The wavelet transform

Multiplication of this 4x8 transformation matrix with the 8x1 column vector of the signal results in 4 wavelet transform coefficients or N/2 coefficients for a data vector of length N. For c, = C2 = Cj = C4 = 1, these wavelet transform coefficients are equivalent to the moving average of the signal over 4 data points. Consequently, 

The coefficients in the H-matrix are the detail coefficients. The output of the H-matrix are the /-components. With N/2 detail components and N/2 approximation components, we are able to reconstruct a signal of length N. 

The discrete wavelet transform can be represented in a vector-matrix notation 

Rows 1-8 are the approximation filter coefficients and rows 9-16 represent the detail filter coefficients. At each next row the two coefficients are moved two positions (shift b equal to 2). This procedure is schematically shown in Fig. 40.43 for a signal consisting of 8 data points. Once W has been defined, the a wavelet transform coefficients are found by solving eq. (40.16), which gives  

The factor Vl/ 2 is introduced to keep the intensity of the signal unchanged. The 8 first wavelet transform coefficients are the a or smooth components. The last eight coefficients are the d or detail components. In the next step, the level 2 components are calculated by applying the transformation matrix, corresponding to the level on the original signal. This transformation matrix contains 4 wavelet filter 

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Best time-frequency representations are performed when the Wavelet Transform is continuous, that is when both parameters a... 

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). 

As approximation schemes, wavelets trivially satisfy the Assumptions 1 and 2 of our framework. Both the Lf and the L°° error of approximation is decreased as we move to higher index spaces. More specifically, recent work (Kon and Raphael, 1993) has proved that the wavelet transform converges uniformly according to the formula... 

A wavelet defined as above is called a first-order wavelet. From Eq. (21) we conclude that the extrema points of the first-order wavelet transform provide the position of the inflexion points of the scaled signal at any level of scale. Similarly, if i/ (f) = d it)/dt, then the zero crossings of the wavelet transform correspond to the inflexion points of the original signal smoothed (i.e., scaled) by the scaling function, tj/it) (Mallat, 1991). 

The combination of PCA and LDA is often applied, in particular for ill-posed data (data where the number of variables exceeds the number of objects), e.g. Ref. [46], One first extracts a certain number of principal components, deleting the higher-order ones and thereby reducing to some degree the noise and then carries out the LDA. One should however be careful not to eliminate too many PCs, since in this way information important for the discrimination might be lost. A method in which both are merged in one step and which sometimes yields better results than the two-step procedure is reflected discriminant analysis. The Fourier transform is also sometimes used [14], and this is also the case for the wavelet transform (see Chapter 40) [13,16]. In that case, the information is included in the first few Fourier coefficients or in a restricted number of wavelet coefficients. 

Fig. 40.43. Waveforms for the discrete wavelet transform using the Haar wavelet for an 8-points long signal with the scheme of Mallat s pyramid algorithm for calculating the wavelet transform coefficients.

Although the Fourier compression method can be effective for reducing data into frequency components, it cannot effectively handle situations where the dominant frequency components vary as a function of position in the spectrum. For example, in Fourier transform near-infrared (FTNIR) spectroscopy, where wavenumber (cm-1) is used as the x-axis, the bandwidths of the combination bands at the lower wavenumbers can be much smaller than the bandwidths of the overtone bands at the higher wavenumbers.31,32 In any such case where relevant spectral information can exist at different frequencies for different positions, it can be advantageous to use a compression technique that compresses based on frequency but still preserves some position information. The Wavelet transform is one such technique.33... 

Starck, J.L., and A. Bijaoui. 1994. A signal processing filtering and deconvolution by the wavelet transform. Signal Proc. 35 195-211. 

A critical difference between the Fourier transform defined in Equation 10.9 and the wavelet transform defined in Equation 10.22 is the fact that the latter permits localization in both frequency and time that is, we can use the equation to determine what frequencies are active at a specific time interval in a sequence. However, we cannot get exact frequency information and exact time information simultaneously because of the Heisenberg uncertainty principle, a theorem that says that for a given signal, the variance of the signal in the time domain a2, and the variance of the signal in the frequency (e.g., Fourier) domain c2p are related... 

We can define the wavelet transform W a, b) as the transformation of a signal /in Cartesian space by the three-dimensional integral... 

Figure 4. Multiscale aspects of oxide surfaces in aqueous environments, (b) represents a single trace across a hematite (001) surface undergoing biotically mediated reductive dissolution (a). Relatively clean steps 2.6 nm in size on the left of the figure give way to a complex surface morphology on the right side of the image where biotic dissolution was pervasive. The wavelet transform, to /2 levels, is shown in (c). (b) provided by Kevin Rosso, Pacific Northwest National Laboratory.

Ismail, A.E., Rutledge, G.C. and Stephanopoulos, G. (2003h) Multiresolution Analysis in Statistical Mechanics — II. The Wavelet Transform as a Basis for Monte Carlo Simulations on Lattices./. Chem. Phys., 118, 4424—4431. 

The Morlet wavelet can be understood to be a linear bandpass filter, centred at frequency m = coo/a with a width of /(aoa). Some Morlet wavelets and their Fourier spectra are illustrated in Fig. 4.4.4. The translation parameter b has been chosen for the wavelet to be centred at time f = 0 (top). With increasing dilatation parameter a the wavelet covers larger durations in time (top), and the centre frequency of the filter and the filter bandwidths become smaller (bottom). Thus depending on the dilatation parameter different widths of the spectrum are preserved in the wavelet transform while other signals in other spectral regions are suppressed. 

By compressing this function in time, Morlet was able to obtain a higher frequency resolution and spread it out to obtain a lower frequency resolution. To localize time, he shifted these waves in time. He called his transform the wavelets of constant shape and today, after a substantial number of studies in its properties, the transform is simply referred to as the Wavelet transform. The Morlet wavelet is defined by two parameters the amount of compression, called the scale, and the location in time. 

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... 

Daubechies Wavelets are basic functions for the wavelet transform, which are selfsimilar and have a fractal structure, used to represent polynomial behavior. 

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