We present a nonparametric algorithm for finding localized energy solutions from limited data. The problem we address is underdetermined, and no prior knowledge of the shape of the region on which the solution is nonzero is assumed. Termed the FOcal Underdetermined System Solver (FOCUSS), the algorithm has two integral parts: a low-resolution initial estimate of the real signal and the iteration process that refines the initial estimate to the final localized energy solution. The iterations are based on weighted norm minimization of the dependent variable with the weights being a function of the preceding iterative solutions. The algorithm is presented as a general estimation tool usable across different applications. A detailed analysis laying the theoretical foundation for the algorithm is given and includes proofs of global and local convergence and a derivation of the rate of convergence. A view of the algorithm as a novel optimization method which combines desirable characteristics of both classical optimization and learning-based algorithms is provided. Mathematical results on conditions for uniqueness of sparse solutions are also given. Applications of the algorithm are illustrated on problems in direction-of-arrival (DOA) estimation and neuromagnetic imaging.

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Signals from eye movements and blinks can be orders of magnitude larger than brain-generated electrical potentials and are one of the main sources of artifacts in electroencephalographic (EEG) data. Rejecting contaminated trials causes substantial data loss, and restricting eye movements/blinks limits the experimental designs possible and may impact the cognitive processes under investigation. This article presents a method based on blind source separation (BSS) for automatic removal of electroocular artifacts from EEG data. BBS is a signal-processing methodology that includes independent component analysis (ICA). In contrast to previously explored ICA-based methods for artifact removal, this method is automated. Moreover, the BSS algorithm described herein can isolate correlated electroocular components with a high degree of accuracy. Although the focus is on eliminating ocular artifacts in EEG data, the approach can be extended to other sources of EEG contamination such as cardiac signals, environmental noise, and electrode drift, and adapted for use with magnetoencephalographic (MEG) data, a magnetic correlate of EEG.

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