MULTIRESOLUTION APPROXIMATION USING SHIFTED SPLINES

We consider the construction of least squares pyramids using shifted p olynomial spline basis functions. We derive the pre and postfilters as a function of the degree n and the shift parameter Delta. We show tha t the underlying projection operator is entirely specified by two tran sfer functions acting on the even and odd signal samples, respectively . We introduce a measure of shift invariance and show that the most fa vorable configuration is obtained when the knots of the splines are ce ntered with respect to the grid points (i.e., Delta = 1/2 when n is od d and Delta = 0 when n is even). The worst case corresponds to the sta ndard multiresolution setting where the spline spaces are nested.