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Multiresolution Approximation Using Shifted Splines

F. Müller, P. Brigger, K. Illgner, M. Unser

IEEE Transactions on Signal Processing, vol. 46, no. 9, pp. 2555–2558, September 1998.

We consider the construction of least squares pyramids using shifted polynomial spline basis functions. We derive the pre- and post-filters as a function of the degree n and the shift parameter Δ. We show that the underlying projection operator is entirely specified by two transfer functions acting on the even and odd signal samples, respectively. We introduce a measure of shift-invariance and show that the most favorable configuration is obtained when the knots of the splines are centered with respect to the grid points (i.e., Δ=1/2 when n is odd, and Δ=0 when n is even). The worst case corresponds to the standard multiresolution setting where the spline spaces are nested.

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