Discrete wavelet transformation

Just as the discrete Fourier transform generates discrete frequencies from sampled data, the discrete wavelet transform (often abbreviated as DWT) uses a discrete sequence of scales aj for j < 0 with a = 21/v, where v is an integer, called the number of voices in the octave. The wavelet support — where the wavelet function is nonzero — is assumed to be -/<72, /<72. For a signal of size N and I < aJ < NIK, a discrete wavelet / is defined by sampling the scale at a] and time (for scale 1) at its integer values, that is... 

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. 

We now focus on the wavelet transform for discrete signals and, more significantly, for finite length (non-infinite) discrete signals since, in practice, most signals are finite. We develop the matrix representation for the discrete wavelet transform. 

Rather than having a continuum of wavelet dilations as in the CWT, the discrete wavelet transform uses discrete dilations that can be thought of as filters of different scales. These act as cutoff frequencies to divide the signal into different frequency bands. Wavelets are actually a pair of filters, called the wavelet and the scaling function, as depicted in Figure 7. To separate each... 

Fig. 6 The ROIs shown in Fig. 2 denoised by discrete wavelet transform, (a) background, (b)...

Step 1. Generate the finite, discrete dyadic wavelet transform of data using Mallat and Zhong s (1992) cubic spline wavelet (Fig. 8c). 

The discrete wavelet transform can be represented in a vector-matrix notation... 

Fig. 40.43. Waveforms for the discrete wavelet transform using the Haar wavelet for an 8-points long signal with the scheme of Mallat s pyramid algorithm for calculating the wavelet transform coefficients.

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... 

Fig. 2.9. Left transient reflectivity change of Bi at various pump densities. Right discrete wavelet transformation spectra obtained for time delay of 0.3 ps (solid line) and 3.0 ps (dotted line). Inset in right panel shows the Aig frequency as a function of the time delay. The dashed line in inset represents the equilibrium frequency. From [36]...

DCT DIC DICOM DIM DWT Discrete Cosine Transformation Differential Interference Contrast Digital Imaging Communications in Medicine Diffraction Imaging Microscopy Discrete Wavelet Transform... 

Both the signal and wavelet are. V-pcriodi/cd. Then, the discrete wavelet transform of t, Wt[n, aJ], is defined by the relation... 

FIGURE 10.23 The cascade of wavelet coefficient vectors output from the wavelet tree filter banks defining the discrete wavelet transform in Figure 10.22. A db-7 mother wavelet was used for the decomposition of the noisy signal in Figure 10.1. 

Figure 10.23 demonstrates one aspect of discrete wavelet transforms that shows similarity to discrete Fourier transforms. Typically, for an. V-point observed signal, the points available to decomposition to approximation and detail representations decrease by (about) a factor of 2 for each increase in scale. As the scale increases, the number of points in the wavelet approximation component decreases until, at very high scales, there is a single point. Also, like a Fourier transform, it is possible to reconstruct the observed signal by performing an inverse wavelet transform,... 

The evaluation of the measurements, the correlation between the medium components and the various ranges of the 2D-fluorescence spectrum was performed by Principal Component Analysis (PCA), Self Organized Map (SOM) and Discrete Wavelet Transformation (DWT), respectively. Back Propagation Network (BPN) was used for the estimation of the process variables [62]. By means of the SOM the courses of several process variables and the CPC concentration were determined. 

The advantage of this transform is that its kernel f(s, x, x) is left unspecified. The discrete wavelet transform was invented by Haar125, used by petroleum geologists to extract meaningful data from noisy seismograms, and later utilized in JPEG2000 pixel compression. 

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