Wavelet packet transform

Selection of the best basis or representation from the WPT hierarchy means choosing a combination of orthogonal, nonredundant coefficients from 

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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. 

B. Walczak, B. van den Bogaert and D.L. Massart, Application of wavelet packet transform in pattern recognition of near-IR data. Anal. Chem., 68 (1996) 1742-1747. 

Walczak, B., van den Bogaert, B. and Massart, D.L., Application of Wavelet Packet Transform in Pattern Recognition of Near-IR Data Anal. Chem. 1996, 68, 1742-1747. 

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

Wavelet Packet Transforms and Best Basis Algorithms... 

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... 

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). 

Wavelet packet transform and joint best-basis... 

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... 

Up to December 1998, more than 30 publications have reported spectroscopic studies with the use of a WT algorithm [9,10], Within this work, WT has been utilized in three major areas that include data denoising, data compression, and pattern recognition. Two classes of wavelet algorithm namely discrete wavelet transform (DWT) and wavelet packet transform (WPT), have been commonly adopted in the computation. The former one is also known as the fast wavelet transform (FWT). The general theory on both FWT and WPT can be found in other Chapters of this book and some chemical journals [16-18], and is not repeated here. In the following sections, selected applications of WT in different spectral techniques will be described. 

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. 

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... 

FAN, X. ZUO, M. J. (2006) Gearbox fault detection using Hilbert and wavelet packet transform. Mechanica S tems ndSiggal rocessing, 20 (4), pp. 966-982. 

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. 

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