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This paper studies the problem of exact localization of sparse (point orextended) objects with noisy data. The crux of the proposed approach consistsof random illumination. Several recovery methods are analyzed: the Lasso, BPDNand the One-Step Thresholding (OST). For independent random probes, it is shownthat both recovery methods can localize exactly $s=\cO(m)$, up to a logarithmicfactor, objects where $m$ is the number of data. Moreover, when the number ofrandom probes is large the Lasso with random illumination has a performanceguarantee for superresolution, beating the Rayleigh resolution limit. Numericalevidence confirms the predictions and indicates that the performance of theLasso is superior to that of the OST for the proposed set-up with randomillumination.

Exact Localization and Superresolution with Noisy Data and Random Illumination