Task-specific wavelets

The Adaptive Wavelet Algorithm for Designing Task Specific Wavelets... 

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... 

Integrated task-specific wavelets and best-basis search for image compression... 

Fig. 12 shows the result when the task-specific wavelet construction is integrated with the best-basis search (with the same threshold cost function). Here, the compression ratio is 9.71, an improvement of 2.2% compared with... 

Fig. 12 Results of the integrated task-specific wavelet construction and hest-hasis search four levels, m = 2, filter length Nf = 12).

Fig. 13 Search space for task-specific wavelet construction (m = 2 and N/ - 6).

The authors would like to thank Dr S. Aeberhard for generating the results for the integrated task-specific wavelet construction and best-basis search. 

The best basis algorithm seeks a basis in the WPT which optimizes some criterion function. Thus, the best basis algorithm is a task-specific algorithm in that the particular basis is dependent upon the role for which it will be used. For example, a basis chosen for compressing data may be quite different from a basis that might be used for classifying or calibrating data, since different criterion functions would be optimized. The wavelet packet coefficients which are resultant of the best basis, may then be used for some specific task such as compression or classification for instance. 

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. 

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. 

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