Wavelet selection

Signal compression is gaining importance in biomedical engineering due to the potential applications in telemedicine. In this work, we propose a novel scheme of signal compression based on signal-dependent wavelets. To adapt the mother wavelet to the signal for the purpose of compression, it is necessary to define (1) a family of wavelets that depend on a set of parameters and (2) a quality criterion for wavelet selection (i.e., wavelet parameter optimization). We propose the use of an unconstrained parameterization of the wavelet for wavelet optimization. A natural performance... 

With the selection of wavelets as the basis functions the learning algorithm can now be finalized. 

Step 1. Select a family of scaling functions and wavelets. 

For the moment, there are no guidelines for the selection of the particular basis functions for any given application. The important issue here is that the properties of the wavelets will be inherited by the approximating... 

In order to compress the measured data through a wavelet-based technique, it is necessary to perform a series of convolutions on the data Becau.se of the finite size of the convolution filters, the data may be decomposed only after enough data has been collected so as to allow convolution and decomposition on a wavelet basis. Therefore, point-bypoint data compression as done by the boxcar or backward slope methods is not possible using wavelets. Usually, a window of data of length 2" m e Z, is collected before decomposition and selection of the appropriate... 

The accuracy of the error equations (Eqs. (22) and (23)] also depends on the selected wavelet. A short and compactly supported wavelet such as the Haar wavelet provides the most accurate satisfaction of the error estimate. For longer wavelets, numerical inaccuracies are introduced in the error equations due to end effects. For wavelets that are not compactly supported, such as the Battle-Lemarie family of wavelets, the truncation of the filters contributes to the error of approximation in the reconstructed signal, resulting in a lower compression ratio for the same approximation error. 

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... 

Several extensions of wavelets have been developed to improve their ability to solve practical problems. Wavelet packets (Coifman and Wick-erhauser, 1992) are a library of basis functions that cover a wide variety of shapes. The library can be searched efficiently to select the best set of... 

The variable selection methods discussed above certainly do not cover all selection methods that have been proposed, and there are several other methods that could be quite effective for PAT applications. These include a modified version of a PLS algorithm that includes interactive variable selection [102], and a combination of GA selection with wavelet transform data compression [25]. 

Usual procedures for the selection of the common best basis are based on maximum variance criteria (Walczak and Massart, 2000). For instance, the variance spectrum procedure computes at first the variance of all the variables and arranges them into a vector, which has the significance of a spectrum of the variance. The wavelet decomposition is applied onto this vector and the best basis obtained is used to transform and to compress all the objects. Instead, the variance tree procedure applies the wavelet decomposition to all of the objects, obtaining a wavelet tree for each of them. Then, the variance of each coefficient, approximation or detail, is computed, and the variance values are structured into a tree of variances. The best basis derived from this tree is used to transform and to compress all the objects. 

Besides Tikhonov regularization, there are numerous other regularization methods with properties appropriate to distinct problems [42, 53,73], For example, an iterated form of Tikhonov regularization was proposed in 1955 [77], Other situations include using different norms instead of the Euclidean norm in Equation 5.25 to obtain variable-selected models [53, 79, 80] and different basis sets such as wavelets [81],... 

The first involves smoothing. If the original data consist of 512 datapoints, and are exactly fitted by 511 wavelets, choose die most significant wavelets (those widi die highest coefficients), e.g. die top 50. In fact, if die nature of the wavelet function is selected with care only a small number of such wavelets may be necessary to model a spectium which, in itself, consists of only a small number of peaks. Replace the spectrum simply with that obtained using the most significant wavelets. 

FIGURE 4.37 Joint time-frequency domains of the AE signal for slurry 1 (a) and slurry 2 (b). The AE signal was filtered by a Debouche 05 wavelet filter. Only middle bands were selected and processed by the marching pursuit joint time-frequency domain algorithm (from Ref 27). 

The hard-thresholding filter, fjj, selects wavelet coefficients that exceed a certain threshold and sets the others to zero ... 

The next example illustrates the different representation of normal and wavelet-transformed RDF descriptors. The first 50 training compounds were selected from a set of 100 benzene derivatives. The remaining 50 compounds were used for the test set. Compounds were encoded as low-pass filtered D20 Cartesian RDF descriptors, each of a length of 64 components, and were divided linearly into eight classes of mean molecular polarizability between 10 and 26 AT... 

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