Microwave imaging has been widely used in the prediction and tracking of hurricanes, typhoons, and tropical storms. Due to the limitations of sensors, the acquired remote sensing data are usually blurry and have relatively low resolution, which calls for the development of fast algorithms for deblurring and enhancing the resolution. We propose an efficient algorithm for simultaneous image deconvolution and upsampling for low-resolution microwave hurricane data. Our model involves convolution, downsampling, and the total variation regularization. After reformulating the model, we are able to apply the alternating direction method of multipliers and obtain three subproblems, each of which has a closed-form solution. We also extend the framework to the multichannel case with the multichannel total variation regularization. A variety of numerical experiments on synthetic and real Advanced Microwave Sounding Unit and Microwave Humidity Sounder data were conducted. The results demonstrate the outstanding performance of the proposed method.
This paper proposes efficient algorithms for group sparse optimization with mixed l2,1-regularization, which arises from the reconstruction of group sparse signals in compressive sensing, and the group Lasso problem in statistics and machine learning. It is known that encoding the group information in addition to sparsity can often lead to better signal recovery/feature selection. The l2,1-regularization promotes group sparsity, but the resulting problem, due to the mixed-norm structure and possible grouping irregularity, is considered more difficult to solve than the conventional l1-regularized problem. Our approach is based on a variable splitting strategy and the classic alternating direction method (ADM). Two algorithms are presented, one derived from the primal and the other from the dual of the l2,1-regularized problem. The convergence of the proposed algorithms is guaranteed by the existing ADM theory. General group configurations such as overlapping groups and incomplete covers can be easily handled by our approach. Computational results show that on random problems the proposed ADM algorithms exhibit good efficiency, and strong stability and robustness.
Compressive sensing encodes a signal into a relatively small number of incoherent linear measurements. In theory,
the optimal incoherence is achieved by completely random measurement matrices. However, such matrices are
often difficult and costly to implement in hardware realizations.
Random Toeplitz and circulant matrices can be easily (or even naturally) realized in various applications. This
paper introduces fast algorithms for reconstructing signals from incomplete Toeplitz and circulant measurements.
Computational results are presented to show that Toeplitz and circulant matrices are not only as effective as
random matrices for signal encoding, but also permit much faster decoding.
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