Wavelet mother function

The basis functions, or wavelets,, are dilated and translated versions of a wavelet mother function. A set of wavelets is specified by a particular set of numbers, called wavelet filter coefficients. To see how a wavelet transform is performed, we will take a closer look at these coefficients that determine the shape of the wavelet mother function. 

Wavelets are a set of basis functions that are alternatives to the complex exponential functions of Fourier transforms which appear naturally in the momentum-space representation of quantum mechanics. Pure Fourier transforms suffer from the infinite scale applicable to sine and cosine functions. A desirable transform would allow for localization (within the bounds of the Heisenberg Uncertainty Principle). A common way to localize is to left-multiply the complex exponential function with a translatable Gaussian window , in order to obtain a better transform. However, it is not suitable when <1) varies rapidly. Therefore, an even better way is to multiply with a normalized translatable and dilatable window, v /yj,(x) = a vl/([x - b]/a), called the analysing function, where b is related to position and 1/a is related to the complex momentum. vl/(x) is the continuous wavelet mother function. The transform itself is now... 

A family of wavelets is a family of functions with all its members derived from the translations (e.g., in time) and dilations of a single, mother function. If iffit) is the mother wavelet, then all the members of the family are given by... 

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. 

An important feature of wavelet analysis is to find the most appropriate mother function. This is not always obvious. The ranges of a and b are flexible, giving rise to continuous wavelets if unlimited [6], or orthonormal discrete wavelets if limited [7]. For the atomic orbital example, above, the authors demonstrated the effect of choosing as a mother function... 

In Table 2 we cite various possible choices of mother wavelet, dual function, and scaling function combinations. Given any two of these, the other is determined. 

Figure 8 mother wavelet y/(t) (left) and wavelet built out of the mother wavelet by time shift b, and dilatation a. Both functions are represented in the time domain and the frequency domain. 

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). 

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. 

It is not a straightforward task to come up with a procedure that would lead to the best mother wavelet for a given class of signals. N-evertheless, exploiting several characteristics of the wavelet function, one can determine which family of wavelets would be more appropriate for a specific application. 

The basic idea of the wavelet transform is to represent any arbitrary function as a superposition of basis functions, the wavelets. As mentioned already, the wavelets P(x) are dilated and translated versions of a mother wavelet Tg. Defining a dilation factor d and a translation factor t, the wavelet function F(x) can be written as... 

The scaling of the mother wavelets Fg is performed by the dilation equation, which is, in fact, a function that is a linear combination of dilated and translated versions of it ... 

The wavelets are scaled and translated copies (known as daughter wavelets ) of a finite-length or fast-decaying oscillating waveform (known as the mother wavelet ). Wavelet transforms have advantages over traditional Fourier transforms for representing functions that have discontinuities and sharp peaks. 

The continuous wavelet transform convolves the function f(t) with translated and dilated versions of a single basis function vj/ft). The basis function v /(t) is often called a mother wavelet. The various translated and dilated versions of the mother wavelet are called children wavelets. The children wavelets have the form j/((t - b)/a), where a is the dilation parameter which squeezes or stretches the window. 

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] ... 

An easy way to construct a multi-dimensional (e.g. 2-D) wavelet transform is, for example, to implement the tensor products of the 1-D counterparts. That is, we apply the 1-D wavelet transform separately along one dimension at a time. This, as we shall see shortly, results in one scaling function and three different mother wavelet functions. 

Unlike the sine and cosine functions in Fourier analysis, which are localised in frequency but not in time, i.e. a small frequency change in the Fourier transform (FT) produces changes everywhere in the time domain, wavelets are localised both in frequency/scale (via dilations of the mother wavelet). 

The CWT, as described in Eq. 9.7, cannot be used in practice because a) the basis functions obtained from the mother wavelet do not form a really orthonormal base, b) translation and scale parameters are continuous variables, which mean that a function f t) might be decomposed in an infinite number of wavelet functions, and c) there is... 

In the present work, the mother wavelet used as activation function corresponds to... 

An intuitive solution to fliis problem would be to allow for sine waves with finite duration to appear as building blocks in the transformed data. This is the basis of wavelet analysis. The wavelet transform is based on such building blocks or elementary functions, which are obtained by dilatations, contractions, and shifts of a unique function called the wavelet prototype (or mother wavelet)... 

The orthogonal basis functions denoted by are obtained by scaling and shifting a prototype wavelet function if t) (also sometimes called a mother wavelet) by scale a and time b, respectively, as shown below ... 

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