Wavelet basis

In order to compress the measured data through a wavelet-based technique, it is necessary to perform a series of convolutions on the data Becau.se of the finite size of the convolution filters, the data may be decomposed only after enough data has been collected so as to allow convolution and decomposition on a wavelet basis. Therefore, point-bypoint data compression as done by the boxcar or backward slope methods is not possible using wavelets. Usually, a window of data of length 2" m e Z, is collected before decomposition and selection of the appropriate... 

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 basis listed in Table 8 is the simplest wavelet basis, and the associated transform is called the Haar transform. Surprisingly, other wavelet bases will lead to the same frequency tiling as we just found, i.e. to Fig. 12(d). The only way we can picture the difference between wavelets is by considering the fact that the tile boundaries are not always as sharp as they are drawn. Depending on the shape of the wavelets, the tiles are more or less blurred. 

For an impulse response that differs from zero on, let us say, four points, several aspects of the pyramid algorithm are less obvious. We need to be able to drop half the points and still represent the signal using the output of the LP and HP filters. In other words, we need to step the linear convolution of signal and impulse response by two points. The Haar wavelet basis is also special in the sense that, as the impulse responses are only two points wide, we do not lose points at the extremes when performing a linear convolution. For wider impulse responses, something has to be done about those extremes, e.g. a circular convolution, which puts an additional constraint on the shapes of those impulse responses. 

A wavelet basis allows a time-frequency analysis similar to that of a short-time Fourier basis. It is different in that its time-localisation is better hence its frequency localisation is worse, for high frequencies than for low fre-... 

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. 

From the full WPT we can generate a large number of possible redundant subtrees, or arbitrary WP trees (called wavelet bases). In fact, the total number of two-way (dyadic) bases is at least (2-)" for a tree depth equal to niev. For example, a dyadic WPT with tree depth niev = 12 has a library of at least 4.2 xlO bases The WPT has an important advantage compared with the WT because fast algorithms exist for the efficient search of the best wavelet basis, based on the minimisation of a cost function such as an entropy criterion. 

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... 

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

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... 

Ip when the SNR value is reduced. Finally, they tested the effect of the number of sampling points from 2 to 2 °. The deviations in Ep and Ip are reduced with a higher value of this parameter. The authors concluded that the third-order B-spline wavelet basis, and truncation frequency L = 3, are the optimum parameters for processing voltammetric signals. 

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 ... 

In the above equation, W, bj and aj denote the weighting factor, translation coefficient and dilation coefficient, respectively, for each wavelet basis. In Liu s work [25], the wavenumber and transmittance quantities of the IR spectrum were used as the input and target output values, respectively, of the network. Their proposed neural network consisted of a single layer network... 

Transform useful for representing the Hartree-Fock operator for solving large chemical systems [3,10,11]. They chose the discrete wavelet basis sets described by Daubechies [12] ... 

P. Fischer and M. Defranceschi, Representation of the Atomic Hartree-Fock Equations in a Wavelet Basis by Means of the BCR Algorithm, in Wavelets Theory, Algorithms, and Applications, (C.K. Chui. L. Montefusco and L. Puccio Eds), Academic Press, New York, (1994), pp. 495-506. 

P. Fischer and M. Defranceschi, Numerical Solution of the Schrodinger Equation in a Wavelet Basis for Hydrogen-like Atoms, SIAM Journal of Numerical Anciivsis, 35 (1998), 1-12. 

Both A and B might be stored in a compact, binary form, as we only need to know whether wavelet basis component is being used or not. That is, one bit... 

The practical implementation of the wavelet transform is not determined yet, since Eqs. 13 and 14 still define a large set of orthonormal wavelet basis. Since admissibility condition states that the behaviour of the Fourier transform of a wavelet is... 

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). 

The wavelet transform ofy could be computed by an orthonormal wavelet basis matrix W, the same approach as used in FT... 

Well-conditioned bases. Condition number of the wavelet bases should be close to the condition number of the spline wavelet bases on the real line. It is well known that the condition number of the wavelet basis on the real line is less or equal to the condition number of the wavelet basis on the interval. 

Step 5 Bandeletize the warped wavelet basis by replacing 7 ( ) 7 wit f 7 (x) 7 (y) through ID discrete wavelet transform. 

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