Wavelet function

Flit) is called the scaled signal and is derived from the filtering of FqU) with the lowpass scaling function. It represents a smoother version of FqU). Diit) is called the detail signal and is derived from the filtering of FqU) with the bandpass wavelet functions. It represents the information that was filtered out of FqU) in producing Fiit). 

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... 

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. 

Just as the discrete Fourier transform generates discrete frequencies from sampled data, the discrete wavelet transform (often abbreviated as DWT) uses a discrete sequence of scales aj for j < 0 with a = 21/v, where v is an integer, called the number of voices in the octave. The wavelet support — where the wavelet function is nonzero — is assumed to be -/<72, /<72. For a signal of size N and I < aJ < NIK, a discrete wavelet / is defined by sampling the scale at a] and time (for scale 1) at its integer values, that is... 

A wavelet is a general function, usually, but by no means exclusively, of time, g(t), which can be modified by translation (b) or dilation (expansion/contraction) (a). The function should add up to 0, and can be symmetric around its mid-point. A very simple example the first half of which has the value +1 and the second half —1. Consider a small spectrum eight datapoints in width. A very simple basic wavelet function consists of four —1 s followed by four —Is. This covers the entire spectrum and is said to be a wavelet of level 0. It is completely expanded and there is no room to translate this function as it covers the entire spectrum. The function can be halved in size (a = 2), to give a wavelet of level 1. This can now be translated (changing b), so there are two possible wavelets of level 1. The wavelets may be denoted by [n, m] where n is the level and m the translation. 

Of course, a function such as that in Table 3.11 is not always ideal or particularly realistic in many cases, so much interest attaches to determining optimum wavelet functions, diere being many proposed and often exotically named wavelets. 

The first involves smoothing. If the original data consist of 512 datapoints, and are exactly fitted by 511 wavelets, choose die most significant wavelets (those widi die highest coefficients), e.g. die top 50. In fact, if die nature of the wavelet function is selected with care only a small number of such wavelets may be necessary to model a spectium which, in itself, consists of only a small number of peaks. Replace the spectrum simply with that obtained using the most significant wavelets. 

Historically, the first wavelet function is attributed to Haar [95] when he replaced the sinusoidal basis functions of FT with an orthonormal function, V (f), given as. 

The best known wavelets are the Daubechies wavelets (dbe) and the Coif-man wavelets (coife). In both cases, e is the number of vanishing moments of the functions. Daubechies also suggested the symlets as the nearly symmetric wavelet family as a modification of the db family. The family Haar is the well-known Haar basis [95]. Figure 6.4 shows a number of wavelet functions. As can be seen, the Haar functions are discontinuous and may not provide good approximation for smooth functions. 

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. 

Nonlinear map from input space X to feature space F A wavelet function... 

A wavelet function with dilation parameter s and translation parameter u... 

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... 

FIGURE 4.5 Shape of Daubechies wavelet and scaling functions with different numbers of coefficients. Both functions become smoother with increasing number of coefficients. With more coefficients, the middle of the wavelet functions and the left side of the scaling function deviate more and more from zero. The number of coefficients defines the filter length and the number of required calculations. 

The scaling function ( ) is determined by the low-pass QMF and thus is associated with the coarse components, or approximations, of the wavelet decomposition. The wavelet function / is determined by the high-pass filter, which also produces the details of the wavelet decomposition. 

By iterative application of the FWT to the high-pass filter coefficients, a shape emerges that is an approximation of the wavelet function. The same applies to the iterative convolution of the low-pass filter that produces a shape approximating the scaling function. Figure 4.6 and Figure 4.7 display the construction of the scaling and wavelet functions, respectively ... 

FIGURE 4.7 Construction of wavelet function / with high-pass D4 filter coefficients (above). Below the functional representation of the high-pass filter coefficients (left) and their refinement by iterative calculation (increasing resolution level j) leading to an approximation of the wavelet function / (right). 

Fig. 5 shows some wavelet functions which are translated and dilated by different amounts. Notice that they all possess the same shape and differ by the amount by which they are translated and dilated. There exist many kinds or families of wavelets. The wavelets shown in Fig. 5 are wavelets from the Daubechies family, named after Ingrid Daubechies. 

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