SPARSE SIGNAL RECONSTRUCTION FROM LIMITED DATA USING FOCUSS - A RE-WEIGHTED MINIMUM NORM ALGORITHM

We present a nonparametric algorithm for finding localized energy solu tions from limited data. The problem we address is underdetermined, an d 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 initi al estimate of the real signal and the iteration process that refines the initial estimate to the final localized energy solution. The itera tions are based on weighted norm minimization of the dependent variabl e with the weights being a function of the preceding iterative solutio ns. The algorithm is presented as a general estimation tool usable acr oss different applications. A detailed analysis laying the theoretical foundation for the algorithm is given and includes proofs of global a nd local convergence and a derivation of the rate of convergence. A vi ew of the algorithm as a novel optimization method which combines desi rable characteristics of both classical optimization and learning-base d algorithms is provided. Mathematical results on conditions for uniqu eness of sparse solutions are also given. Applications of the algorith m are illustrated on problems in direction-of-arrival (DOA) estimation and neuromagnetic imaging.