Journal ArticleDOI
Exact Reconstruction of Sparse Signals via Nonconvex Minimization
TLDR
It is shown that by replacing the lscr1 norm with theLscrp norm, exact reconstruction is possible with substantially fewer measurements, and a theorem in this direction is given.Abstract:
Several authors have shown recently that It is possible to reconstruct exactly a sparse signal from fewer linear measurements than would be expected from traditional sampling theory. The methods used involve computing the signal of minimum lscr1 norm among those having the given measurements. We show that by replacing the lscr1 norm with the lscrp norm with p < 1, exact reconstruction is possible with substantially fewer measurements. We give a theorem in this direction, and many numerical examples, both in one complex dimension, and larger-scale examples in two real dimensions.read more
Citations
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Journal ArticleDOI
Enhancing Sparsity by Reweighted ℓ 1 Minimization
TL;DR: A novel method for sparse signal recovery that in many situations outperforms ℓ1 minimization in the sense that substantially fewer measurements are needed for exact recovery.
Journal ArticleDOI
Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization
TL;DR: It is shown that if a certain restricted isometry property holds for the linear transformation defining the constraints, the minimum-rank solution can be recovered by solving a convex optimization problem, namely, the minimization of the nuclear norm over the given affine space.
Journal Article
Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization
TL;DR: In this paper, it was shown that if a certain restricted isometry property holds for the linear transformation defining the constraints, the minimum-rank solution can be recovered by solving a convex optimization problem, namely, the minimization of the nuclear norm over the given affine space.
Journal ArticleDOI
Message-passing algorithms for compressed sensing
TL;DR: A simple costless modification to iterative thresholding is introduced making the sparsity–undersampling tradeoff of the new algorithms equivalent to that of the corresponding convex optimization procedures, inspired by belief propagation in graphical models.
Journal ArticleDOI
Image reconstruction in circular cone-beam computed tomography by constrained, total-variation minimization
Emil Y. Sidky,Xiaochuan Pan +1 more
TL;DR: An iterative algorithm, based on recent work in compressive sensing, that minimizes the total variation of the image subject to the constraint that the estimated projection data is within a specified tolerance of the available data and that the values of the volume image are non-negative is developed.
References
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Book
Compressed sensing
TL;DR: It is possible to design n=O(Nlog(m)) nonadaptive measurements allowing reconstruction with accuracy comparable to that attainable with direct knowledge of the N most important coefficients, and a good approximation to those N important coefficients is extracted from the n measurements by solving a linear program-Basis Pursuit in signal processing.
Journal ArticleDOI
Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information
TL;DR: In this paper, the authors considered the model problem of reconstructing an object from incomplete frequency samples and showed that with probability at least 1-O(N/sup -M/), f can be reconstructed exactly as the solution to the lscr/sub 1/ minimization problem.
Journal ArticleDOI
Atomic Decomposition by Basis Pursuit
TL;DR: Basis Pursuit (BP) is a principle for decomposing a signal into an "optimal" superposition of dictionary elements, where optimal means having the smallest l1 norm of coefficients among all such decompositions.
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Decoding by linear programming
Emmanuel J. Candès,Terence Tao +1 more
TL;DR: F can be recovered exactly by solving a simple convex optimization problem (which one can recast as a linear program) and numerical experiments suggest that this recovery procedure works unreasonably well; f is recovered exactly even in situations where a significant fraction of the output is corrupted.
Journal ArticleDOI
Stable signal recovery from incomplete and inaccurate measurements
TL;DR: In this paper, the authors considered the problem of recovering a vector x ∈ R^m from incomplete and contaminated observations y = Ax ∈ e + e, where e is an error term.