Non-uniform Random Sampling and Reconstruction in Signal Spaces with Finite Rate of Innovation

Abstract

We consider non-uniform random sampling in a signal space with finite rate of innovation $V^{2}(\varLambda,\varPhi) \subset{\mathrm {L}}^{2}(\mathbb {R}^{d})$ generated by a series of functions $\varPhi=(\phi_{\lambda})_{\lambda \in\varLambda}$ . A subset $V_{R,\delta}^{2}(\varLambda,\varPhi)$ of $V^{2}(\varLambda,\varPhi)$ is consisting of functions concentrates at least $1-\delta$ of the whole energy in a cube with side lengths $R$ . Under mild assumptions on the generators and the probability distribution, we show that for $R$ sufficiently large, taking $O(R^{d} \log(R^{d}))$ many samples with such the non-uniform distribution yields a sampling set for $V_{R,\delta}^{2}(\varLambda,\varPhi)$ with high probability. We impose compact support on the generators as an additional constraint for obtaining a reconstruction algorithm from non-uniform random sampling with high probability.