Quantitative control of the error bounds of a fast super-resolution technique for microscopy and astronomy

While the registration step is often problematic for superresolution, many microscopes and telescopes are now equipped with a piezoelectric mechanical system which permits to accurately control their motion (down to nanometers). Therefore one can use such devices to acquire multiple images of the same scene at various controlled positions. Then a fast super-resolution algorithm [1] can be used for efficient super-resolution. However the minimal use of r² images for a resolution enhancement factor r is generally not sufficient to obtain good results. We propose to take several images at positions randomly distributed close to each reference position. We study the number of images necessary to control the error resulting from the super-resolution algorithm by [1] due to the uncertainty on positions. The main result is a lower bound on the number of images to respect a given error upper bound with probability higher than a desired confidence level.