Discretized Continuous Wavelet

Handy, C. R. and Murenzi, M., (1999) Moment Quantization and (A-adic) Discrete-Continuous Wavelet Transform Theory , J. Phys. A Math. Gen. 32, 8111. 

Wavelet transforms (WT) are classified into continuous wavelet transforms (CWTs) and discrete wavelet transforms (DWTs). Wavelet is defined as the dilation and translation of the basis function /(t), and the continuous wavelet transforms is defined as [Shao, Leung et al, 2003]... 

Wavelets and the wavelet transformation refer the representation of a spectral data set in terms of a finite spectral range or a rapidly decaying oscillating waveform. This waveform is scaled and translated to match the original spectmm. Wavelet transformation may be considered to calculate the time-frequency representation, related to the subject of harmonic analysis. The projection of a spectmm on a single wavelet or a series of wavelets reduces the dimensionality of the data set. Wavelet transforms are broadly divided into three classes, the continuous wavelet transform, the discrete wavelet transform and multiresolution-based wavelet transforms. Each class has advantages and disadvantages in terms of the wanted information. 

The main difference between the continuous wavelet transform and the discrete wavelet transform (of continuous functions) is that the wavelet is stretched or dilated by 2 j for some integer), and translated by 2 k for some integer k. For example if j = 2, the children wavelets will be dilated by and translated by k. 

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. 

An important feature of wavelet analysis is to find the most appropriate mother function. This is not always obvious. The ranges of a and b are flexible, giving rise to continuous wavelets if unlimited [6], or orthonormal discrete wavelets if limited [7]. For the atomic orbital example, above, the authors demonstrated the effect of choosing as a mother function... 

Since the original formulation of Continuous Wavelet Transform (CWT) theory by Grossmann and Morlet (1984), including its orthonormal discretization by Daubechies (1988), wavelets have become an important tool in the multiresolution analysis of signals and images, as well as the mathematical and computational study of ordinary and partial differential equations. 

There are various kinds of wavelets, including continuous wavelets, discrete orthogonal wavelets, spline wavelets, local cosines, and wavelet packets. Different applications... 

Raw nuclear magnetic resonance (NMR) data consist of damped oscillations produced by choruses of nuclei with different resonant frequencies. Analysis consists of determining those resonances, and is traditionally done with the discrete Fourier transform. However, the time variation of the amplitudes introduces errors, and the weakness of the resonances forces many repetitions to get adequate height in the peaks. The continuous wavelet transform spectral estimation algorithm has recently been used instead, to avoid both these problems. Without the need for repetitions, it is possible to analyze chemical reactions at time steps comparable with nuclear spin relaxation times. 

Multiresolution analysis (MRA) [7,8,9] provides a concise framework for explaining many aspects of wavelet theory such as how wavelets can be constructed [1,10]. MRA provides greater insight into the representation of functions using wavelets and helps establish a link between the discrete wavelet transform of continuous functions and discrete signals. MRA also allows for an efficient algorithm for implementing the discrete wavelet transform. This is called the fast wavelet transform and follows a pyramidal... 

These operations can be considered equivalent to the discrete wavelet transform of a continuous function using higher multiplicity wavelets. 

We have demonstrated that it is possible to obtain the discrete wavelet transform of both continuous functions and discrete data points without having to construct the scaling or wavelet functions. We only need to work with the filter coefficients. One may begin to wonder where the filter coefficients actually come from. Basically, wavelets with special characteristics such as orthogonality, can be determined by placing restrictions on the filter coefficients. 

When discrete wavelets are used to transform a continuous signal, the resulting set of coefficients is called the wavelet series decomposition. For this transformation to be useful it must be invertible, and for synthesis to take place, Eq. 9.11 must be satisfied... 

Heil, C.E., Walnut, D.F. Continuous and discrete wavelet transforms. SIAM Review 31, 628-... 

The discrete wavelet l/(k) can be, but not neeessarily is, a sampled version of a continuous counterpart. 

To avoid the evaluation of these quantities, the continuous phase equilibrium function is expressed in terms of the single-space basis e.g. K T,p, F ) G Vn- Using the fast Wavelet transform T the single-scale representation of the trial solution is obtained and the residuum of (1.3) can be formulated in terms of the single-scale basis The residuum is thus expressed by means of scaling functions of common level n. A subsequent fast wavelet transform provides the residuum of (1.3) in the multiscale representation, whose inner products with the weight functions can easily be evaluated due to orthogonality. This approach exactly recovers the discrete model without any deviations due to the continuous model formulation if the Haar basis is used. 

WT methods can be categorized into two main classes (1) continuous WT (CWTs) and (2) discrete wavelet transforms (DWTs). Each is discussed below. 

By a wavelet transform it becomes possible to detect unusual as well as periodical signals. Several kinds of wavelet transforms have been proposed, including continuous and discrete transforms. Examples are summarized in Table 13. 

Another method used for ECG peaks detection is wavelet transform. It is normally used for analyzing heart rate fluctuations due to its ability processing data at different scales and resolutions. Besides that, wavelets are normally used to represent data and other functions whenever the equations satisfy certain mathematical expressions. Basically, a wavelet equation depends on two parameters, scale a, and position T. These parameters vary continuously over the real numbers. If scale o = 2 (jez, z is an integer set), then the wavelet is called dyadic wavelet and its corresponding transform is called Discrete Wavelet Transform (DWT). The related equation [6] is... 

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