DeepFRI: A Deep Plug-and-Play Technique for Finite-Rate-of-Innovation Signal Reconstruction

The finite-rate-of-innovation (FRI) sampling framework is a sample-efficient and power-efficient model for analog-to-digital conversion. It can be interpreted as a framework for performing continuous-domain sparse deconvolution starting from discrete measurements. The promise of the FRI framework is its ability to resolve time delays beyond conventional theoretical limits, while acquiring measurements at the rate of innovation. In the current state-of-the-art, application of the FRI framework to real-world problems is challenging due to its limited performance in the presence of noise. In this paper, we consider signal reconstruction in the Fourier domain and propose a new optimization formulation that solves for the Fourier coefficients. We employ the proximal gradient method, and analyze the role of the denoiser in a plug-and-play (PnP) setting. Within the proposed framework, it is sufficient for the denoiser to be Lipschitz continuous, thus motivating the application of a deep PnP denoising neural network with a continuous piecewise-linear architecture. Such a neural network is interpretable and possesses similar theoretical guarantees as model-based techniques, while obtaining superior performance in the estimation of signal parameters when the signal-to-noise ratio (SNR) is low. Since the technique is derived from an optimization algorithm, we use the ensemble strategy to combine the Cadzow denoiser, which is widely used in FRI problems, and the deep PnP denoiser in order to achieve perfect reconstruction in the high SNR regime. The resulting method is called <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">DeepFRI</i>. On synthetically generated signals, the proposed technique offers up to one order of improvement in estimating the signal parameters in the low SNR regime compared with the benchmark techniques, while performing on par with them in the high SNR regime. We demonstrate an application to real-world ultrasound signals and show that the proposed technique offers superior reconstruction performance with respect to the benchmarks.