250 research outputs found

    Compressed Sensing with General Frames via Optimal-dual-based β„“1\ell_1-analysis

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    Compressed sensing with sparse frame representations is seen to have much greater range of practical applications than that with orthonormal bases. In such settings, one approach to recover the signal is known as β„“1\ell_1-analysis. We expand in this article the performance analysis of this approach by providing a weaker recovery condition than existing results in the literature. Our analysis is also broadly based on general frames and alternative dual frames (as analysis operators). As one application to such a general-dual-based approach and performance analysis, an optimal-dual-based technique is proposed to demonstrate the effectiveness of using alternative dual frames as analysis operators. An iterative algorithm is outlined for solving the optimal-dual-based β„“1\ell_1-analysis problem. The effectiveness of the proposed method and algorithm is demonstrated through several experiments.Comment: 34 pages, 8 figures. To appear in IEEE Transactions on Information Theor

    Sampling in the Analysis Transform Domain

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    Many signal and image processing applications have benefited remarkably from the fact that the underlying signals reside in a low dimensional subspace. One of the main models for such a low dimensionality is the sparsity one. Within this framework there are two main options for the sparse modeling: the synthesis and the analysis ones, where the first is considered the standard paradigm for which much more research has been dedicated. In it the signals are assumed to have a sparse representation under a given dictionary. On the other hand, in the analysis approach the sparsity is measured in the coefficients of the signal after applying a certain transformation, the analysis dictionary, on it. Though several algorithms with some theory have been developed for this framework, they are outnumbered by the ones proposed for the synthesis methodology. Given that the analysis dictionary is either a frame or the two dimensional finite difference operator, we propose a new sampling scheme for signals from the analysis model that allows to recover them from their samples using any existing algorithm from the synthesis model. The advantage of this new sampling strategy is that it makes the existing synthesis methods with their theory also available for signals from the analysis framework.Comment: 13 Pages, 2 figure
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