FRI Sampling on Structured Nonuniform Grids-Application to Super-Resolved Optical Imaging

The annihilating filter forms the core of many state-of-the-art techniques for finite-rate-of-innovation (FRI) signal reconstruction and spectral estimation. In its standard manifestation, the annihilating filter is setup in the discrete-time domain with uniform sampling, and is used to estimate the parameters of a signal that is expressible as a sum of exponentials. In this paper, we consider annihilation in the continuous domain using translation operators (codenamed OperA) and show that FRI signals can be reconstructed using measurements taken on a structured nonuniform sampling (SNS) grid. The parameters are estimable uniquely under certain conditions. The transition from uniform sampling to SNS comes with specific advantages: the measurements could be made in bursts from high signal-to-noise ratio (SNR) regions, which directly enhances parameter estimation accuracy; the sampling interval could be made larger than in the uniform case, which leads to a reduction in the mean-squared error (MSE); and closely spaced exponential parameters could be resolved better. We present theoretical developments related to the SNS grid and demonstrate accurate parameter estimation. Monte Carlo performance analysis in the presence of noise shows that the OperA with the SNS approach yields a significant decrease in MSE up to 10 to 20 dB for certain SNR ranges compared with the uniform sampling scenario. Considering an application to frequency-domain optical-coherence tomography, we show that the proposed methodology results in higher resolution and higher reconstruction accuracy than the standard techniques