  Finite Rate of Innovation (FRI) sampling theory enables reconstruction of
classes of continuous non-bandlimited signals that have a small number of free
parameters from their low-rate discrete samples. This task is often translated
into a spectral estimation problem that is solved using methods involving
estimating signal subspaces, which tend to break down at a certain peak
signal-to-noise ratio (PSNR). To avoid this breakdown, we consider alternative
approaches that make use of information from labelled data. We propose two
model-based learning methods, including deep unfolding the denoising process in
spectral estimation, and constructing an encoder-decoder deep neural network
that models the acquisition process. Simulation results of both learning
algorithms indicate significant improvements of the breakdown PSNR over
classical subspace-based methods. While the deep unfolded network achieves
similar performance as the classical FRI techniques and outperforms the
encoder-decoder network in the low noise regimes, the latter allows to
reconstruct the FRI signal even when the sampling kernel is unknown. We also
achieve competitive results in detecting pulses from in vivo calcium imaging
data in terms of true positive and false positive rate while providing more
precise estimations.
