Wavelets analysis

It is difficult to give an introduction to wavelet analysis without reference to Fourier analysis, which is discussed in Section 8.2. 

An intuitive solution to fliis problem would be to allow for sine waves with finite duration to appear as building blocks in the transformed data. This is the basis of wavelet analysis. The wavelet transform is based on such building blocks or elementary functions, which are obtained by dilatations, contractions, and shifts of a unique function called the wavelet prototype (or mother wavelet) 

There are four different types of wavelet transforms  

The continuous wavelet transform. This transform is given as 

It has a parallel in the Fourier transform, and the variable t, scale a, and shift t are all continuous. 

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LASAYGUES (P.) and LEFEBRE (J.P.), Applications of wavelets analysis in ultrasonic tomography, Traitement du signal, 1995, 12, 373. 

Successive PCA and Wavelet analysis processes improve small flaw detection (figure 14), because small size involves linear physical processes, where PCA is efficient. 

Lasaygues, P.,. Lefebvre, J.P., and Mensah S., Deconvolution and Wavelet Analysis on Ultrasonic Reflection Tomography, III International Workshop, Advances in Signal Processing for Non Destructive Evaluation of Materials, Quebec, Canada, (1997). 

L. Eriksson, J. Trygg, E. Johansson, R. Bro, S. Wold, Orthogonal signal correction, wavelet analysis, and multivariate calibration of complicated process fluorescence data. Anal. Chim. Acta, 2000 420, 181-195. 

The common oscillations were consistent during the whole 45 years period, as revealed by cross-wavelet analysis (Fig. 5). Analysis for streamflow, ENSO index, and AF indicates a persistent common power for oscillations with period between 30 and 65 months, which appears to vary according to the intensification and weakening of the quasibieimial and quasiquadrennial band of ENSO. 

Fig. 5 Cross-wavelet analysis for Streamflow vs. ENSO (upper panel), Streamflow vs. AF (middle panel), and ENSO vs. AE (bottom panel). The thick black contour designates the 95% signiflcance level against red noise. Shaded areas represent the cone of influence, where interpretations should be cautious...

Since in many applications minor absorption changes have to be detected against strong, interfering background absorptions of the matrix, advanced chemometric data treatment, involving techniques such as wavelet analysis, principle component analysis (PCA), partial least square (PLS) methods and artificial neural networks (ANN), is a prerequisite. 

Steel EA, Lange IA (2007) Using wavelet analysis to detect changes in water temperature regimes at multiple scales effects of multi-purpose dams in the Willamette River basin. River Res Appl 23 351-359... 

A. Arneodo, E. Bacry, P. V. Graves, and J.-F Muzy, Characterizing long-range correlations in DNA sequences from wavelet analysis. Phys. Rev. Lett. 74, 3293-3296 (1995). 

Buranachai C, Kamiyama D, Chiba A et al (2008) Rapid frequency-domain FLIM spinning disk confocal microscope lifetime resolution, image improvement and wavelet analysis. JFluoresc 18 929-942... 

Now, thanks to the development of the local wavelet analysis [3], it is possible to represent mathematically, according to the observations, finite waves with a well-defined energy. Therefore it is possible to represent the 0 wave, the extended yet localized part of the particle, with a defined energy by a wavelet. The wavelets, or finite waves, were developed in geophysics by Morlet in the early 1980s, to avoid some shortcomings of the nonlocal Fourier analysis. 

If, instead of the nonlocal Fourier analysis, one uses the local wavelet analysis to represent a quantum particle, the uncertainty relationships may change in form. On the other hand, this process has the advantage of containing the usual uncertainty relations when the size of the basic gaussian wavelet increases indefinitely. 

From Fig. 18 it is seen that the new the uncertainty relations derived with the local wavelet analysis exhibit the form... 

It is easily seen from (90) that when the size of the basic wavelet Axo is large enough, the new relation turns itself into the old, usual Heisenberg relations, which is a very satisfactory result. This situation corresponds to the limiting case when the wavelet analysis transforms into the nonlocal Fourier analysis. 

G. R. Fleming Sure the wavepacket spreads, but not as much as you would think on this time scale. What can be done experimentally to get precise data is to do wavelet analysis to see what shape it had. That is a realistic goal for a simple system using solid-state lasers. 

P. Gaspard As far as I know, the wavelet analysis of spectra has not yet been done and would be very interesting to develop. A remark is that the vibrogram also depends on the width e of the Gaussian window, which may be varied to construct another kind of plot. 

One of the more challenging unsolved problems is the representation of transient events, such as attacks in musical percussive sounds and plosives in speech, which are neither quasi-periodic nor random. The residual which results from the deterministic/stochastic model generally contains everything which is not deterministic, i.e., everything that is not sine-wave-like. Treating this residual as stochastic when it contains transient events, however, can alter the timbre of the sound, as for example in time-scale expansion. A possible approach to improve the quality of such transformed sounds is to introduce a second layer of decomposition where transient events are separated and transformed with appropriate phase coherence as developed in section 4.4. One recent method performs a wavelet analysis on the residual to estimate and remove transients in the signal [Hamdy et al., 1996] the remainder is a broadband noise-like component. 

Chui, 1992a] Chui, C. K. (1992a). Wavelet Analysis and its Applications, Volume 1 An Introduction to Wavelets. Academic Press, inc. 

Figure 12.2c shows the temporal variation of the instantaneous frequencies for the two modes. It is interesting to observe how the frequency of the fast mode is modulated in a fairly regular manner. With about 17 modulation cycles for fjast during the 500 s of observation time, we conclude that the frequency of the fast mode is modulated by the presence of the slow mode, indicating that the two modes interact with one another. If one compares the phase of the tubular pressure variations in Fig. 12.2a with the phase of the frequency modulation in Fig. 12.2c it appears that the maximum of ffast occurs about 60° after the maximum of Pt. It is important to note, however, that the various steps of our wavelet analysis may have introduced a certain phase lag. We are presently trying to correct for such effects in order to obtain a better understanding of the instantaneous relation between the two variables. 

In a recent study [34], we made use of wavelet and double-wavelet analysis to examine the relative occurrence of various states of synchronization in pairs of interacting nephrons. We showed that both full and partial synchronization occur for normotensive as well as for hypertensive rats, and that the partial synchronization can involve only the slow oscillations or only the fast oscillations. We also used... 

Wavelet analysis takes Gabor s idea one step further it defines a windowing transform technique with variably sized window regions. The continuous wavelet transform of the sequence h(t) is defined by Equation 10.23... 

To compare the changes in correlation between the NIN03 index and the Danube discharge seasonal anomalies, we used cross-wavelet transform software, produced by Aslak Grinsted. The details of cross-wavelet analysis methodology are given in [19-21]. 

Transformation — Several approaches are available for transformation of time domain data into the - frequency domain, including - Fourier transformation, the maximum entropy method (MEM) [i], and wavelet analysis [ii]. The latter two methods are particularly useful for nonstationary signals whose spectral composition vary over long periods of time or that exhibit transient or intermittent behavior or for time records with unevenly sampled data. In contrast to Fourier transformation which looks for perfect sine... 

Wavelet analysis is a rather new mathematical tool for the frequency analysis of nonstationary time series signals, such as ECN data. This approach simulates a complex time series by breaking up the ECN data into different frequency components or wave packets, yielding information on the amplitude of any periodic signals within the time series data and how this amplitude varies with time. This approach has been applied to the analysis of ECN data [v, vi]. Since electrochemical noise is 1/f (or flicker) noise, the new technique of -> flicker noise spectroscopy may also find increasing application. 

Wavelet analysis has great potential in image processing applications of interest in mineralogy (see Moktadir and Sato (2000) for an illustrative example for silicon). As an illustration, in Figure 4 we show a version of Equation (3) over a one-dimensional trace across a two-dimensional AFM image of a hematite surface where there are some traces of bacterially mediated reduction reactions. One-dimensional wavelets with the second-derivative of the Gaussian function, also known as Mexican-hat wavelets because of their... 

Gibbs GV (1982) Molecules as models for bonding in silicates. Am Mineral 67 421-450 Gibbs GV, Hill FC, Boisen MB, Downs RT (1998) Power law relationships between bond length, bond strength and electron density distributions. Phys Chem Minerals 25 585-590 Gibert D, Holschneider M, Le Mouel J-L (1998) Wavelet analysis of the Chandler Wobble. J Geophys Res... 

Resnikoff HL, Weiss Jr RO (1998) Wavelet Analysis The Scalable Structure of Information. New York Springer-Verlag... 

Turning Point Quantization and Scalet-Wavelet Analysis (C. R. Handy)... 

Exploiting the potential of three dimensional spatial wavelet analysis to explore nesting of temporal oscillations and spatial variance in simultaneous EEG-fMRI data. Prog Biophys Mol Biol 105 67-79... 

B. J. West, A. Maciejewsk, M. Latka, T. Sebzda, and Z. Swierczynski, Wavelet analysis of scaling properties of gastric electrical activity. To appear in Am. J. Physiol. 

Dealing with boundaries becomes an issue in wavelet analysis of finite array of data. These edge effects or singularities can be avoided by using adaptive filters near the signal boundaries. 

AG Bruce, DL Donoho, HY Gao, and RD Martin. Denoising and robust nonlinear wavelet analysis. In SPIE Proceedings - Wavelet Applications, page 2242, Orlando, FL, 1994. 

Transformations of a set of molecular descriptors are often performed when there is the need of a —> variable reduction or the need to modify binary vectors, such as site and substituent-oriented variables, into real-valued variable vectors. The milestone of these techniques is the —> Principal Component Analysis (PCA), but also —> Fourier analysis and —> Wavelet analysis are often used, especially for spectra descriptors compression. 

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