Finding and Designing a Wavelet
It is possible that for certain applications we do not find a suitable wavelet among the known options. It is then natural to try to produce a new wavelet adapted to the specific problem being treated. In this chapter we are interested in two wavelet construction processes: one is useful for continuous analysis and the other is linked to the discrete case.
In the first part of this chapter we consider the construction of wavelets usable for continuous analysis. This problem does not present great difficulty as the requirements to obtain such wavelets are relatively limited. We proceed in the following manner: starting from a pattern which, in general, is not a wavelet, we seek the wavelet nearest to the given form in the least squares sense. The usefulness of the construction process is demonstrated when this is applied to a detection problem. The task consists of employing the adapted wavelet to identify in a signal the patterns stemming from the basic form via translation and dilation.
In the second part of this chapter we tackle the construction of wavelets for discrete analyses, which is, in turn, more delicate. Until a few years ago, it was still only a subject for specialist discussion. Recently, the lifting method developed by Sweldens (see [SWE 98]) has made this task easier. It makes it possible to construct an infinite number of biorthogonal wavelet bases starting from a given biorthogonal wavelet base.
In fact, ...