Self-Calibration via Linear Least Squares.

Whenever we use devices to take measurements, calibration is indispensable. While the purpose calibration is to reduce bias uncertainty in the measurements, it can quite difficult, expensive sometimes even impossible to implement. We study challenging problem called self-calibration, i.e., the task designing algorithm for devices so that the algorithm is able to perform calibration automatically. More precisely, we consider the setup $boldsymbol{y} = mathcal{A}(boldsymbol{d}) boldsymbol{x} + boldsymbol{epsilon}$ where only partial information about the sensing matrix $mathcal{A}(boldsymbol{d})$ is known where $mathcal{A}(boldsymbol{d})$ linearly depends on $boldsymbol{d}$. The goal is to estimate the calibration parameter $boldsymbol{d}$ (resolve the uncertainty in the sensing process) the signal/object of interests $boldsymbol{x}$ simultaneously. For three different models practical relevance we show how such a bilinear inverse problem, including blind deconvolution as an important example, can be solved via a simple linear least squares approach. As consequence, the proposed algorithms numerically extremely efficient, thus allowing for real-time deployment. Explicit theoretical guarantees stability theory derived and the number sampling complexity is nearly optimal (up to a poly-log factor). Applications in imaging sciences signal processing discussed numerical simulations are presented to demonstrate the effectiveness efficiency our approach.