TABLE OF CONTENTS The rest of this chapter is dedicated to the construction of wavelets for discrete analysis. First of all, we widen the framework passing by the two-channel filter banks, whose properties are examined. In particular, we determine the conditions enabling perfect reconstruction. This classical signal processing approach [EST 77] leads to what is called quadrature mirror filters (QMF) [MIN 85] or also conjugated mirror filters [SMI 86], which are also often referred to as QMF.
A perfect reconstruction filter bank is known as "biorthogonal" and the associated filters as biorthogonal. Lightly constraining conditions make it possible to obtain such filter banks without too much difficulty. With the "lifting" method it is then possible to construct an infinite number thereof starting from a biorthogonal bank. Moreover, thanks to the technique known as "polyphase", we show [DAU 97] that all the biorthogonal transformations can be decomposed into elementary lifting steps.
Finally, the link with wavelets is the subject of the last section. By starting from a biorthogonal filter bank with some additional conditions we can construct biorthogonal wavelet bases. In particular, on the basis of a biorthogonal filter bank stemming from wavelets, we can construct an infinite number of filter banks of the same type, among which it is possible to distinguish those associated with biorthogonal wavelets.
Within the framework of orthogonal wavelets the decomposition-reconstruction algorithm uses operations of filtering by convolution, down-sampling and up-sampling.
