Wavelet transforms applications

J. Kautsky, An Algebraic Construction of Discrete Wavelet Transforms, Applications of Mathematics 3 (1993), 169-193. 

P. Guillemain, R. Kronland-Martinet and B. Martens. Estimation of Spectral Line with the Help of the Wavelet Transform - Application in NMR Spectroscopy, ir Wavelets and Applications, Proceedings of the Second International Conference or Wavelets and Their Applications, Marseilles, France. May 1989. (Y. Meyer Ed.) Springer-Verlag, Paris, 1992, pp. 38-60. 

Describing below very briefly the results of the wavelet transform application to the Granat ART-P data, we are willing to demonstrate that this method can be also beneficially used for the analysis of X-ray images obtained with coded-mask telescopes. 

F. Ehrentreich, Anal. Bioanal. Chem., 372,115-121 (2002). Wavelet Transform Applications... 

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... 

D. Juan-Rimbaud, B. Walczak, R.J. Poppi, O.E. de Noord and D.L. Massart, Application of wavelet transform to extract the relevant component from spectral data for multivariate calibration. Anal. Chem., 69 (1997) 4317-4323. 

Kazemeini H., Juhlin C., et al. Application of the continuous wavelet transform on seismic data for mapping of channel deposits and gas detection at the C02SINK site, Ketzin, Germany. 2008 Geophysics Prospect 57 111-123. 

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]. 

Generally, chemometrics handles data sets constituted by many objects described by the same variables. In this perspective, the application of wavelet transform should be performed, obtaining, for all the objects, a single basis formed by the same coefficients, the so-called common best basis. 

While we have discussed pre-processing steps in a sequential manner, newer approaches are capable of integrating all the steps into single transformation approaches. For example, applications of wavelet transforms [77] can... 

Barclay, V.J., Bonner, R.F., and Hamilton, I.P., Application of wavelet transform to experimental spectra smoothing, denoising, and data set compression, Anal. Chem., 69, 78, 1997. 

Chau, F.-T. and Leung, A.K.-M., Application of wavelet transform in processing chromatographic data, Data Handling Sci. Technol., 22, 205-223, 2000. 

R277 F.-T. Chau and A. K.-M. Leung, Applications of Wavelet Transform in Spectroscopic Studies , Data Handl Sci. TechnoL, 2000, 22, 241... 

Wavelet transforms are a quite new field of data processing, but they have proven to be a valuable addition to the analyst s collection of tools. Therefore, they should be introduced here a detailed discussion of how the transformation procedure is applied to descriptors can be found in the next chapter. A general overview of wavelet transformations is described by Strang [57] the mathematical details are published by DeVore and Lucier [58] and a review on applications in chemistry is given by Leung et al. [59]. 

Principles and applications of wavelet transformation to chemometrics. Anal. Chim. Acta, 420, 169-180. 

Qin X, Shen LS, Wavelet transform and its application in spectral analysis, Spectroscopy and Spectral Analysis, 2000, 20, 892-897. 

This vast spectral bandwidth illustrates the necessity of a reliable scale and time resolved decomposition of available observations to separate and describe single processes as individual parts of the whole system. Often, the comlex interplay between climate subsystems plays an essential role and the understanding of coupling mechanisms is of crucial importance for the study and prediction of at first sight independent phenomena. Continuous wavelet transformation (CWT) is the prototypic instrument to address these tasks As an important application, it transforms time series to the time/scale domain for estimating the linear non-stationary spectral properties of the underlying process. 

As stated previously, with most applications in analytical chemistry and chemometrics, the data we wish to transform are not continuous and infinite in size but discrete and finite. We cannot simply discretise the continuous wavelet transform equations to provide us with the lattice decomposition and reconstruction equations. Furthermore it is not possible to define a MRA for discrete data. One approach taken is similar to that of the continuous Fourier transform and its associated discrete Fourier series and discrete Fourier transform. That is, we can define a discrete wavelet series by using the fact that discrete data can be viewed as a sequence of weights of a set of continuous scaling functions. This can then be extended to defining a discrete wavelet transform (over a finite interval) by equating it to one period of the data length and generating a discrete wavelet series by its infinite periodic extension. This can be conveniently done in a matrix framework. 

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