Numerical Stability of Biorthogonal Wavelet Transforms

For orthogonal wavelets, the discrete wavelet and wave packet transforms and their inverses are orthogonal operators with perfect numerical stability. For biorthogonal wavelets, numerical instabilities can occur. We derive bounds for the 2-norm and average 2-norm of these transforms, including eecient numerical estimates if the number L of decomposition levels is small, as well as growth estimates for L ! 1. These estimates allow easy determination of numerical stability directly from the wavelet coeecients. Examples show that many biorthogonal wavelets are in fact numerically well behaved. 1. Introduction The discrete wavelet transform and wave packet transform have become well established in many applications, such as signal processing. Originally derived for orthogonal wavelets, they can equally well be based on biorthogonal wavelets. Numerical experiments with various types of biorthogonal wavelets show that in some cases considerable roundoo error is accumulated during decomposition and reconstruction. Strangely enough, these wavelets still perform well in certain applications such as signal compression (see 8]). Other types of biorthogonal wavelets have excellent stability behavior. That is, the round-oo error remains close to machine accuracy even after extensive calculations. We denote by W L , P L the wavelet and wave packet transforms through L levels of decom