Learning to Sample: Data-Driven Sampling and Reconstruction of FRI Signals

Finite-rate-of-innovation (FRI) signal model is well suited for time-of-flight imaging applications such as ultrasound, lidar, sonar, radar, and more. Due to their finite degrees of freedom, the sub-Nyquist sampling framework is used for FRI signals. In this framework, sampling is achieved by using appropriate sampling kernels. Reconstruction is performed by first computing Fourier samples of the FRI signal and then applying sparse-recovery algorithms. The choice of the Fourier samples and reconstruction method plays a crucial role in the reconstruction quality. In this paper, we consider jointly optimizing the choice of Fourier samples and reconstruction parameters. Our framework combines a greedy subsampling algorithm and a learning-based sparse recovery method. The combination has three distinct advantages. First, the network does not require knowledge of the FRI pulse shape, which is not the case with existing approaches. Second, the network does not suffer from differentiability issues during training which is common in sampling networks. Further, the proposed algorithm can flexibly handle changes in the sampling rate. Numerical results show that, for a given number of samples, the proposed joint design leads to lower reconstruction error for FRI signals than independent data-driven design methods for noisy and clean samples. We also propose a way to extend the approach for large-scale problems. Our learning-to-sample approach can be readily applied to other sampling setups, including compressed sensing problems.