Wavelet basis functions

Wavelet Transform is mathematical method to linear operation that decomposes a function into a continuous spectrum of its frequency components. Wavelet basis functions are localized in space and frequency. 

The wavelet basis functions are derived by translating and dilating one basic wavelet, called a mother wavelet. The dilated and translated wavelet basis functions are called children wavelets. The wavelet coefficients are the coefficients in the expansion of the wavelet basis functions. The wavelet transform is the procedure for computing the wavelet coefficients. The wavelet coefficients convey information about the weight that a wavelet basis function contributes to the function. Since the wavelet basis functions are localised and have varying scale, the wavelet coefficients therefore provide information about the frequency-like behaviour of the function. 

Nofe For clarification of terminology, we refer to the wavelet transform" as being the procedure for producing the wavelet coefficients. When the function f(t) is represented as a linear combination of the wavelet coefficients and wavelet basis functions this is referred to as the "wavelet series representation" or "wavelet decomposition" of f(t). This is discussed in greater detail in Section 5. 

Thus we have arrived at the wavelet series representation of f(t) (also called the wavelet decomposition of f(t)). Alternatively, one could write f(t) as a linear combination of scaling and wavelet basis functions as follows... 

Often there is a finite number of non-zero filter coefficients. We use the notation Nf to denote the number of non-zero filter coefficients. Values for the filter coefficients appear in several texts, see for example [7]. Each set of filter coefficients defines the corresponding scaling and wavelet basis functions. Whilst it is possible to use off-the-shelf wavelets, in Chapter 8 we suggest a possible approach for designing your own wavelets. 

Transform the noisy signal into the time-frequency domain by decomposing the signal on a set of orthonormal wavelet basis functions. 

Unlike representation of functions using wavelet basis functions, there are many different combinations of wavelet packet basis functions that can be used in signal representation. Hence there is some degree of redundancy. With redundancy comes choice, and Section 3 discusses one approach for selecting a set of basis functions. 

The function f(t) can be written as a linear combination of wavelet basis functions... 

In recent years, the development of wavelet transform (WT) theory in different fields of science has been growing very rapidly. The WT has two major characteristics, in that the basis functions of WT are localized in both the time and frequency domain, and there are a number of possible wavelet basis functions available. Such properties have attracted analytical chemists to... 

As suggested in reference [25], the traditional sigmoidal function can be replaced with the Morlet wavelet basis function Fqwt in neural network analysis (Fig. 4(b)). When a spectral data, X, is applied to this WNN system, a response or an output value Ydwt >s obtained as follows ... 

A second approach to data compression is to compress infrared spectra with a construct called a wavelet neural network (WNN). The WNN approach stores large amounts of infrared data for fast archiving of spectral data. It is achieved by modifying the machine learning technique of artificial neural networks (ANNs) to capture the shape of infrared spectra using wavelet basis functions. The WNN approach is similar to another approach... 

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

Wavelet transformation (analysis) is considered as another and maybe even more powerful tool than FFT for data transformation in chemoinetrics, as well as in other fields. The core idea is to use a basis function ("mother wavelet") and investigate the time-scale properties of the incoming signal [8], As in the case of FFT, the Wavelet transformation coefficients can be used in subsequent modeling instead of the original data matrix (Figure 4-7). 

The framework, however, as introduced so far is of little help for our purpose since the shift from any subspace to its immediate in hierarchy would require to change entirely the set of basis functions. Although j x) are all created by the same function, they are different functions and, consequently, the approximation problem has to be solved from scratch with any change of subspace. The theory of wavelets and its relation to multiresolution analysis provides the ladder that allows the transition from one space to the other. 

With the selection of wavelets as the basis functions the learning algorithm can now be finalized. 

For the moment, there are no guidelines for the selection of the particular basis functions for any given application. The important issue here is that the properties of the wavelets will be inherited by the approximating... 

Bakshi, B., Koulouris, A., and Stepanopoulos, G., Learning at multiple resolutions Wavelets as basis functions in artificial nemal networks and inductive decision trees. In Wavelet Applications in Chemical Engineering (R. L. Motard and B. Joseph, eds.) Kluwer Academic Publishers, Dordrecht/Norwell, MA, p. 139 (1994). 

A popular technique for approximation consists of representing the data as a weighted sum of a set of basis functions. As described in Section III, A, wavelets form a convenient set of basis functions to represent signals consisting of a variety of features. A signal decomposed on an... 

Fixed-shape basis functions. The basis functions are of a fixed shape, such as linear, sigmoid, Gaussian, wavelet, or sinusoid. Adjusting the... 

The multiscale basis functions capture the fast changes in coefficients corresponding to the fine-scale basis functions, while the slower changes are captured by the coarse-scale basis functions. Thus, the wavelet thresholding method adapts its resolution to the nature of the signal features and reduces the contribution of errors with minimum distortion of the features retained in the rectified signal. 

Several extensions of wavelets have been developed to improve their ability to solve practical problems. Wavelet packets (Coifman and Wick-erhauser, 1992) are a library of basis functions that cover a wide variety of shapes. The library can be searched efficiently to select the best set of... 

One problem encountered in solving Eq. (11.12) is the modeling of the prior distribution P x. It is assumed that this distribution is not known in advance and must be calculated from historical data. Several methods for estimating the density function of a set of variables are presented in the literature. Among these methods are histograms, orthogonal estimators, kernel estimators, and elliptical basis function (EBF) estimators (see Silverman, 1986 Scott, 1992 Johnston and Kramer, 1994 Chen et al., 1996). A wavelet-based density estimation technique has been developed by Safavi et al. (1997) as an alternative and superior method to other common density estimation techniques. Johnston and Kramer (1998) have proposed the recursive state... 

A disadvantage of Fourier compression is that it might not be optimal in cases where the dominant frequency components vary across the spectrum, which is often the case in NIR spectroscopy [40,41], This leads to the wavelet compression [26,27] method, which retains both position and frequency information. In contrast to Fourier compression, where the full spectral profile is fit to sine and cosine functions, wavelet compression involves variable-localized fitting of basis functions to various intervals of the spectrum. The... 

In general, wavelet functions are chosen such that they and their compressed representations are orthogonal to one another. As a result, the basis functions in Wavelet compression, like those in PCA and Fourier compression, are completely independent of one another. Several researchers have found that representation of spectral data in terms... 

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