpoint of numerical stabihty, ideal wavelet bases are orthogonal wavelet bases. However, they are... 

If Wj is the orthogonal complement we obtain orthogonal wavelets [9]. In this case the transformation T , which relates the single and the multiscale coefficients is the Fast Wavelet Transform [5]. If V is a multiresolution analysis, then a refinement equation is valid for < ... 

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... 

Discrete analog of wavelet transform (orthogonal wavelet basis functions by dilating and translating in discrete steps Pressure fluctuation Signal denoising Roy et al. (1999)... 

H.Li, M.Takei, M.Ochi, Y.Saito, and K.Horii, Application of Two-dimensional Orthogonal Wavelets to Multiresolution Image Analysis of a Turbulent Jet, Transactions of the Japan Society for Aeronautical and Space Sciences, Vol.42, No. 137, ppl20-127 (1999)... 

There are various kinds of wavelets, including continuous wavelets, discrete orthogonal wavelets, spline wavelets, local cosines, and wavelet packets. Different applications... 

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