We study the problem of estimating a $p$-dimensional $s$-sparse vector in a linear model with Gaussian design. In the case where the labels are contaminated by at most $o$ adversarial outliers, we prove that the $\ell_1$-penalized Huber's $M$-estimator based on $n$ samples attains the optimal rate of convergence $(s/n)^{1/2} + (o/n)$, up to a logarithmic factor. For more general design matrices, our results highlight the importance of two properties: the transfer principle and the incoherence property. These properties with suitable constants are shown to yield the optimal rates of robust estimation with adversarial contamination.

The aim of this work is to exploit the MUSIC algorithm performance in order to enhance target separability in range and azimuth, i.e. achieve point targets separation inside a resolution cell. Simulations have been done in order to plan and check the feasibility of a super-resolution experiment that took place in September 2006 on the test site of Oberpfaffenhofen (Germany). The data set has been acquired with the E-SAR system of the DLR in X-band. The targets to be separated were seven small corner reflectors that have been placed in a way that their response falls in one or, at maximum, two resolution cells of the standard Fourier SAR image. A post-processing implementation of the MUSIC algorithm has been proposed allowing, in the already focused SAR image, to retrieve the targets geometry. Conditions and analysis of the results have been carried out.