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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 Smn ⊂ l2. 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 s ∈ l2 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. 

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TITLE="Discrete Spline Filters for Multiresolution and Wavelets of
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JOURNAL="{SIAM} Journal on Mathematical Analysis",
YEAR="1994",
volume="25",
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