Exact interpolation and iterative subdivision schemes

We examine the circumstances under which a discrete-time signal can be exactly interpolated given only every Mth sample. After pointing out the connection between designing an M-fold interpolator and the construction of an M-channel perfect reconstruction filter bank, we derive necessary and sufficient conditions on the signal under which exact interpolation is possible. Bandlimited signals are one obvious example, but numerous others exist. We examine these and show how the interpolators may be constructed. A main application is to iterative interpolation schemes, used for the efficient generation of smooth curves. We show that conventional bandlimited interpolators are not suitable in this context. A better criterion is to use interpolators that are exact for polynomial functions. We demonstrate that these interpolators converge when iterated, and show how these may be designed for any polynomial degree N and any interpolation factor M. This makes it possible to design interpolators for iterative schemes to make best use of the resolution available in a given display medium