Abstract
Wavelet transforms, and in particular multiresolution analyses, have become a popular technique for signal decomposition. This talk presents a new approach to the use of multiresolution analysis with sampled functions.

In its most general form, multiresolution analysis provides a computation- ally-effective structure for doing (continuous-time) signal decomposition with a broad class of wavelet functions, namely, compactly-supported biorthogonal wavelets. By exploiting the connection between multiresolution analysis and perfect-reconstruction filterbanks, we derive a novel explicit representation for all filterbanks which may be associated with biorthogonal wavelet bases, and hence a way to generate arbitrary multiresolution analyses for continuous- time signals.

In practice, these algorithms are applied to sampled-data signals. By restricting our attention correspondingly, we show that rather than represent- ing signals with wavelet functions, the practical use of multiresolution algorithms leads to signal expansions in terms of a new class of functions (dubbed "cascadelets") which share all the important properties of wavelets but are not actually wavelets. Furthermore, these representations are exact, finite, simpler than traditional multiresolution analysis, and free of some of the theoretical problems that can arise in traditional multiresolution analysis.