  We report on image processing techniques and experimental procedures to
determine the lattice-site positions of single atoms in an optical lattice with
high reliability, even for limited acquisition time or optical resolution.
Determining the positions of atoms beyond the diffraction limit relies on
parametric deconvolution in close analogy to methods employed in
super-resolution microscopy. We develop a deconvolution method that makes
effective use of the prior knowledge of the optical transfer function, noise
properties, and discreteness of the optical lattice. We show that accurate
knowledge of the image formation process enables a dramatic improvement on the
localization reliability. This allows us to demonstrate super-resolution of the
atoms' position in closely packed ensembles where the separation between
particles cannot be directly optically resolved. Furthermore, we demonstrate
experimental methods to precisely reconstruct the point spread function with
sub-pixel resolution from fluorescence images of single atoms, and we give a
mathematical foundation thereof. We also discuss discretized image sampling in
pixel detectors and provide a quantitative model of noise sources in electron
multiplying CCD cameras. The techniques developed here are not only beneficial
to neutral atom experiments, but could also be employed to improve the
localization precision of trapped ions for ultra precise force sensing.
