Detail coefficients

The coefficients in the H-matrix are the detail coefficients. The output of the H-matrix are the /-components. With N/2 detail components and N/2 approximation components, we are able to reconstruct a signal of length N. 

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... 

We examine a few particular radial coefficients in detail. Coefficients Mq pertaining to adiabatic corrections in Y R) appear in only an expression for auxiliary coefficient Zq q, explicitly in this form [55] ... 

Due to the down-sampling procedure during decomposition, the number of resulting wavelet coefficients (i.e., approximations and details) at each level is exactly the same as the number of input points for this level. It is sufficient to keep all detail coefficients and the final approximation coefficient (at the coarsest level) to be able to reconstruct the original data. The... 

Example The DWT of the two signals defined in Eq. 6.6 and Eq. 6.7 are shown in Figures 6.8 and 6.9, respectively. Some random noise (zero mean, unit variance) is also added to each signal to distort the original features slightly. Figures show the approximate and the detail coefficients of the MSD for a three level decomposition. Following observations are made ... 

The detail coefficients show the strength of the signal component removed at each scale level. Especially in Figure 6.9, one can clearly see how the first level removes the noise components followed by signal components with distinct frequency behavior. 

Each additional resolution — up to the highest resolution level J—decomposes the coarse coefficients and leaves the detail coefficients unchanged. The remaining coarse coefficient cannot be decomposed further it consists of just four components. J is determined by the size n of the original vector with J = log2(n) - 2. Consequently, a wavelet-transformed descriptor can be represented by either single-level (j = 1) or multilevel (/< = /) decomposition. 

The resolution level can be chosen arbitrarily between 1 and J. Any of the valid combinations of coarse and detail coefficients at a certain resolution level that lead to a descriptor of the same size are possible. For example, an original RDF descriptor with 256 components (i.e., / = 6) can be decomposed up to the resolution level j = 3, and the Wavelet transform (WLT) can be represented as + D -1- + )< ( This... 

Combining coarse- and detail-filtered complete decomposition results in a combination of the coarse coefficients of the last resolution level and all the detail coefficients D - ... 

Figure 5.21 displays the combination of the coarse and the detail coefficients filtered up to the highest possible resolution level (/ = 6 with 256 components in the original RDF descriptor). In this case, the final (and only) coarse coefficient... 

User-defined combination of coarse and detail coefficients. 

A 3-band DWT for the spectrum x = (xq.xi,. .., xg) is shown in Fig. 1. There is one low pass and two high pass filters producing one set of scaling (or smoothed) coefficients and two sets of wavelet (or detailed) coefficients. As before, to go from one level to the next, only the scaling coefficients are filtered and the number of coefficients in each band is reduced by one third when moving from one level to the next. We have presented a transform with two levels (niev = 2)... 

Let us consider the discrete wavelet transform with the Haar wavelet. If the DWT with the Haar wavelet is applied to a time series the detail coefficients supply information about the temporal change of the time series. The detail coefficients of different levels correspond to changes on different time scales. Hence, these coefficients may serve as a measure of change of a time series. If a transition occurs in a time series the detail coefficients take values that are larger than the values they take during stable states. Below it is described how this property is used for detection of transitions. 

Since online applications are considered in this chapter only the most recent development of a time series is of interest for our problem. This means that for a DWT down to level j the detail coefficients that measure the most recent change are the j — 1 coefficients dj.k where k takes the values 2J 2,2 2, ..., 2 These are the detail coefficients of the different levels that measure the most recent change of the time series (k = 2> corresponds to the last approximation coefficient, a time series of length 2 is assumed). 

Note that if transitions of a maximum length of 2 points in time are considered for online analysis it is only necessary to calculate the DWT for the latest 2 measurements as this is the maximum number of necessary measurements for J applications of the DWT (the detail coefficients that correspond to the largest time scales will then consist of only one number). Hence, a window of length 2 is moved across the data. The right edge of this window is the last measurement. For simplicity of notation these coefficients will in the following be denoted dh with h = 1.J (see also Fig. 

As it is usually not known in advance on what time scales transitions can occur different values h in general an interval has to be considered. If the window on the data is chosen such that a DWT up to level J can be performed there are J different detail coefficients/levels di.dj. Therefore two... 

Wavelet-based software packages generate the sub-bands in the form of a grouped display of the smoothed and detailed sub-bands. For example, the grouped display of the sub-bands in a single decomposition step as generated by S + WAVELETS is shown in Fig. 4. Consistent with the notation used in this book, the cl and dl labels represent the smoothed and detailed coefficients, respectively, so that cl-cl corresponds to the smoothed (Il,l) wavelet coefficients, whereas cl-dl , dl-cl and dl-dl correspond to the detailed wavelet coefficients (Il.h, Ih.l and Ih.h sub-bands, respectively). NB In some wavelet software packages (e.g. S -l- WAVELETS ), the... 

For each decomposition level f a faithful reconstruction of the original signal is possible using the inverse discrete wavelet transform (IDWT) and the set of approximation coefficients obtained at level altogether with all sets of detail coefficients from level f until level 1. IDWT is achievable by upsampling the coefficients obtained at level j and applying Eq. 9.18 ... 

In DWT, the digital filters are repeatedly applied to the sets of approximation and detail coefficients until a series of wavelet components obtained at a certain decomposition level is chosen as the result. 

Figure 9 This series illustrates multiresolution analysis, separating out the high-frequency information at each level of transformation in the pyramid algorithm (illustrated in Figure 8). Note that the approximation (a,) signal in higher level iterations contain much less detailed information, because this has been removed and encoded into wavelet detail coefficients at each DWT deconstruction step. To reconstruct the original signal, the inverse DWT needs the wavelet coefficients of a given approximation level / (ai) and all detail information leading to that level (di i).

MR A analysis, the corresponding frequency bands are separated as illustrated in Figure 4.7b. By DWT, the signal f t) is decomposed into two parts low-pass approximation coefficients and high-pass detail coefficients. The next step then decomposes the new approximation coefficients. 

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