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Discrete Spline Filters for Multiresolution and Wavelets of l2

A. Aldroubi, M. Eden, M. Unser

SIAM Journal on Mathematical Analysis, vol. 25, no. 5, pp. 1412-1432, September 1994.

The authors consider the problem of approximation by B-spline functions, using a norm compatible with the discrete sequence-space l2 instead of the usual norm L2. 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 Smnl2. For n odd, Smn is partitioned into sets of multiresolution and wavelet spaces of l2. It is shown that the least squares approximation in Smn of a sequence sl2 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 l2-optimal and the stepwise l2-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 l2 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.

AUTHOR="Aldroubi, A. and Eden, M. and Unser, M.",
TITLE="Discrete Spline Filters for Multiresolution and Wavelets of
JOURNAL="{SIAM} Journal on Mathematical Analysis",

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