Wavelet transform detail-filtered

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... 

FIGURE 5.21 Combination coarse- and detail-filtered transformed RDF descriptor performed at the highest resolution level (J = 6). The transform is an alternative representation of an RDF in the wavelet domain. This signal consists of the coefficients G ) + D -1- 0(S)... 

The coarse- or detail-filtered wavelet transform or a combination of the coefficients for an RDF descriptor at a certain resolution level is done right before any postprocessing, like normalization. 

The WPT has a tree-like structure, where each band in the transform produces two new children bands at the next lower level. The tree-like structure occurs because now the detailed (or wavelet) coefficients are filtered through a low-pass and a high-pass filter to obtain the next lower level of the WPT. 

A 3-band DWT for the spectrum x = (xq.xi,. .., xg) is shown in Fig. 1. There is one low pass and two high pass filters producing one set of scaling (or smoothed) coefficients and two sets of wavelet (or detailed) coefficients. As before, to go from one level to the next, only the scaling coefficients are filtered and the number of coefficients in each band is reduced by one third when moving from one level to the next. We have presented a transform with two levels (niev = 2)... 

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