Compressive Signal Recovery Under Sensing Matrix Errors Combined With Unknown Measurement Gains

Compressed Sensing assumes a linear model for acquiring signals however imperfections may arise in the specification of the `ideal' measurement model. We present the first study which considers the case of two such common calibration issues: (a) unknown measurement scaling (sensor gains) due to hardware vagaries or due to unknown object motion in MRI scanning, \emph{in conjunction with} (b) unknown offsets to measurement frequencies in case of a Fourier measurement matrix. We propose an alternating minimisation algorithm for on-the-fly signal recovery in the case when errors (a) and (b) occur \emph{jointly}. We show simulation results over a variety of situations that outperform the baselines of signal recovery by ignoring either or both types of calibration errors. We also show theoretical results for signal recovery by introducing a perturbed version of the well-known Generalized Multiple Measurement Vectors (GMMV) model.