DISCRETE SPLINE FILTERS FOR MULTIRESOLUTIONS AND WAVELETS OF L(2)

The authors consider the problem of approximation by B-spline function s, using a norm compatible with the discrete sequence-space l2 instead of the usual norm L2. This setting is natural for digital signal/imag e processing and for numerical analysis. To this end, sampled B-spline s are used to define a family of approximation spaces S(m)n subset-of l2. For n odd, S(m)n is partitioned into sets of multiresolution and w avelet spaces Of l2. It is shown that the least squares approximation in S(m)n of a sequence s is-an-element-of l2 is obtained using transla tion-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: th e l2-optimal and the stepwise l2-optimal pyramids, the advantage of th e latter being that it can be computed by the repetitive application o f a single procedure. Finally, a step by step discrete wavelet transfo rm Of l2 is derived that is based on the stepwise optimal representati on. As an application, these representations are implemented and compa red with the Gaussian/Laplacian pyramids that are widely used in compu ter vision.