Discrete Spline Filters for Multiresolutions and Wavelets of $l_2 $

Abstract

The authors consider the problem of approximation by B-spline functions, using a norm compatible with the discrete sequence-space $l_2 $ instead of the usual norm $L_2 $. This setting is natural for digital signal/image processing and for numerical analysis. To this end, sampled B-splines are used to define a family of approximation spaces ${\bf S}_m^n \subset l_2 $. For n odd, ${\bf S}_m^n $ is partitioned into sets of multiresolution and wavelet spaces of $l_2 $. It is shown that the least squares approximation in ${\bf S}_m^n $ of a sequence $s \in l_2 $ is obtained using translation-invariant filters. The authors study the asymptotic properties of these filters and provide the link with Shannon’s sampling procedure. Two pyramidal representations of signals are derived and compared: the $l_2 $-optimal and the stepwise $l_2 $-optimal pyramids, the advantage of the latter being that it can be computed by the repetitive application of a single procedure. Finally, a step by step discrete wavelet transform of $l_2 $ is derived that is based on the stepwise optimal representation. As an application, these representations are implemented and compared with the Gaussian/Laplacian pyramids that are widely used in computer vision.