Simultaneous estimation of a system matrix by compressed sensing and finding optimal regularization parameters for the inversion problem

This paper deals with the problem of measuring the system matrix of a linear system model with the help of test signals and using the estimated matrix within an inverse problem. In some cases, such as medical imaging, the process of measuring the system matrix can be very time and memory consuming. Fortunately, the underlying physical relationships often have a sparse representation, and in such situations, compressed-sensing techniques may be used to predict the system matrix. However, since there may be systematic errors inside the predicted matrix, its inversion can cause significant noise amplification and large errors on the reconstructed quantities. To combat this, regularization methods are often applied. In this paper, based on the singular value decomposition, the minimum mean square error estimator, and Stein's unbiased risk estimate, we show how optimal regularization parameters can be obtained from a few number of measurements. The efficiency of our approach is shown for two different systems.