The B-spline functions are used to develop recursive algorithms for the efficient implementation of two-dimensional linear digital image filters. These filters may be spatially varying. The B-splines are used in a representation of the desired point spread function. We show that this leads to recursive algorithms and hardware implementations which are more efficient than either direct spatial domain filter realizations or FFT implementations. The Z-transform is used to develop a discrete version of Duhamel's theorem. A computer architecture for B-spline image filters is proposed and a complexity analysis and comparison to other approaches is provided.