Blind calibration for compressed sensing by convex optimization

We consider the problem of calibrating a compressed sensing mea-surement system under the assumption that the decalibration consists in unknown gains on each measure. We focus on blind calibration, us-ing measures performed on a few unknown (but sparse) signals. A naive formulation of this blind calibration problem, using `1 minimization, is reminiscent of blind source separation and dictionary learning, which are known to be highly non-convex and riddled with local minima. In the considered context, we show that in fact this formulation can be exactly expressed as a convex optimization problem, and can be solved using off-the-shelf algorithms. Numerical simulations demonstrate the effectiveness of the approach even for highly uncalibrated measures, when a sufficient number of (unknown, but sparse) calibrating signals is provided. We ob-serve that the success/failure of the approach seems to obey sharp phase transitions.