The adaptive wavelet algorithm for designing task specific wavelets

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, Coifiets, 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.\ud \ud In this chapter, we describe one method for generating your own set of filter coefficients. Here we demonstrate how wavelets can be designed to suit almost any general application, but in this chapter we concentrate on designing wavelets for the classification of spectral data. In Chapter 18, we extend the principle of the adaptive wavelet algorithm to regression and classification. Since wavelets can be derived from their respective filter coefficients, we generate the filter coefficients which optimize a relevant criterion function. We introduce a wavelet matrix called A which stores both the low-pass and high-pass filter coefficients. Instead of optimizing over each element in A, we make use of the factorized form [1] of a wavelet matrix and the conditions placed therein to reduce the number of parameters to be optimized. Since the filter coefficients gradually adapt to the application at hand, the procedure for designing the task specific filter coefficients is referred to as the adaptive wavelet algorithm (A W A). This should not be confused with the adaptive wavelets of Coifman and Wickerhauser who refer to adaptive wavelets as a procedure for constructing a best basis [2] (see Chapter 6).\ud \ud There exist other applications involving the optimization of wavelets. This includes the work performed by Telfer et al. [3] and Szu et al. [4]. In [3] the shift and dilation parameters of the discretization of a chosen wavelet transform are optimized, while [4] sought the optimal linear combination of predefined wavelet bases for the classification of speech signals. In both papers, the wavelet features are updated by adaptively computing the wavelet parameters and shape. This is a form of integrated feature extraction which also makes use of neural networks. Sweldens [5] also discusses a lifting scheme for the construction of biorthogonal second generation wavelets. The main distinction between [3,4,5] and our algorithm is that the filter coefficients are generated from first principles - without any reference to predefined families. Our approach also allows for the general m-band wavelet transform to be utilized, as well as the more common 2-band wavelet transform. The adaptive wavelet algorithm presented in this chapter is an extension of the material presented in [6] who introduced adaptive wavelets for the detection and removal of disturbances from signals.\ud \ud Before describing how wavelets can be designed for a specific task, we first discuss the idea of higher multiplicity wavelets and the m-band discrete wavelet transform. This is done in Sections 2 and 3, respectively. Basically, higher multiplicity wavelets consider dilating the wavelet functions by integers greater than or equal to two. This can be likened to down-sampling discrete signals by integer amounts greater than or equal to 2. We let m equal the amount by which we dilate or down-sample. Consequently, the m-band discrete wavelet transform has m bands. One band contains the scaling coefficients, and the remaining m - 1 bands contain wavelet coefficients. In Section 4 we discuss conditions which can be placed on the filter coefficients so that a multi resolution analysis (MRA) and wavelet basis exist. These coefficients are stored in a matrix called a filter coefficient matrix. Section 5 shows how we can factorize the filter coefficient matrix, thereby allowing us to search for a filter coefficient matrix which optimizes some cost function relevant to the task at hand. Section 6 summarises the adaptive wavelet algorithms. Section 7 discusses various criterion functions. The chapter concludes in Section 8 where key issues arising from the implementation and interpretation of adaptive wavelets are discussed