Wavelet transform discrete

The CWT results in wavelet coefficients at every possible scale. Thus, there is a significant amount of redundancy in the computation. But there 

In this representation, integer j indexes the scale or resolution of analysis, i.e., smaller j corresponds to a higher resolution, and jo indicates the coarsest scale or the lowest resolution, k indicates the time location of the analysis. For a wavelet 4 t) centered at time zero and frequency u o, the wavelet coefficient dj k measures the signal content around time 2 k and frequency 2 uio- The scaling coefficient Ck measures the local mean around time 2 °k. The DWT represents a function by a countable set of wavelet coefficients, which correspond to points on a 2-D grid of discrete points in the scale-time domain. 

One can show that the relationship between high-pass and low-pass finite impulse response (FIR) filters and the corresponding wavelet and 

This approach greatly facilitates the calculation of wavelet and scaling coefficients as typically implemented in the Matlab Wavelet Toolbox [199]. One can associate the scaling coefficients with the signal approximation, and the wavelet coefficients as the signal detail. 

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 

Similar to the DFT, the DWT can be defined as the sum over all distances r of the descriptor g r) (expressed in its discrete form g[x]) multiplied by individually scaled basis functions 

The basis functions, or wavelets,, are dilated and translated versions of a wavelet mother function. A set of wavelets is specified by a particular set of numbers, called wavelet filter coefficients. To see how a wavelet transform is performed, we will take a closer look at these coefficients that determine the shape of the wavelet mother function. 

The basic idea of the wavelet transform is to represent any arbitrary function as a superposition of basis functions, the wavelets. As mentioned already, the wavelets P(x) are dilated and translated versions of a mother wavelet Tg. Defining a dilation factor d and a translation factor t, the wavelet function F(x) can be written as 

For efficient calculations dyadic dilations d = 2 ) of an integer dilation level j (usually called a resolution level) and dyadic translations (t = kd) with an integer translation level k are used. Rearranging Equation 4.62 yields 

The scaling of the mother wavelets Fg is performed by the dilation equation, which is, in fact, a function that is a linear combination of dilated and translated versions of it  

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. 

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

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... 

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... 

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. 

Like the FFT, the fast wavelet transform (FWT) is a fast, linear operation that operates on a data vector in which the length is an integer power of two (i.e., a dyadic vector), transforming it into a numerically different vector of the same length. Like the FFT, the FWT is invertible and in fact orthogonal that is, the inverse transform, when viewed as a matrix, is simply the transpose of the transform. Both the FFT and the discrete wavelet transform (DWT) can be regarded as a rotation in function... 

Wavelet analysis was also proposed for variable reduction problems and, in particular, the wavelet coefficients obtained from discrete wavelet transforms (DWT) were proposed as a molecular representation in PEST descriptor methodology and their sums as molecular descriptors [Breneman, Sundhng et al., 2003 Lavine, Davidson et al, 2003]. 

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. 

When combining the power spectra of the first difference operator and the moving average over two, we see that the two spectra have a constant sum. This corresponds with what we already knew, the information passed by the two operators is complementary. The moving average over 2 and the first difference operator form a special pair of an LP filter and an FIP filter that divide the frequency domain in two, right in the middle of the domain. This type of filter pair plays an important role in the discrete wavelet transform. 

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... 

The fast wavelet transform provides an efficient algorithm for computing the discrete wavelet transform. We will show that provided we know some function fy, then the scaling and wavelet coefficients can be calculated in the absence of the scaling and wavelet functions, avoiding the integral expression in Eqs. (5) and (6). An expression for the scaling coefficients will be derived first, an expression for the wavelet coefficients then follows. 

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