This paper studies the problem of exact localization of sparse (point or extended)
objects with noisy data. The crux of the proposed approach consists of random illumination.
Several recovery methods are analyzed: the Lasso, BPDN and the One-Step Thresholding (OST).
For independent random probes, it is shown that both recovery methods can localize exactly
$s=\cO(m)$, up to a logarithmic factor, objects where $m$ is the number of data. Moreover,
when the number of random probes is large the Lasso with random illumination has a
performance guarantee for superresolution, beating the Rayleigh resolution limit. Numerical
evidence confirms the predictions and indicates that the performance of the Lasso is
superior to that of the OST for the proposed set-up with random illumination.