Wavelet coefficients

Unfortunately, the requirements for translational invariance of the wavelet decomposition are difficult to satisfy. Consequently, for either discretization scheme, comparison of the wavelet coefficients for two signals may mislead us into thinking that the two trends are different, when in fact one is simply a translation of the other. 

Fig. 14. Esxtracting distinguishing features from noise pulse signal. Wavelet coefficients in shaded regions represent stable extrema, (a) Wavelet decomposition of noisy pulse signal (b) wavelet decomposition of pulse signal. (Reprinted from Bakshi and Stephanopoulos, Representation of process trends. Part III. Computers and Chemical Engineering, 18(4), p. 267, Copyright (1994), with kind permission from Elsevier Science Ltd., The Boulevard, Langford Lane, Kidlington 0X5 1GB, UK.)...

Utilizing the wavelet coefficients of the stable extrema we can reconstruct a signal that contains the distinguished features at several scales. In Fig. 14, by utilizing the wavelet coefficients of the shaded region, we can completely reconstruct the pulse with minimum quantitative distortion. 

Step 3. For each scaling level, m, and for each scaling episode, (a) determine the range of stability and (b) collect the wavelet coefficients at all scales in the range of the episode s stability. 

Step 4. Reconstruct the signal using the last scaled signal and the wavelet coefficients collected in step 3. 

Fig. 16. Stable trend of the signal in Fig. 1, generated through the use of Witkin s stability criterion, and the wavelet coefficients shown in Fig. 13b.

Compression may be achieved if some regions of the time-frequency space in which the data are decomposed do not contain much information. The square of each wavelet coefficient is proportional to the least-squares error of approximation incurred by neglecting that coefficient in the reconstruction ... 

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. 

From the wavelet coefficients Ttl,[x](a, b) one can calculate the energy density Ethree-dimensional surface E(l,[x](a, t). Sections of this surface at fixed time moments t = b define the local energy spectrum Ev[x](f, t) with / = a 1. Finally, in order to obtain the mean spectral distribution of the time series x t) we may consider a so-called scalogram, i.e., the time-averaged energy spectrum. This is analogous to the classic Fourier spectrum. 

FIGURE 10.23 The cascade of wavelet coefficient vectors output from the wavelet tree filter banks defining the discrete wavelet transform in Figure 10.22. A db-7 mother wavelet was used for the decomposition of the noisy signal in Figure 10.1. 

FIGURE 10.25 Denoising of the wavelet components of the noisy signal in Figure 10.1 using an entropy threshold and db-7 wavelets. Thresholds for the decomposition are shown as dashed lines. Wavelet coefficients contained inside the thresholds are set to zero in this form of denoising. 

The CWT results in wavelet coefficients at every possible scale. Thus, there is a significant amount of redundancy in the computation. But there... 

In this representation, integer j indexes the scale or resolution of analysis, i.e., smaller j corresponds to a higher resolution, and jo indicates the coarsest scale or the lowest resolution, k indicates the time location of the analysis. For a wavelet 4> t) centered at time zero and frequency u>o, the wavelet coefficient dj k measures the signal content around time 2 k and frequency 2 uio- The scaling coefficient Ck measures the local mean around time 2 °k. The DWT represents a function by a countable set of wavelet coefficients, which correspond to points on a 2-D grid of discrete points in the scale-time domain. 

This approach greatly facilitates the calculation of wavelet and scaling coefficients as typically implemented in the Matlab Wavelet Toolbox [199]. One can associate the scaling coefficients with the signal approximation, and the wavelet coefficients as the signal detail. 

Due to the down-sampling procedure during decomposition, the number of resulting wavelet coefficients (i.e., approximations and details) at each level is exactly the same as the number of input points for this level. It is sufficient to keep all detail coefficients and the final approximation coefficient (at the coarsest level) to be able to reconstruct the original data. The... 

The hard-thresholding filter, fjj, selects wavelet coefficients that exceed a certain threshold and sets the others to zero ... 

The soft-thresholding filter, Fl, is similar to the hard-thresholding filter, but it also shrinks the wavelet coefficients above the threshold,... 

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