Wavelet equations

Applying the filter coefficients to the scaling function (Equation 4.64) leads to the wavelet equations... 

In general, we choose compact wavelets (i.e. only a finite number of coefficients for the dilation and wavelet equations are non-zero) and therefore only a finite number of Aj blocks are non-zero. In this case the matrix W is sparse and banded (see Section 4.2.2). Compactly supported wavelets have good localisation properties but may not always have a high degree of smoothness (e.g. the Haar wavelet). 

We have also two scaling equations, one for the primal scaling function a the second for the dual. Then wavelets can be determined by wavelet equations as follows... 

The scaling and wavelet equation are key components to derive the discrete orthogonal wavelet transform. If/eZ,2(IK), and if we denote scaling coefficients of function/ by yj, and wavelet coefficients of function / by then... 

These conditions are very restrictive and moreover usually lead to poor regularity of constructed wavelets. The same coefficients as in the scaling equation are used in reverse with alternate signs to produce the corresponding wavelet equation, i. e. 

Substituting the scaling and wavelet equation into the previous relations and we simply obtain... 

Then the primal wavelet can be again determined from the wavelet equation... 

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... 

Strang. G., Wavelets and dilation equations A brief introduction. SIAM Rev. 31, 614 (1989). Ungar, L. H., Powell, B. A., and Kamens, S. N., Adaptive Networks for fault diagnosis and process control. Comput. Chem. Eng. 14, 561 (1990). 

Using these projections over all wavelets with (s, u) e R, we can reconstruct the function by the following equation ... 

Equation (6a) implies that the scale (dilation) parameter, m, is required to vary from - ac to + =. In practice, though, a process variable is measured at a finite resolution (sampling time), and only a finite number of distinct scales are of interest for the solution of engineering problems. Let m = 0 signify the finest temporal scale (i.e., the sampling interval at which a variable is measured) and m = Lbe coarsest desired scale. To capture the information contained at scales m > L, we define a scaling function, (r), whose Fourier transform is related to that of the wavelet, tf/(t), by... 

As r 0, the error of approximation tends to the L norm. Equations (22) and (23) are useful for data compaction with orthonormal wavelets. 

The accuracy of the error equations (Eqs. (22) and (23)] also depends on the selected wavelet. A short and compactly supported wavelet such as the Haar wavelet provides the most accurate satisfaction of the error estimate. For longer wavelets, numerical inaccuracies are introduced in the error equations due to end effects. For wavelets that are not compactly supported, such as the Battle-Lemarie family of wavelets, the truncation of the filters contributes to the error of approximation in the reconstructed signal, resulting in a lower compression ratio for the same approximation error. 

Strang, G., Wavelets and dilation equations a brief introduction. SIAM Review, 31(4), 614—... 

Besides Tikhonov regularization, there are numerous other regularization methods with properties appropriate to distinct problems [42, 53,73], For example, an iterated form of Tikhonov regularization was proposed in 1955 [77], Other situations include using different norms instead of the Euclidean norm in Equation 5.25 to obtain variable-selected models [53, 79, 80] and different basis sets such as wavelets [81],... 

Wavelet analysis takes Gabor s idea one step further it defines a windowing transform technique with variably sized window regions. The continuous wavelet transform of the sequence h(t) is defined by Equation 10.23... 

A critical difference between the Fourier transform defined in Equation 10.9 and the wavelet transform defined in Equation 10.22 is the fact that the latter permits localization in both frequency and time that is, we can use the equation to determine what frequencies are active at a specific time interval in a sequence. However, we cannot get exact frequency information and exact time information simultaneously because of the Heisenberg uncertainty principle, a theorem that says that for a given signal, the variance of the signal in the time domain a2, and the variance of the signal in the frequency (e.g., Fourier) domain c2p are related... 

Note that this equation follows after two steps in which the Stokes parameters for one wavelet and an ensemble of wavelets are used first, the differential cross section is calculated for one wavelet which is completely elliptically polarized and, second, the presence of differently polarized wavelets in the incident light is taken into account using equ. (1.42). 

We now will treat the rarefaction as if it were a shock that is, we will apply the jump equations such that we will let the high-pressure material Jump down to a lower pressure state. We also are going to allow this to happen in two steps. The first step, or rarefaction wavelet, relieves the material from state P, v (the shock pressure) to 2, 2 (half way down to ambient). 

Wavelet analysis has great potential in image processing applications of interest in mineralogy (see Moktadir and Sato (2000) for an illustrative example for silicon). As an illustration, in Figure 4 we show a version of Equation (3) over a one-dimensional trace across a two-dimensional AFM image of a hematite surface where there are some traces of bacterially mediated reduction reactions. One-dimensional wavelets with the second-derivative of the Gaussian function, also known as Mexican-hat wavelets because of their... 

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 ... 

---