Optimal wavelet

Choosing optimal wavelet bases. Unlike the Fourier transform there are several basis functions to select from when using wavelet transforms. This means there must be a criterion for choosing the optimal wavelet. A reasonable criterion is to use the compression ability of the analysing wavelet. This means that the optimal wavelet is defined to be the one that produces the smallest number of coefficients needed to describe the data. The following algorithm is here used ... 

The sum of all the elements in each of the normalised u vectors is calculated. This gives an indication of the area Ek below each curve. These curves are monotonously decreasing with only positive values and the one with the smallest area Ek corresponds to the optimal wavelet for a particular data set. For all data sets in this chapter the following six wavelets were tested for Haar, Beylkin, Coiffet, Daubechies, Symmlet and Vai-dyanathan with varying number of vanishing moments. 

Using the method described earlier it was found that the optimal wavelet for this data set was Coiflet 5. The results from the multiscale cluster analysis of this data set are summarised in Fig. 21. 

Eubact results. For this data set DFA was used in an unsupervised mode. The optimal wavelet was found to be Symmlet 9. Fig. 23 shows the results from the multiscale cluster analysis. 

As stated in the previous section, optimal wavelet image compression can be achieved by integrating the process of wavelet construction with best-basis search. The best-basis search using a standard off-the-shelf Coifman wavelet (Nf = 2) for four levels is shown in Fig. II. In this case the threshold cost function used was simply the constant value 0.2. The resulting compression ratio obtained was 9.50. 

Optimal wavelets for biomedical signal compression(Nielsen M,... 

Wang G, Wang Z, Chen W et al. (2006) Classification of surface EMG signals using optimal wavelet packet method based on Davies-Bouldin criterion. Med Biol Eng Comput 44 865-872... 

The space-frequency localization of wavelets has lead other researchers as well (Pati, 1992 Zhang and Benveniste, 1992) in considering their use in a NN scheme. In their schemes, however, the determination of the network involves the solution of complicated optimization problem where not only the coefficients but also the wavelet scales and positions in the input space are unknown. Such an approach evidently defies the on-line character of the learning problem and renders any structural adaptation procedure impractical. In that case, those networks suffer from all the deficiencies of NNs for which the network structure is a static decision. 

A disadvantage of Fourier compression is that it might not be optimal in cases where the dominant frequency components vary across the spectrum, which is often the case in NIR spectroscopy [40,41], This leads to the wavelet compression [26,27] method, which retains both position and frequency information. In contrast to Fourier compression, where the full spectral profile is fit to sine and cosine functions, wavelet compression involves variable-localized fitting of basis functions to various intervals of the spectrum. The... 

Figure 5 Use of metabolite basis functions to fit clinical MRS data. (A) Final metabolite + baseline fit (black) overlaid on raw data (grey). (B) Non-parametric baseline signal estimation (based on wavelet filtering). (C) Metabolite basis functions modulated via scaling, B0 shift, lineshape and phase 0 and phase 1 to optimally fit raw data. (D) Residual spectrum of metabolite + baseline minus the raw data.

XJ Jiao, MS Davies, and GA Dumont. Wavelet packet analysis of paper machine data for control assessment and trim loss optimization. Pulp Paper Canada, 105(9) T208-211, 2004. 

When both the scaling and wavelet coefficients are filtered there is a surplus of information stored in the wavelet packet tree. An advantage of this redundant information is that it provides greater freedom in choosing an orthogonal basis. The best basis algorithm is a routine which endeavours to find a basis in the WPT which optimizes some criterion. 

The best basis algorithm seeks a basis in the WPT which optimizes some criterion function. Thus, the best basis algorithm is a task-specific algorithm in that the particular basis is dependent upon the role for which it will be used. For example, a basis chosen for compressing data may be quite different from a basis that might be used for classifying or calibrating data, since different criterion functions would be optimized. The wavelet packet coefficients which are resultant of the best basis, may then be used for some specific task such as compression or classification for instance. 

If some stopping criterion has been reached, then the algorithm proceeds to Step 10 where the Lawton matrix condition is verified. Provided Conditions 1 and 2 of Section 4 hold, then the Lawton matrix condition will not be satisfied for exceptional degenerate cases, thus the Lawton matrix is verified after the adaptive wavelet has been found. Finally, the multivariate statistical procedure can be performed using the coefficients X " (to). The optimizer used in the adaptive wavelet algorithm is the default unconstrained MAT-LAB optimizer [12]. 

The adaptive wavelet algorithm outlined in Section 6 can be used for a variety of situations, and its goal is reflected by the particular criterion which is to be optimized. In this chapter, we apply the filter coefficients produced from the adaptive wavelet algorithm for discriminant analysis. It was stated earlier that the dimensionality is reduced by selecting some band(jg,xg) of wavelet coefficients from the discrete wavelet transform. It then follows that the criterion function will be based on the same coefficients i.e. Xl " (xg). 

Fig. 4 The CVQPM for the coefficients at initialization and termination of the adaptive wavelet algorithm. Optimization was based on (a) the eoefficients A (0) and (h) the...

For the Wilk s Lambda criterion, optimization was based on band(3,3), while the entropy criterion optimized over band(3,2). The CVQPM criterion optimized over the scaling band(3,0). Some features which we might expect from the adaptive wavelet algorithm, is that at termination, the band on which optimization was based would outperform the other bands, at least in... 

Constrained optimization versus unconstrained optimization. In the adaptive wavelet algorithm, it was possible to avoid using constraints which ensured orthogonality. This is due to some clever algebraic factorizations of the wavelet matrix for which much credit is due to [6]. However, one constraint which we have not discussed in very much... 

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