Wavelet families

As a result of the dyadic discretization in dilation and translation, the members of the wavelet family are given by... 

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. 

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 Haar, Daubechies, symmlets and coiflets are wavelet families which exhibit orthogonality and compact support (see Fig. 12). Criteria which the scaling (j)(t) and wavelet vl/(t) must satisfy for orthogonality were discussed in Section 5. Also, in this section the term compact support was briefly mentioned. A wavelet is compactly supported if it is nonzero over a finite interval and zero outside this interval. Such wavelets include the Haar, Daubechies, symmlets and coiflets. 

There exists many different kinds or families of wavelets. These wavelet families are defined by their respective filter coefficients which are readily available for the situation when m = 2, and include for example the Daubechies wavelets, Coiflets, Symlets and the Meyer and Haar wavelets. One basic issue to overcome is deciding which set (or family) of filter coefficients will produce the best results for a particular application. It is possible to trial different sets of filter coefficients and proceed with the family of filter coefficients which produces the most desirable results. It can be advantageous however, to design your own task specific filter coefficients rather than using a predefined set. 

In this section, we design our own task specific filter coefficients using the adaptive wavelet algorithm of Chapter 8. The idea behind the adaptive wavelet algorithm is to avoid the decision of which set of filter coefficients and hence the wavelet family which would be best suited to our data. Instead, we basis design our own wavelets or more specifically, the filter coefficients which define the wavelet and scaling function. This is done to suit the current task at hand, which in this case is discriminant analysis. 

Although many wavelet applications use orthogonal wavelet basis, others work better with redundant wavelet families. The redundant representation offered by wavelet frames has demonstrated to be good both in signal denoising and compaction (Daubechies et al. 1986, 1992). 

A first implementation of a dynamic model for a separation process highlights the practical applicability of the approach. The technique outlined can be generalized to accommodate any problem specific wavelet family and allows for a model reduction by tresholding of the multi-scale representation of the trial solution. Further work will address these issues. 

The restrictions placed on the mother wavelets for multiresolution analysis do not limit the variety of shapes that can be used as mother wavelets different researchers have proposed several different wavelet functions, each with benefits and drawbacks. The wavelet shape tradeoff is between how compactly it can be localized in space and its level of smoothness. For example, the Haar wavelet, which is the simplest wavelet and was identified almost 100 years ago, is well localized in space, but it has an unnatural square-wave oscillation (see Figure 10). Many related wavelets exist, collectively referred to as wavelet families some of these families include the Meyer wavelet, Coiflet wavelet, spline wavelet, orthogonal wavelet, symmlet wavelet, and local cosine basis. Figure 10 depicts several of these wavelets and... 

Step 1. Select a family of scaling functions and wavelets. 

A family of wavelets is a family of functions with all its members derived from the translations (e.g., in time) and dilations of a single, mother function. If iffit) is the mother wavelet, then all the members of the family are given by... 

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. 

Fig. 40.42. A family of Morlet wavelets with various dilation values.

In brief, wavelets are a family of functions of constant shape that are localized in both time and frequency. A family of discrete dyadic wavelets is represented as... 

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... 

It is not a straightforward task to come up with a procedure that would lead to the best mother wavelet for a given class of signals. N-evertheless, exploiting several characteristics of the wavelet function, one can determine which family of wavelets would be more appropriate for a specific application. 

An important property of wavelet bases is their lack of translational invariance. In other words, when a pattern is translated, its descriptors are not only translated but also modified. This is a direct consequence of the down-sampling procedure and leads to distorted reconstruction of the underlying signal features. A possible solution is to omit down-sampling, resulting in a redundant family of coefficients. 

According to a proposal of Ingrid Daubechies, the notation DK will be used for a Daubechies Wavelet transform with K coefficients. Actually, D2 is identical to the simplest Wavelet of all, the so-called Haar Wavelet, and, thus, is not originally a member of the Daubechies family. 

The Chebychev polynomial is just one possibility to construct a fixed orthogonal basis for n-dimensional space. There are many others. Interesting members of the family are the Hermite polynomial. Fourier and Wavelets. As there are many, the question arises how to choose between them. Before we are able to answer that question, we need to deal with another, more fundamental one why do a basis transformation in the first place ... 

Fig. 5 shows some wavelet functions which are translated and dilated by different amounts. Notice that they all possess the same shape and differ by the amount by which they are translated and dilated. There exist many kinds or families of wavelets. The wavelets shown in Fig. 5 are wavelets from the Daubechies family, named after Ingrid Daubechies. 

Fig. 5 An example of dilating and Iran.dating wavelets from the Daubechies family.

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