Atomic Decomposition by Basis Pursuit
SIAM Journal on Scientific Computing
Inpainting and Zooming Using Sparse Representations
The Computer Journal
Stable recovery of sparse overcomplete representations in the presence of noise
IEEE Transactions on Information Theory
IEEE Transactions on Information Theory
Near-Optimal Signal Recovery From Random Projections: Universal Encoding Strategies?
IEEE Transactions on Information Theory
An EM algorithm for wavelet-based image restoration
IEEE Transactions on Image Processing
IEEE Transactions on Information Theory
Journal of Mathematical Imaging and Vision
Foundations and Trends® in Machine Learning
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This work focuses on several optimization problems involved in recovery of sparse solutions of linear inverse problems. Such problems appear in many fields including image and signal processing, and have attracted even more interest since the emergence of the compressed sensing (CS) theory. In this paper, we formalize many of these optimization problems within a unified framework of convex optimization theory, and invoke tools from convex analysis and maximal monotone operator splitting. We characterize all these optimization problems, and to solve them, we propose fast iterative convergent algorithms using forward-backward and/or Peaceman/Douglas-Rachford splitting iterations. With nondifferentiable sparsity-promoting penalties, the proposed algorithms are essentially based on iterative shrinkage. This makes them very competitive for large-scale problems. We also report some experiments on image reconstruction in CS to demonstrate the applicability of the proposed framework.