The divergent k-plane transform

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Biorthogonal Wavelets For Fast Matrix Computations

. In [1], Beylkin et al. introduced a wavelet-based algorithm that approximates integral or matrix operators of a certain type by highly sparse matrices, as the basis for efficient approximate calculations. The wavelets best suited for achieving the highest possible compression with this algorithm are Daubechies wavelets, while Coiflets lead to a faster decomposition algorithm at slightly lesser compression. We observe that the same algorithm can be based on biorthogonal instead of orthogonal wavelets, and derive two classes of biorthogonal wavelets that achieve high compression and high decomposition speed, respectively. In numerical experiments, these biorthogonal wavelets achieved both higher compression and higher speed than their wavelet counterparts, at comparable accuracy. 1. Introduction In [1], Beylkin et al. observed that wavelet decomposition can be used to approximate linear operators of a certain type by highly sparse matrices. The decomposition and subsequent calculation..

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 efficient 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 coefficients. 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 roundoff error is accumulated during deco..