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Asymptotic Properties of Least Squares Spline Filters and Applications to Multi-Scale Decomposition of Signals

A. Aldroubi, M. Unser, M. Eden

Proceedings of the International Symposium on Information Theory and Its Applications (ISITA'90), Honolulu HI, USA, November 27-30, 1990, pp. 271-274.



We use B-spline functions to define a family of sequence-spaces 𝕊mn included in the finite energy and discrete space ℓ2. We derive invariant filters that operate on finite energy signals to output their least squares approximations in 𝕊mn. We obtain results on the convergence of the various filters to the ideal lowpass filter providing the link with Shannon's sampling theorem. As an application, we derive pyramidal representations of signals that can be implemented with fast algorithms and compare these representations with the Gaussian/Laplacian pyramid which is widely used in signal processing and computer vision.


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