Wavelet bases

Kovacevic, J., and Vetterli, M., Nonseparable multiresolutional perfect reconstruction banks and wavelet bases for R". IEEE Trans. Inf. Theory, 38, 533 (1992). 

Let us now see how the theory of the wavelet-based decomposition and reconstruction of discrete-time functions can be converted into an efficient numerical algorithm for the multiscale analysis of signals. From Eq. (6b) it is easy to see that, given a discrete-time signal, FqU) we have... 

In Section II we defined the trend of a measured variable as a strictly ordered sequence of scaling episodes. Since each scaling episode is defined by its bounding inflexion points, it is clear that the extraction of trends necessitates the localization of inflexion points of the measured variable at various scales of the scale-space image. Finally, the interval tree of scale (see Section II) indicates that there is a finite number of distinct sequences of inflexion points, implying a finite number of distinct trends. The question that we will try to answer in this section is, How can you use the wavelet-based decomposition of signals in order to identify the distinct sequences of inflexion points and thus of the signal s trends ... 

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... 

One problem encountered in solving Eq. (11.12) is the modeling of the prior distribution P x. It is assumed that this distribution is not known in advance and must be calculated from historical data. Several methods for estimating the density function of a set of variables are presented in the literature. Among these methods are histograms, orthogonal estimators, kernel estimators, and elliptical basis function (EBF) estimators (see Silverman, 1986 Scott, 1992 Johnston and Kramer, 1994 Chen et al., 1996). A wavelet-based density estimation technique has been developed by Safavi et al. (1997) as an alternative and superior method to other common density estimation techniques. Johnston and Kramer (1998) have proposed the recursive state... 

Safavi, A., Chen, J., and Romagnoli, J. A. (1997). Wavelet-based density estimation and application to process monitoring. AIChE J. 43, 1227-1241. 

Wang, D., and Romagnoli, J. A. (1998). Wavelet Based Robust Estimation, Internal Rep. PSE-I-No. 5. University of Sydney, Laboratory of Process Systems Engineering, Sydney, Australia. 

A. A Wavelet-Based Method to Detect Putative Replication Origins... 

A. Arneodo, B. Audit, N. Decoster, J.-F. Muzy, and C. Vaillant, Wavelet based multifractal formalism Application to DNA sequences, satellite images of the cloud structure and stock market data, in The Science of Disasters Climate Disruptions, Heart Attacks, and Market Crashes, Springer-Verlag, Berlin, 2002, pp. 26-102. 

A. Arneodo, Y. d Aubenton-Carafa, E. Bacry, P. V. Graves, J.-F. Muzy, and C. Thermes, Wavelet based fractal analysis of DNA sequences. Physica D 96, 291-320 (1996). 

Filterbanks. There is still continued research on filter banks for high quality audio coding. Topics include wavelet based filter banks, low delay filter banks [Schuller, 1995] or variable filter banks allowing a higher degree of variability than classic window switching [Princen and Johnston, 1995],... 

Wavelet based filter banks. In the last few years a number of audio coding systems have been proposed using wavelet based filters [Sinha and Tewfik, 1993]. A thorough description of the theory of wavelet based filter banks can be found in [Vetterli and Kovacevic., 1995],... 

Ganesan R, Das T, Sikder A, Kumar A. Wavelet-based identification of delamination defect in CMP (Cu-low k) using nonstationary acoustic emission signal. IEEE Trans Semicond Mannf 2003 16(4) 677-685. 

Wallace G. S. and Bergantz G. W. (2002) Wavelet-based correlation (WBC) of zoned crystal populations and magma mixing. Earth Planet. Sci. Lett. 202, 133—145. 

An important property of wavelet bases is their lack of translational invariance. In other words, when a pattern is translated, its descriptors are not only translated but also modified. This is a direct consequence of the down-sampling procedure and leads to distorted reconstruction of the underlying signal features. A possible solution is to omit down-sampling, resulting in a redundant family of coefficients. 

MS Crouse, RD Nowak, and RG Baraniuk. Wavelet-based statistical signal pocessing using hidden Markov models. IEEE Trans, on Signal Processing, 46 886-902, 1998. 

F Doymaz, A Bakhtazad, JA Romagnoli, and A Palazoglu. Wavelet-based robust filtering of process data. Comput. Chem. Engg., 25 1549-1559, 2001. 

Cocchi, M., Corbellini, M., Foca, G., Ludsano, M., Pagani, M.A., Tassi, L. and Ulrid, A. (2005) Classiflcation of bread wheat flours in different quality categories by a wavelet-based feature selection/dassiflcation algorithm on NIR spectra. Anal. Chim. Acta, 544, 100—107. 

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