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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