Separable wavelets

The one-dimensional (1-D) discrete wavelet transform (DWT) defined in the first part of the book can be generalised to higher dimensions. The most general case has been studied by Lawton and Resnikoff [1]. An N-dimen-sional (N-D) DWT is described also in [2]. The separable extension of the wavelet transform (WT) to three dimensions, for example, is explained in [2, 3,4]. In this chapter, for simplicity and because of the problems studied, only the theory of the 2-D and 3-D DWT will be outlined, and only separable 2-D and 3-D wavelets will be considered. These wavelets are constructed from one-dimensional wavelets. Separable wavelets are most frequently used in practice, since they lead to significant reduction in the computational complexity. 

B. A. Hess Dr. Gaspard has introduced the vibrogram as a tool to extract periodic orbits from the spectrum by means of a windowed Fourier transform. This raises the question whether other recent techniques of signal analysis like multiresolution analysis or wavelet transforms of the spectrum could be used to separate the time scales and thus to disentangle the information pertinent to the quasiclassical, semiclassical, and long-time regimes. Has this ever been tried ... 

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. 

The use of multiple sample injections (up to a maximum of three) was found to enhance S/N, i.e., the S/N is slightly higher than the square root of the number of injected sample plugs. In addition, multipoint detections of a two-component sample [700,701] or four-component sample [699] were also achieved, but the separation resolution was not as good as that obtained from the conventional single-point detection [701]. Besides Fourier transform, wavelet transform was also used in multipoint fluorescent detection to retain some time information in addition to the frequency information [702]. 

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. 

To our knowledge, Torrence and Compo [19] were the first to establish significance tests for wavelet spectral measures. They assumed a reasonable background spectrum for the null hypothesis and tested for every point in the time/scale plane separately (i.e. pointwise) whether the power exceeded a certain critical value corresponding to the chosen significance level. Since the critical values of the background model are difficult to be accessed analytically [11], they need to be estimated based on a parametric bootstrap ... 

There are several items regarding the adaptive wavelet algorithm which warrant further discussion. These items are now considered separately. 

Validation without an independent test set. Each application of the adaptive wavelet algorithm has been applied to a training set and validated using an independent test set. If there are too few observations to allow for an independent testing and training data set, then cross validation could be used to assess the prediction performance of the statistical method. Should this be the situation, it is necessary to mention that it would be an extremely computational exercise to implement a full cross-validation routine for the AWA. That is. it would be too time consuming to leave out one observation, build the AWA model, predict the deleted observation, and then repeat this leave-one-out procedure separately. In the absence of an independent test set, a more realistic approach would be to perform cross-validation using the wavelet produced at termination of the AWA, but it is important to mention that this would not be a full validation. 

Then, the wavelet de-noising process was performed separately on Sa and Sb. and the de-noised signals were then recombined to regenerate the signal in the original domain. 

Further developments [3] lead naturally to improved solutions of the Schrodinger equation, at least at the Hartree-Fock limit (which approximates the multi-electron problem as a one-electron problem where each electron experiences an average potential due to the presence of the other electrons.) The authors apply a continuous wavelet mother. v (x), to both sides of the Hartree-Fock equation, integrate and iteratively solve for the transform rather than for the wavefunction itself. In an application to the hydrogen atom, they demonstrate that this novel approach can lead to the correct solution within one iteration. For example, when one separates out the radial (one-dimensional) component of the wavefunction, the Hartree-Fock approximation as applied to the hydrogen atom s doubly occupied orbitals is, in spherical coordinates. 

Where in the spectral domain is the feature located is responsible for the separation of cluster 1. 2 and 3 after adding scale 3 By employing the method of systematic variation of mask vectors to scale 3. it is found that of 2 = 256 possible masking vectors there is only non-overlap between the clusters in 48 of the combinations. For these combinations it was observed that wavelet variable 5 in scale 3 (covers region 2024-1534 cm in the wavenumber domain) was always selected. 

After adding scale 3 it was observed above that cluster 2 separates out from all the others. The next problem is to localise the region(s) that is(are) responsible. Since scale 3 has 2 = 8 wavelet coefficients, there are in total 2 = 256 different masking vectors possible to test out. For each of these a DFA is performed and the overlap of cluster 2 with the other clusters is recorded. In 74 combinations a non-overlap situation was observed. In all ol these combinations, wavelet coefficient no. 3 in scale 3 was always present. 

Discriminant plots were obtained for the adaptive wavelet coefficients which produced the results in Table 2. Although the classifier used in the AWA was BLDA, it was decided to supply the coefficients available upon termination of the AWA to Fisher s linear discriminant analysis, so we could visualize the spatial separation between the classes. The discriminant plots are produced using the testing data only. There is a good deal of separation for the seagrass data (Fig. 5), while for the paraxylene data (Fig. 6) there is some overlap between the objects of class I and 3. Quite clearly, the butanol data (Fig. 7) post a challenge in discriminating between the two classes. 

An easy way to construct a multi-dimensional (e.g. 2-D) wavelet transform is, for example, to implement the tensor products of the 1-D counterparts. That is, we apply the 1-D wavelet transform separately along one dimension at a time. This, as we shall see shortly, results in one scaling function and three different mother wavelet functions. 

They show that the assumption above cannot be confirmed. This seems to be in correspondence with the results reported by Wang and Huang [17], which were obtained for the compression of medical images using a separable 3-D wavelet transform. 

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