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

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.