Wavelet transforms

Wavelet transforms are normally applied to datasets whose size is a power of two, for example consisting of 512 or 1024 datapoints. If a spectrum or chromatogram is longer, it is conventional simply to clip the data to a conveniently sized window. 

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. 

Seven wavelets for an eight point series are presented in Table 3.11. The smallest is a two point wavelet. It can be seen that for a series consisting of 2N points, 

The key to the usefulness of wavelet transforms is that it is possible to express the data in terms of a sum of wavelets. For a spectrum 512 datapoints long, there will be 511, plus associated scaling factors. This transform is sometimes expressed by 

It is beyond the scope of diis text to provide details as to how to obtain W, but many excellent papers exist on diis topic. 

---

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. 

---