Wavelet filter coefficients

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. 

From Chapter 8 we note that the construction of a task-specific wavelet proceeds by generating a normalised vector v of dimensionality m-1 and a further Nf/m-1 normalised vectors Uj, each of length m (Nf is the number of wavelet filter coefficients). The total number of free parameters required to construct the wavelet is therefore... 

Rows 1-8 are the approximation filter coefficients and rows 9-16 represent the detail filter coefficients. At each next row the two coefficients are moved two positions (shift b equal to 2). This procedure is schematically shown in Fig. 40.43 for a signal consisting of 8 data points. Once W has been defined, the a wavelet transform coefficients are found by solving eq. (40.16), which gives ... 

The factor Vl/ 2 is introduced to keep the intensity of the signal unchanged. The 8 first wavelet transform coefficients are the a or smooth components. The last eight coefficients are the d or detail components. In the next step, the level 2 components are calculated by applying the transformation matrix, corresponding to the level on the original signal. This transformation matrix contains 4 wavelet filter... 

FWTs can be implemented quite efficiently the calculation time of algorithms performing wavelet transformations increases only linearly with the length of the transformed vector. A special kind of wavelet was developed by Ingrid Daubechies [63]. Daubechies wavelets are base functions of finite length and represent sharp edges by a small number of coefficients. They have a compact support that is, they are zero outside a specific interval. There are many Daubechies wavelets, which are characterized by the length of the analysis and synthesis filter coefficients. 

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

By iterative application of the FWT to the high-pass filter coefficients, a shape emerges that is an approximation of the wavelet function. The same applies to the iterative convolution of the low-pass filter that produces a shape approximating the scaling function. Figure 4.6 and Figure 4.7 display the construction of the scaling and wavelet functions, respectively ... 

FIGURE 4.7 Construction of wavelet function / with high-pass D4 filter coefficients (above). Below the functional representation of the high-pass filter coefficients (left) and their refinement by iterative calculation (increasing resolution level j) leading to an approximation of the wavelet function / (right). 

Having the three-dimensional coordinates of atoms in the molecules, we can convert these into Cartesian RDF descriptors of 128 components (B = 100 A ). To simplify the descriptor we can exclude hydrogen atoms, which do not essentially contribute to the skeleton structure. Finally, a wavelet transform can be applied using a Daubechies wavelet with 20 filter coefficients (D20) to compress the descriptor. A low-pass filter on resolution level 1 results in vectors containing 64 components. These descriptors can be encoded in binary format to allow fast comparison during descriptor search. 

The descriptors are transformed by Daubechies wavelet decomposition with 20 filter coefficients (D20). 

Often there is a finite number of non-zero filter coefficients. We use the notation Nf to denote the number of non-zero filter coefficients. Values for the filter coefficients appear in several texts, see for example [7]. Each set of filter coefficients defines the corresponding scaling and wavelet basis functions. Whilst it is possible to use off-the-shelf wavelets, in Chapter 8 we suggest a possible approach for designing your own wavelets. 

Provided we know the scaling coefficients at some resolution level j, the remaining scaling coefficients and wavelet coefficients can be found by the pyramidal filtering algorithm without even having to construct a wavelet or scaling function. We need only work with the filter coefficients Ik and hk. 

The first row in the wavelet matrix A, simply eontains the low-pass filter coefficients. The second row in the wavelet matrix A, contains the high-pass filter coefficients. As shown above, it is sometimes more convenient to store the filter eoefficients in the matrix A as a sequence of sub-blocks Aj j —. ..,-2,-1,0,1,2,...). The sub-blocks are simply the filter coefficients found in the lattice decomposition and reconstruction equations ar-... 

In Fig. 1 the number of bands doubles from one level to the next (lower) level, since each of the bands in the previous level are passed through a low-pass and a high-pass filter. At the next level, there will be four bands of wavelet packet coefficients which are obtained by... 

Perform the wavelet packet transform using filter coefficients associated with the Haar wavelet and scaling functions, then, compute the wavelet packet coefficients associated with the best basis using the entropy cost function for the signal x = (0.0000,0.0491.0.1951,0.4276,0.7071,0.9415,0.9808,0.6716). 

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. 

We have demonstrated that it is possible to obtain the discrete wavelet transform of both continuous functions and discrete data points without having to construct the scaling or wavelet functions. We only need to work with the filter coefficients. One may begin to wonder where the filter coefficients actually come from. Basically, wavelets with special characteristics such as orthogonality, can be determined by placing restrictions on the filter coefficients. 

The restrictions which are imposed on the filter coefficients so that an MRA and orthogonal wavelet basis exist are summarized as follows [6]... 

If more sophisticated wavelet and scaling functions are required, then more constraints need to be placed on the filter coefficients. 

The adaptive wavelet algorithm outlined in Section 6 can be used for a variety of situations, and its goal is reflected by the particular criterion which is to be optimized. In this chapter, we apply the filter coefficients produced from the adaptive wavelet algorithm for discriminant analysis. It was stated earlier that the dimensionality is reduced by selecting some band(jg,xg) of wavelet coefficients from the discrete wavelet transform. It then follows that the criterion function will be based on the same coefficients i.e. Xl " (xg). 

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. 

One might be interested in how the adaptive wavelet performs against predefined filter coefficients. In this section, we perform the 2-band DWT on each data set using filter coefficients from the Daubechies family with Nf = 16. The coefficients from some band (j, x) are supplied to BLDA. We consider four bands - band(3,0), band(3,l), band(4,0) and band(4,l). The results for the training and testing data are displayed in Table 3. The test CCR rates are the same for the seagrass and butanol data, but the AWA clearly produces superior results for the paraxylene data. 

Table 3. Classification results for wavelet and scaling coefficients produced using filter coefficients from the Daubechies family with...

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