A note on construction of biorthogonal multi-scaling functions
Vasily Strela · Contemporary mathematics - American Mathematical Society · 1998
. In many applications biorthogonal wavelets prove to be more efficient than orthogonal ones. In this note we present a procedure for constructing biorthogonal multi-scaling functions with any given approximation order, starting from a low pass multi-filter H 0 . We show that dual low pass multifilter F 0 can be found if det(H 0 (z)) and det(H 0 (\\Gammaz)) do not have common roots. We also suggest "two-scale balancing" as a way to enhance approximation properties of F 0 (z). As an illustration to our technique we construct multi-scaling functions with various smoothness biorthogonal to Hermite cubics and a pair of low-pass biorthogonal multi-filters based on the well known GHM orthogonal example. 1. Introduction Most wavelet applications consist of three parts: decomposition (analysis) of the signal using a wavelet basis, processing of the decomposed data, and reconstruction (synthesis) of the signal. Generally, analysis and synthesis steps use different wavelets w(t) and e w(t). A ...