Wavelets transform

Wavelet transform (WT), a modern tool of applied mathematics, is a signal processing technique that has shown higher performance compared to Fourier transform and Short Time Fourier Transform in analyzing non-stationary signals. These advantages are due to its good localization properties in both, the time- and frequency domain. 

For a wave-like function to be considered a wavelet it must fulfil certain mathematical conditions. The most important conditions are admissibility and regularity (Mallat 1999). Admissibility condition, expressed as 

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 

When discrete wavelets are used to transform a continuous signal, the resulting set of coefficients is called the wavelet series decomposition. For this transformation to be useful it must be invertible, and for synthesis to take place, Eq. 9.11 must be satisfied 

The family of daughter wavelets y/j. that satisfy Eq. 9.11 forms a. frame with 

Like the FFT, the fast wavelet transform (FWT) is a fast, linear operation that operates on a data vector in which the length is an integer power of two (i.e., a dyadic vector), transforming it into a numerically different vector of the same length. Like the FFT, the FWT is invertible and in fact orthogonal that is, the inverse transform, when viewed as a matrix, is simply the transpose of the transform. Both the FFT and the discrete wavelet transform (DWT) can be regarded as a rotation in function 

Space from the input space or time domain — where the basis functions are the unit vectors or Dirac delta functions in the continuum limit — to a different domain. For the FFT, this new domain has sinusoid and cosinusoid basis functions. In the wavelet domain, the basis functions, or wavelets, are somewhat more complicated. 

FWTs can be implemented quite efficiently the calculation time of algorithms performing wavelet transformations increases only linearly with the length of the transformed vector. A special kind of wavelet was developed by Ingrid Daubechies [63]. Daubechies wavelets are base functions of finite length and represent sharp edges by a small number of coefficients. They have a compact support that is, they are zero outside a specific interval. There are many Daubechies wavelets, which are characterized by the length of the analysis and synthesis filter coefficients. 

Wavelets transforms are useful for compression of descriptors for searches in binary descriptor databases and as alternative representations of molecules for neural networks in classification tasks. 

The WT provides an entirely new perspective on traditional signal processing techniques (i.e., FT methods) for breaking up a signal into its component parts. Literally, the term wavelet means little wave. More specifically, a wavelet is a function that satisfies the following two conditions  

Given a particular wavelet function (a particular basis function), the wavelet transform operates in a manner similar to the FT by using its basis function to convert a signal from one domain to another. The WT 

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P. Simard M. Piriou B. Benoist, A. Masia. Wavelet transformation Filtering of eddy current signals. In l th International Conference on NDe in the nuclear and Pressure Vessel Industries, pages 313-317, 1997. 

Signal analysis using Continuous Wavelet Transform... 

Among these techniques, the Continuous Wavelet Transform (CWT) is particularly well suited to the eddy current signal coming from the tube control, as shown in this paper, and provides efficient detection results. 

As for the Fourier Transform (FT), the Continuous Wavelet Transform (CWT) is expressed by the mean of an inner product between the signal to analyze s(t) and a set of analyzing function ... 

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

Fig. 6 The ROIs shown in Fig. 2 denoised by discrete wavelet transform, (a) background, (b)...

Furthermore, one may need to employ data transformation. For example, sometimes it might be a good idea to use the logarithms of variables instead of the variables themselves. Alternatively, one may take the square roots, or, in contrast, raise variables to the nth power. However, genuine data transformation techniques involve far more sophisticated algorithms. As examples, we shall later consider Fast Fourier Transform (FFT), Wavelet Transform and Singular Value Decomposition (SVD). 

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). 

Widely used methods of data transformation are Fast Fourier and Wavelet Transformations or Singular Value Decomposition... 

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... 

Fig, y. Resolution in scale space of (a) window Fourier transform and (b) wavelet transform. 

The projection of Fit) to all wavelets of the above form with m Z and u e R, yields the so-called dyadic wavelet transform of (/), with the following components ... 

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). 

Step 1. Generate the finite, discrete dyadic wavelet transform of data using Mallat and Zhong s (1992) cubic spline wavelet (Fig. 8c). 

Mailat, S, G., Zero crossing of a wavelet transform. IEEE Trans. Inf. Theory IT-37(4), 1019-1033 (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. 

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 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... 

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.

Having a closer look at the pyramid algorithm in Fig. 40.43, we observe that it sequentially analyses the approximation coefficients. When we do analyze the detail coefficients in the same way as the approximations, a second branch of decompositions is opened. This generalization of the discrete wavelet transform is called the wavelet packet transform (WPT). Further explanation of the wavelet packet transform and its comparison with the DWT can be found in [19] and [21]. The final results of the DWT applied on the 16 data points are presented in Fig. 40.44. The difference with the FT is very well demonstrated in Fig. 40.45 where we see that wavelet describes the locally fast fluctuations in the signal and wavelet a the slow fluctuations. An obvious application of WT is to denoise spectra. By replacing specific WT coefficients by zero, we can selectively remove... 

I. Daubechies, S. Mallat and A.S. Willsky, Special issue on wavelet transforms and multiresolution signal analysis. IEEE Trans. Info Theory, 38 (1992) 529-531. 

R. Kronland-Martinet, J. Morlet and A. Grossmann, Analysis of sound patterns through wavelet transforms. Int. J. Pattern Recogn. Artif. Intell., 1 (1987) 273-302. 

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