Multiresolution approximation using shifted splines

We consider the construction of least squares pyramids using shifted polynomial spline basis functions. We derive the pre and postfilters as a function of the degree n and the shift parameter . 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., = 1=2 when n is odd and = 0 when n is even). The worst case corresponds to the standard multiresolution setting where the spline spaces are nested.