  The aim of this paper is to investigate superresolution in deconvolution
driven by sparsity priors. The observed signal is a convolution of an original
signal with a continuous kernel.With the prior knowledge that the original
signal can be considered as a sparse combination of Dirac delta peaks, we seek
to estimate the positions and amplitudes of these peaks by solving a finite
dimensional convex problem on a computational grid. Because, the support of the
original signal may or may not be on this grid, by studying the discrete
deconvolution of sparse peaks using L1-norm sparsity prior, we confirm recent
observations that canonically the discrete reconstructions will result in
multiple peaks at grid points adjacent to the location of the true peak. Owning
to the complexity of this problem, we analyse carefully the de-convolution of
single peaks on a grid and gain a strong insight about the dependence of the
reconstructed magnitudes on the exact peak location. This in turn allows us to
infer further information on recovering the location of the exact peaks i.e. to
perform super-resolution. We analyze in detail the possible cases that can
appear and based on our theoretical findings, we propose an self-driven
adaptive grid approach that allows to perform superresolution in
one-dimensional and multi-dimensional spaces. With the view that the current
study can provide a further step in the development of more robust algorithms
for the detection of single molecules in fluorescence microscopy or
identification of characteristic frequencies in spectral analysis, we
demonstrate how the proposed approach can recover sparse signals using
simulated clusters of point sources (peaks) of low-resolution in one and
two-dimensional spaces.
