The wavelet packet transform

A wavelet basis allows a time-frequency analysis similar to that of a short-time Fourier basis. It is different in that its time-localisation is better hence its frequency localisation is worse, for high frequencies than for low fre- 

Being able to zoom in is a nice feature, but what if one does not know what to zoom in on, which is the most likely situation in chemical applications. We do not usually know what tiling of the time-frequency domain is most suited for, let us say, our NIR spectrum. Fortunately, one does not need to know this in advance. Techniques exist that select the best basis for a particular situation from the wealth of bases offered by wavelets and wavelet packets. 

In practice, we may not even need a basis. A basis allows us to go back to the domain of the original signal, but sometimes there is no need to go back, e.g. when we want to extract features from our signals. 

Mallet, O. de Vel and D. Coomans Statistics and Intelligent Data Analysis Group, 

School of Computer Science, Mathematics and Physics, James Cook University. Townsville, Australia 

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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... 

B. Walczak and D.L.Massart, Tutorial Noise suppression and signal compression using the wavelet packet transform. Chemom. Intell. Lab. Syst., 36 (1997) 81-94. 

Fig. 2 Example of the wavelet packet transform applied to a simulated signal.

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). 

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. 

Two special applications of WT to chromatographic studies have been reported in recent years. Collantes et al. [44] proposed the employment of the wavelet packets transform (WPT) for pre-processing HPLC results by an artificial neural network. The application of WPT for data processing in chemistry is very rare. These authors aimed to evaluate several artificial... 

B. Walczak and D.L. Massart, Noise Suppression and Signal Compression using the Wavelet Packet Transform. Chcmometric Intelligent Laboratory Systems. 36... 

The invertible transformation stage uses a different mathematical basis of features in an attempt to decorrelate the data. The resulting data will have a set of features that capture most of the independent features in the original data set. Typical features used include frequency and spatial location. The transformation is nearly loss-less as it is implemented using real arithmetic and is subject to (small) truncation errors. Examples of invertible transforms include the discrete cosine transform (DCT), the discrete wavelet transform (DWT) and the wavelet packet transform (WPT). We will investigate these transforms later. 

Improvements in Ultrasonic Testing of Adhesive Bonds using the Wavelet Packet Transform (S. Tavroul and C. Jones, eds.), School of Engineering and Science, School of Mathematical Sciences, Swinburne University of Technology, Melbonme, Australia, 2006. 

Suppression and Signal Compression Using the Wavelet Packet Transform - A Tutorial. 

Theorem 1 The principle component load vector for WpX is same as the principle component load vector for X, and the principle component score vector for WpX is the wavelet packet transform of the principle component score vector for X. 

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. 

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