Wavelet packet coefficients

In Fig. 1 the number of bands doubles from one level to the next (lower) level, since each of the bands in the previous level are passed through a low-pass and a high-pass filter. At the next level, there will be four bands of wavelet packet coefficients which are obtained by... 

The same procedure may continue until there is one wavelet packet coefficient in each of the bands. As for the DWT. there can be a maximum of J levels in the WPT, the main difference is that the WPT has bands at each level j e J, J - I,..., 0. ... 

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. 

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... 

Fig. 4 describes the best basis algorithm, or more specifically how to find the wavelet packet coefficients from the best basis algorithm. Step 1 performs the WPT to some prespecified level jo as described previously. Step 2 then initializes a current best basis or best set of bands. Initially, the best set of bands (BB) is simply all the bands at level Jq in the WPT. Steps 3-9 begin to compare the cost measure of the parent nodes against the current best of bands which are descendants of the parent nodes being examined. Here, the aim is to minimize the cost measure. 

Obtaining the wavelet packet coefficients From the best basis algorithm... 

One commonly used cost function particularly in data compression is entropy. If we let Wj t j denote the ith wavelet packet coefficient band(j, t) of the wavelet packet transform, then the entropy-like criterion for band(j, t) is defined as follows ... 

Perform the wavelet packet transform using filter coefficients associated with the Haar wavelet and scaling functions, then, compute the wavelet packet coefficients associated with the best basis using the entropy cost function for the signal x = (0.0000,0.0491.0.1951,0.4276,0.7071,0.9415,0.9808,0.6716). 

Expanding m vectors into wavelet packet coefficients... 

Use retain score matrixes and load matrixes to rebuild wavelet packet coefficients matrixes... 

It shows that the principle components score vectors of the matrix WpX are the wavelet packet transformations of the principle components score vectors of the matrix X. Done. Theorem 2 When no principle component is ignored in any scales and no wavelet packet coefficient are eliminate by the threshold value, the result of WPPCA is equal to the result of PCA. 

Proving From the properties of wavelet packet transformation and wavelet packet reconstruction we will see that if the wavelet packet coefficients are not treated, the wavelet packet reconstructed signals are the same as the original signals. When all principle components are reserved, the reconstructed data matrix is the same with the original data matrix. Done. 

The notation which characterises each wavelet packet Wy reflects the scale 2 and location 2> k. The wavelet packet coefficients are then produced from the integral,... 

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... 

This is done in the same way that the smoothed (or scaling coefficients) are filtered. Fig. 1 presents the structure of a wavelet packet transform for some... 

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. 

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. 

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 ... 

For these coefficients, respectively use wavelet packet de-noise limit method to process these coefficients and get de-noising coefficients ... 

In which WPL,2. i=0,l,...,L) denotes the wavelet packet filter coefficient matrix. The relations between principle component analysis for X and the principle component analysis for WpX are given by following two theorems ... 

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