Recently, iterative reweighted methods have attracted much interest in compressed sensing, since they perform better than unweighted ones in most cases. Currently, weights are chosen heuristically in existing iterative reweighted methods, and nding an optimal weight is an open problem since we do not know the exact support set beforehand. In this paper, we present a novel weighted l1-norm minimization problem for the sparsest solution of underdetermined linear equations, whose solution is also the sparsest under some given conditions. We propose an iterative weighted thresholding method for the weighted l1-norm minimization problem, where the weight w and variable x are optimized simultaneously, and prove that the iteration process will converge eventually. Moreover, we enhance the performance of our iterative weighted thresholding method using the homotopy technique. Extensive computational experiments show that our method performs better both in running time and recovery accuracy comparing with some state-of-the-art methods.
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