Vanishing moments

FIGURE 10.19 Some example mother wavelet functions. From left to right a coiflet (coif), a symlet (sym), and two Daubechies (db) wavelets. The numbers relate to the number of vanishing moments of the wavelet. 

There are several families of wavelets, proposed by different authors. Those developed by Daubechies [46] are extensively used in engineering applications. Wavelets from these families are orthogonal and compactly supported, they possess different degrees of smoothness and have the maximum number of vanishing moments for a given smoothness. In particular, a function f t) has e vanishing moments if... 

The best known wavelets are the Daubechies wavelets (dbe) and the Coif-man wavelets (coife). In both cases, e is the number of vanishing moments of the functions. Daubechies also suggested the symlets as the nearly symmetric wavelet family as a modification of the db family. The family Haar is the well-known Haar basis [95]. Figure 6.4 shows a number of wavelet functions. As can be seen, the Haar functions are discontinuous and may not provide good approximation for smooth functions. 

The electric moments are not invariant under a shift of the origin of the coordinate system. The first non-vanishing moment is however usually independent of the choice of the coordinates. The origin of coordinate system is thus often chosen as the center of gravity or the center of the charge distribution. 

This chapter has briefly eluded to two wavelet families, the Haar and Daubechies wavelets. In fact when Nf = 2 the Daubechies wavelet is identical to the Haar wavelet. In this section we would like to discuss in greater detail more about these wavelet families and other wavelet families. We will also provide a brief comparison between the different properties possessed by these wavelets and other wavelet families. This is important because depending on your application, you may need to choose a wavelet that satisfies special properties. We first review the terms orthogonal and compact support. Following this, we will introduce some more properties, namely smoothness and symmetry of wavelets and also discuss the term vanishing moments. 

The decision to use a particular wavelet may extend to other properties besides orthogonality and compact support. Other properties include symmetry, smoothness, and vanishing moments. 

So that a wavelet function can efficiently represent characteristics of an underlying function, it is necessary that the wavelet be sufficiently smooth [14], If one considers the Haar wavelet depicted in Fig. 11, one can clearly see that this wavelet suffers from a lack of smoothness. Smoothness is closely related to how many times a wavelet can be differentiated and to the number of vanishing moments possessed by the wavelet vj/(t). One measure of smoothness is the Holder exponent [14], which is defined to equal = q + Y where and Y are the largest values such that... 

A wavelet is M — 1 times differentiable if it has M vanishing moments. A wavelet has M vanishing moments if... 

Vanishing moments are a necessary condition for vj/(t) to be M times differentiable. More precisely, a wavelet is M times differentiable only if v /(t) has M vanishing moments. (Note that the reverse is not true.)... 

When the wavelet v /(t), has M vanishing moments then the polynomials of form 5 olinearly combining scaling... 

If the task at hand is to compress large amounts of data, then one may be more inclined to choose a wavelet that has a high number of vanishing moments. If the task was related to image processing, then wavelets with smooth properties would be preferred. 

The sum of all the elements in each of the normalised u vectors is calculated. This gives an indication of the area Ek below each curve. These curves are monotonously decreasing with only positive values and the one with the smallest area Ek corresponds to the optimal wavelet for a particular data set. For all data sets in this chapter the following six wavelets were tested for Haar, Beylkin, Coiffet, Daubechies, Symmlet and Vai-dyanathan with varying number of vanishing moments. 

Sometimes real, symmetric wavelets or wavelets with more vanishing moments and at the same time with smaller support than corresponding orthogonal wavelets or simply more regular", in closed form defined wavelets are required. One way to obtain them is to construct two sets of biorthogonal wavelets r/r and its dual x). One of these two biorthogonal wavelets is consequently used to decompose the signal and the second one to reconstruct it. The numbers M,N of... 

In Fig. 1 is displayed the measured signal and its discrete wavelet transform with the biorthogonal wavelet of order 3 with 5 vanishing moments (CDF35) designed by (Cohen et al. 1992). It seems that oscillations and noise are quite well removed. However as ean be seen on Fig. 2, the disadvantage of this... 

The regularity condition requires that the mother wavelet has to be locally smooth and concentrated in both the time and frequency domains. To ascertain a good local property in the frequency domain, the W f a,b) is required to decay rapidly with decrease of scale a. As a result this requires the first m moments of x) equal to zero. These are the vanishing moments. If the Fourier transform of the wavelet is M times differentiable the wavelet has M vanishing moments. 

As a result, from the function approximation point of view, we hope the wavelet with smaller support R and higher vanish moment M. At the same time, with smaller R, we get a better time resolution. According to the uncertainty principle, the frequency resolution decreases. 

Thus it is unwise to attempt to obtain distributed multipole models solely by reference to experimental data, especially as experimental data for the multipole moments are usually very unreliable for all but the first one or two non-vanishing moments. There is no difficulty in obtaining the information needed for the distributed multipole description from ab initio calculations the distributed multipole analysis takes a fraction of the time... 

An important feature of the electric multipole moments, as defined in Eqs. (4.4)-(4.6) and (4.8), is that the first non-vanishing moment of a charge distribution is independent of the choice of the origin Rq- However, all the higher moments depend on this origin. Thus, the dipole moment of a neutral molecule or the quadrupole moment of a neutral and non-polar molecule are both independent of the origin Rq, whereas the dipole moment of an ion or the quadrupole moment of a neutral but polar molecule will depend on the origin Rq [see Exercise 4.2]. 

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