Linear inverse problems such as that of compressive sensing are never perfectly defined in practice. Hence the system needs to be calibrated for imperfections in the linear transfer function. However this calibration can rarely be done offline, hence one needs perform calibration while solving the inverse problem, blindly.
In this work, we have solved the blind calibration problem for the specific case of calibrating for unknown (complex valued) gains. This problem is very interesting because it is seen in various applications (deconvolution in frequency domain, distributed sensor networks, radio spectrometry imaging, etc.). Our approach solves this problem (and shows when it is solveable) by convex optimization.
In this work we have used our own Matlab implementations of the Alternating Direction Method of Multipliers (ADMM) algorithm for the convex optimization.