Sampling piecewise smooth signals and its application to image up-sampling

In this paper we consider the problem of sampling piecewise smooth signals. The classical sampling theory is not able to sample them efficiently since the signals are neither bandlimited nor live in a shift-invariant subspace. We propose a sampling scheme which still use the set of samples from a classical sampling set-up with an arbitrary acquisition device but achieves accurate reconstruction by making use of the non-linear approximate FRI reconstruction method in addition to the classical linear reconstruction. Specifically, we see the class of piecewise smooth signal as the sum of a piecewise polynomial, which is a signal that is fully specified by finite number of parameters and can be recovered with FRI non-linear methods, and a globally smooth term, which can be accurately recovered by linear reconstruction. We also show that our proposed scheme based on combining linear and non-linear reconstruction methods can be employed for resolution enhancement of discrete-time piece-wise smooth signals and images.