Superresolution method for data deconvolution by superposition of point sources

Abstract: In this work we present a new algorithm for data deconvolution that allows the retrieval of the target function with super-resolution with a simple approach that after a precis e measurement of the instrument response function (IRF), the measured data are fit by a superposition of point sources (SUPPOSe) of equal intensity. In this manner only the positions of the sources need to be determined by an algorithm that minimizes the norm of the difference between the measured data and the convolution of the superposed point sources with the IRF. An upper bound for the uncertainty in the position of the sources was derived and two very different experimental situations were used for the test (an optical spectrum and fluorescent microscopy images) showing excellent reconstructions and agreement with the predicted uncertainties, achieving {\lambda}/10 resolution for the microscope and a fivefold improvement in the spectral resolution for the spectrometer. The method also provides a way to determine the optimum number of sources to be used for the fit.