Wavelet packets decomposition

The first step in obtaining the wavelet packet coefficients from the best basis is to liroduce the wavelet packet decomposition tree to some level jo- A criterion measure for each of the wavelet packet coefficients in each node (or band) in the wavelet packet decomposition is calculated and is denoted by... 

For each variable, use the same best full wavelet packet base algorithm to process wavelet packet decomposition tree and find best wavelet packet decomposition 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... 

Fig. 2 Dyadic wavelet packet transform (WPT) decomposition scheme showing time-frequency segmentation and wavelet packet tree (depth = 3).

It is also possible to apply a filter F to decompose all m signals, using the Wavelet Packet Transform (WPT). For each signal, a matrix is obtained that contains the wavelets coefficients (see Fig. 6). Element denotes the ith wavelet coefficient at the jth level in the r band of the kth signal decomposition. 

Whilst the 2D-DWT provides an efficient space-frequency characterisation of a given image, it only uses a fixed decomposition of the pixel space. As in the case of the 1-D wavelet packet transform, we can extend the wavelet packets to two dimensions. That is, the 2D wavelet packet transform (2D-WPT) generates a more general, full m -ary tree representation with a total of m + m + m sub-bands for h levels. Each sub-band in a given level of the tree splits into a smoothed sub-band and m"-l detailed sub-bands, resulting in a tree that resembles an m-way pyramidal stack of sub-bands. For the case of a dyadic decomposition scheme, this corresponds to a pyramidal sub-band structure where each sub-band is decomposed into 2 = 4 sub-bands at each successive (higher) level (see Fig. 2). Fig. 7 shows results of the third level of the 2D WPT for the dyadic case - a total of = 64... 

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