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

Finite-rate-of-innovation (FRI) signals are ubiquitous in applications such as radar, ultrasound, and time of flight imaging. Due to their finite degrees of freedom, FRI signals can be sampled at sub-Nyquist rates using appropriate sampling kernels and reconstructed using sparse-recovery algorithms. Typically, Fourier samples of the FRI signals are used for reconstruction. The reconstruction quality depends on the choice of Fourier samples and recovery method. In this paper, we consider to jointly optimize the choice of Fourier samples and reconstruction parameters. Our framework is a combination of a greedy subsampling algorithm and a learning-based sparse recovery method. Unlike existing techniques, the proposed algorithm can flexibly handle changes in the sampling rate and does not suffer from differentiability issues during training. Importantly, exact knowledge of the FRI pulse is not required. Numerical results show that, for a given number of samples, the proposed joint design leads to lower reconstruction error for FRI signals compared to independent data-driven design methods for both noisy and clean samples. Our learning to sample approach can be readily applied to other sampling setups as well including compressed sensing problems.