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

  • Yancheng Lu
  • Jun XianEmail author
Article
  • 12 Downloads

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.

Keywords

Random sampling Non-uniform sampling Spaces with finite rate of innovation Non-uniform distribution Reconstruction algorithm 

Mathematics Subject Classification (2000)

94A20 42C15 60E15 62M30 

Notes

Acknowledgements

The authors would like to thank the reviewers for their valuable comments and suggestions that led to the improvement of this paper. The corresponding author Jun Xian is partially supported by the National Natural Science Foundation of China (11422102, 11631015, 11871481); the Guangdong Provincial Government of China through the Computational Science Innovative Research Team program, China; and the Guangdong Province Key Laboratory of Computational Science, China.

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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of MathematicsSun Yat-sen UniversityGuangzhouChina
  2. 2.Guangdong Province Key Laboratory of Computational ScienceSun Yat-sen UniversityGuangzhouChina

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